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d oes matter differ from vacuum?

1001

quantum effects, such as particle–antiparticle creation or annihilation. Table 74 shows that the approach distance is near or smaller than the Compton wavelength only in the microscopic world, so that such effects are not observed in everyday life. We do not therefore need quantum field theory to describe common observations. The combined concepts of quantum field theory and general relativity are required in situations in which both conditions are satisfied simultaneously. The necessary approach distance for such situations is calculated by setting r S = 2λ C (the factor 2 is introduced for simplicity). We find that this is the case when lengths or times are (of the order of) l Pl =

¼

ħG~c 3 ¼ t Pl = ħG~c 5

Ref. 1018

= 5.4 ë 10−44 s , the Planck time.

(666)

Whenever we approach objects at these scales, both general relativity and quantum mechanics play a role; at these scales effects of quantum gravity appear. Because the values of the Planck dimensions are extremely small, this level of sophistication is unnecessary in everyday life, in astronomy and even in particle physics. However, to answer the questions posted at the beginning of the book – why do we live in three dimensions and why is the proton 1836.15 times heavier than the electron? – we require a precise and complete description of nature. The contradictions between quantum mechanics and general relativity appear to make the search for answers impossible. However, while the unified theory describing quantum gravity is not yet complete, we can already get a few glimpses at its implications from its present stage of development. Note that the Planck scales specify one of only two domains of nature where quantum mechanics and general relativity apply at the same time. (What is the other?) As Planck scales are the easier of the two to study, they provide the best starting point for the following discussion. When Max Planck discovered the existence of Planck scales or Planck units, he was interested in them mainly as natural units of measurement, and that is what he called them. However, their importance in nature is much more widespread, as we will shall see in the new section. We will discover that they determine what is commonly called quantum geometry. Farewell to instants of time

Ref. 1019, Ref. 1020

The appearance of the quantum of action in the description of motion leads to quantum limits to all measurements. These limits have important consequences at Planck dimensions. Measurement limits appear most clearly when we investigate the properties of clocks and metre rules. Is it possible to construct a clock that is able to measure time intervals shorter than the Planck time? Surprisingly, the answer is no, even though the time–energy indeterminacy relation ∆E∆t E ħ seems to indicate that by making ∆E arbitrary large, we can make ∆t arbitrary small. Every clock is a device with some moving parts. Parts can be mechanical wheels, particles of matter in motion, changing electrodynamic fields, i.e. photons, or decaying

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Time is composed of time atoms ... which in fact are indivisible. Moses Maimonides, 12th century.

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Challenge 1426 n

= 1.6 ë 10−35 m, the Planck length,

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1002

Ref. 1021, Ref. 1022 Ref. 1023

xi general rel ativity versus quantum mechanics

radioactive particles. For each moving component of a clock, such as the two hands, the indeterminacy principle applies. As discussed most clearly by Michael Raymer, the indeterminacy relation for two non-commuting variables describes two different, but related situations: it makes a statement about standard deviations of separate measurements on many identical systems; and it describes the measurement precision for a joint measurement on a single system. Throughout this article, only the second situation is considered. For any clock to work, we need to know both the time and the energy of each hand. Otherwise it would not be a recording device. Put more generally, a clock must be a classical system. We need the combined knowledge of the non-commuting variables for each moving component of the clock. Let us focus on the component with the largest time indeterminacy ∆t. It is evident that the smallest time interval δt that can be measured by a clock is always larger than the quantum limit, i.e. larger than the time indeterminacy ∆t for the most ‘uncertain’ component. Thus we have ħ , ∆E

(667)

where ∆E is the energy indeterminacy of the moving component, and this energy indeterminacy ∆E must be smaller than the total energy E = mc 2 of the component itself.* Furthermore, a clock provides information and thus signals have to be able to leave it. To make this possible, the clock must not be a black hole and its mass m must therefore be smaller than the Schwarzschild mass for its size, i.e. m D c 2 l~G, where l is the size of the clock (neglecting factors of order unity). Finally, for a sensible measurement of the time interval δt, the size l of the clock must be smaller than c δt itself, because otherwise different parts of the clock could not work together to produce the same time display.** If we combine all these conditions, we get

or

δt E δt E

ħG = t Pl . c5

(668)

(669)

In summary, from three simple properties of any clock, namely that there is only a single clock, that we can read its dial and that it gives sensible read-outs, we get the general conclusion that clocks cannot measure time intervals shorter than the Planck time. Note that this argument is independent of the nature of the clock mechanism. Whether the clock is powered by gravitational, electrical, plain mechanical or even nuclear means, the limit still applies.*** The same result can also be found in other ways. For example, any clock small enough to measure small time intervals necessarily has a certain energy indeterminacy due to * Physically, this condition means being sure that there is only one clock; the case ∆E A E would mean that it is impossible to distinguish between a single clock and a clock–anticlock pair created from the vacuum, or a component plus two such pairs, etc. ** It is amusing to explore how a clock larger than c δt would stop working, as a result of the loss of rigidity

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Ref. 1027

¾

ħG c 5 δt

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δt E ∆t E

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Page 410 Ref. 1016

1003

the indeterminacy relation. At the same time, on the basis of general relativity, any energy density induces a deformation of space-time and signals from the deformed region arrive with a certain delay due to that deformation. The energy indeterminacy of the source leads to an indeterminacy in the deformation and thus in the delay. The expression from general relativity for the deformation of the time part of the line element due to a mass m is δt = mG~l c 3 . From the mass–energy relation, an energy spread ∆E produces an indeterminacy ∆t in the delay ∆E G ∆t = . (670) l c5

Farewell to points in space Ref. 1028

Furthermore, the measurement cannot be performed if signals cannot leave the objects; thus they may not be black holes. Therefore their masses must be small enough for their Schwarzschild radius r S = 2Gm~c 2 to be smaller than the distance δl separating them. Challenge 1427 n

Ref. 1025 1024 Ref. 1026

in its components. *** Note that gravitation is essential here. The present argument differs from the well-known study on the limitations of clocks due to their mass and their measuring time which was published by Salecker and Wigner and summarized in pedagogical form by Zimmerman. Here, both quantum mechanics and gravity are included, and therefore a different, lower and much more fundamental limit is found. Note also that the discovery of black hole radiation does not change the argument; black hole radiation notwithstanding, measurement devices cannot exist inside black holes.

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In a similar way, we can deduce that it is impossible to make a metre rule or any other length measuring device that is able to measure lengths shorter than the Planck length. Obviously, we can already deduce this from l Pl = c t Pl , but a separate proof is also possible. The straightforward way to measure the distance between two points is to put an object at rest at each position. In other words, joint measurements of position and momentum are necessary for every length measurement. Now, the minimal length δl that can be measured must be larger than the position indeterminacy of the two objects. From the indeterminacy principle it is known that each object’s position cannot be determined with a precision ∆l better than that given by the indeterminacy relation ∆l ∆p = ħ, where ∆p is the momentum indeterminacy. The requirement that there is only one object at each end, i.e. avoiding pair production from the vacuum, means that ∆p < mc; together, these requirements give ħ . (671) δl E ∆l E mc

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This indeterminacy determines the precision of the clock. Furthermore, the energy indeterminacy of the clock is fixed by the indeterminacy relation for time and energy ∆E E ħ~∆t, in turn fixed by the precision of the clock. Combining all this, we again find the relation δt E t Pl for the minimum measurable time. We are forced to conclude that in nature there is a minimum time interval. In other words, at Planck scales the term ‘instant of time’ has no theoretical or experimental basis. It therefore makes no sense to use the term.

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xi general rel ativity versus quantum mechanics

Again omitting the factor of 2, we get δl E

Ref. 1006, Ref. 1016 Ref. 1028, Ref. 1029, Ref. 1030

Page 349

(672)

Another way to deduce this limit reverses the roles of general relativity and quantum theory. To measure the distance between two objects, we have to localize the first object with respect to the other within a certain interval ∆x. The corresponding energy indeterminacy obeys ∆E = c(c 2 m 2 + (∆p)2 )1~2 E cħ~∆x. However, general relativity shows that a small volume filled with energy changes the curvature of space-time, and thus changes the metric of the surrounding space. For the resulting distance change ∆l, compared to empty space, we find the expression ∆l  G∆E~c 4 . In short, if we localize the first particle in space with a precision ∆x, the distance to a second particle is known only with precision ∆l. The minimum length δl that can be measured is obviously larger than either of these quantities; inserting the expression for ∆E, we find again that the minimum measurable length δl is given by the Planck length. We note that, as the Planck length is the shortest possible length, it follows that there can be no observations of quantum mechanical effects for situations in which the corresponding de Broglie or Compton wavelength is smaller than the Planck length. In proton– proton collisions we observe both pair production and interference effects. In contrast, the Planck limit implies that in everyday, macroscopic situations, such as car–car collisions, we cannot observe embryo–antiembryo pair production and quantum interference effects. In summary, from two simple properties common to all length measuring devices, namely that they can be counted and that they can be read out, we arrive at the conclusion that lengths smaller than the Planck length cannot be found in measurements. Whatever method is used, be it a metre rule or time-of-flight measurement, we cannot overcome this fundamental limit. It follows that the concept of a ‘point in space’ has no experimental basis. In the same way, the term ‘event’, being a combination of a ‘point in space’ and an ‘instant of time’, also loses its meaning for the description of nature. A simple way to deduce the minimum length using the limit statements which structure this ascent is the following. General relativity is based on a maximum force in nature, or alternatively, on a minimum mass change per time; its value is given by dm~dt = c 3 ~4G. Quantum theory is based on a minimum action in nature, given by L = ħ~2. Since a distance d can be expressed like L d2 = , (673) dm~dt one sees directly that a minimum action and a maximum mass rate imply a minimum distance. In other words, quantum theory and general relativity, when put together, imply a minimum distance. These results are often expressed by the so-called generalized indeterminacy principle ∆p∆x E ħ~2 + f

G (∆p)2 c3

(674)

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Ref. 1034

ħG = l Pl . c3

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Ref. 1031, Ref. 1032, Ref. 1033

¾

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d oes matter differ from vacuum?

or

Challenge 1428 e

Ref. 1034 Ref. 1035

Ref. 1037, Ref. 1038, Ref. 1039

Ref. 1040

2 l Pl (∆p)2 , ħ

(675)

where f is a numerical factor of order unity. A similar expression holds for the time– energy indeterminacy relation. The first term on the right hand side is the usual quantum mechanical indeterminacy. The second term, negligible for everyday life energies, plays a role only near Planck energies and is due to the changes in space-time induced by gravity at these high energies. You should be able to show that the generalized principle (674) automatically implies that ∆x can never be smaller than f 1~2 l Pl . The generalized indeterminacy principle is derived in exactly the same way in which Heisenberg derived the original indeterminacy principle ∆p∆x E ħ~2, namely by studying the deflection of light by an object under a microscope. A careful re-evaluation of the process, this time including gravity, yields equation (674). For this reason, all approaches that try to unify quantum mechanics and gravity must yield this relation; indeed, it appears in the theory of canonical quantum gravity, in superstring theory and in the quantum group approach. We remember that quantum mechanics starts when we realize that the classical concept of action makes no sense below the value of ħ~2; similarly, unified theories start when we realize that the classical concepts of time and length make no sense below Planck values. However, the usual description of space-time does contain such small values; the usual description involves the existence of intervals smaller than the smallest measurable one. Therefore, the continuum description of space-time has to be abandoned in favour of a more appropriate description. The new indeterminacy relation appearing at Planck scales shows that continuity cannot be a good description of space-time. Inserting c∆p E ∆E E ħ~∆t into equation (674), we get ∆x∆t E ħG~c 4 = t Pl l Pl , (676)

Farewell to the space-time manifold The consequences of the Planck limits for measurements of time and space can be taken much further. It is commonplace to say that given any two points in space or any two instants of time, there is always a third in between. Physicists sloppily call this property continuity, while mathematicians call it denseness. However, at Planck dimensions this

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which of course has no counterpart in standard quantum mechanics. It shows that spacetime events do not exist. A final way to convince oneself that points have no meaning is that a point is an entity with vanishing volume; however, the minimum volume possible 3 in nature is the Planck volume VPl = l Pl . While space-time points are idealizations of events, this idealization is incorrect. The use of the concept of ‘point’ is similar to the use of the concept of ‘aether’ a century ago: it is impossible to detect and it is only useful for describing observations until a way to describe nature without it has been found. Like ‘aether´, also ‘point´ leads reason astray. In other words, the Planck units do not only provide natural units, they also provide — within a factor of order one — the limit values of space and time intervals.

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Ref. 1036

∆p∆x E ħ~2 + f

1005

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1006

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Ref. 1028, Ref. 1042

property cannot exist, since intervals smaller than the Planck time can never be found. Thus points and instants are not dense, and between two points there is not always a third. This means that space and time are not continuous. Of course, at large scales they are – approximately – continuous, in the same way that a piece of rubber or a liquid seems continuous at everyday dimensions, even though it is not at a small scale. All paradoxes resulting from the infinite divisibility of space and time, such as Zeno’s argument on the impossibility to distinguish motion from rest, or the Banach–Tarski paradox, are now avoided. We can dismiss the paradoxes straight away because of their incorrect premises concerning the nature of space and time. But let us go on. Special relativity, quantum mechanics and general relativity all rely on the idea that time can be defined for all points of a given reference frame. However, two clocks a distance l apart cannot be synchronized with arbitrary precision. Since the distance between two clocks cannot be measured with an error smaller than the Planck length l Pl , and transmission of signals is necessary for synchronization, it is not possible to synchronize two clocks with a better precision than the time l Pl ~c = t Pl , the Planck time. Because it is impossible to synchronize clocks precisely, a single time coordinate for a whole reference frame is only an approximation, and this idea cannot be maintained in a precise description of nature. Moreover, since the time difference between events can only be measured within a Planck time, for two events distant in time by this order of magnitude, it is not possible to say with complete certainty which of the two precedes the other! This is an important result. If events cannot be ordered, the concept of time, which was introduced into physics to describe sequences, cannot be defined at all at Planck scales. In other words, after dropping the idea of a common time coordinate for a complete frame of reference, we are forced to drop the idea of time at a single ‘point’ as well. Therefore, the concept of ‘proper time’ loses its meaning at Planck scales. It is straightforward to use the same arguments to show that length measurements do not allow us to speak of continuous space, but only of approximately continuous space. As a result of the lack of measurement precision at Planck scales, the concepts of spatial order, translation invariance, isotropy of vacuum and global coordinate systems have no experimental basis. But there is more to come. The very existence of a minimum length contradicts special relativity theory, in which it is shown that lengths undergo Lorentz contraction when the frame of reference is changed. A minimum length thus cannot exist in special relativity. But we just deduced that there must be such a minimum distance in nature. There is only one conclusion: special relativity cannot be correct at smallest distances. Thus, space-time is neither Lorentz invariant nor diffeomorphism invariant nor dilatation invariant at Planck dimensions. All symmetries that are at the basis of special and general relativity are thus only approximately valid at Planck scales. As a result of the imprecision of measurement, most familiar concepts used to describe spatial relations become useless. For example, the concept of metric loses its usefulness at Planck scales. Since distances cannot be measured with precision, the metric cannot be determined. We deduce that it is impossible to say precisely whether space is flat or curved. In other words, the impossibility of measuring lengths exactly is equivalent to fluctuations of the curvature, and thus equivalent to fluctuations of gravity. In addition, even the number of spatial dimensions makes no sense at Planck scales.

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Ref. 1041

xi general rel ativity versus quantum mechanics

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d oes matter differ from vacuum?

Let us remind ourselves how to determine this number experimentally. One possible way is to determine how many points we can choose in space such that all the distances between them are equal. If we can find at most n such points, the space has n − 1 dimensions. We can see that if reliable length measurement at Planck scale is not possible, there is no way to determine reliably the number of dimensions of a space with this method. Another way to check for three spatial dimensions is to make a knot in a shoe string and glue the ends together: since it stays knotted we know that space has three dimensions, because there is a mathematical theorem that in spaces with greater or fewer than three dimensions, knots do not exist. Again, at Planck dimensions the errors in measurement do not allow to say whether a string is knotted or not, because measurement limits at crossings make it impossible to say which strand lies above the other; in short, at Planck scales we cannot check whether space has three dimensions or not. There are many other methods for determining the dimensionality of space.* All these methods start from the definition of the concept of dimensionality, which is based on a precise definition of the concept of neighbourhood. However, at Planck scales, as just mentioned, length measurements do not allow us to say whether a given point is inside or outside a given volume. In short, whatever method we use, the lack of reliable length measurements means that at Planck scales, the dimensionality of physical space is not defined. It should therefore not come as a surprise that when we approach these scales, we may get a scale-dependent answer for the number of dimensions, that may be different from three. The reason for the problems with space-time become most evident when we remember Euclid’s well-known definition: ‘A point is that which has no part.’ As Euclid clearly understood, a physical point, and here the stress is on physical, cannot be defined without some measurement method. A physical point is an idealization of position, and as such includes measurement right from the start. In mathematics, however, Euclid’s definition is rejected; mathematical points do not need a metric for their definition. Mathematical points are elements of a set, usually called a space. In mathematics, a measurable or metric space is a set of points equipped afterwards with a measure or a metric. Mathematical points do not need a metric for their definition; they are basic entities. In contrast to the mathematical situation, the case of physical space-time, the concepts of measure and of metric are more fundamental than that of a point. The difficulties distinguishing physical and mathematical space and points arise from the failure to distinguish a mathematical metric from a physical length measurement.**

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* For example, we can determine the dimension using only the topological properties of space. If we draw a so-called covering of a topological space with open sets, there are always points that are elements of several sets of the covering. Let us call p the maximal number of sets of which a point can be an element in a given covering. This number can be determined for all possible coverings. The minimum value of p, minus one, gives the dimension of the space. In fact, if physical space is not a manifold, the various methods may give different answers for the dimensionality. Indeed, for linear spaces without norm, a unique number of dimensions cannot be defined. The value then depends on the specific definition used and is called e.g. fractal dimension, Lyapunov dimension, etc. ** Where does the incorrect idea of continuous space-time have its roots? In everyday life, as well as in physics, space-time is introduced to describe observations. Space-time is a book-keeping device. Its properties are extracted from the properties of observables. Since observables can be added and multiplied, we extrapolate that they can take continuous values. This extrapolation implies that length and time intervals can take continuous values, and, in particular, arbitrary small values. From this result we get the possibil-

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Ref. 1043

1007

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1008

Ref. 1044

xi general rel ativity versus quantum mechanics

There is nothing in the world but matter in motion, and matter in motion cannot move otherwise than in space and time. Lenin

To complete this review of the situation, if space and time are not continuous, no quantities defined as derivatives with respect to space or time are precisely defined. Velocity, acceleration, momentum, energy, etc., are only well-defined under the assumption of continuous space and time. That important tool, the evolution equation, is based on derivatives and thus can no longer be used. Therefore the Schrödinger or the Dirac equation ity of defining points and sets of points. A special field of mathematics, topology, shows how to start from a set of points and construct, with the help of neighbourhood relations and separation properties, first a topological space. Then, with the help of a metric, a metric space can be built. With the appropriate compactness and connectedness relations, a manifold, characterized by its dimension, metric and topology, can be constructed. * A manifold is what locally looks like an Euclidean space. The exact definition can be found in Appendix D.

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Farewell to observables and measurements

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Perhaps the most beautiful way to make this point is the Banach–Tarski theorem, which clearly shows the limits of the concept of volume. The theorem states that a sphere made up of mathematical points can be cut into five pieces in such a way that the pieces can be put together to form two spheres, each of the same volume as the original one. However, the necessary cuts are ‘infinitely’ curved and detailed: they are wildly disconnected. For physical matter such as gold, unfortunately – or fortunately – the existence of a minimum length, namely the atomic distance, makes it impossible to perform such a cut. For vacuum, the puzzle reappears: for example, the energy of zero-point fluctuations is given by the density times the volume; following the Banach–Tarski theorem, the zero point energy content of a single sphere should be equal to the zero point energy of two similar spheres each of the same volume as the original one. The paradox is solved by the Planck length, because it also provides a fundamental length scale for vacuum, thus making infinitely complex cuts impossible. Therefore, the concept of volume is only well defined at Planck scales if a minimum length is introduced. To sum up, physical space-time cannot be a set of mathematical points. But the surprises are not finished. At Planck dimensions, since both temporal and spatial order break down, there is no way to say if the distance between two space-time regions that are close enough together is space-like or time-like. Measurement limits make it impossible to distinguish the two cases. At Planck scales, time and space cannot be distinguished from each other. In addition, it is impossible to state that the topology of space-time is fixed, as general relativity implies. The topology changes – mentioned above – required for particle reactions do become possible. In this way another of the contradictions between general relativity and quantum theory is resolved. In summary, space-time at Planck scales is not continuous, not ordered, not endowed with a metric, not four-dimensional and not made up of points. If we compare this with the definition of the term manifold,* not one of its defining properties is fulfilled. We arrive at the conclusion that the concept of a space-time manifold has no backing at Planck scales. This is a strong result. Even though both general relativity and quantum mechanics use continuous space-time, the combination of both theories does not.

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1009

Ref. 1045 Ref. 1046 Ref. 1047 Ref. 1049

Discretization of space-time has been studied already since 1940s. Recently, the idea that space-time could be described as a lattice has also been explored most notably by David Finkelstein and by Gerard ’t Hooft. The idea of space-time as a lattice is based on the idea that, if a minimum distance exists, then all distances are a multiple of this minimum. It is generally agreed that, in order to get an isotropic and homogeneous situation for large,

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Can space-time be a lattice? – A glimpse of quantum geometry

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lose their basis. Concepts such as ‘derivative’, ‘divergence-free’, ‘source free’ etc., lose their meaning at Planck scales. In fact, all physical observables are defined using length and time measurements. A list of physical units shows that each is a product of powers of length, time (and mass) units. (Even though in the SI system electrical quantities have a separate base quantity, the ampere, the argument still holds; the ampere is itself defined in terms of a force, which is measured using the three base units length, time and mass.) Since time and length are not continuous, observables themselves are not defined, because their value is not fixed. This means that at Planck scales, observables cannot be described by real numbers. In addition, if time and space are not continuous, the usual expression for an observable field A, namely A(t, x), does not make sense: we have to find a more appropriate description. Physical fields cannot exist at Planck scales. The consequences for quantum mechanics are severe. It makes no sense to define multiplication of observables by real numbers, thus by a continuous range of values, but only by a discrete set of numbers. Among other implications, this means that observables do not form a linear algebra. We recognize that, because of measurement errors, we cannot prove that observables do form such an algebra. This means that observables are not described by operators at Planck scales. And, because quantum mechanics is based on the superposition principle, without it, everything comes crumbling down. In particular, the most important observables are the gauge potentials. Since they do not now form an algebra, gauge symmetry is not valid at Planck scales. Even innocuous looking expressions such as [x i , x j ] = 0 for x i x x j , which are at the root of quantum field theory, become meaningless at Planck scales. Since at those scales also the superposition principle cannot be backed up by experiment, even the famous Wheeler–DeWitt equation, often assumed to describe quantum gravity, cannot be valid. Similarly, permutation symmetry is based on the premise that we can distinguish two points by their coordinates, and then exchange particles between those two locations. As we have just seen, this is not possible if the distance between the two particles is small; we conclude that permutation symmetry has no experimental basis at Planck scales. Even discrete symmetries, like charge conjugation, space inversion and time reversal cannot be correct in this domain, because there is no way to verify them exactly by measurement. CPT symmetry is not valid at Planck scales. Finally we note that all types of scaling relations do not work at smallest scales. As a result, renormalization symmetry is also destroyed at Planck scales. All these results are consistent: if there are no symmetries at Planck scales, there are also no observables, since physical observables are representations of symmetry groups. In fact, the limits on time and length measurements imply that the concept of measurement has no significance at Planck scales.

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1010

Ref. 1048

Farewell to particles In every example of motion, some object is involved. One of the important discoveries of the natural sciences was that all objects are composed of small constituents, called elementary particles. Quantum theory shows that all composite, non-elementary objects

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everyday scales, the structure of space-time cannot be periodic, but must be random. In addition, any fixed structure of space-time violates the result that there are no lengths smaller than the Planck length: as a result of the Lorentz contraction, any moving observer would find lattice distances smaller than the Planck value. Worse still, the lattice idea conflicts with general relativity, in particular with the diffeomorphism invariance of vacuum. Finally, where would a particle be during the jump from one lattice point to the next? Thus, in summary, space-time cannot be a lattice. A minimum distance does exist in nature; however, the hope that all other distances are simple multiples of the smallest distance is not correct. We will discover more evidence for this later on. If space-time is not a set of points or events, it must be something else. Three hints already appear at this stage. The first step necessary to improve the description of motion is the recognition that abandoning ‘points’ means abandoning the local description of nature. Both quantum mechanics and general relativity assume that the phrase ‘observable at a point’ has a precise meaning. Because it is impossible to describe space as a manifold, this expression is no longer useful. The unification of general relativity and quantum physics forces the adoption of a non-local description of nature at Planck scales. The existence of a minimum length implies that there is no way to physically distinguish locations that are even closer together. We are tempted to conclude therefore that no pair of locations can be distinguished, even if they are one metre apart, since on any path joining two points, no two locations that are close together can be distinguished. This situation is similar to the question about the size of a cloud or of an atom. If we measure water density or electron density, we find non-vanishing values at any distance from the centre of the cloud or the atom; however, an effective size can still be defined, because it is very unlikely that the effects of the presence of a cloud or of an atom can be seen at distances much larger than this effective size. Similarly, we can guess that two points in space-time at a macroscopic distance from each other can be distinguished because the probability that they will be confused drops rapidly with increasing distance. In short, we are thus led to a probabilistic description of space-time. Space-time becomes a macroscopic observable, a statistical or thermodynamic limit of some microscopic entities. We note that a fluctuating structure for space-time would also avoid the problems of fixed structures with Lorentz invariance. This property is of course compatible with a statistical description. In summary, the experimental observations of special relativity, i.e. Lorentz invariance, isotropy and homogeneity, together with that of a minimum distance, point towards a fluctuating description of space-time. Research efforts in quantum gravity, superstring theory and quantum groups have confirmed independently of each other that a probabilistic and non-local description of space-time at Planck dimensions, resolves the contradictions between general relativity and quantum theory. This is our first result on quantum geometry. To clarify the issue, we have to turn to the concept of the particle.

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Ref. 1050

xi general rel ativity versus quantum mechanics

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have a finite, non-vanishing size. This property allows us to determine whether a particle is elementary or not. If it behaves like a point particle, it is elementary. At present, only the leptons (electron, muon, tau and the neutrinos), the quarks and the radiation quanta of the electromagnetic, weak and strong nuclear interactions (the photon, the W and Z bosons, the gluons) have been found to be elementary. A few more elementary particles are predicted by various refinements of the standard model. Protons, atoms, molecules, cheese, people, galaxies etc., are all composite, as shown in Table 74. Elementary particles are characterized by their vanishing size, their spin and their mass. Even though the definition of ‘elementary particle’ as point particle is all we need in the following argument, it is not complete, because it seems to leave open the possibility that future experiments could show that electrons or quarks are not elementary. This is not so! In fact, any particle smaller than its own Compton wavelength is elementary. If it were composite, there would be a lighter component inside it and this lighter particle would have a larger Compton wavelength than the composite particle. This is impossible, since the size of a composite particle must be larger than the Compton wavelength of its components. (The possibility that all components are heavier than the composite, which would avoid this argument, does not lead to satisfying physical properties; for example, it leads to intrinsically unstable components.) The size of an object, such as those given in Table 74, is defined as the length at which differences from point-like behaviour are observed. This is the way in which, using alpha particle scattering, the radius of the atomic nucleus was determined for the first time in Rutherford’s experiment. In other words, the size d of an object is determined by measuring how it scatters a beam of probe particles. In daily life as well, when we look at objects, we make use of scattered photons. In general, in order to make use of scattering, the effective wavelength λ = ħ~mv of the probe must be smaller than the object size d to be determined. We thus need d A λ = ħ~(mv) E ħ~(mc). In addition, in order to make a scattering experiment possible, the object must not be a black hole, since, if it were, it would simply swallow the approaching particle. This means that its mass m must be smaller than that of a black hole of the same size; in other words, from equation (664) we must have m < dc 2 ~G. Combining this with the previous condition we get ħG = l Pl . c3

(677)

In other words, there is no way to observe that an object is smaller than the Planck length. There is thus no way in principle to deduce from observations that a particle is point-like. In fact, it makes no sense to use the term ‘point particle’ at all! Of course, there is a relation between the existence of a minimum length for empty space and a minimum length for objects. If the term ‘point of space’ is meaningless, then the term ‘point particle’ is also meaningless. As in the case of time, the lower limit on length results from the combination of quantum mechanics and general relativity.* The size d of any elementary particle must by definition be smaller than its own Compton wavelength ħ~(mc). Moreover, the size of a particle is always larger than the * Obviously, the minimum size of a particle has nothing to do with the impossibility, in quantum theory, of localizing a particle to within less than its Compton wavelength.

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dA

¾

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Ref. 1051

1011

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xi general rel ativity versus quantum mechanics

1012

Planck length: d A l Pl . Combining these two requirements and eliminating the size d we get the condition for the mass m of any elementary particle, namely ħ m< = c l Pl

Ref. 1052

Ref. 1054

Ref. 1055

ħc = m Pl = 2.2 ë 10−8 kg = 1.2 ë 1019 GeV~c2 . G

(678)

The limit m Pl , the so-called Planck mass, corresponds roughly to the mass of a human embryo that is ten days old, or equivalently, to that of a small flea. In short, the mass of any elementary particle must be smaller than the Planck mass. This fact was already noted as ‘well-known’ by Andrei Sakharov* in 1968; he explains that these hypothetical particles are sometimes called ‘maximons’. And indeed, the known elementary particles all have masses well below the Planck mass. (In fact, the question why their masses are so incredibly much smaller than the Planck mass is one of the most important questions of high-energy physics. We will come back to it.) There are many other ways to arrive at the mass limit for particles. For example, in order to measure mass by scattering – and that is the only way for very small objects – the Compton wavelength of the scatterer must be larger than the Schwarzschild radius; otherwise the probe will be swallowed. Inserting the definition of the two quantities and neglecting the factor 2, we get again the limit m < m Pl . (In fact it is a general property of descriptions of nature that a minimum spacetime interval leads to an upper limit for elementary particle masses.) The importance of the Planck mass will become clear shortly. Another property connected with the size of a particle is its electric dipole moment. It describes the deviation of its charge distribu- Andrei Sakharov tion from spherical. Some predictions from the standard model of elementary particles give as upper limit for the electron dipole moment d e a value of (679)

Sd e S < 3 ë 10−21 m , e

(680)

where e is the charge of the electron. This value is ten thousand times smaller than the Planck length l Pl e. Since the Planck length is the smallest possible length, we seem to have a potential contradiction here. However, a more recent prediction from the standard model is more careful and only states

which is not in contradiction with a minimal length in nature. The issue is still not settled. We will see below that the experimental limit is expected to allow to test these predictions * Andrei Dmitrievich Sakharov, famous Soviet nuclear physicist (1921–1989). One of the keenest thinkers in physics, Sakharov, among others, invented the Tokamak, directed the construction of nuclear bombs, and explained the matter-antimatter asymmetry of nature. Like many others, he later campaigned against nuclear weapons, a cause for which he was put into jail and exile, together with his wife, Yelena Bonner. He received the Nobel Peace Prize in 1975.

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Sd e S < 10−39 m , e

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Ref. 1053

¾

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Farewell to mass The Planck mass divided by the Planck volume, i.e. the Planck density, is given by ρ Pl =

(681)

and is a useful concept in the following. If we want to measure the (gravitational) mass M enclosed in a sphere of size R and thus (roughly) of volume R 3 , one way to do this is to put a test particle in orbit around it at that same distance R. Universal gravitation then gives for the mass M the expression M = Rv 2 ~G, where v is the speed of the orbiting test particle. From v < c, we thus deduce that M < c 2 R~G; since the minimum value for R is the Planck distance, we get (neglecting again factors of order unity) a limit for the mass density ρ, namely ρ < ρ Pl . (682)

In other words, the Planck density is the maximum possible value for mass density. Unsurprisingly, a volume of Planck dimensions cannot contain a mass larger than the Planck mass. Interesting things happen when we start to determine the error ∆M of a mass measurement in a Planck volume. Let us return to the mass measurement by an orbiting probe.

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Challenge 1429 e

c5 = 5.2 ë 1096 kg~m3 G2ħ

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in the foreseeable future. Planck scales have other strange consequences. In quantum field theory, the difference between a virtual particle and a real particle is that a real particle is ‘on shell’, obeying E 2 = m 2 c 4 + p2 c 2 , whereas a virtual particle is ‘off shell’, obeying E 2 x m 2 c 4 + p2 c 2 . Because of the fundamental limits of measurement precision, at Planck scales we cannot determine whether a particle is real or virtual. However, that is not all. Antimatter can be described as matter moving backwards in time. Since the difference between backwards and forwards cannot be determined at Planck scales, matter and antimatter cannot be distinguished at Planck scales. Particles are also characterized by their spin. Spin describes two properties of a particle: its behaviour under rotations (and if the particle is charged, its behaviour in magnetic fields) and its behaviour under particle exchange. The wave function of particles with spin 1 remains invariant under a rotation of 2π, whereas that of particles with spin 1/2 changes sign. Similarly, the combined wave function of two particles with spin 1 does not change sign under exchange of particles, whereas for two particles with spin 1/2 it does. We see directly that both transformations are impossible to study at Planck scales. Given the limit on position measurements, the position of a rotation axis cannot be well defined, and rotations become impossible to distinguish from translations. Similarly, position imprecision makes impossible the determination of precise separate positions for exchange experiments. In short, spin cannot be defined at Planck scales, and fermions cannot be distinguished from bosons, or, phrased differently, matter cannot be distinguished from radiation at Planck scales. We can thus easily see that supersymmetry, a unifying symmetry between bosons and fermions, somehow becomes natural at Planck dimensions. But let us now move to the main property of elementary particles.

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xi general rel ativity versus quantum mechanics

M

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R

F I G U R E 381 A Gedanken experiment showing that

From the relation GM = rv 2 we deduce by differentiation that G∆M = v 2 ∆r + 2vr∆v A 2vr∆v = 2GM∆v~v. For the error ∆v in the velocity measurement we have the indeterminacy relation ∆v E ħ~(m∆r) + ħ~(MR) E ħ~(MR). Inserting this in the previous inequality, and forgetting again the factor of 2, we find that the mass measurement error ∆M of a mass M enclosed in a volume of size R is subject to the condition ∆M E

ħ . cR

(683)

Copyright © Christoph Schiller November 1997–May 2006

Note that for everyday situations, this error is extremely small, and other errors, such as the technical limits of the balance, are much larger. To check this result, we can explore another situation. We even use relativistic expressions, in order to show that the result does not depend on the details of the situation or the approximations. Imagine having a mass M in a box of size R and weighing the box with a scale. (It is assumed that either the box is massless or that its mass is subtracted by the scale.) The mass error is given by ∆M = ∆E~c 2 , where ∆E is due to the indeterminacy in the kinetic energy of the mass inside the box. Using the expression E 2 = m 2 c 4 + p2 c 2 , we get that ∆M E ∆p~c, which again reduces to equation (683). Now that we are sure of the result, let us continue. From equation (683) we deduce that for a box of Planck dimensions, the mass measurement error is given by the Planck mass. But from above we also know that the mass that can be put inside such a box must not be larger than the Planck mass. Therefore, for a box of Planck dimensions, the mass measurement error is larger than (or at best equal to) the mass contained in it: ∆M E M Pl . In other words, if we build a balance with two boxes of Planck size, one empty and the other full, as shown in Figure 381, nature cannot decide which way the balance should hang! Note that even a repeated or a continuous measurement will not resolve the situation: the balance will only randomly change inclination, staying horizontal on average. The argument can be rephrased as follows. The largest mass that we can put in a box

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at Planck scales, matter and vacuum cannot be distinguished

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of size R is a black hole with a Schwarzschild radius of the same value; the smallest mass present in such a box – corresponding to what we call vacuum – is due to the indeterminacy relation and is given by the mass with a Compton wavelength that matches the size of the box. In other words, inside any box of size R we have a mass m, the limits of which are given by: ħ c2 R AmA (empty box) . (684) (full box) G cR

∆M ∆δλ = . M δλ

(685)

Copyright © Christoph Schiller November 1997–May 2006

In order to determine the mass in a Planck volume, the probe has to have a wavelength of the Planck length. But we know from above that there always is a minimum wavelength indeterminacy, given by the Planck length l Pl . In other words, for a Planck volume the mass error is always as large as the Planck mass itself: ∆M E M Pl . Again, this limit is a direct consequence of the limit on length and space measurements. This result has an astonishing consequence. In these examples, the measurement error is independent of the mass of the scatterer, i.e. independent of whether or not we start with a situation in which there is a particle in the original volume. We thus find that in a volume of Planck size, it is impossible to say whether or not there is something there when we probe it with a beam! In short, all arguments lead to the same conclusion: vacuum, i.e. empty space-time, cannot be distinguished from matter at Planck scales. Another, often used way to express this state of affairs is to say that when a particle of Planck energy travels through space it will be scattered by the fluctuations of space-time itself, thus making it impossible to say whether it was scattered by empty space-time or by matter. These surprising results rely on a simple fact: whatever definition of mass we use, it is always measured via combined length and time measurements. (This is even the case for normal weighing scales: mass is measured by the displacement of some part of the machine.) The error in these measurements makes it impossible to distinguish vacuum from matter. We can put this result in another way. If on one hand, we measure the mass of a piece

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We see directly that for sizes R of the order of the Planck scale, the two limits coincide; in other words, we cannot distinguish a full box from an empty box in that case. To be sure of this strange result, we check whether it also occurs if, instead of measuring the gravitational mass, as we have just done, we measure the inertial mass. The inertial mass for a small object is determined by touching it, i.e. physically speaking, by performing a scattering experiment. To determine the inertial mass inside a region of size R, a probe must have a wavelength smaller than R, and thus a correspondingly high energy. A high energy means that the probe also attracts the particle through gravity. (We thus find the intermediate result that at Planck scales, inertial and gravitational mass cannot be distinguished. Even the balance experiment shown in Figure 381 illustrates this: at Planck scales, the two types of mass are always inextricably linked.) Now, in any scattering experiment, e.g. in a Compton-type experiment, the mass measurement is performed by measuring the wavelength change δλ of the probe before and after the scattering experiment. The mass indeterminacy is given by

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Ref. 1056

of vacuum of size R, the result is always at least ħ~cR; there is no possible way to find a perfect vacuum in an experiment. On the other hand, if we measure the mass of a particle, we find that the result is size dependent; at Planck dimensions it approaches the Planck mass for every type of particle, be it matter or radiation. If we use another image, when two particles approach each other to a separation of the order of the Planck length, the indeterminacy in the length measurements makes it impossible to say whether there is something or nothing between the two objects. In short, matter and vacuum are interchangeable at Planck dimensions. This is an important result: since both mass and empty space-time cannot be differentiated, we have confirmed that they are made of the same ‘fabric’. This approach, already suggested above, is now commonplace in all attempts to find a unified description of nature. This approach is corroborated by the attempts to apply quantum mechanics in highly curved space-time, where a clear distinction between vacuum and particles is impossible. This has already been shown by Fulling–Davies–Unruh radiation. Any accelerated observer and any observer in a gravitational field detects particles hitting him, even if he is in vacuum. The effect shows that for curved space-time the idea of vacuum as a particlefree space does not work. Since at Planck scales it is impossible to say whether space is flat or not, it again follows that it is impossible to say whether it contains particles or not. Curiosities and fun challenges on Planck scales The strange results at Planck scales imply many other consequences. ** Observes are made of matter. Observer are thus biased, because they take a specific standpoint. But at Planck scale, vacuum, radiation and matter cannot me distinguished. Two conclusions result: first, only at those scales would a description be free of any bias in favour of matter; but secondly, observers do not exist at all at Planck energy. **

Challenge 1430 n Challenge 1431 n

The Planck energy is rather large. Imagine that we want to impart this amount of energy to protons using a particle accelerator. How large would that accelerator have to be? In contrast, in everyday life, the Planck energy is rather small. Measured in litres of gasoline, how much fuel does it correspond to?

The usual concepts of matter and of radiation are not applicable at Planck dimensions. Usually, it is assumed that matter and radiation are made up of interacting elementary particles. The concept of an elementary particle is one of an entity that is countable, pointlike, real and not virtual, that has a definite mass and a definite spin, that is distinct from its antiparticle, and, most of all, that is distinct from vacuum, which is assumed to have zero mass. All these properties are found to be incorrect at Planck scales. At Planck dimensions, it does not make sense to use the concepts of ‘mass’, ‘vacuum’, ‘elementary particle’, ‘radiation’ and ‘matter’. **

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**

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d oes matter differ from vacuum?

Challenge 1432 n

1017

Do the large mass measurement errors make it possible to claim that mass can be negative at Planck energy? ** We now have a new answer to the old question: why is there anything rather than nothing? Well, we now see that at Planck scales there is no difference between anything and nothing. In addition, we now can honestly say about ourselves that we are made of nothing.

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** If vacuum and matter or radiation cannot be distinguished, then it is incorrect to claim that the universe appeared from nothing. The impossibility of making this distinction thus shows that naive creation is a logical impossibility. Creation is not a description of reality. The term ‘creation’ turns out to be a result of lack of imagination.

Challenge 1433 r Ref. 1058

Special relativity implies that no length or energy can be invariant. Since we have come to the conclusion that the Planck energy and the Planck length are invariant, there must be deviations from Lorentz invariance at high energy. Can you imagine what the effects would be? In what experiment could they be measured? If you find an answer, publish it; you might get known. First attempts are appearing in the research papers. We return to the issue in the third part, with some interesting insights. **

Ref. 1057 Ref. 1038

Challenge 1434 ny

Quantum mechanics alone gives, via the Heisenberg indeterminacy relation, a lower limit on the spread of measurements, but strangely enough not on their precision, i.e. not on the number of significant digits. Jauch gives the example that atomic lattice constants are known much more precisely than the position indeterminacy of single atoms inside the crystal. It is sometimes claimed that measurement indeterminacies smaller than the Planck values are possible for large enough numbers of particles. Can you show why this is incorrect, at least for space and time? **

**

Challenge 1435 n

A Planck energy particle falling in a gravitational field would gain energy. However, this is impossible, as the Planck energy is the highest energy in nature. What does this imply for this situation? **

Ref. 1027

One way to generalize the results presented here is to assume that, at Planck energy, nature is event symmetric, i.e. nature is symmetric under exchange of any two events. This idea,

Copyright © Christoph Schiller November 1997–May 2006

Of course, the idea that vacuum is not empty is not new. More than two thousand years ago, Aristotle argued for a filled vacuum, even though he used incorrect arguments, as seen from today’s perspective. In the fourteenth century the discussion on whether empty space was composed of indivisible entities was rather common, but died down again later.

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**

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developed by Phil Gibbs, provides an additional formulation of the strange behaviour of nature at extreme scales. ** Page 489

Because there is a minimum length in nature, so-called naked singularities do not exist. The issue, so hotly debated in the twentieth century, becomes uninteresting, thus ending decades of speculation.

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** Page 870

** Page 37

If vacuum and matter cannot be distinguished, we cannot distinguish between objects and their environment. However, this was one the starting points of our journey. Some interesting adventures thus still await us! **

Page 676

We have seen earlier that characterizing nature as made up of particles and vacuum creates problems when interactions are included, since on one hand interactions are the difference between the parts and the whole, while on the other hand, according to quantum theory, interactions are exchanges of particles. This apparent contradiction can be used to show either that vacuum and particles are not the only components of nature, or that something is counted twice. However, since matter and space-time are both made of the same ‘stuff,’ the contradiction is resolved. ** Is there a smallest possible momentum? And a smallest momentum error? **

Challenge 1437 n

There is a maximum acceleration in nature. Can you deduce the value of this so-called Planck acceleration? Does it require quantum theory? ** Given that time becomes an approximation at Planck scales, can we still say whether nature is deterministic? Let us go back to the beginning. We can define time, because in nature change is not random, but gradual. What is the situation now that we know that time is only approximate? Is non-gradual change possible? Is energy conserved? In other words, are surprises possible?

Copyright © Christoph Schiller November 1997–May 2006

Challenge 1436 d

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Ref. 1059

Since mass density and thus energy density are limited, we know that the number of degrees of freedom of any object of finite volume is finite. The entropy of black holes has shown us already that entropy values are always finite. This implies that perfect baths do not exist. Baths play an important role in thermodynamics (which is thus found to be only an approximation) and also in recording and measuring devices: when a device measures, it switches from a neutral state to a state in which it shows the result of the measurement. In order to avoid the device returning to the neutral state, it must be coupled to a bath. Without a bath, a reliable measuring device cannot be made. In short, perfect clocks and length measuring devices do not exist because nature puts a limit on their storage ability.

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Challenge 1438 n

1019

To say that time is not defined at Planck scales and that therefore determinism is an undefinable concept is correct, but not a satisfying answer. What happens at daily life scales? The first answer is that at our everyday scales, the probability of surprises is so small that the world indeed is effectively deterministic. The second answer is that nature is not deterministic, but that the difference is not measurable, since every measurement and observation, by definition, implies a deterministic world. The lack of surprises would be due to the limitations of our human nature, and more precisely to the limitations of our senses and brain. The third answer is that the lack of surprises is only apparent, and that we have not yet experienced them yet. Can you imagine any other possibility? To be honest, it is not possible to answer at this point. But we need to keep the alternatives in mind. We have to continue searching, but with every step we take, we have to consider carefully what we are doing.

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**

**

Ref. 1060

Vacuum has zero mass density at large scales, but Planck mass density at Planck scales. Cosmological measurements show that the cosmos is flat or almost flat at large scales, i.e. its energy density is quite low. In contrast, quantum field theory maintains that vacuum has a high energy density (or mass density) at small scales. Since mass is scale dependent, both viewpoints are right, providing a hint to the solution of what is usually called the cosmological constant problem. The contradiction is only apparent; more about this issue later on. **

Challenge 1439 n

When can matter and vacuum be distinguished? At what energy? **

** Can we distinguish between liquids and gases by looking at a single atom? No, only by looking at many. In the same way, we cannot distinguish between matter and vacuum by looking at one point, but only by looking at many. We must always average. However, even averaging is not completely successful. Distinguishing matter from vacuum is like distinguishing clouds from the clear sky; like clouds, matter also has no defined boundary. ** In our exploration we have found that there is no argument which shows that space and

Copyright Š Christoph Schiller November 1997â&#x20AC;&#x201C;May 2006

If matter and vacuum cannot be distinguished, there is a lack of information, which in turn produces an intrinsic basic entropy associated with any part of the universe. We will come back to this topic shortly, in the discussion of the entropy of black holes.

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If matter and vacuum cannot be distinguished, matter and vacuum each has the properties of the other. For example, since space-time is an extended entity, matter and radiation are also extended entities. Furthermore, as space-time is an entity that reaches the borders of the system under scrutiny, particles must also do so. This is the first hint at the extension of matter; in the following, we will examine this argument in more detail.

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xi general rel ativity versus quantum mechanics

time are either continuous or made up of points. Indeed, in contrast, we have found that the combination of relativity and quantum theory makes this impossible. In order to proceed in our ascent of Motion Mountain, we need to leave behind us the usual concept of space-time. At Planck dimensions, the concept of ‘space-time point’ or ‘mass point’ is not applicable in the description of nature. Farewell to the big bang A minimum length, or equivalently,* a minimum action, both imply that there is a maximum curvature for space-time. Curvature can be measured in several ways; for example, surface curvature is an inverse area. A minimum length thus implies a maximum curvature. Within a factor of order one, we find

Challenge 1440 n

c3 = 0.39 ë 1070 m−2 . Għ

(686)

as limit for surface curvature K in nature. In other words, the universe never has been a point, never had zero age, never had infinite density and never had infinite curvature. It is not difficult to get a similar limit for temperature or any other physical quantity. In short, since events do not exist, also the big bang cannot have been an event. There never was an initial singularity or a beginning of the universe. The baggage left behind

Page 1068

Ref. 1013

* The big bang section was added in summer 2002.

Copyright © Christoph Schiller November 1997–May 2006

Ref. 1012

In this rapid journey, we have destroyed all the experimental pillars of quantum theory: the superposition principle, space-time symmetry, gauge symmetry, renormalization symmetry and permutation symmetry. We also have destroyed the foundations of general relativity, namely the existence of the space-time manifold, the field concept, the particle concept and the concept of mass. We have even seen that matter and space-time cannot be distinguished. All these conclusions can be drawn in a simpler manner, by using the minimum action of quantum theory and the maximum force of general relativity. All the mentioned results above are confirmed. It seems that we have lost every concept used for the description of motion, and thus made its description impossible. We naturally ask whether we can save the situation. First of all, since matter is not distinguishable from vacuum, and since this is true for all types of particles, be they matter or radiation, we have an argument which demonstrates that the quest for unification in the description of elementary particles is correct and necessary. Moreover, since the concepts ‘mass’, ‘time’ and ‘space’ cannot be distinguished from each other, we also know that a new, single entity is necessary to define both particles and space-time. To find out more about this new entity, three approaches are being pursued at the beginning of the twenty-first century. The first, quantum gravity, especially the approach using the loop representation and Ashtekar’s new variables, starts by generalizing space-time symmetry. The second, string theory, starts by generalizing gauge symmetries

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K<

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some experimental predictions

Ref. 1014

1021

and interactions, while the third, the algebraic quantum group approach, looks for generalized permutation symmetries. We will describe these approaches in more detail later on. Before we go on however, we should check with experiments what we have deduced so far.

S ome experimental predictions

Ref. 1061

Ref. 1063, Ref. 1064

Ref. 1065, Ref. 1064

Ref. 1066

Ref. 1067

ħ(ω 1 − ω 2 )d . c∆t

(687)

This energy value is 8 ë 1016 GeV for the best measurement to date. This value is not far from the Planck energy; in fact, it is even closer when the missing factors of order unity are included. It is expected that the Planck scale will be reached in a few years, so that tests will become possible on whether the quantum nature of space-time influences the dispersion of light signals. Planck scale effects should produce a minimum dispersion, different from zero. Detecting it would confirm that Lorentz symmetry is not valid at Planck scales. Another candidate experiment is the direct detection of distance fluctuations between bodies. Gravitational wave detectors are sensitive to extremely small noise signals in length measurements. There should be a noise signal due to the distance fluctuations induced near Planck energies. The length indeterminacy with which a length l can be measured is predicted to be l Pl 2~3 δl E‹  (688) l l

The expression is deduced simply by combining the measurement limit of a ruler in quantum theory with the requirement that the ruler cannot be a black hole. We will discuss this result in more detail in the next section. The sensitivity to noise of the detectors might reach the required level in the early twenty-first century. The noise induced by quantum gravity effects is also predicted to lead to detectable quantum decoherence and vacuum fluctuations. * As more candidates appear, they will be added to this section.

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Page 1036

E char =

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Ref. 1062

At present, there is a race going on both in experimental and in theoretical physics: which will be the first experiment that will detect quantum gravity effects, i.e. effects sensitive to the Planck energy?* One might think that the fluctuations of space and time might make images from far away galaxies unsharp or destroy the phase relation between the photons. However, this effect has been shown to be unmeasurable in all possible cases. A better candidate is measurement of the speed of light at different frequencies in far away light flashes. There are natural flashes, called gamma ray bursts, which have an extremely broad spectrum, from 100 GeV down to visible light of about 1 eV. These flashes often originate at cosmological distances d. From the difference in arrival time ∆t for two frequencies we can define a characteristic energy by setting

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1022

Ref. 1064

Ref. 1068

Ref. 1069 Ref. 1067

xi general rel ativity versus quantum mechanics

Copyright © Christoph Schiller November 1997–May 2006

* This subsection, in contrast to the ones so far, is speculative; it was added in February 2001. ** The entropy of a black hole is thus given by the ratio between its horizon and º half the minimum area. Of course, a detailed investigation also shows that the Planck mass (divided by 8 ) is the limit for elementary particles from below and for black holes from above. For everyday systems, there is no limit.

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A third candidate for measurable quantum gravity is the detection of the loss of CPT symmetry at high energies. Especially in the case of the decay of certain elementary particles, such as neutral kaons, the precision of experimental measurement is approaching the detection of Planck scale effects. A fourth candidate is the possibility that quantum gravity effects may change the threshold energy at which certain particle reactions become possible. It may be that extremely high energy photons or cosmic rays will make it possible to prove that Lorentz invariance is indeed broken near the Planck scale. A fifth candidate is the possibility that the phase of light that travels over long distances gets washed out. However, the first tests show that this is not the case; light form extremely distant galaxies still interferes. The precise prediction of the phase washing effect is still in discussion; most probably the effect is too small to be measured. In the domain of atomic physics, it has also been predicted that quantum gravity effects will induce a gravitational Stark effect and a gravitational Lamb shift in atomic transitions. Either effect could be measurable. A few candidates for quantum gravity effects have also been predicted by the author. To get an overview, we summarize and clarify the results found so far.* Special relativity starts with the discovery that observable speeds are limited by the speed of light c. Quantum theory starts with the result that observable actions are limited by ħ~2. Gravitation shows that for every system with length L and mass M, the observable ratio L~M is limited by the constant 4G~c 2 . Combining these results, we have deduced that all physical observables are bound, namely by what are usually called the Planck values, though modified by a factor of square root of 2 (or several of them) to compensate for the numerical factors omitted from the previous sentence. We need to replace ħ by ħ~2 and G by 4G in all the defining expressions for Planck quantities, in order to find º the corresponding measurement limits. In particular, the limit 2 times the Planck value and the limit for energy is the Planck for lengths and times is º value divided by 8 .** Interestingly, the existence of bounds on all observables makes it possible to deduce several experimentally testable predictions for the unification of quantum theory and general relativity. These predictions do not depend on the detailed final theory. However, first we need to correct the argument that we have just presented. The argument is only half the story, because we have cheated. The (corrected) Planck values do not seem to be the actual limits to measurements. The actual measurement limits are even stricter still. First of all, for any measurement, we need certain fundamental conditions to be realized. Take the length measurement of an object. We need to be able to distinguish between matter and radiation, since the object to be measured is made up of matter, and since radiation is the measurement tool that is used to read distances from the ruler. For a measurement process, we need an interaction, which implies the use of radiation. Note that even the use of particle scattering to determine lengths does not invalidate this general requirement.

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some experimental predictions

Page 1104 Ref. 1070

In addition, for the measurement of wavelengths we need to distinguish between matter and radiation, because matter is necessary to compare two wavelengths. In fact, all length measurements require the distinction between matter and radiation.* However, this distinction is impossible at the energy of grand unification, when the electroweak and the strong nuclear interactions are unified. At and above this energy, particles of matter and particles of radiation transform into each other; in practice they cannot be distinguished from each other. If all matter and radiation particles were the same or mixtures of each other, mass could not be defined. Similarly, spin, charge or any other quantum numbers could be defined. To sum up, no measurement can be performed at energies equal to or greater than the GUT unification energy. In other words, the particle concept (and thus the matter concept) does not run into trouble at the Planck scale, it has already done so at the unification scale. Only below the unification scale do our standard particle and space-time concepts apply. Only below the unification scale can particles and vacuum be effectively distinguished. º As a result, the smallest length in nature º is 2 times the Planck length reduced by the ratio between the maximal energy E Pl ~ 8 and the unification energy E GUT . Present estimates give E GUT = 1016 GeV, implying that the smallest accessible length L min is L min =

º E Pl 2 l Pl º  10−32 m  800 l Pl . 8 E GUT

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d A d min = e L min = e 10−32 m = 1.5 ë 10−51 Cm .

Page 1012 Ref. 1072

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We saw that this result is in contradiction with one of the predictions deduced from the standard model, but not with others. More interestingly, the prediction is in the reach of future experiments. This improved limit may be the simplest possible measurement of yet unpredicted quantum gravity effects. Measuring the dipole moment could thus be a way to determine the unification energy (the factor 800) independently of high-energy physics experiments and possibly to a higher precision. * To speak in modern high energy concepts, all measurements require broken supersymmetry.

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It is unlikely that measurements at these dimensions will ever be possible. Anyway, the smallest measurable length is significantly larger than the Planck scale of nature discussed above. The reason for this is that the Planck scale is that length for which particles and vacuum cannot be distinguished, whereas the minimal measurable length is the distance at which particles of matter and particles of radiation cannot be distinguished. The latter happens at lower energy than the former. We thus have to correct our previous statement to: the minimum measurable length cannot be smaller than L min . The experimentally determined factor of about 800 is one of the great riddles of physics. It is the high-energy equivalent of the quest to understand why the electromagnetic coupling constant is about 1~137, or more simply, why all things have the colours they have. Only the final theory of motion will provide the answer. In particular, the minimum length puts a bound on the electric dipole moment d of elementary particles, i.e. on any particles without constituents. We get the limit

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Ref. 1071

1023

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1024 Inverse of coupling

electromagnetism

Dvipsbugw 1/26

weak interaction

105

1010

1015 GUT energy

"1020" Planck energy

Energy in GeV

F I G U R E 382 Coupling constants and their spread as a function of energy

Interestingly, the bound on the measurability of observables also puts a bound on the measurement precision for each observable. This bound is of no importance in everyday life, but it is important at high energy. What is the precision with which a coupling constant can be measured? We can illustrate this by taking the electromagnetic coupling constant as an example. This constant α, also called the fine structure constant, is related to the charge q by ¼ q=

4πε 0 ħcα .

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q ma = . q unit m unit a unit

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Therefore any error in mass and acceleration measurements implies errors in measurements of charge and the coupling constant. We found in the part on quantum theory that the electromagnetic, the weak and the strong interactions are characterized by coupling constants, the inverse of which depend linearly on the logarithm of the energy. It is usually assumed that these three lines meet at the unification energy already mentioned. Measurements put the unification coupling

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Now, any electrical charge itself is defined and measured by comparing, in an electrical field, the acceleration to which the charged object is subjected with the acceleration of some unit charge q unit . In other words, we have

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strong interaction

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some experimental predictions

Ref. 1071

value at about 1/26. We know from the above discussions that the minimum measurement error for any energy measurement at high energies is given by the ratio between the energy to be measured and the limit energy. Inserting this into the graph of the coupling constants ‘running’ with energy – as physicist like to say – we get the result shown in Figure 382. The search for the consequences of this fan-out effect is delightful. One way to put the result is to say that coupling constants are by definition subject to an error. However, all measurement devices, be they clocks, metre rules, scales or any other device, use electromagnetic effects at energies of around 1 eV plus the electron rest energy. This is about 10−16 times the GUT energy. As a consequence, the measurement precision of any observable is limited to about 19 digits. The maximum precision presently achieved is 15 digits, and, for the electromagnetic coupling constant, about 9 digits. It will thus be quite some time before this prediction can be tested. The fun is thus to find a system in which the spreading coupling constant value appears more clearly in the measurements. For example, it may be that high-precision measurements of the д-factor of elementary particles or of high-energy cosmic ray reactions will show some effects of the fan-out. The lifetime of elementary particles could also be affected. Can you find another effect? In summary, the experimental detection of quantum gravity effects should be possible, despite their weakness, at some time during the twenty-first century. The successful detection of any such effect will be one of the highlights of physics, as it will challenge the usual description of space and time even more than general relativity did. We now know that the fundamental entity describing space-time and matter that we are looking for is not point-like. What does it look like? To get to the top of Motion Mountain as rapidly as possible, we will make use of some explosive to blast away a few disturbing obstacles.

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Challenge 1441 n

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Biblio graphy 1003 Steven Weinberg, The cosmological constant problem, Reviews of Modern Physics 61, 1004 1005 1006 1007

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1010

1011 1012

1013

1015 1016 1017

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pp. 1–23, 1989. Cited on page 996. Steven Weinberg, The quantum theory of fields, Cambridge University Press, volume I, 1995, and II, 1996. Cited on page 996. The difficulties are summarised by B.S. de Witt, Quantum field theory in curved spacetime, Physics Reports 19, pp. 295–357, 1975. Cited on page 997. C.W. Misner & K.S. Thorne, J.A. Wheeler, Gravitation, Freeman, 1973. Cited on pages 998, 1000, and 1004. J.A. Wheeler, in Relativity, Groups and Topology, edited by C. DeWitt & B. DeWitt, Gordon and Breach, 1994. See also J.A. Wheeler, Physics at the Planck length, International Journal of Modern Physics A 8, pp. 4013–4018, 1993. Cited on page 998. J.L. Friedman & R.D. Sorkin, Spin 1/2 from gravity, Physical Review Letters 44, pp. 1100–1103, 1980. Cited on page 998. A.P. Bal achandran, G. Bimonte, G. Marmo & A. Simoni, Topology change and quantum physics, Nuclear Physics B 446, pp. 299–314, 1995, http://www.arxiv.org/abs/ hep-th/9503046. Cited on page 998. Many themes discussed here resulted from an intense exchange with Luca Bombelli, now at the Department of Physics and Astronomy of the University of Mississippi. These pages are a reworked version of those published in French as C. Schiller, Le vide diffère-t-il de la matière? in E. Gunzig & S. Diner, editeurs, Le vide – Univers du tout et du rien – Des physiciens et des philosophes s’interrogent, Les Éditions de l’Université de Bruxelles, 1998. An older English version is also available as C. Schiller, Does matter differ from vacuum? http://www.arxiv.org/abs/gr-qc/9610066. Cited on page 998. See for example R ichard P. Feynman, R.B. Leighton & M. Sands, The Feynman Lectures on Physics, Addison Wesley, 1977. Cited on page 998. A. Ashtekar, Recent mathematical developments in quantum general relativity, preprint found at http://www.arxiv.org/abs/gr-qc/9411055. C. Rovelli, Ashtekar Formulation of general relativity and loop-space non-perturbative quantum gravity: a report, Classical and Quantum Gravity 8, pp. 1613–1675, 1991. Cited on pages 999 and 1020. M.B. Green, J.H. Schwarz & E. Witten, Superstring theory, volume 1 and 2 (Cambridge University Press, 1987). G. Veneziano, A stringy nature needs just two constants, Europhysics Letters 2, pp. 199–204, 1986. Cited on pages 999 and 1020. S. Majid, Introduction to Braided Geometry and q-Minkowski Space, preprint downloadable at http://www.arxiv.org/abs/hep-th/9410241. See also S. Majid, Beyond supersymmetry and quantum symmetry (an introduction to braided groups and braided matrices), in Quantum Groups, Integrable Statistical Models and Knot Theory, edited by M.-L. Ge & H.J. de Vega, World Scientific, 1993, pp. 231–282. Cited on pages 999 and 1021. C.J. Isham, Prima facie questions in quantum gravity, http://www.arxiv.org/abs/gr-qc/ 9310031. Cited on page 999. Steven Weinberg, Gravitation and Cosmology, Wiley, 1972. Cited on pages 1000, 1003, and 1004. The observations of black holes at the centre of galaxies and elsewhere are summarised by R. Bl andford & N. Gehrels, Revisiting the black hole, Physics Today 52, June 1999.

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Cited on page 1000. M. Pl anck, Über irreversible Strahlungsvorgänge, Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften zu Berlin, pp. 440–480, 1899. Today it is commonplace to use Dirac’s ħ instead of Planck’s h, which Planck originally called b. Cited on page 1001. The argument is given e.g. in E.P. Wigner, Relativistic invariance and quantum phenomena, Reviews of Modern Physics 29, pp. 255–258, 1957. Cited on page 1001. The starting point for the following arguments is taken from M. Schön, Operative time definition and principal indeterminacy, preprint http://www.arxiv.org/abs/gr-qc/9304024, and from T. Padmanabhan, Limitations on the operational definition of space-time events and quantum gravity, Classical and Quantum Gravity 4, pp. L107–L113, 1987; see also Padmanabhan’s earlier papers referenced there. Cited on page 1001. W. Heisenberg, Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Zeitschrift für Physik 43, pp. 172–198, 1927. Cited on page 1002. E.H. Kennard, Zur Quantenmechanik einfacher Bewegungstypen, Zeitschrift für Physik 44, pp. 326–352, 1927. Cited on page 1002. M.G. R aymer, Uncertainty principle for joint measurement of noncommuting variables, American Journal of Physics 62, pp. 986–993, 1994. Cited on page 1002. H. Salecker & E.P. Wigner, Quantum limitations of the measurement of space-time distances, Physical Review 109, pp. 571–577, 1958. Cited on page 1003. E.J. Zimmerman, The macroscopic nature of space-time, American Journal of Physics 30, pp. 97–105, 1962. Cited on page 1003. S.W. Hawking, Particle creation by black holes, Communications in Mathematical Physics 43, pp. 199–220, 1975; see also S.W. Hawking, Black hole thermodynamics, Physical Review D13, pp. 191–197, 1976. Cited on page 1003. P. Gibbs, The small structure of space-time: a bibliographical review, unpublished preprint found at http://www.arxiv.org/abs/hep-th/9506171. Cited on pages 1002 and 1017. C.A. Mead, Possible connection between gravitation and fundamental length, Physical Review B 135, pp. 849–862, 1964. Cited on pages 1003, 1004, 1006, and 1027. P.K. Townsend, Small-scale structure of space-time as the origin of the gravitational constant, Physical Review D 15, pp. 2795–2801, 1977. Cited on page 1004. M.-T. Jaekel & S. R enaud, Gravitational quantum limit for length measurement, Physics Letters A 185, pp. 143–148, 1994. Cited on page 1004. D.V. Ahluwalia, Quantum measurement, gravitation and locality, Physics Letters B 339, pp. 301–303, 1994, also preprint http://www.arxiv.org/abs/gr-qc/9308007. Cited on page 1004. L. Garay, Quantum gravity and minimum length, International Journal of Modern Physics A 10, pp. 145–165, 1995, also preprint http://www.arxiv.org/abs/gr-qc/9403008; this paper also includes an extensive bibliography. Cited on page 1004. G. Amelino–Camelia, Limits on the measurability of space-time distances in (the semiclassical approximation of) quantum gravity, Modern Physics Letters A 9, pp. 3415– 3422, 1994, also preprint http://www.arxiv.org/abs/gr-qc/9603014. See also Y.J. Ng & H. Van Dam, Limit to space-time measurement, Modern Physics Letters A 9, pp. 335–340, 1994. Cited on page 1004. The generalised indeterminacy relation is an implicit part of Ref. 1028, especially on page 852, but the issue is explained rather unclearly. Obviously the author found the result too simple to be mentioned explicitly. Cited on pages 1004 and 1005.

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1023

1027

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1035 C. Rovelli & L. Smolin, Discreteness of area and volume in quantum gravity, Nuclear

Physics B 442, pp. 593–619, 1995. R. Loll, The volume operator in discretized quantum gravity, preprint http://www.arxiv.org/abs/gr-qc/9506014. Cited on page 1005. 1036 D. Amati, M. Ciafaloni & G. Veneziano, Superstring collisions at Planckian en-

ergies, Physics Letters B 197, pp. 81–88, 1987. D.J. Gross & P.F. Mende, The high energy behavior of string scattering amplitudes, Physics Letters B 197, pp. 129–134, 1987. K. Konishi, G. Paffu ti & P. Provero, Minimum physical length and the generalised uncertainty principle, Physics Letters B 234, pp. 276–284, 1990. P. Aspinwall, Minimum Distances in Non-Trivial String Target Spaces, Nuclear Physics B 431, pp. 78–96, 1994, also preprint http://www.arxiv.org/abs/hep-th/9404060. Cited on page 1005.

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1037 M. Maggiore, A generalised uncertainty principle in quantum mechanics, Physics Let-

ters B 304, pp. 65–69, 1993. Cited on page 1005. 1038 A simple approach is S. Doplicher, K. Fredenhagen & J.E. Roberts, Space-time

1039 A. Kempf, Uncertainty relation in quantum mechanics with quantum group symmetry,

Journal of Mathematical Physics 35, pp. 4483–4496, 1994. A. Kempf, Quantum Groups and quantum field theory with Nonzero minimal uncertainties in positions and momenta, Czechoslovak Journal of Physics 44, pp. 1041–1048, 1994. Cited on page 1005. 1040 E.J. Hellund & K. Tanaka, Quantized space-time, Physical Review 94, pp. 192–195,

1954. Cited on page 1005. 1041 The impossibility to determine temporal ordering in quantum theory is discussed by J.

Oppenheimer, B. R eznik & W.G. Unruh, Temporal ordering in quantum mechanics, Journal of Physics A 35, pp. 7641–7652, 2001, or http://www.arxiv.org/abs/quant-ph/ 0003130. Cited on page 1006. 1042 A. Peres & N. Rosen, Quantum Limitations on the measurement of gravitational fields,

Physical Review 118, pp. 335–336, 1960. Cited on page 1006. 1043 It is the first definition in Euclid’s Elements, c. 300 bce. For an English translation see T.

Heath, The Thirteen Books of the Elements, Dover, 1969. Cited on page 1007. 1044 A beautiful description of the Banach–Tarski paradox is the one by Ian Stewart, Para-

dox of the spheres, New Scientist, 14 January 1995, pp. 28–31. Cited on page 1008. 1045 H.S. Snyder, Quantized space-time, Physical Review 71, pp. 38–41, 1947. H.S. Snyder,

1046 D. Finkelstein, ‘Superconducting’ causal nets, International Journal of Theoretical Phys-

ics 27, pp. 473–519, 1985. Cited on page 1009. 1047 N.H. Christ, R. Friedberg & T.D. Lee, Random lattice field theory: general formu-

lation, Nuclear Physics B 202, pp. 89–125, 1982. G. ’t Hooft, Quantum field theory for elementary particles – is quantum field theory a theory?, Physics Reports 104, pp. 129–142, 1984. Cited on page 1009. 1048 See for example L. B ombelli, J. Lee, D. Meyer & R.D. Sorkin, Space-time as

a causal set, Physical Review Letters 59, pp. 521–524, 1987. G. Brightwell & R.

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The electromagnetic field in quantised of space-time, Physical Review 72, pp. 68–74, 1947. A. Schild, Discrete space-time and integral Lorentz transformations, Physical Review 73, pp. 414–415, 1948. E.L. Hill, Relativistic theory of discrete momentum space and discrete space-time, Physical Review 100, pp. 1780–1783, 1950. H.T. Flint, The quantization of space-time, Physical Review 74, pp. 209–210, 1948. A. Das, Cellular space-time and quantum field theory, Il Nuovo Cimento 18, pp. 482–504, 1960. Cited on page 1009.

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quantization induced by classical gravity, Physics Letters B 331, pp. 39–44, 1994, Cited on pages 1005 and 1017.

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Gregory, Structure of random space-time, Physical Review Letters 66, pp. 260–263, 1991. Cited on page 1010. For a discussion, see R. Sorabji, Time, Creation and the continuum: theories in antiquity and the early Middle Ages, Duckworth, London 1983. Cited on page 1009. For quantum gravity, see for example A. Ashtekar, Quantum geometry in action: big bang and black holes, http://www.arxiv.org/abs/math-ph/0202008. Cited on page 1010. The false belief that particles like quarks or electrons are composite is slow to die out. See, e.g, S. Fredriksson, Preon prophecies by the standard Model, http://www.arxiv.org/ abs/hep-ph/0309213. Preon models gained popularity in the 1970s and 1980s, in particular through the papers by J.C. Pati & A. Sal am, Physical Review D 10,, p. 275, 1974, H. Harari, Physics Letters B 86,, p. 83, 1979, M.A. Shupe, Physics Letters B 86,, p. 87, 1979, and H. Fritzsch & G. Mandelbaum, Physics Letters B 102,, p. 319, 1981. Cited on page 1011. A.D. Sakharov, Vacuum quantum fluctuations in curved space and the theory of gravitation, Soviet Physics – Doklady, 12, pp. 1040–1041, 1968. Cited on page 1012. C. Wolf, Upper limit for the mass of an elementary particle due to discrete time quantum mechanics, Il Nuovo Cimento B 109, pp. 213–218, 1994. Cited on page 1012. N.F. R amsey & A. Weis, Suche nach permanenten elektrischen Dipolmomenten: ein Test der Zeitumkehrinvarianz, Physikalische Blätter 52, pp. 859–863, 1996. The paper by E.D. Commins, S.B. Ross, D. DeMille & B.C. R egan, Improved experimental limit on the electric dipole moment of the electron, Physical Review A 50, p. 2960, 1994, gives an upper experimental limit to dipole moment of the electron of 3.4 ë 10−29 e m. W. Bernreu ther & M. Suzuki, The electric dipole moment of the electron, Reviews of Modern Physics 63, pp. 313–340, 1991. See also the musings in Hans Dehmelt, Is the electron a composite particle?, Hyperfine Interactions 81, pp. 1–3, 1993. Cited on page 1012. K. Akama, T. Hattori & K. Katsuura, Naturalness bounds on dipole moments from new physics, http://www.arxiv.org/abs/hep-ph/0111238 Cited on page 1012. W.G. Unruh, Notes on black hole evaporation, Physical Review D 14, pp. 870–875, 1976. W.G. Unruh & R.M. Wald, What happens when an accelerating observer detects a Rindler particle, Physical Review D 29, pp. 1047–1056, 1984. Cited on page 1016. W. Jauch, Heisenberg’s uncertainty relation and thermal vibrations in crystals, American Journal of Physics 61, pp. 929–932, 1993. Cited on page 1017. One way out is proposed in J. Magueijo & L. Smolin, Lorentz invariance with an invariant energy scale, Physical Review Letters 88, p. 190403, 26 April 2002. They propose a modification of the mass energy relation of the kind γmc 2 1+

γmc 2 E Pl

and

p=

γmv 1+

γmc 2 E Pl

.

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Another, similar approach of recent years, with a different proposal, is called doubly special relativity. A recent summary is G. Amelino–Camelia, Doubly-special relativity: first results and key open problems, http://www.arxiv.org/abs/gr-qc/0210063. The paper shows how conceptual problems hinder the advance of the field. Cited on page 1017. 1059 H.D. Zeh, On the interpretation of measurement in quantum theory, Foundations of Physics 1, pp. 69–76, 1970. Cited on page 1018. 1060 Steven Weinberg, The cosmological constant problem, Reviews of Modern Physics 61, pp. 1–23, 1989. Cited on page 1019.

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E=

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1053

1029

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1061 See Y.J. Ng, W.A. Christiansen & H. van Dam, Probing Planck-scale physics with

1062 1063

1065

1066

1067

1069

1070

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1068

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extragalactic sources?, Astrophysical Journal 591, pp. L87–L90, 2003, also downloadable electronically at http://www.arxiv.org/abs/astro-ph/0302372; D.H. Coule, Planck scale still safe from stellar images, Classical and Quantum Gravity 20, pp. 3107–3112, 2003, also available as electronic preprint at http://www.arxiv.org/abs/astro-ph/0302333. Negative experimental results (and not always correct calculations) are found in R. Lieu & L. Hillman, The phase coherence of light from extragalactic sources - direct evidence against first order Planck scale fluctuations in time and space, Astrophysical Journal 585, pp. L77–L80, 2003, and R. R agazzoni, M. Turatto & W. Gaessler, The lack of observational evidence for the quantum structure of spacetime at Planck scales, Astrophysical Journal 587, pp. L1–L4, 2003. Cited on page 1021. B.E. Schaefer, Severe limits on variations of the speed of light with frequency, Physical Review Letters 82, pp. 4964–4966, 21 June 1999. Cited on page 1021. G. Amelino–Camelia, J. Ellis, N.E. Mavromatos, D.V. Nanopoulos & S. Sakar, Potential sensitivity of gamma-ray-burster observations to wave dispersion in vacuo, Nature 393, p. 763, 1998, and preprint http://www.arxiv.org/abs/astro-ph/9712103. Cited on page 1021. G. Amelino–Camelia, Phenomenological description of space-time foam, preprint found at http://www.arxiv.org/abs/gr-qc/0104005, 2001. The paper includes a clearly written overview of present experimental approaches to detecting quantum gravity effects. See also his update G. Amelino–Camelia, Quantum-gravity phenomenology: status and prospects, preprint found at http://www.arxiv.org/abs/gr-qc/0204051, 2002. Cited on pages 1021 and 1022. G. Amelino–Camelia, An interferometric gravitational wave detector as a quantum gravity apparatus, Nature 398, p. 216, 1999, and preprint http://www.arxiv.org/abs/gr-qc/ 9808029. Cited on page 1021. F. Karolyhazy, Gravitation and quantum mechanics of macroscopic objects, Il Nuovo Cimento A42, pp. 390–402, 1966, Y.J. Ng & H. van Dam, Limit to space-time measurement, Modern Physics Letters A 9, pp. 335–340, 1994, Y.J. Ng & H. Van Dam, Modern Physics Letters A Remarks on gravitational sources, 10, pp. 2801–2808, 1995. The discussion is neatly summarised in Y.J. Ng & H. van Dam, Comment on ‘Uncertainty in measurements of distance’, http://www.arxiv.org/abs/gr-qc/0209021. See also Y.J. Ng, Spacetime foam, preprint found at http://www.arxiv.org/abs/gr-qc/0201022. Cited on page 1021. L.J. Garay, Spacetime foam as a quantum thermal bath, Physics Review Letters 80, pp. 2508–2511, 1998, also downloadable as http://www.arxiv.org/abs/gr-qc/9801024. Cited on pages 1021 and 1022. G. Amelino–Camelia & T. Piran, Planck scale-deformation of Lorentz symmetry as a solution to the UHECR and the TeV γ paradoxes, preprint http://www.arxiv.org/ astro-ph/0008107, 2000. Cited on page 1022. Experimental tests were described by R. Lieu & L.W. Hillman, The phase coherence of light from extragalactic sources: direct evidence against first-order Planck-scale fluctuations in time and space, Astrophysical Journal 585, pp. L77–L80, 2003, and R. R agazzoni & al., The lack of observational evidence for the quantum structure of spacetime at Planck scales, Astrophysical Journal 587, pp. L1–L4, 2003, Discussions on the predicted phase values are found in http://www.arxiv.org/abs/astro-ph/0302372 and http:// www.arxiv.org/abs/astro-ph/0302333. Cited on page 1022. At the unification scale, supersymmetry is manifest; all particles can transform into each other. See for example ... Cited on page 1023.

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1071 See the paper by U. Amaldi, W. de B oer & H. Fürstenau, Comparison of grand

unified theories with elektroweak and strong coupling constants measured at LEP, Physics Letters 260, pp. 447–455, 1991. This is the standard reference on the topic. Cited on pages 1023 and 1025. 1072 M.V. Romalis, W.C. Griffith, J.P. Jacobs & E.N. Fortson, New limit on the permanent electric dipole moment of 199 Hg, Physical Review Letters 86, pp. 2505–2508, 19 March 2001, or http://www.arxiv.org/abs/hep-ex/0012001. Their upper limit on the dipole moment of mercury atoms is 2.1 ë 1030 e m. Research for better measurement methods is progressing; a modern proposal is by S.K. L amoreaux, Solid state systems for electron electric dipole moment and other fundamental measurements, http://www.arxiv.org/abs/ nucl-ex/0109014. Cited on page 1023.

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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006

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1032

34.

nature at l arge scales – is the universe something or nothing?

“ Ref. 1073

Die Grenze ist der eigentlich fruchtbare Ort der Erkenntnis.* Paul Tillich, Auf der Grenze.

T

Cosmological scales

Hic sunt leones.*** Antiquity

d  r S = 2Gm~c 2 .

Challenge 1442 n

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It is straightforward to confirm that, with the usually quoted mass m and size d of everything visible in the universe, this condition is indeed fulfilled. We do need general relativity and thus curved space-time when talking about the whole of nature. * ‘The frontier is the really productive place of understanding’. Paul Tillich (1886–1965), German theologian, socialist and philosopher. ** Written between June and December 2000. *** ‘Here are lions.’ Written across unknown and dangerous regions on ancient maps.

Copyright © Christoph Schiller November 1997–May 2006

The description of motion requires the application of general relativity whenever the scale d of the situation are of the order of the Schwarzschild radius, i.e. whenever

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his strange question is the topic of the current leg of our mountain ascent. In he last section we explored nature in the vicinity of Planck dimensions; but he other limit, namely to study the description of motion at large, cosmological scales, is equally fascinating. As we proceed, many incredible results will appear, and at the end we will discover a surprising answer to the question in the section title. This section is not standard textbook material; a large part of it is original** and thus speculative and open to question. Even though this section aims at explaining in simple words the ongoing research in the domains of quantum gravity and superstring theory, be warned. With every sentence of this section you will find at least one physicist who disagrees! We have discussed the universe as a whole several times already. In classical physics we enquired about the initial conditions of the universe and whether it is isolated. In the first intermezzo we asked whether the universe is a set or a concept and indeed, whether it exists at all. In general relativity we gave the classical definition of the term, as the sum of all matter and space-time, we studied the expansion of the universe and we asked about its size and topology. In quantum theory we asked whether the universe has a wave function, whether it is born from a quantum fluctuation, and whether it allows the number of particles to be defined. Here we will settle all these issues by combining general relativity and quantum theory at cosmological scales. That will lead us to some of the strangest results we will encounter in our journey.

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1033

Similarly, quantum theory is required for the description of motion of an object whenever we approach it within a distance d of the order of the Compton wavelength λ C , i.e. whenever h . (695) d  λC = mc

Maximum time

Challenge 1443 n

Does the universe have a definite age? One should never trust a woman who tells one her real age. A woman who would tell one that, would tell one anything. Oscar Wilde

* This conclusion implies that so-called ‘oscillating’ universe models, in which it is claimed that ‘before’ the big bang there were other phenomena, cannot be based on nature or on observations. They are based on beliefs.

Copyright © Christoph Schiller November 1997–May 2006

Is it possible to measure time intervals of any imaginable size? General relativity shows that in nature there is a maximum time interval, with a value of about fourteen thousand million years or 430 Ps, providing an upper limit to the measurement of time. It is called the ‘age’ of the universe and has been deduced from two sets of measurements: the expansion of space-time and the age of matter. We all know of clocks that have been ticking for a long time: the hydrogen atoms in our body. All hydrogen atoms were formed just after the big bang. We can almost say that the electrons in these atoms have been orbiting the nuclei since the dawn of time. In fact, inside the protons in these atoms, the quarks have been moving already a few hundred thousand years longer. Anyway, we thus get a common maximum time limit for any clock made of atoms. Even ‘clocks’ made of radiation (can you describe one?) yield a similar maximum time. Also the study of the spatial expansion of the universe leads to the same maximum. No real or imaginable clock or measurement device was ticking before this maximum time and no clock could provide a record of having done so. In summary, it is not possible to measure time intervals greater than the maximum one, either by using the history of space-time or by using the history of matter or radiation.* The maximum time is thus rightly called the ‘age’ of the universe. Of course, all this is not new, although looking at the issue in more detail does provide some surprises.

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Obviously, for the total mass of the universe this condition is not fulfilled. However, we are not interested in the motion of the universe itself; we are interested in the motion of its components. In the description of these components, quantum theory is required whenever pair production and annihilation play a role. This is especially the case in the early history of the universe and near the horizon, i.e. for the most distant events that we can observe in space and time. We are thus obliged to include quantum theory in any precise description of the universe. Since at cosmological scales we need both quantum theory and general relativity, we start our investigation with the study of time, space and mass, by asking at large scales the same questions that we asked above at Planck scales.

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How precise can age measurements be?

Ref. 1075

Ref. 1074

The first way to measure the age of the universe* is to look at clocks in the usual sense of the term, namely at clocks made of matter. As explained in the part on quantum theory, Salecker and Wigner showed that a clock built to measure a total time T with a precision * Note that the age t 0 is not the same as the Hubble time T = 1~H 0 . The Hubble time is only a computed quantity and (almost) always larger than the age; the relation between the two depends on the value of the cosmological constant, on the density and on other parameters of the universe. For example, for the standard hot big bang scenario, i.e. for the matter-dominated Einstein–de Sitter model, we have the simple relation T = (3~2) t 0 .

Copyright © Christoph Schiller November 1997–May 2006

No woman should ever be quite accurate about her age. It looks so calculating. Oscar Wilde

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Asking about the age of the universe may seem a silly question, because we have just discussed it. Furthermore, the value is found in many books and tables, including that of Appendix B, and its precise determination is actually one of the most important quests in modern astrophysics. But is this quest reasonable? In order to measure the duration of a movement or the age of a system, we need a clock. The clock has to be independent of that movement or system and thus has to be outside the system. However, there are no clocks outside the universe and, inside it, a clock cannot be independent. In fact we have just seen that inside the universe, no clock can run throughout its complete history. Indeed, time can be defined only once it is possible to distinguish between matter and space-time. Once this distinction can be made, only the two possibilities just discussed remain: we can either talk about the age of space-time, as is done in general relativity, by assuming that matter provides suitable and independent clocks; or we can talk about the age of matter, such as stars or galaxies, by assuming that the extension of space-time or some other matter provides a good clock. Both possibilities are being explored experimentally in modern astrophysics; both give the same result of about fourteen thousand million years that was mentioned previously. However, for the universe as a whole, an age cannot be defined. The issue of the starting point of time makes this difficulty even more apparent. We may imagine that going back in time leads to only two possibilities: either the starting instant t = 0 is part of time or it is not. (Mathematically, this means that the segment describing time should be either closed or open.) Both cases assume that it is possible to measure arbitrarily small times, but we know from the combination of general relativity and quantum theory that this is not the case. In other words, neither possibility is incorrect: the beginning cannot be part of time, nor can it not be part of it. To this situation there is only one solution: there was no beginning at all. In other words, the situation is consistently muddled. Neither the age of the universe nor its origin makes sense. What is going wrong? Or, more correctly, how are things going wrong? In other words, what happens if instead of jumping directly to the big bang, we approach it as closely as possible? The best way to clarify the issue is to ask about the measurement error we are making when we say that the universe is fourteen thousand million years old. This turns out to be a fascinating topic.

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1035

∆t has a minimum mass m given by mA

Ref. 1076

ħ T . c 2 (∆t)2

(696)

A simple way to incorporate general relativity into this result was suggested by Ng and van Dam. Any clock of mass m has a minimum resolution ∆t due to the curvature of space that it introduces, given by Gm ∆t A 3 . (697) c

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If m is eliminated, these two results imply that any clock with a precision ∆t can only measure times T up to a certain maximum value, namely

Challenge 1444 e

Challenge 1445 ny

(∆t)3 2 t Pl

,

(698)

» where t Pl = ħG~c 5 = 5.4 ë 10−44 s is the already familiar Planck time. (As usual, we have omitted factors of order one in this and in all the following results of this section.) In other words, the higher the accuracy of a clock, the shorter the time during which the clock works dependably! The precision of a clock is not (only) limited by the budget spent on building it, but by nature itself. Nevertheless, it does not take much to check that for clocks in daily life, this limit is not even remotely approached. For example, you may want to deduce how precisely your own age can be specified. As a consequence of (698), a clock trying to achieve an accuracy of one Planck time can do so for at most one single Planck time! Simply put, a real clock cannot achieve Planck time accuracy. If we try to go beyond limit (698), fluctuations of space-time hinder the working of the clock and prevent higher precision. With every Planck time that passes, the clock accumulates a measuring error of at least one Planck time. Thus, the total measurement error is at least as large as the measurement itself. The conclusion is also valid for clocks based on radiation, for example on background radiation. In short, measuring age with a clock always involves some errors; whenever we try to reduce these errors to Planck level, the clock becomes so imprecise that age measurements become impossible.

“ Page 45

Time is waste of money. Oscar Wilde

From the origins of physics onwards, the concept of ‘time’ has designated what is measured by a clock. Since equation (698) expresses the non-existence of perfect clocks, it also implies that time is only an approximate concept, and that perfect time does not exist. Thus there is no ‘idea’ of time, in the Platonic sense. In fact, all discussion in the previous and present sections can be seen as proof that there are no perfect or ‘ideal’ examples of any classical or everyday concept.

Copyright © Christoph Schiller November 1997–May 2006

Does time exist?

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T<

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Challenge 1446 e

Page 1004 Challenge 1447 e

Time does not exist. Despite this conclusion, time is obviously a useful concept in everyday life. A simple explanation is provided when we focus on the importance of energy. Any clock, in fact any system of nature, is characterized by a simple number, namely the highest ratio of the kinetic energy to the rest energy of its components. In daily life, this fraction is about 1 eV~10 GeV = 10−10 . Such low-energy systems are well suited to building clocks. The more precisely the motion of the main moving part – the pointer of the clock – can be kept constant and can be monitored, the higher the precision of the clock becomes. To achieve the highest possible precision, the highest possible mass of the pointer is required; indeed, both the position and the speed of the pointer must be measured, and the two measurement errors are related by the quantum mechanical indeterminacy relation ∆v ∆x A ħ~m. High mass implies low intrinsic fluctuations. In order to screen the pointer from outside influences, even more mass is needed. This connection might explain why better clocks are usually more expensive than less accurate ones. The usually quoted indeterminacy relation is valid only at everyday energies. Increasing the mass does not allow to reach arbitrary small time errors, since general relativity changes the indeterminacy relation to ∆v ∆x A ħ~m + G(∆v)2 m~c 3 . The additional term on the right hand side, negligible at everyday scales, is proportional to energy. Increasing it by too large an amount limits the achievable precision of the clock. The smallest measurable time interval turns out to be the Planck time. In summary, time exists as a good approximation only for low-energy systems. Any increase in precision beyond a certain limit will require an increase in the energy of the components; at Planck energy, this energy increase will prevent an increase in precision. What is the error in the measurement of the age of the universe?

Challenge 1448 e

Applying the discussion about the measurement of time to the age of the universe is now straightforward. Expression (698) implies that the highest precision possible for a clock is about 10−23 s, or about the time light takes to move across a proton. The finite age of the universe also yields a maximum relative measurement precision. Expression (698) can be written as t Pl ∆t A ( )2~3 (699) T T

and thus decreases with increasing measurement energy. For high energies, however, the error is given by gravitational effects as ∆t E meas  T E Pl

(701)

Copyright © Christoph Schiller November 1997–May 2006

which shows that no time interval can be measured with a precision of more than about 40 decimals. In order to clarify the issue, we can calculate the error in measurement as a function of the observation energy E meas . There are two limit cases. For small energies, the error is given by quantum effects as 1 ∆t  (700) T E meas

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Ref. 1073

xi general rel ativity versus quantum mechanics

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Relative measurement error 1

quantum error

total error quantum gravity error

∆Emin E Energy

Eopt

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EPl

F I G U R E 383 Measurement errors as a function of measurement energy

d . v

(702)

This is in simple words the method used to determine the age of the universe from the expansion of space-time, for galaxies with red-shifts below unity.* Of interest in the fol* At higher red-shifts, the speed of light, as well as the details of the expansion, come into play; if we continue

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T=

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so that the total result is as shown in Figure 383. In particular, energies that are too high do not reduce measurement errors, because any attempt to reduce the measurement error for the age of the universe below 10−23 s would require energies so high that the limits of space-time would be reached, making the measurement itself impossible. We reached this conclusion through an argument based on clocks made of particles. Below we will find out that even by determining the age of the universe using space-time expansion leads to the same limit. Imagine to observe a tree which, as a result of some storm or strong wind, has fallen towards second tree, touching it at the very top, as shown in Figure 384. It is possible to determine the height of both trees by measuring their separation and v the angles at the base. The error in height will depend on the errors in separation and angles. Similarly, the age of the universe follows from the present distance and speed of objects – such as galaxies – observed in the night sky. The present distance d T corresponds to separation of the trees at ground level and the speed v to the angle between the two trees. The Hubble time T of the universe – which, as has already been mentioned, is usud ally assumed to be larger than the age of the universe – then corresponds to the height at which the two trees meet. This time since the universe ‘started’, in a naive sense since the galaxies F I G U R E 384 Trees and galaxies ‘separated’, is then given, within a factor of order one, by

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xi general rel ativity versus quantum mechanics

lowing is the (positive) measurement error ∆T, which becomes ∆T ∆d ∆v = + . T d v

(703)

2~3

∆d l Pl A 2~3 T d

Challenge 1449 e

Challenge 1450 e

Ref. 1074

(704)

with the image of inclined trees, we find that the trees are not straight all the way up to the top and that they grow on a slope, as shown in Figure 385.

Copyright © Christoph Schiller November 1997–May 2006

thus giving the same indeterminacy in the age of the universe as found above in the case of material clocks. We can try to reduce this error in two ways: by choosing objects at either small or large distances. Let us start with the smallest possible distances. In order to get high precision at small distances, we need high observation energies. It is fairly obvious that at observation energies near the Planck value, the value of ∆T~T approaches unity. In fact, both terms on the right-hand side of expression (703) become of order one. At these energies, ∆v approaches c and the maximum value for d approaches the Planck length, for the same reason that at Planck energies the maximum measurable time is the Planck time. In short, at Planck scales it is impossible to say whether the universe is old or young. Let us continue with the other extreme, namely objects extremely far away, say with a red-shift of z Q 1. Relativistic cosmology requires the diagram of Figure 384 to be replaced by the more realistic diagram of Figure 385. The ‘light onion’ replaces the familiar light cone of special relativity: light converges near the big bang. In this case the measurement error for the age of the universe also depends on the distance and velocity errors. At the largest possible distances, the signals an object must send out must be of high energy, because the emitted wavelength must be smaller than the universe itself. Thus, inevitably we reach Planck energies. However, we saw that in such high-energy situations, the emitted radiation, as well as the object itself, are indistinguishable from the space-time background. In other words, the red-shifted signal we would observe today would have a wavelength as large as the size of the universe, with a correspondingly small frequency.

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Exploring this in more detail is worthwhile. For any measurement of T we have to choose the object, i.e. a distance d, as well as an observation time ∆t, or, equivalently, an observation energy ∆E = 2πħ~∆t. We will now investigate the consequences of these choices for expression (703), always taking into account both quantum theory and general relativity. At everyday energies, the result of the determination of the age t 0 is about 14  2 ë 109 Ga. This value is deduced by measuring red-shifts, i.e. velocities, and distances, using stars and galaxies in distance ranges from some hundred thousand light years up to a red-shift of about 1. Measuring red-shifts does not produce large velocity errors. The main source of experimental error is the difficulty in determining the distances of galaxies. What is the smallest possible error in distance? Obviously, equation (699) implies

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nature at l arge scales – is the universe something or nothing?

Another way to describe the situation is the folbig bang space lowing. At Planck energies or near the horizon, the original signal has an error of the same size as the signal itself. When measured at the present time, the red-shifted signal still has an error of the same 4ct0/9 size as the signal. As a result, the error on the holight cone: what we rizon distance becomes as large as the value to be can see measured. time In short, even if space-time expansion and large scales are used, the instant of the so-called beginother galaxies in the night sky ning of the universe cannot be determined with an error smaller than the age of the universe itself, a t0 result we also found at Planck distances. Whenever our galaxy we aim for perfect precision, we find that the universe is 14  14 thousand million years old! In other F I G U R E 385 The speed and distance of words, at both extremal situations it is impossible to remote galaxies say whether the universe has a non-vanishing age. We have to conclude that the anthropomorphic concept of ‘age’ does not make any sense for the universe as a whole. The usual textbook value is useful only for domains in time, space and energy for which matter and space-time are clearly distinguished, namely at everyday, human-scale energies; however, this anthropocentric value has no overall meaning. By the way, you may like to examine the issue of the fate of the universe using the same arguments. In the text, however, we continue on the path outlined at the start of this section; the next topic is the measurement of length. Maximum length

Page 460

˙ the * In cosmology, we need to distinguish between the scale factor R, the Hubble radius c~H = cR~R, horizon distance h and the size d of the universe. The Hubble radius is a computed quantity giving the distance at which objects move away with the speed of light. It is always smaller than the horizon distance, at which in the standard Einstein–de Sitter model, for example, objects move away with twice the speed of light. However, the horizon itself moves away with three times the speed of light.

Copyright © Christoph Schiller November 1997–May 2006

Ref. 1074

General relativity shows that in the standard cosmological model, for hyperbolic (open) and parabolic (marginal) evolutions of the universe, the actual size of the universe is infinite. It is only the horizon distance, i.e. the distance of objects with infinite red-shift, which is finite. In a hyperbolic or parabolic universe, even though the size is infinite, the most distant visible events (which form the horizon) are at a finite distance.* For elliptical evolution, the total size is finite and depends on the curvature. However, in this case also the present measurement limit yields a minimum size for the universe many times larger than the horizon distance. At least, this is what general relativity says. On the other hand, quantum field theory is based on flat and infinite space-time. Let us see what happens when the two theories are combined. What can we say about measurements of length in this case? For example, would it be possible to construct and use a metre rule to measure lengths larger than the distance to the horizon? It is true that we would have no time to push it up to there, since in the standard Einstein–de Sitter big

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Challenge 1451 ny

1039

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1040

Ref. 1074

Ref. 1074

xi general rel ativity versus quantum mechanics

bang model the horizon moves away from us faster than the speed of light. We should have started using the metre rule right at the big bang. For fun, let us assume that we have actually managed to do this. How far away can we read off distances? In fact, since the universe was smaller in the past and since every observation of the sky is an observation of the past, Figure 385 shows that the maximum spatial distance an object can be seen away from us is only (4~9)ct 0 . Obviously, for spacetime intervals, the maximum remains ct 0 . Thus, in all cases it turns out to be impossible to measure lengths larger than the horizon distance, even though general relativity predicts such distances. This unsurprising result is in obvious agreement with the existence of a limit for measurements of time intervals. The real surprises come now.

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Is the universe really a big place? Ref. 1077

Astronomers and Hollywood movies answer this question in the affirmative. Indeed, the distance to the horizon of the universe is usually included in tables. Cosmological models specify that the scale factor R, which fixes the distance to the horizon, grows with time t; for the case of the usually assumed mass-dominated Einstein–de Sitter model, i.e. for a vanishing cosmological constant and flat space, we have R(t) = C t 2~3 ,

where the numerical constant C relates the commonly accepted horizon distance to the commonly accepted age. Indeed, observation shows that the universe is large and is still getting larger. But let us investigate what happens if we add to this result from general relativity the limitations of quantum theory. Is it really possible to measure the distance to the horizon? We look first at the situation at high energies. We saw above that space-time and matter are not distinguishable at Planck scales. Therefore, at Planck energies we cannot state whether objects are localized or not. At Planck scales, a basic distinction of our thinking, namely the one between matter and vacuum, becomes obsolete. Equivalently, it is not possible to claim that space-time is extended at Planck scales. Our concept of extension derives from the possibility of measuring distances and time intervals, and from observations such as the ability to align several objects behind one another. Such observations are not possible at Planck scales. In fact, none of the observations in daily life from which we deduce that space is extended are possible at Planck scales. At Planck scales, the basic distinction between vacuum and matter, namely the opposition between extension and localization, disappears. As a consequence, at Planck energies the size of the universe cannot be measured. It cannot even be called larger than a match box. At cosmological distances, the situation is even easier. All the arguments given above on the errors in measurement of the age can be repeated for the distance to the horizon. Essentially, at the largest distances and at Planck energies, the measurement errors are of the same magnitude as the measured value. All this happens because length measurements become impossible at nature’s limits. This is corroborated by the lack of any standard with which to compare the size of the universe. Studying the big bang also produces strange results. At Planck energies, whenever we

Copyright © Christoph Schiller November 1997–May 2006

Challenge 1452 ny

(705)

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Page 1167

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nature at l arge scales – is the universe something or nothing?

Challenge 1453 ny

1041

The boundary of space-time – is the sky a surface?

Challenge 1454 ny

* In addition, the measurement errors imply that no statement can be made about translational symmetry at cosmological scales. Are you able to confirm this? In addition, at the horizon it is impossible to distinguish between spacelike and timelike distances. Even worse, concepts such as ‘mass’ or ‘momentum’ are muddled at the horizon. This means that, as at Planck energies, we are unable to distinguish between objects and the background, and between state and intrinsic properties. We will come back to this important point shortly.

Copyright © Christoph Schiller November 1997–May 2006

Challenge 1455 ny

The horizon of the universe, essentially the black part of the night sky, is a fascinating entity. Everybody interested in cosmology wants to know what happens there. In newspapers the horizon is sometimes called the boundary of space-time. Some surprising insights, not yet common in newspapers, appear when the approaches of general relativity and quantum mechanics are combined. We saw above that the errors in measuring the distance of the horizon are substantial. They imply that we cannot pretend that all points of the sky are equally far away from us. Thus we cannot say that the sky is a surface. There is even no way to determine the dimensionality of the horizon or the dimensionality of space-time near the horizon.* Measurements thus do not allow us to determine whether the boundary is a point, a surface, or a line. It may be an arbitrary complex shape, even knotted. In fact, quantum theory tells us that it must be all of these from time to time, in short, that the sky fluctuates in height and shape. In short, it is impossible to determine the topology of the sky. But that is nothing new. As is well known, general relativity is unable to describe pair creation particles with spin 1/2. The reason for this is the change in space-time topology required by the process. On the other hand, the universe is full of such processes, implying that it is impossible to define a topology for the universe and, in particular, to talk of the topology of the horizon itself. Are you able to find at least two other arguments to show this? Worse still, quantum theory shows that space-time is not continuous at a horizon, as

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try to determine the size of the big bang, we cannot claim that the universe was smaller than the present size. At Planck energies, there is no way to distinguish length values. Somehow, Planck dimensions and the size of the universe get confused. There are also other confirmations. Let us go back to the example above. If we had a metre rule spanning the whole universe, even beyond the horizon, with zero at the place where we live, what measurement error would it produce for the horizon? It does not take long to discover that the expansion of space-time from Planck scales to the present also expands the indeterminacy in the Planck size into one of the order of the distance to the horizon. The error is as large as the measurement result. Since this also applies when we try to measure the diameter of the universe instead of its radius, it becomes impossible to state whether the antipodes in the sky really are distant from each other! We can summarize the situation by noting that anything said about the size of the universe is as limited as anything said about its age. The height of the sky depends on the observation energy. At Planck energies, it cannot be distinguished from the Planck length. If we start measuring the sky at standard observation energies, trying to increase the precision of measurement of the distance to the horizon, the measurement error increases beyond all bounds. At Planck energies, the volume of the universe is indistinguishable from the Planck volume!

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Ref. 1073

xi general rel ativity versus quantum mechanics

can easily be deduced by applying the Planck-scale arguments from the previous section. Time and space are not defined there. Finally, there is no way to decide whether the various boundary points are different from each other. The distance between two points in the night sky is undefined. In other words, it is unclear what the diameter of the horizon is. In summary, the horizon has no specific distance or shape. The horizon, and thus the universe, cannot be shown to be manifolds. This leads to the next question:

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Does the universe have initial conditions? Ref. 1078

The number of stars, about 10231 , is included in every book on cosmology, as it is in the table of Appendix B.1167 A subset of this number can be counted on clear nights. If we ask the same question about particles instead of stars, the situation is similar. The commonly quoted number of baryons is 10811 , together with 10901 photons. However, this does not settle the issue. Neither quantum theory nor general relativity alone make predictions

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Does the universe contain particles and stars?

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Ref. 1073

One often reads about the quest for the initial conditions of the universe. But before joining this search, we should ask whether and when such initial conditions make any sense. Obviously, our everyday description of motion requires them. Initial conditions describe the state of a system, i.e. all those aspects that differentiate it from a system with the same intrinsic properties. Initial conditions, like the state of a system, are attributed to a system by an outside observer. More specifically, quantum theory tells us that initial conditions or the state of a system can only be defined by an outside observer with respect to an environment. It is already difficult to be outside the universe. In addition, quite independently of this issue, even inside the universe a state can only be defined if matter can be distinguished from vacuum. However, this is impossible at Planck energies, near the big bang, or at the horizon. Thus the universe has no state. No state also means that there is no wave function of the universe. The limits imposed by the Planck values confirm this conclusion in other ways. First of all, they show that the big bang was not a singularity with infinite curvature, density or temperature, because infinitely large values do not exist in nature. Second, since instants of time do not exist, it is impossible to define the state of any system at a given time. Third, as instants of time do not exist, neither do events exist, and thus the big bang was not an event, so that for this more prosaic reason, neither an initial state nor an initial wave function can be ascribed to the universe. (Note that this also means that the universe cannot have been created.) In short, there are no initial conditions for the universe. Initial conditions make sense only for subsystems and only far away from Planck scales. Thus, for initial conditions to exist, the system must be far away from the horizon and it must have evolved for some time ‘after’ the big bang. Only when these two requirements are fulfilled can objects move in space. Of course, this is always the case in everyday life. At this point in our mountain ascent, where time and length are unclearly defined at cosmological scales, it should come as no surprise that there are similar difficulties concerning the concept of mass.

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Page 773

Challenge 1457 ny

Does the universe contain masses and objects? Page 1167

The average density of the universe, about 10−26 kg~m3 , is frequently cited in texts. Is it different from a vacuum? Quantum theory shows that, as a result of the indeterminacy relation, even an empty volume of size R has a mass. For a zero-energy photon inside such a vacuum, we have E~c = ∆p A ħ~∆x, so that in a volume of size R, we have a minimum mass of at least m mi n (R) = h~cR. For a spherical volume of radius R there is

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about the number of particles, either inside or outside the horizon. What happens if we combine them? In order to define the number of particles in a region, quantum theory first of all requires a vacuum state to be defined. The number of particles is defined by comparing the system with the vacuum. If we neglect or omit general relativity by assuming flat spacetime, this procedure poses no problem. However, if we include general relativity and thus a curved space-time, especially one with such a strangely behaved horizon as the one we have just found, the answer is simple: there is no vacuum state with which we can compare the universe, for two reasons. First, nobody can explain what an empty universe would look like; second, and more importantly, there is no way to define a state of the universe at all. The number of particles in the universe thus becomes undefinable. Only at everyday energies and for finite dimensions are we able to speak of an approximate number of particles. Comparison between a system and the vacuum is also impossible in the case of the universe for purely practical reasons. The requirement for such a comparison effectively translates into the requirement that the particle counter be outside the system. (Can you confirm the reason for this connection?) In addition, it is impossible to remove particles from the universe. The impossibility of defining a vacuum state, and thus the number of particles in the universe, is not surprising. It is an interesting exercise to investigate the measurement errors that appear when we try to determine the number of particles despite this fundamental impossibility. Can we count the stars? In principle, the same conclusion applies as for particles. However, at everyday energies the stars can be counted classically, i.e. without taking them out of the volume in which they are enclosed. For example, this is possible if the stars are differentiated by mass, colour or any other individual property. Only near Planck energies or near the horizon are these methods inapplicable. In short, the number of stars is only defined as long as the observation energy is low, i.e. as long as we stay away from Planck energies and from the horizon. In short, despite what appears to be the case on human scales, there is no definite number of particles in the universe. The universe cannot be distinguished from vacuum by counting particles. Even though particles are necessary for our own existence and functioning, a complete count of them cannot be made. This conclusion is so strange that we should not accept it too easily. Let us try another method of determining the content of matter in the universe: instead of counting particles, let us weigh them.

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thus a minimal mass density given approximately by ρ min 

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For the universe, if the standard horizon distance R 0 of 14 000 million light years is inserted, the value becomes about 10−142 kg~m3 . This describes the density of the vacuum. In other words, the universe, with a density of about 10−26 kg~m3 , seems to be clearly different from vacuum. But are we sure? We have just deduced that the radius of the horizon is undefined: depending on the observation energy, it can be as small as the Planck length. This implies that the density of the universe lies somewhere between the lowest possible value, given by the density of vacuum just mentioned, and the highest possible one, namely the Planck density.* In short, relation (706) does not really provide a clear statement. Another way to measure the mass of the universe would be to apply the original definition of mass, as given by Mach and modified by special relativity. Thus, let us try to collide a standard kilogram with the universe. It is not hard to see that whatever we do, using either low or high energies for the standard kilogram, the mass of the universe cannot be constrained by this method. We would need to produce or to measure a velocity change ∆v for the rest of the universe after the collision. To hit all the mass in the universe at the same time, we need high energy; but then we are hindered by Planck energy effects. In addition, a properly performed collision measurement would require a mass outside the universe, a rather difficult feat to achieve. Yet another way to measure the mass would be to determine the gravitational mass of the universe through straightforward weighing. But the lack of balances outside the universe makes this an impractical solution, to say the least. A way out might be to use the most precise definition of mass provided by general relativity, the so-called ADM mass. However, for definition this requires a specified behaviour at infinity, i.e. a background, which the universe lacks. We are then left with the other general relativistic method: determining the mass of the universe by measuring its average curvature. Let us take the defining expressions for average curvature κ for a region of size R, namely

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1 2 r curvature

=

3 4πR 2 − S 15 4πR 3 ~3 − V = . 4π R4 4π R5

(708)

We have to insert the horizon radius R 0 and either its surface area S 0 or its volume V0 . However, given the error margins on the radius and the volume, especially at Planck energies, for the radius of curvature we again find no reliable result. * In fact, at everyday energies the density of the universe lies almost exactly between the two values, yielding the strange relation 2 2 2 2 4 2 m 0 ~R 0  m Pl ~R Pl = c ~G . (707)

But this is nothing new. The approximate equality can be deduced from equation 16.4.3 (p. 620) of Steven Weinberg, Gravitation and Cosmology, Wiley, 1972, namely Gn b m p = 1~t 02 . The relation is required by several cosmological models.

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Copyright © Christoph Schiller November 1997–May 2006

κ=

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(706)

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m mi n (R) ħ = 4 . 3 R cR

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An equivalent method starts with the usual expression for the indeterminacy ∆κ in the scalar curvature for a region of size R provided by Rosenfeld, namely ∆κ A

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2 16πl Pl . R4

(709)

However, this expression also shows that the error in the radius of curvature behaves like the error in the distance to the horizon. In summary, at Planck energies, the average radius of curvature of nature turns out to lie between infinity and the Planck length. This implies that the density of matter lies between the minimum value and the Planck value. There is thus no method to determine the mass of the universe at Planck energies. (Can you find one?) The concept of mass cannot be applied to the universe as a whole. Thus, the universe has no mass.

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Does the universe have a boundary? It is common to take ‘boundary’ and ‘horizon’ as synonyms in the case of the universe, because they are the same for all practical purposes. To study this concept, knowledge of mathematics does not help us; the properties of mathematical boundaries, e.g. that they themselves have no boundary, are not applicable in the case of nature, since space-time is not continuous. We need other, physical arguments. The boundary of the universe is obviously supposed to represent the boundary between something and nothing. This gives three possibilities:

Copyright © Christoph Schiller November 1997–May 2006

We have already seen that at the horizon, space-time translation symmetry breaks down. Let us have a quick look at the other symmetries. What happens to permutation symmetry? Exchange is an operation on objects in space-time. Exchange thus automatically requires a distinction between matter, space and time. If we cannot distinguish positions, we cannot talk about exchange of particles. However, this is exactly what happens at the horizon. In short, general relativity and quantum theory together make it impossible to define permutation symmetry at the horizon. CPT symmetry suffers the same fate. As a result of measurement errors or of limiting maximum or minimum values, it is impossible to distinguish between the original and the transformed situations. It is therefore impossible to maintain that CPT is a symmetry of nature at horizon scales. In other words, matter and antimatter cannot be distinguished at the horizon. The same happens with gauge symmetry, as you may wish to check in detail yourself. For its definition, the concept of gauge field requires a distinction between time, space and mass; at the horizon this is impossible. We therefore also deduce that at the horizon also concepts such as algebras of observables cannot be used to describe nature. Renormalization also breaks down. All symmetries of nature break down at the horizon. The complete vocabulary we use when we talk about observations, including terms such as such as magnetic field, electric field, potential, spin, charge, or speed, cannot be used at the horizon. And that is not all.

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Do symmetries exist in nature?

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— ‘Nothing’ could mean ‘no matter’. But we have just seen that this distinction cannot be made at Planck scales. As a consequence, the boundary will either not exist at all or it will encompass the horizon as well as the whole universe. — ‘Nothing’ could mean ‘no space-time’. We then have to look for those domains where space and time cease to exist. These occur at Planck scales and at the horizon. Again, the boundary will either not exist or it will encompass the whole universe. — ‘Nothing’ could mean ‘neither space-time nor matter.’ The only possibility is a boundary that encloses domains beyond the Planck scale and beyond the horizon; but again, such a boundary would also encompass all of nature.

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Is the universe a set? – Again

Copyright © Christoph Schiller November 1997–May 2006

We are used to calling the universe the sum of all matter and all space-time. In other words, we imply that the universe is a set of components, all different from each other. This idea was introduced in three situations: it was assumed that matter consists of particles, that space-time consists of events (or points) and that the set of states consists of different initial conditions. However, our discussion so far shows that the universe is not a set of such distinguishable elements. We have encountered several proofs: at the horizon, at the big bang and at Planck scales distinction between events, between particles, between observables and between space-time and matter becomes impossible. In those domains, distinctions of any kind become impossible. We have found that any distinction between two entities, such as between a toothpick and a mountain, is only approximately possible. The approximation is possible because we live at energies much smaller than the Planck energy. Obviously, we are able to distinguish cars from people and from toothpicks; the approximation is so good that we do not notice the error when we perform it. Nevertheless, the discussion of the situation at Planck energies shows that a perfect distinction is impossible in principle. It is impossible to split the universe into separate entities. Another way to reach this result is the following. Distinguishing between two entities requires different measurement results, such as different positions, masses, sizes, etc. Whatever quantity we choose, at Planck energies the distinction becomes impossible. Only at everyday energies is it approximately possible. In short, since the universe contains no distinguishable entities, the universe is not a

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Ref. 1080

This result is puzzling. When combining quantum theory and relativity, we do not seem to be able to find a conceptual definition of the horizon that distinguishes the horizon from what it includes. In fact, if you find one, publish it! A distinction is possible in general relativity; and equally, a distinction is possible in quantum theory. However, as soon as we combine the two, the boundary becomes indistinguishable from its content. The interior of the universe cannot be distinguished from its horizon. There is no boundary. A difficulty to distinguish the horizon and its contents is definitely interesting; it suggests that nature may be symmetric under transformations that exchange interiors and boundaries. This connection is nowadays called holography because it vaguely recalls the working of credit card holograms. It is a busy research field in present-day high-energy physics. However, for the time being, we shall continue with our original theme, which directly leads us to ask:

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* Some people knew this long before physicists; for example, the belief that the universe is or contains information was ridiculed most thoroughly in the popular science fiction parody by Douglas Adams, The Hitchhiker’s Guide to the Galaxy, 1979, and its sequels.

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Copyright © Christoph Schiller November 1997–May 2006

set. We have already envisaged this possibility in the first intermezzo; now it is confirmed. The concept of ‘set’ is already too specialized to describe the universe. The universe must be described by a mathematical concept that does not contain any set. This is a powerful result: it means that the universe cannot be described precisely if any of the concepts used for its description presuppose the use of sets. But all concepts we have used so far to describe nature, such as space-time, phase space, Hilbert space and its generalizations, namely Fock space and particle space, are based on sets. They all must be abandoned at Planck energies, as well as in any precise description. Furthermore, many speculations about unified descriptions do not satisfy the criterion that sets must not be included. In particular, all studies of quantum fluctuations, mathematical categories, posets, complex mathematical spaces, computer programs, Turing machines, Gödel’s theorem, creation of any sort, space-time lattices, quantum lattices and Bohm’s unbroken wholeness fail to satisfy this requirement. In addition, almost none of the speculations about the origin of the universe can be correct. For example, you may wish to check the religious explanations you know against this result. In fact, no approach used by theoretical physics in the year 2000 satisfies the requirement that sets must be abandoned; perhaps a future version of string or M theory will do so. The task is not easy; do you know of a single concept not based on a set? Note that this radical conclusion is deduced from only two statements: the necessity of using quantum theory whenever the dimensions are of the order of the Compton wavelength, and the necessity to use general relativity whenever the dimensions are of the order of the Schwarzschild radius. Together, they mean that any precise description of nature cannot contain sets. We have reached this result after a long and interesting, but in a sense unnecessary, digression. The difficulties in complying with this result may explain why the unification of the two theories has not so far been successful. Not only does unification require that we stop using space, time and mass for the description of nature; it also requires that all distinctions, of any kind, should be only approximate. But all physicists have been educated on the basis of exactly the opposite creed! Note that, because it is not a set, the universe is not a physical system. Specifically, it has no state, no intrinsic properties, no wave function, no initial conditions, no density, no entropy and no cosmological constant. The universe is thus neither thermodynamically closed nor open; and it contains no information. All thermodynamic quantities, such as entropy, temperature and free energy, are defined using ensembles. Ensembles are limits of systems which are thermodynamically either open or closed. As the universe is neither of these two, no thermodynamic quantity can be defined for it.* All physical properties are defined only for parts of nature which are approximated or idealized as sets, and thus are physical systems.

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Curiosities and fun challenges about the universe Insofern sich die Sätze der Mathematik auf die Wirklichkeit beziehen, sind sie nicht sicher, und sofern sie sicher sind, beziehen sie sich nicht auf die Wirklichkeit.* Albert Einstein

The contradictions between the term ‘universe’ and the concept of ‘set’ lead to numerous fascinating issues. Here are a few.

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**

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In mathematics, 2 + 2 = 4. This statement is an idealization of statements such as ‘two apples plus two apples makes four apples.’ However, we now know that at Planck energies this is not a correct statement about nature. At Planck energies, objects cannot be counted. Are you able to confirm this?

Ref. 1081

In 2002, Seth Lloyd estimated how much information the universe can contain, and how many calculations it has performed since the big bang. This estimate is based on two ideas: that the number of particles in the universe is a well-defined quantity, and that the universe is a computer, i.e. a physical system. We now know that neither assumption is correct. This example shows the power of the criteria that we deduced for the final description of motion. **

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People take pictures of the cosmic background radiation and its variations. Is it possible that the photographs will show that the spots in one direction of the sky are exactly the same as those in the diametrically opposite direction? **

Ref. 1082 Challenge 1468 n

In 1714, Leibniz published his Monadologie. In it he explores what he calls a simple substance, which he defined to be a substance that has no parts. He called it a monade and explores some of its properties. However, due mainly to its incorrect deductions, the term has not been taken over by others. Let us forget the strange deductions and focus only on the definition: what is the physical concept most related to that of monade?

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We usually speak of the universe, implying that there is only one of them. Yet there is a simple case to be made that ‘universe’ is an observer-dependent concept, since the idea of ‘all’ is observer-dependent. Does this mean that there are many universes? **

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If all particles would be removed – assuming one would know where to put them – there wouldn’t be much of a universe left. True? * In so far as mathematical statements describe reality, they are not certain, and as far as they are certain, they are not a description of reality.

Copyright © Christoph Schiller November 1997–May 2006

**

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**

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** Challenge 1471 ny

At Planck energies, interactions cannot be defined. Therefore, ‘existence’ cannot be defined. In short, at Planck energies we cannot say whether particles exist. True? Hilbert’s sixth problem settled

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Drum hab ich mich der Magie ergeben, [...] Daß ich erkenne, was die Welt Im Innersten zusammenhält.* Goethe, Faust.

Is the universe really the sum of matter–energy and space-time? Or of particles and vacuum? We have heard this so often up to now that we may be lulled into forgetting to * Thus I have devoted myself to magic, [...] that I understand how the innermost world is held together.

Copyright © Christoph Schiller November 1997–May 2006

Does the universe make sense?

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Page 157

In the year 1900, David Hilbert gave a famous lecture in which he listed 23 of the great challenges facing mathematics in the twentieth century. Most of these problems provided challenges to many mathematicians for decades afterwards. A few are still unsolved, among them the sixth, which challenged mathematicians and physicists to find an axiomatic treatment of physics. Since the universe is not even a set, we can deduce that such an axiomatic description of nature is impossible. The reasoning is simple; all mathematical systems, be they algebraic systems, order systems or topological systems, are based on sets. Mathematics does not have axiomatic systems that do not contain sets. The reason for this is simple: any (mathematical) concept contains at least one set. However, nature does not. The impossibility of an axiomatic system for physics is also confirmed in another way. Physics starts with a circular definition: space-time is defined with the help of objects and objects are defined with the help of space-time. Physics thus has never been modelled on the basis of mathematics. Physicists have always had to live with logical problems. The situation is similar to a child’s description of the sky as ‘made of air and clouds’. Looking closely, we discover that clouds are made up of water droplets. However, there is air inside clouds, and there is also water vapour elsewhere in the air. When clouds and air are viewed through a microscope, there is no clear boundary between the two. We cannot define either of the terms ‘cloud’ and ‘air’ without the other. No axiomatic definition is possible. Objects and vacuum also behave in this way. Virtual particles are found in vacuum, and vacuum is found inside objects. At Planck scales there is no clear boundary between the two; we cannot define either of them without the other. Despite the lack of a precise definition and despite the logical problems that ensue, in both cases the description works well at large, everyday scales. We note that, since the universe is not a set and since it contains no information, the paradox of the physics book containing a full description of nature disappears. Such a book can exist, as it does not contradict any property of the universe. But then a question arises naturally:

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TA B L E 75 Physical statements about the universe

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* There is also another well-known, non-physical concept about which nothing can be said. Many scholars have explored it in detail. Can you see what it is?

Copyright © Christoph Schiller November 1997–May 2006

Page 644

check the statement. To find the answer, we do not need magic, as Faust thought; we only need to list what we have found so far, especially in this section, in the section on Planck scales, and in the intermezzo on brain and language. Table 75 shows the result. Not only are we unable to state that the universe is made of space-time and matter; in fact, we are unable to say anything about the universe at all!* It is not even possible to say that it exists, since it is impossible to interact with it. The term ‘universe��� does not allow us to make a single sensible statement. (Can you find one?) We are only able to say which properties it does not have. We are unable to find any property the universe does have. Thus, the universe has no properties! We cannot even say whether the universe is something or nothing. The universe isn’t anything in particular. In other words, the term ‘universe’ is not useful at all for the description of motion. We can obtain a confirmation of this strange conclusion from the first intermezzo. There we found that any concept needs defined content, defined limits and a defined domain of application. In this section, we have found that for the term ‘universe’, not one of these three aspects is defined; there is thus no such concept. If somebody asks: ‘why does the universe exist?’ the answer is: not only does the use of ‘why’ wrongly suggest that something may exist outside the universe, providing a reason for it and thus contradicting the definition of the term ‘universe’ itself; most importantly of all, the universe simply does not exist. In summary, any sentence containing the word universe makes no sense.

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The universe has no age. The universe has no beginning. The universe has no size. The universe has no volume. The universe has no shape. The universe’s particle number is undefined. The universe has no mass. The universe has no energy. The universe has no density. The universe contains no matter. The universe has no cosmological constant. The universe has no initial conditions. The universe has no state. The universe has no wave function. The universe is not a physical system. The universe contains no information. The universe is not isolated. The universe is not open. The universe has no boundaries. The universe does not interact. The universe cannot be measured. The universe cannot be said to exist. The universe cannot be distinguished from The universe cannot be distinguished from a nothing. single event. The universe contains no moments. The universe is not composite. The universe is not a set. The universe is not a concept. The universe cannot be described. There is no plural for ‘universe’. The universe cannot be distinguished from va- The universe was not created. cuum.

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The term ‘universe’ only seems to express something, even if it doesn’t.* The conclusion that the term universe makes no sense may be interesting, even strangely beautiful; but does it help us to understand motion more precisely? Interestingly so, it does. A concept without a set eliminates contradictions

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In the chapter Quantum physics in a nutshell we listed all the unexplained properties of nature left open either by general relativity or by quantum theory. The present conclusions provide us with new connections among them. Indeed, many of the cosmological results of this section sound surprisingly familiar; let us compare them systematically with those of the section on Planck scales. Both sections explored topics – some in more detail than others – from the list of unexplained properties of nature. First, Table 76 shows that none of the unexplained properties makes sense at both limits of nature, the small and the large. All open questions are open at both extremes. Second and more importantly, nature behaves in the same way at horizon scales and at Planck scales. In fact, we have not found any difference between the two cases. (Are you able to discover one?) We are thus led to the hypothesis that nature does not distinguish between the large and the small. Nature seems to be characterized by extremal identity. Is extremal identity a principle of nature? The principle of extremal identity incorporates some rather general points:

* Of course, the term ‘universe’ still makes sense if it is defined more restrictively, such as ‘everything interacting with a particular human or animal observer in everyday life’. But such a definition is not useful for our quest, as it lacks the precision required for any description of motion.

Copyright © Christoph Schiller November 1997–May 2006

— all open questions about nature appear at its two extremes; — a description of nature requires both general relativity and quantum theory; — nature or the universe is not a set; — initial conditions and evolution equations make no sense at nature’s limits; — there is a relation between local and global issues in nature; — the concept of ‘universe’ has no content.

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The discussion about the term ‘universe’ also shows that the term does not contain any set. In other words, this is the first term that will help us on the way to a precise description of nature. We will see later on how this happens. By taking into account the limits on length, time, mass and all other quantities we have encountered, we have deduced a number of almost painful conclusions about nature. However, we also received something in exchange: all the contradictions between general relativity and quantum theory that we mentioned at the beginning of this chapter are now resolved. Although we have had to leave behind us many cherished habits, in exchange we have the promise of a description of nature without contradictions. But we get even more.

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TA B L E 76 Properties of nature at maximal, everyday and minimal scales

At every day scale

At P l a n c k scale

requires quantum theory and relativity intervals can be measured precisely length and time intervals are space-time is not continuous points and events cannot be distinguished space-time is not a manifold space is 3 dimensional space and time are indistinguishable initial conditions make sense space-time fluctuates Lorentz and Poincaré symmetry CPT symmetry renormalization permutation symmetry interactions number of particles algebras of observables matter indistinguishable from vacuum boundaries exist nature is a set

true false limited true true true false true false true does not apply does not apply does not apply does not apply do not exist undefined undefined true false false

false true unlimited false false false true false true false applies applies applies applies exist defined defined false true true

true false limited true true true false true false true does not apply does not apply does not apply does not apply do not exist undefined undefined true false false

Extremal identity thus looks like a good candidate tool for use in the search for a unified description of nature. To be a bit more provocative, it may be the only known principle incorporating the idea that the universe is not a set, and thus might be the only candidate tool for use in the quest of unification. Extremal identity is beautiful in its simplicity, in its unexpectedness and in the richness of its consequences. You might enjoy exploring it by yourself. The study of the consequences of extremal identity is currently the focus of much activity in high energy particle physics, although often under different names. The simplest approach to extremal identity – in fact one that is too simple to be correct – is inversion. It looks as if extremal identity implies a connection such as r

2 l Pl r

or

xµ 

2 l Pl xµ xµ x µ

(710)

relating distances r or coordinates x µ with their inverse values using the Planck length l Pl . Can this mapping, called inversion, be a symmetry of nature? At every point of space? For

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Ref. 1084

At h ori zon scale

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P h ys i c a l p r o p e rt y o f n at u r e

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example, if the horizon distance is inserted, equation (710) implies that lengths smaller than l Pl ~1061  10−96 m never appear in physics. Is this the case? What would inversion imply for the big bang? Numerous fascinating questions are contained in the simple hypothesis of extremal identity. They lead to two main directions for investigation. First, we have to search for some stronger arguments for the validity of extremal identity. We will discover a number of simple arguments, all showing that extremal identity is indeed a property of nature and producing many beautiful insights. The other quest then follows. We need to find the correct version of equation (710). That oversimplified expression is neither sufficient nor correct. It is not sufficient because it does not explain any of the issues left open by general relativity and quantum theory. It only relates some of them, thus reducing their number, but it does not solve any of them. You may wish to check this for yourself. In other words, we need to find the precise description of quantum geometry and of elementary particles. However, inversion is also simply wrong. Inversion is not the correct description of extremal identity because it does not realize a central result discovered above: it does not connect states and intrinsic properties. Inversion keeps them distinct. This means that inversion does not take interactions into account. And most open issues at this point of our mountain ascent are properties of interactions.

Copyright © Christoph Schiller November 1997–May 2006

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Biblio graphy 1073 Available at http://www.dse.nl/motionmountain/C10-QMGR.pdf. Those pages are a re-

worked version of the ones published in French as C. Schiller, Le vide diffère-t-il de la matière? in E. Gunzig & S. Diner, editeurs, Le vide – Univers du tout et du rien – Des physiciens et des philosophes s’interrogent, Les Éditions de l’Université de Bruxelles, 1998. See also C. Schiller, Does matter differ from vacuum? http://www.arxiv.org/abs/gr-qc/ 9610066. Cited on pages 1032, 1036, and 1042.

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1074 See the lucid discussion by G.F.R. Ellis & T. Rothman, Lost horizons, American

Journal of Physics 61, pp. 883–893, 1993. Cited on pages 1034, 1038, 1039, and 1040. 1075 H. Salecker & E.P. Wigner, Quantum limitations of the measurement of space-time

distances, Physical Review 109, pp. 571–577, 1958. See also the popular account of this paper by E.J. Zimmerman, The macroscopic nature of space-time, American Journal of Physics 30, pp. 97–105, 1962. Cited on page 1034. Cimento A42, pp. 390–402, 1966, Y.J. Ng & H. Van Dam, Limit to space-time measurement, Modern Physics Letters A 9, pp. 335–340, 1994, Y.J. Ng & H. Van Dam, Remarks on gravitational sources, Modern Physics Letters A 10, pp. 2801–2808, 1995. Newer summaries are Y.J. Ng & H. van Dam, Comment on “Uncertainty in measurements of distance”, http://www.arxiv.org/abs/gr-qc/0209021 and Y.J. Ng, Spacetime foam, http://www.arxiv. org/abs/gr-qc/0201022. Cited on page 1035. 1077 See, for example, the Hollywood movie Contact, based on the book by Carl Sagan,

Contact, 1985. Cited on page 1040. 1078 See, for example, the international bestseller by S. Hawking, A Brief History of Time –

From the Big Bang to Black Holes, 1988. Cited on page 1042. 1079 L. Rosenfeld, in H.-J. Treder, Entstehung, Entwicklung und Perspektiven der Ein-

steinschen Gravitationstheorie, Springer Verlag, 1966. Cited on page 1045. 1080 Holography in high-energy physics is connected with the work by ’t Hooft and by Susskind.

1081 S. Lloyd, Computational capacity of the universe, Physical Review Letters 88, p. 237901,

2002. Cited on page 1048. 1082 Gottfried Wilhelm Leibniz, La Monadologie, 1714. Written in French, it is avail-

able freely at http://www.uqac.uquebec.ca/zone30/Classiques_des_sciences_sociales and in various other languages on other websites. Cited on page 1048. 1083 See, for example, H. Wussing & P.S. Alexandrov, (eds.), Die Hilbertschen Probleme,

Akademische Verlagsgesellschaft Geest & Portig, 1983, or Ben H. Yandell, The Honours Class: Hilbert’s Problems and their Solvers, A.K. Peters, 2002. Cited on page 1049. 1084 Large part of the study of dualities in string and M theory can be seen as investigations into

the detailed consequences of extremal identity. For a review of duality, see ... A classical version of duality is discussed by M.C.B. Abdall a, A.L. Gadelka & I.V. Vancea, Du-

Copyright © Christoph Schiller November 1997–May 2006

See for example G. ’t Hooft, Dimensional reduction in quantum gravity, pp. 284–296, in A. Ali, J. Ellis & S. R andjbar-Daemi, Salaamfeest, 1993, or the often cited paper by L. Susskind, The world as a hologram, Journal of Mathematical Physics 36, pp. 6377– 6396, 1995, or as http://www.arxiv.org/abs/hep-th/9409089 A good modern overview is R. B ousso, The holographic principle, Review of Modern Physics 74, pp. 825–874, 2002, also available as http://www.arxiv.org/abs/hep-th/0203101, even though it has some argumentation errors, as explained on page 871. Cited on page 1046.

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1076 F. Karolyhazy, Gravitation and quantum mechanics of macroscopic objects, Il Nuovo

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1055 bibliography

ality between coordinates and the Dirac field, http://www.arxiv.org/abs/hep-th/0002217. Cited on page 1052.

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Copyright © Christoph Schiller November 1997–May 2006

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35.

the physics of love – a summary of the first t wo and a half parts

Sex is the physics urge sublimated.

M

Graffito

Love is communication. Love is tiring. Love is an interaction between moving bodies. Love takes time. Love is attractive. Love is repulsive. Love makes noise. In love, size matters. Love is for reproduction. Love can hurt. Love needs memory. Love is Greek. Love uses the sense of sight. Love is animalic. Love is motion. Love is holy. Love is based on touch. Love uses motion again. Love is fun. Love is private. Love makes one dream.

Let us see what these observations imply for the description of nature. **

Ref. 1085

* Here we deduce physics from love. We could also deduce physics from sexuality. The modern habit of saying ‘sex’ instead of ‘sexuality’ mixes up two completely different concepts. In fact, studying the influences of sex on physics is almost fully a waste of time. We avoid it. Maybe one day we shall understand why there do not seem to be any female crackpots proposing pet physical theories.

Copyright © Christoph Schiller November 1997–May 2006

Love is communication. Communication is possible because nature looks similar from different standpoints and because nature shows no surprises. Without similarity we could not understand each other, and a world of surprises would even make thinking impossible; it would not be possible to form concepts to describe observations. But fortunately, the world is regular; it thus allows to use concepts such as time and space for its description.

Dvipsbugw

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aybe you have once met a physicist who has told you, in one of those oments of confidentiality, that studying physics is even more beautiful than aking love. At this statement, many will simply shake their head in disbelief and strongly disapprove. In this section we shall argue that it is possible to learn so much about physics while making love that discussions about their relative beauty can be put aside altogether. Imagine to be with your partner on a beautiful tropical island, just after sunset, and to look together at the evening sky. Imagine as well that you know little of what is taught at school nowadays, e.g. that your knowledge is that of the late Renaissance, which probably is a good description of the average modern education level anyway. Imagine being busy enjoying each other’s company. The most important results of physics can be deduced from the following experimental facts:*

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**

Challenge 1479 n

Love is an interaction between moving bodies. Together with the previous result, this implies that we can and need to describe moving bodies with mass, energy and momentum. That is not a small feat. For example, it implies that the Sun will rise tomorrow if the sea level around the island is the usual one. ** Love is attractive. When feeling attracted to your partner, you may wonder if this attraction is the same which keeps the Moon going around the Earth. You make a quick calculation and find that applying the expression for universal gravity E pot = −

GMm r

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(711)

** Love makes noise. That is no news. However, even after making love, even when everybody and everything is quiet, in a completely silent environment, we do hear something. The noises we hear are produced within the ear, partly by the blood flowing through the head, partly by the electrical noise generated in the nerves. That is strange. If matter were continuous, there would be no noise even for low signal levels. In fact, all proofs for the discreteness of matter, of electric current, of energy, or of light are based on the increase of fluctuations with the smallness of systems under consideration. The persistence of noise thus makes us suspect that matter is made of smallest entities. Making love confirms this suspicion in several ways. **

Copyright © Christoph Schiller November 1997–May 2006

Love is for reproduction. Love is what we owe our life to, as we all are results of reproduction. But the reproduction of a structure is possible only if it can be constructed, in other words if the structure can be built from small standard entities. Thus we again suspect ourselves to be made of smallest, discrete entities. Love is also a complicated method of reproduction. Mathematics provides a much simpler one. If matter objects were not made of particles, but were continuous, it would be possible to perform reproduction by cutting and reassembling. A famous mathematical theorem by Banach and Tarski proves that it is possible to take a continuous solid, cut it into five pieces and rearrange the pieces in such a way that the result are two copies of the same size and volume as the original. In fact, even volume increases can be produced in this way, thus realizing growth without any need for food. Mathematics thus provides

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to both of you, the involved energy is about as much as the energy added by the leg of a fly on the skin. (M and m are your masses, r your distance, and the gravitational constant has the value G = 6.7 ë 10−11 m3 ~kg s2 .) In short, your partner teaches you that in nature there are other attractive interactions apart from gravity; the average modern education is incomplete. Nevertheless, this first equation is important: it allows to predict the position of the planets, the time of the tides, the time of eclipses, the return of comets, etc., to a high accuracy for thousands of years in advance.

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xi general rel ativity versus quantum mechanics

some interesting methods for growth and reproduction. However, they assume that matter is continuous, without a smallest length scale. The observation that these methods do not work in nature is compatible with the idea that matter is not continuous. ** Ref. 1086

Challenge 1480 n

Love needs memory. If you would not recognize your partner among all possible ones, your love life would be quite complicated. A memory is a device which, in order to store information, must have small internal fluctuations. Obviously, fluctuations in systems get smaller as their number of components increase. Since our memory works so well, we can follow that we are made of a large number of small particles. In summary, love shows that we are made of some kind of lego bricks: depending on the level of magnification, these bricks are called molecules, atoms, or elementary particles. It is possible to estimate their size using the sea around the tropical island, as well as a bit of oil. Can you imagine how?

Ref. 1087

Love uses the sense of sight. Seeing each other is only possible because we are cold whereas the Sun is hot. If we and our environment all had the same temperature as the Sun, we would not see each other. This can be checked experimentally by looking into a hot oven: Inside a glowing oven filled with glowing objects it is impossible to discern them against the background. ** Love is motion. Bodies move against each other. Moreover, their speed can be measured. Since measurement is a comparison with a standard, there must be a velocity standard in nature, some special velocity standing out. Such a standard must either be the minimum or the maximum possible value. Now, daily life shows that for velocity a finite minimum value does not exist. We are thus looking for a maximum value. To estimate the value of the maximum, just take your mobile phone and ring home from the island to your family. From the delay in the line and the height of the satellite, you can deduce the telephone speed c and get 3 ë 108 m~s. The existence of a maximum speed c implies that time is different for different observers. Looking into the details, we find that this effect becomes noticeable at energies (712)

where m is the mass of the object. For example, this applies to electrons inside a television tube. **

Challenge 1481 n

Love is based on touching. When we touch our partner, sometimes we get small shocks. The energies involved are larger than than those of touching fly legs. In short, people are electric. In the dark, we observe that discharges emit light. Light is thus related to electricity. In addition, touching proves that light is a wave: simply observe the dark lines between two fingers near your eye in front of a bright background. The lines are due to interference

Copyright © Christoph Schiller November 1997–May 2006

E different time  mc 2 ,

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**

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the physics of love – a summary of the first two and a half parts

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effects. Light thus does not move with infinite speed. In fact, it moves with the same speed as that of telephone calls. **

Page 210

Love is fun. People like to make love in different ways, such as in a dark room. But rooms get dark when the light is switched off only because we live in a space of odd dimensions. In even dimensions, a lamp would not turn off directly after the switch is flipped, but dim only slowly. Love is also fun because with our legs, arms and bodies we can make knots. Knots are possible only in three dimensions. In short, love is real fun only because we live in 3 dimensions.

Dvipsbugw

**

R horizon  Gm~c 2 .

(713)

For example, only due a horizon, albeit one appearing in a different way, the night sky is dark. **

Ref. 1091

Love takes time. It is known that men and women have different opinions on durations. It is also known that love happens between your ears. Indeed, biological research has shown that we have a clock inside the brain, due to circulating electrical currents. This clock provides our normal sense of time. Since such a brain clock can be built, there must be a time standard in nature. Again, such a standard must be either a minimum or a maximum time interval. We shall discover it later on. **

** Love can hurt. For example, it can lead to injuries. Atoms can get ripped apart. That hap* Note that the classical electron radius is not an exception: it contains the elementary charge e, which contains a length scale, as shown on page 492.

Copyright © Christoph Schiller November 1997–May 2006

Love is repulsive. And in love, size matters. Both facts turn out to be the two sides of the same coin. Love is based on touch, and touch needs repulsion. Repulsion needs a length scale, but neither gravity nor classical electrodynamics provide one. Classical physics only allows for the measurement of speed. Classical physics cannot explain that the measurement of length, time, or mass is possible.* Classically, matter cannot be hard; it should be possible to compress it. But love shows us that this is not the case. Love shows us that lengths scales do exist in nature and thus that classical physics is not sufficient for the description of nature.

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Love is tiring. The reason is gravity. But what is gravity? A little thinking shows that since there is a maximum speed, gravity is the curvature of space-time. Curved space also means that a horizon can appear, i.e. a largest possible visible distance. From equations (711) and (712), we deduce that this happens when distances are of the order of

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xi general rel ativity versus quantum mechanics

pens when energies are concentrated on small volumes, such as a few aJ per atom. Investigating such situations more precisely, we finds that strange phenomena appear at distances r if energies exceed the value E

ħc r

;

(714)

r pair production 

ħ . mc

(715)

At such small distances we cannot avoid using the quantum description of nature. ** Love is not only Greek. The Greeks were the first to make theories above love, such as Plato in his Phaedrus. But they also described it in another way. Already before Plato, Democritus said that making love is an example of particles moving and interacting in vacuum. If we change ‘vacuum’ to ‘curved 3+1-dimensional space’ and ‘particle’ to ‘quantum particle’, we do indeed make love in the way Democritus described 2500 years ago. It seems that physics has not made much progress in the mean time. Take the statement made in 1939 by the British astrophysicist Arthur Eddington:

Ref. 1088

I believe there are 15,747,724,136,275,002,577,605,653,961,181,555,468,044,717,914,527,116,709,366,231,425,076,185,631,031,296 protons in the universe and the same number of electrons.

Baryons in the universe: 10811 ; total charge: near zero.

The second is more honest, but which of the two is less sensible? Both sentences show that there are unexplained facts in the Greek description nature, in particular the number of involved particles. ** Love is animalic. We have seen that we can learn a lot about nature from the existence of love. We could be tempted to see this approach of nature as a special case of the so-called

Copyright © Christoph Schiller November 1997–May 2006

Compare it with the version of 2006:

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in particular, energy becomes chunky, things become fuzzy, boxes are not tight, and particles get confused. These are called quantum phenomena. The new constant ħ = 10−34 Js is important: it determines the size of things, because it allows to define distance and time units. In other words, objects tear and break because in nature there is a minimum action, given roughly by ħ. If even more energy is concentrated in small volumes, such as energies of the order of mc 2 per particle, one even observes transformation of energy into matter, or pair production. From equations (712) and (714), we deduce that this happens at distances of

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the physics of love – a summary of the first two and a half parts

1061

** Ref. 1089

Copyright © Christoph Schiller November 1997–May 2006

Love is holy. Following the famous definition by the theologian Rudolf Otto, holiness results from a mixture of a mysterium tremendum and a mysterium fascinans. Tremendum means that it makes one tremble. Indeed, love produces heat and is a dissipative process. All systems in nature which produce heat have a finite lifetime. That is true for machines, stars, animals, lightning, fire, lamps and people. Through heat, love shows us that we are going to die. Physicists call this the second principle of thermodynamics. But love also fascinates. Everything which fascinates has a story. Indeed, this is a principle of nature: every dissipative structure, every structure which appears or is sustained through the release of energy, tells us that it has a story. Take atoms, for example. All the protons we are made of formed during the big bang. Most hydrogen we are made of is also that old. The other elements were formed in stars and then blown into the sky during nova or supernova explosions. They then regrouped during planet formation. We truly are made of stardust. Why do such stories fascinate? If you only think about how you and your partner have met, you will discover that it is through a chain of incredible coincidences. If only one of all these coincidences had not taken place, you and your partner would not be together. And of course, we all owe our existence to such a chain of coincidences, which brought our parents together, our grandparents, and made life appear on Earth. The realization of the importance of coincidences automatically produces two kinds of questions: why? and what if? Physicists have now produced a list of all the answers to repeated why questions and many are working at the list of what-if questions. The first list, the why-list of Table 77, gives all facts still unexplained. It can also be called the complete list of all surprises in nature. (Above, it was said that there are no surprises in nature about what happens. However, so far there still are a handful of surprises on how all these things happen.)

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anthropic principle. However, some care is required here. In fact, we could have learned exactly the same if we had taken as starting point the observation that apes or pigs have love. There is no ‘law’ of nature which distinguishes between them and humans. In fact, there is a simple way to determine whether any ‘anthropic’ statement makes sense: the reasoning must be equally true for humans, apes, and pigs. A famous anthropic deduction was drawn by the British astrophysicist Fred Hoyle. While studying stars, he predicted a resonance in the carbon-12 nucleus. If it did not exist, he argued, stars could not have produced the carbon which afterwards was spread out by explosions into interstellar space and collected on Earth. Also apes or pigs could reason this way; therefore Hoyle’s statement does make sense. On the other hand, claiming that the universe is made especially for people is not sensible: using the same arguments, pigs would say it is made for pigs. The existence of either requires all ‘laws’ of nature. In summary, the anthropic principle is true only in so far as its consequences are indistinguishable from the porcine or the simian principle. In short, the animalic side of love puts limits to the philosophy of physics.

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TA B L E 77 Everything quantum field theory and general relativity do not explain; in other words, a list of

the only experimental data and criteria available for tests of the unified description of motion

O b s e r va b l e P r o p e r t y u n e x p l a i n e d s o f a r Local quantities, from quantum theory the low energy value of the electromagnetic coupling constant the low energy value of the weak coupling constant the low energy value of the strong coupling constant the values of the 6 quark masses the values of 3 lepton masses the values of the independent mass of the W vector boson the value of the Weinberg angle three mixing angles the value of the CP parameter the value of the strong topological angle the number of particle generations the value of the observed vacuum energy density or cosmological constant the number of space and time dimensions

Global quantities, from general relativity 1.2(1) ë 1026 m ? 1082 ? A 1092 ?

the distance of the horizon, i.e. the ‘size’ of the universe (if it makes sense) the number of baryons in the universe, i.e. the average matter density in the universe (if it makes sense) the initial conditions for more than 1092 particle fields in the universe, including those at the origin of galaxies and stars (if they make sense)

Local structures, from quantum theory

Global structures, from general relativity maybe R  S3 ?

the unknown topology of the universe (if it makes sense)

This why-list fascinates through its shortness, which many researchers are still trying

Copyright © Christoph Schiller November 1997–May 2006

the origin of particle identity, i.e. of permutation symmetry the renormalization properties, i.e. the existence of point particles the origin of Lorentz (or Poincaré) symmetry (i.e. of spin, position, energy, momentum) C‡ the origin of the algebra of observables Gauge group the origin of gauge symmetry (and thus of charge, strangeness, beauty, etc.) in particular, for the standard model: U(1) the origin of the electromagnetic gauge group (i.e. of the quantization of electric charge, as well as the vanishing of magnetic charge) SU(2) the origin of weak interaction gauge group SU(3) the origin of strong interaction gauge group S(n) Ren. group SO(3,1)

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α em αw αs mq ml mW θW β1 , β2 , β3 θ CP θ st 3 0.5 nJ~m3 3+1

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1063

to reduce. But it is equally interesting to study what consequences appear if any of the values from Table 77 were only a tiny bit different. It is not a secret that small changes in nature would lead to completely different observations, as shown in Table 78. TA B L E 78 A small selection of the consequences when changing aspects of nature

R e su lt

Moon size

smaller

Moon size

larger

Jupiter

smaller

Jupiter

larger

Oort belt

smaller

small Earth magnetic field; too much cosmic radiation; widespread child cancers. large Earth magnetic field; too little cosmic radiation; no evolution into humans. too many comet impacts on Earth; extinction of animal life. too little comet impacts on Earth; no Moon; no dinosaur extinction. no comets, no irregular asteroids, no Moon; still dinosaurs. irregular planet motion; supernova dangers. proton decay; leucemia.

Star distance smaller Strong coupling smaller constant

The large number of coincidences of life force our mind to realize that we are only a tiny part of nature. We are a small droplet shaken around in the ocean of nature. Even the tiniest changes in nature would prevent the existence of humans, apes and pigs. In other words, making love tells us that the universe is much larger than we are and tells us how much we are dependent and connected to the rest of the universe. **

**

Copyright © Christoph Schiller November 1997–May 2006

We said above that love uses motion. It contains a remarkable mystery, worth a second look: - Motion is the change of position with time of some bodies. - Position is what we measure with a ruler. Time is what we measure with a clock. Both rulers and clocks are bodies. - A body is an entity distinct from its environment by its shape or its mass. Shape is the extension of a body in space (and time). Mass is measured by measuring speed or acceleration, i.e. by measuring space and time. This means that we define space-time with bodies – as done in detail in general relativity – and that we define bodies with space-time – as done in detail in quantum theory. This circular reasoning shows that making love is truly a mystery. The circular reasoning has not yet been eliminated yet; at present, modern theoretical physicists are busy attempting to do so. The most promising approach seems to be M-theory, the modern extension of string theory. But any such attempt has to overcome important difficulties which can also be experienced while making love.

Dvipsbugw

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O b s e r va b l e C h a n g e

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M

Dvipsbugw R

R

F I G U R E 386 A Gedanken experiment showing that

Ref. 1090

Love is private. But is it? Privacy assumes that a person can separate itself from the rest, without important interactions, at least for a given time, and come back later. This is possible if the person puts enough empty space between itself and others. In other words, privacy is based on the idea that objects can be distinguished from vacuum. Let us check whether this is always possible. What is the smallest measurable distance? This question has been almost, but only almost answered by Max Planck in 1899. The distance δl between two objects of mass m is surely larger than their position indeterminacy ħ~∆p; and the momentum indeterminacy must be smaller that the momentum leading to pair production, i.e. ∆p < mc. This means that ħ . (716) δl E ∆l E mc

δl E

¾

ħG = l Pl = 1.6 ë 10−35 m . c3

(717)

This expression defines a minimum length in nature, the so-called Planck length. Every other Gedanken experiment leads to this characteristic length as well. In fact, this minimum distance (and the corresponding minimum time interval) provides the measurement standard we were looking for at the beginning of our musings about length and time measurements. A more detailed discussion shows that the smallest measurable distance is somewhat larger, a multiple of the Planck length, as measurements require the distinction of matter and radiation. This happens at scales about 800 times the Planck length. In other words, privacy has its limits. In fact, the issue is even more muddled when

Copyright © Christoph Schiller November 1997–May 2006

In addition, the measurements require that signals leave the objects; the two masses must not be black holes. Their masses must be so small that the Schwarzschild radius is smaller than the distance to be measured. This means that r S  Gm~c 2 < δl or that

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at Planck scales, matter and vacuum cannot be distinguished

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the physics of love – a summary of the first two and a half parts

1065

we explore the consequences for bodies. A body, also a human one, is something we can touch, throw, hit, carry or weigh. Physicists say that a body is something with energy or momentum. Vacuum has none of it. In addition, vacuum is unbounded, whereas objects are bounded. What happens if we try to weigh objects at Planck scales? Quantum theory makes a simple prediction. If we put an object of mass M in a box of size R onto a scale – as in Figure 386 – equation (714) implies that there is a minimal mass error ∆M given by ∆M 

ħ . cR

(719)

How large is the mass we can put into a box of Planck size? Obviously it is given by the maximum possible mass density. To determine it, imagine a planet and put a satellite in orbit around it, just skimming its surface. The density ρ of the planet with radius r is given by v2 M (720) ρ 3 = 2 . r Gr

Using equation (716) we find that the maximum mass density in nature, within a factor of order one, is the so-called Planck density, given by ρ Pl =

Ref. 1090

(721)

Therefore the maximum mass that can be contained inside a Planck box is the Planck mass. But that was also the measurement error for that situation. This implies that we cannot say whether the original box we measured was empty or full: vacuum cannot be distinguished from matter at Planck scales. This astonishing result is confirmed by every other Gedanken experiment exploring the issue. It is straightforward to deduce with similar arguments that objects are not bound in size at Planck scales, i.e. that they are not localized, and that the vacuum is not necessarily extended at those scales. In addition, the concept of particle number cannot be defined at Planck scales. So, why is there something instead of nothing? Making love shows that there is no difference between the two options! ** Love makes us dream. When we dream, especially at night, we often look at the sky. How far is it away? How many atoms are enclosed by it? How old is it? These questions have an answer for small distances and for large distances; but for the whole of the sky or the whole of nature they cannot have one, as there is no way to be outside of the sky in order to measure it. In fact, each of the impossibilities to measure nature at smallest distances

Copyright © Christoph Schiller November 1997–May 2006

Challenge 1482 n

c5 = 5.2 ë 1096 kg~m3 . G2ħ

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If the box has Planck size, the mass error is the Planck mass » ∆M = M Pl = ħc~G  22 µg .

Dvipsbugw (718)

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1066

Ref. 1092

xi general rel ativity versus quantum mechanics

are found again at the largest scales. There seems to be a fundamental equivalence, or, as physicists say, a duality between the largest and the smallest distances. The coming years will hopefully show how we can translate these results into an even more precise description of motion and of nature. In particular, this description should allow us to reduce the number of unexplained properties of nature. ** In summary, making love is a good physics lesson. Enjoy the rest of your day.

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bibliography

1067

Biblio graphy 1085 An attempt to explain the lack of women in physics is made in Margaret Wertheim,

1086 1087

1088

1089

Pythagoras’ Trousers – God, Physics and the Gender Wars, Fourth Estate, 1997. Cited on page 1056. The consequences of memory loss in this case are already told by Voltaire, Aventure de la mémoire, 1775. Cited on page 1058. A picture of objects in a red hot oven and at room temperature is shown in C.H. Bennett, Demons, Engines and the Second Law, Scientific American 255, pp. 108–117, November 1987. Cited on page 1058. The famous quote is found at the beginning of chapter XI, ‘The Physical Universe’, in Arthur Eddington, The Philosophy of Physical Science, Cambridge, 1939. Cited on page 1060. See the first intermezzo for details and references. Cited on page 1061.

in French as C. Schiller, Le vide diffère-t-il de la matière? in E. Gunzig & S. Diner, editeurs, Le vide – Univers du tout et du rien – Des physiciens et des philosophes s’interrogent, Les Éditions de l’Université de Bruxelles, 1998. Cited on pages 1064 and 1065. 1091 An introduction to the sense of time as result of signals in the ganglia and in the temporal lobe of the brain is found in R.B. Ivry & R. Spencer, The neural representation of time, Current Opinion in Neurobiology 14, pp. 225–232, 2004. Cited on page 1059. 1092 For details of fthe arguments leading to duality, see section 34. It also includes suggestions supporting the notion that the universe is not a even a set, thus proposing a solution for Hilbert’s sixth problem. Cited on page 1066.

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1090 Details can be found in Chapter XI of this text, a reworked version of the pages published

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maximum force and minimum distance – physics in limit statements

Fundamental limits to all observables

Special relativity in one statement The step from everyday or Galilean physics to special relativity can be summarized in a single limit statement on motion. It was popularized by Hendrik Antoon Lorentz: There is a maximum energy speed in nature. For all physical systems and all observers, vDc.

(722)

* These limits were given for the first time in 2003, in this section of the present text. ** Stimulating discussions with Saverio Pascazio, Corrado Massa and Steve Carlip helped shaping this section.

Copyright © Christoph Schiller November 1997–May 2006

At dinner parties physicists are regularly asked to summarize physics in a few sentences. It is useful to be able to present a few simple statements answering the request.** Here we propose such a set of statements. We are already familiar with each statement from what we have found out so far in our exploration of motion. But by putting them all together we will get direct insight into the results of modern research on unification. The main lesson of modern physics is the following: Nature limits the possibilities of motion. These limits are the origin of special relativity, general relativity and quantum theory. In fact, we will find out that nature poses limits to every aspect of motion; but let us put first things first.

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The principle of maximum force or power allows us to summarize special relativity, quantum theory and general relativity in one fundamental limit principle each. The three principles are fully equivalent to the standard formulation of the theories. In particular, we show that the maximum force c 4 ~4G implies and is equivalent to the field equations of general relativity. With the speed limit of special relativity and the action–angular momentum limit of quantum theory, the three fundamental principles imply a bound for every physical observable, from acceleration to size. The new, precise limit values differ from the usual Planck values by numerical prefactors of order unity.* Continuing this approach, we show that every observable in nature has both a lower and an upper limit value. As a result, a maximum force and thus a minimum length imply that the noncontinuity of space-time is an inevitable consequence of the unification of quantum theory and relativity. Furthermore, the limits are shown to imply the maximum entropy bound, including the correct numerical prefactor. The limits also imply a maximum measurement precision possible in nature, thus showing that any description by real numbers is approximate. Finally, the limits show that nature cannot be described by sets. As a result, the limits point to a solution to Hilbert’s sixth problem. The limits also show that vacuum and matter cannot be fully distinguished in detail; they are two limit cases of the same entity. These fascinating results provide the basis for any search for a unified theory of motion.

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fundamental limits to all observables

Page 282

1069

This size limit is induced by the speed of light c; it is also valid for the displacement d of a system, if the acceleration measured by an external observer is used. Finally, the limit implies an ‘indeterminacy’ relation ∆l ∆a D c 2

Challenge 1483 n

(724)

for the length and acceleration indeterminacies. You might want to take a minute to deduce it from the time-frequency indeterminacy. All this is standard knowledge.

Ref. 1093 Page 704

In the same way, the difference between Galilean physics and quantum theory can be summarized in a single statement on motion, due to Niels Bohr: There is a minimum action in nature. For all physical systems and all observers, SE

ħ . 2

(725)

* A physical system is a region of space-time containing mass–energy, the location of which can be followed over time and which interacts incoherently with its environment. With this definition, images, geometrical points or incomplete entangled situations are excluded from the definition of system.

Copyright © Christoph Schiller November 1997–May 2006

Quantum theory in one statement

Dvipsbugw

Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

A few well-known remarks set the framework for the discussion that follows. The speed v is smaller than or equal to the speed of light c for all physical systems;* in particular, this limit is valid both for composed systems as well as for elementary particles. The speed limit statement is also valid for all observers; no exception to the statement is known. Indeed, only a maximum speed ensures that cause and effect can be distinguished in nature, or that sequences of observations can be defined. The opposite statement, implying the existence of (real) tachyons, has been explored and tested in great detail; it leads to numerous conflicts with observations. The maximum speed forces us to use the concept of space-time to describe nature. Maximum speed implies that space and time mix. The existence of a maximum speed in nature also implies observer-dependent time and space coordinates, length contraction, time dilation, mass–energy equivalence and all other effects that characterize special relativity. Only the existence of a maximum speed leads to the principle of maximum ageing that governs special relativity; only a maximum speed leads to the principle of least action at low speeds. In addition, only a finite speed limit makes it to define a unit of speed that is valid at all places and at all times. If a global speed limit did not exist, no natural measurement standard for speed, independent of all interactions, would exist in nature; speed would not then be a measurable quantity. Special relativity also limits the size of systems, independently of whether they are composed or elementary. Indeed, the limit speed implies that acceleration a and size l cannot be increased independently without bounds, because the two ends of a system must not interpenetrate. The most important case concerns massive systems, for which we have c2 . (723) lD a

Dvipsbugw


1070

Page 704

dE

Page 708

ħ . 2mc

(726)

In other words, (half) the (reduced) Compton wavelength of quantum theory is recovered as lower limit on the displacement of a system, whenever gravity plays no role. Since the quantum displacement limit applies in particular to an elementary system, the limit is also valid for the size of a composite system. However, the limit is not valid for the size of elementary particles. The action limit of quantum theory also implies Heisenberg’s well-known indeterminacy relation for the displacement d and momentum p of systems:

Dvipsbugw

Copyright © Christoph Schiller November 1997–May 2006

Half the Planck constant ħ is the smallest observable action or angular momentum. This statement is valid for both composite and elementary systems. The action limit is used less frequently than the speed limit. It starts from the usual definition of the action, S = ∫ (T − U)dt, and states that between two observations performed at times t and t + ∆t, even if the evolution of a system is not known, the action is at least ħ~2. Physical action is defined to be the quantity that measures the amount of change in the state of a physical system. In other words, there is always a minimum change of state taking place between two observations of a system. In this way, the quantum of action expresses the well-known fundamental fuzziness of nature at a microscopic scale. It can easily be checked that no observation results in a smaller value of action, irrespective of whether photons, electrons or macroscopic systems are observed. No exception to the statement is known. A minimum action has been observed for fermions, bosons, laser beams and matter systems and for any combination of these. The opposite statement, implying the existence of change that is arbitrary small, has been explored in detail; Einstein’s long discussion with Bohr, for example, can be seen as a repeated attempt by Einstein to find experiments that make it possible to measure arbitrarily small changes in nature. In every case, Bohr found that this aim could not be achieved. The minimum value of action can be used to deduce the indeterminacy relation, the tunnelling effect, entanglement, permutation symmetry, the appearance of probabilities in quantum theory, the information theory aspect of quantum theory and the existence of elementary particle reactions. The minimum value of action implies that, in quantum theory, the three concepts of state, measurement operation and measurement result need to be distinguished from each other; a so-called Hilbert space needs to be introduced. The minimal action is also part of the Einstein–Brillouin–Keller quantization. The details of these connections can be found in the chapter on quantum theory. Obviously, the existence of a quantum of action has been known right from the beginning of quantum theory. The quantum of action is at the basis of all descriptions of quantum theory, including the many-path formulation and the information-theoretic descriptions. The existence of a minimum quantum of action is completely equivalent to all standard developments of quantum theory. We also note that only a finite action limit makes it possible to define a unit of action. If an action limit did not exist, no natural measurement standard for action would exist in nature; action would not then be a measurable quantity. The action bound S D pd D mcd, together with the quantum of action, implies a limit on the displacement d of a system between any two observations:

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Ref. 1094

xi general rel ativity versus quantum mechanics

Dvipsbugw


fundamental limits to all observables

∆d ∆p E

1071

ħ . 2

(727)

This is valid for both massless and massive systems. All this is textbook knowledge, of course. General relativity in one statement Least known of all is the possibility of summarizing the step from Galilean physics to general relativity in a single statement on motion: There is a maximum force or power in nature. For all physical systems and all observers,

Ref. 1095 Page 349

Challenge 1484 e

Page 349

PD

c5 = 9.1 ë 1051 W . 4G

(728)

The limit statements contain both the speed of light c and the constant of gravitation G; they thus indeed qualify as statements concerning relativistic gravitation. Like the previous limit statements, they are valid for all observers. This formulation of general relativity is not common; in fact, it seems that it was only discovered 80 years after the theory of general relativity had first been proposed.* A detailed discussion is given in the chapter on general relativity. The value of the force limit is the energy of a Schwarzschild black hole divided by its diameter; here the ‘diameter’ is defined as the circumference divided by π. The power limit is realized when such a black hole is radiated away in the time that light takes to travel along a length corresponding to the diameter. Force is change of momentum; power is the change of energy. Since momentum and energy are conserved, force or power can be pictured as the flow of momentum or energy through a given surface. The value of the maximum force is the mass–energy of a black hole divided by its diameter. It is also the surface gravity of a black hole times its mass. The force limit thus means that no physical system of a given mass can be concentrated in a region of space-time smaller than a (non-rotating) black hole of that mass. In fact, the mass–energy concentration limit can be easily transformed by simple algebra into the force limit; both are equivalent. It is easily checked that the maximum force is valid for all systems observed in nature, whether they are microscopic, macroscopic or astrophysical. Neither the ‘gravitational force’ (as long as it is operationally defined) nor the electromagnetic or the nuclear interactions are ever found to exceed this limit. The next aspect to check is whether a system can be imagined that does exceed the limit. An extensive discussion shows that this is impossible, if the proper size of observers or test masses is taken into account. Even for a moving observer, when the force value is increased by the (cube of the) relativistic dilation factor, or for an accelerating observer, when the observed acceleration is increased by the acceleration of the observer itself, the force limit must still hold. However, no situations allow the limit to be exceeded, because for high accelerations a, horizons appear at distance a~c 2 ; since a mass m has a minimum * It might be that the present author was the first to point it out, in this very textbook. Also Gary Gibbons found this result independently.

Copyright © Christoph Schiller November 1997–May 2006

Challenge 1485 e

c4 = 3.0 ë 1043 N or 4G

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FD

Dvipsbugw

Dvipsbugw


1072

xi general rel ativity versus quantum mechanics

diameter given by l E 4Gm~c 2 , we are again limited by the maximum force. The exploration of the force and power limits shows that they are achieved only at horizons; the limits are not reached in any other situation. The limits are valid for all observers and all interactions. As an alternative to the maximum force and power limits, we can use as basic principle the statement: There is a maximum mass change in nature. For all systems and observers, one has dm c 3 D = 1.0 ë 1035 kg~s . (729) dt 4G

Dvipsbugw

Equivalently, we can state: There is a maximum mass per length ratio in nature. For all systems and observers, one has

Page 349 Page 349

(730)

In detail, both the force and the power limits state that the flow of momentum or of energy through any physical surface – a term defined below – of any size, for any observer, in any coordinate system, never exceeds the limit values. Indeed, as a result to the lack of nearby black holes or horizons, neither limit value is exceeded in any physical system found so far. This is the case at everyday length scales, in the microscopic world and in astrophysical systems. In addition, even Gedanken experiments do not allow to the limits to be exceeded. However, the limits become evident only when in such Gedanken experiments the size of observers or of test masses is taken into account. If this is not done, apparent exceptions can be constructed; however, they are then unphysical. Deducing general relativity*

Ref. 1096

* This section can be skipped at first reading.

Copyright © Christoph Schiller November 1997–May 2006

In order to elevate the force or power limit to a principle of nature, we have to show that, in the same way that special relativity results from the maximum speed, general relativity results from the maximum force. The maximum force and the maximum power are only realized at horizons. Horizons are regions of space-time where the curvature is so high that it limits the possibility of observation. The name ‘horizon’ is due to a certain analogy with the usual horizon of everyday life, which also limits the distance to which one can see. However, in general relativity horizons are surfaces, not lines. In fact, we can define the concept of horizon in general relativity as a region of maximum force; it is then easy to prove that it is always a two-dimensional surface, and that it has all properties usually associated with it. The connection between horizons and the maximum force or power allows a simple deduction of the field equations. We start with a simple connection. All horizons show energy flow at their location. This implies that horizons cannot be planes. An infinitely extended plane would imply an infinite energy flow. To characterize the finite extension of a given horizon, we use its radius R and its total area A. The energy flow through a horizon is characterized by an energy E and a proper length L of the energy pulse. When such an energy pulse flows perpendicularly through a hori-

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dm c 2 D = 3.4 ë 1026 kg~m . dl 4G

Dvipsbugw


fundamental limits to all observables

zon, the momentum change dp~dt = F is given by F=

E . L

1073

(731)

For a horizon, we need to insert the maximum possible values. With the horizon area A and radius R, we can rewrite the limit case as E 1 c4 = 4πR 2 4G A L

Ref. 1097

1 a2 A L . 4πG

(733)

Special relativity shows that horizons limit the product aL between proper length and acceleration to the value c 2 ~2. This leads to the central relation for the energy flow at horizons: c2 E= aA. (734) 8πG

This equation makes three points. First, the energy flowing through a horizon is limited. Second, this energy is proportional to the area of the horizon. Third, the energy flow is proportional to the surface gravity. These results are fundamental statements of general relativity. No other part of physics makes comparable statements. For the differential case the last relation can be rewritten as δE =

(735)

In this way, the result can also be used for general horizons, such as horizons that are curved or time-dependent.* In a well known paper, Jacobson has given a beautiful proof of a simple connection: if energy flow is proportional to horizon area for all observers and all horizons, then * Relation (735) is well known, though with different names for the observables. Since no communication is possible across a horizon, the detailed fate of energy flowing through a horizon is also unknown. Energy whose detailed fate is unknown is often called heat. Relation (735) therefore states that the heat flowing through a horizon is proportional to the horizon area. When quantum theory is introduced into the discussion, the area of a horizon can be called ‘entropy’ and its surface gravity can be called ‘temperature’; relation (735) can then be rewritten as δQ = T δS . (736) However, this translation of relation (735), which requires the quantum of action, is unnecessary here. We only cite it to show the relation between horizon behaviour and quantum gravity.

Copyright © Christoph Schiller November 1997–May 2006

Ref. 1098

c2 a δA . 8πG

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where the maximum force and the maximum possible area 4πR 2 of a horizon of (maximum local) radius R were introduced. The fraction E~A is the energy per area flowing through the horizon. Often, horizons are characterized by the so-called surface gravity a instead of the radius R. In the limit case, two are related by a = c 2 ~2R. This leads to E=

Dvipsbugw

(732)

Dvipsbugw


1074

xi general rel ativity versus quantum mechanics

general relativity holds. To see the connection to general relativity, we generalize relation (735) to general coordinate systems and general energy-flow directions. This is achieved by introducing tensor notation. To realize this at horizons, one introduces the general surface element dΣ and the local boost Killing vector field k that generates the horizon (with suitable norm). Jacobson uses them to rewrite the left hand side of relation (735) as δE =

∫T

ab k

a

dΣ b ,

(737)

Dvipsbugw

where Tab is the energy–momentum tensor. This rewrites the energy for arbitrary coordinate systems and arbitrary energy flow directions. Jacobson’s main result is that the essential part of the right hand side of relation (735) can be rewritten, using the (purely geometric) Raychaudhuri equation, as

∫R

ab k

a

dΣ b ,

(738)

where R ab is the Ricci tensor describing space-time curvature. Combining these two steps, we find that the energy–area relation (735) for horizons can be rewritten as c4 Tab k a dΣ b = R ab k a dΣ b . (739) 8πG

Jacobson shows that this equation, together with local conservation of energy (i.e., vanishing divergence of the energy–momentum tensor), can only be satisfied if Tab =

c4 1 ‹R ab − ( R + Λ)дab  , 8πG 2

(740)

Copyright © Christoph Schiller November 1997–May 2006

where Λ is a constant of integration whose value is not specified by the problem. These are the full field equations of general relativity, including the cosmological constant Λ. The field equations are thus shown to be valid at horizons. Since it is possible, by choosing a suitable coordinate transformation, to position a horizon at any desired space-time event, the field equations must be valid over the whole of space-time. It is possible to have a horizon at every event in space-time; therefore, at every event in nature there is the same maximum possible force (or power). This maximum force (or power) is a constant of nature. In other words, we just showed that the field equations of general relativity are a direct consequence of the limited energy flow at horizons, which in turn is due to the existence of a maximum force or power. Maximum force or power implies the field equations. One can thus speak of the maximum force principle. In turn, the field equations imply maximum force. Maximum force and general relativity are equivalent. The bounds on force and power have important consequences. In particular, they imply statements on cosmic censorship, the Penrose inequality, the hoop conjecture, the non-existence of plane gravitational waves, the lack of space-time singularities, new experimental tests of the theory, and on the elimination of competing theories of relativistic

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aδA = c 2

Dvipsbugw


fundamental limits to all observables

Ref. 1099

1075

gravitation. These consequences are presented elsewhere. Deducing universal gravitation

Challenge 1486 e

The size of physical systems in general relativity General relativity, like the other theories of modern physics, provides a limit on the size of systems: there is a limit to the amount of matter that can be concentrated into a small volume. 4Gm lE 2 . (741) c

Ref. 1100

The size limit is only achieved for black holes, those well-known systems which swallow everything that is thrown into them. It is fully equivalent to the force limit. All composite systems in nature comply with the lower size limit. Whether elementary particles fulfil or even achieve this limit remains one of the open issues of modern physics. At present, neither experiment nor theory allow clear statements on their size. More about this issue below. General relativity also implies an ‘indeterminacy relation’: ∆E c4 D . ∆l 4G

(742)

A mechanical analogy for the maximum force The maximum force is central to the theory of general relativity. That is the reason why the value of the force (adorned with a factor 2π) appears in the field equations. The importance of a maximum force becomes clearer when we return to our old image of space-time as a deformable mattress. Like any material body, a mattress is characterized by a material constant that relates the deformation values to the values of applied energy. Similarly, a mattress, like any material, is characterized by the maximum stress it can bear before

Copyright © Christoph Schiller November 1997–May 2006

Since experimental data are available only for composite systems, we cannot say yet whether this inequality also holds for elementary particles. The relation is not as popular as the previous. In fact, testing the relation, for example with binary pulsars, may lead to new tests that would distinguish general relativity from competing theories.

Dvipsbugw

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Universal gravitation can be derived from the force limit in the case where forces and values. The first condition implies º speeds are2 much smaller than the maximum 4GMa  c , the second v  c and al  c 2 . Let us apply this to a specific case. We study a satellite circling a central mass M at distance R with acceleration a. This system, with length l = 2R, has only one characteristic speed. Whenever º this speed v is much 2 smaller than c, v must be proportional both to al = 2aR and to 4GMa . If they are taken together, they imply that a = f GM~R 2 , where the numerical factor f is not yet fixed. A quick check, for example using the observed escape velocity values, shows that f = 1. Forces and speeds much smaller than the limit values thus imply that the inverse square law of gravity describes the interaction between systems. In other words, nature’s limit on force implies the universal law of gravity, as is expected.

Dvipsbugw


xi general rel ativity versus quantum mechanics

1076

breaking. Like mattresses, crystals also have these two values. In fact, for perfect crystals (without dislocations) these two material constants are the same. Empty space-time somehow behaves like a perfect crystal or a perfect mattress: it has a deformation-energy constant that at the same time is the maximum force that can be applied to it. The constant of gravitation thus determines the elasticity of space-time. Now, crystals are not homogeneous, but are made up of atoms, while mattresses are made up of foam bubbles. What is the corresponding structure of space-time? This is a central question in the rest of our adventure. One thing is sure: vacuum has no preferred directions, in complete contrast to crystals. In fact, all these analogies even suggest that the appearance of matter might be nature’s way of preventing space-time from ripping apart. We have to patient for a while, before we can judge this option. A first step towards the answer to the question appears when we put all limits together.

Dvipsbugw

Units and limit values for all physical observables

S E

ħ 2 v Dc

quantum theory on action: special relativity on speed:

F D

general relativity on force:

Challenge 1487 e

c4 . 4G

(743)

The limits are valid for all physical systems, whether composed or elementary, and are valid for all observers. We note that the limit quantities of special relativity, quantum theory and general relativity can also be seen as the right-hand sides of the respective indeterminacy relations. Indeed, the set (724, 727, 742) of indeterminacy relations or the set (723, 726, 741) of length limits is each fully equivalent to the three limit statements (743). Each set of limits can be taken as a (somewhat selective) summary of twentieth century physics. If the three fundamental limits are combined, a limit on a number of physical observables arise. The following limits are valid generally, for both composite and elementary systems:

time distance product: acceleration: angular frequency:

tE

2Għ c5 2Għ td E 4 c ¾ c7 aD 2Għ ¾ c5 ω D 2π 2Għ

= 7.6 ë 10−44 s

= 1.7 ë 10−78 sm

= 4.0 ë 1051 m~s2 = 8.2 ë 1043 ~s

(744) (745) (746) (747)

Copyright © Christoph Schiller November 1997–May 2006

time interval:

¾

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The existence of a maximum force in nature is equivalent to general relativity. As a result, physics can now be seen as making three simple statements on motion that is found in nature:

Dvipsbugw


units and limit values for all physical observables

1077

With the additional knowledge that, in nature, space and time can mix, we get distance: area: volume curvature: mass density:

Ref. 1103

2Għ c3 2Għ AE 3 c 2Għ 3~2 V E‹ 3  c c3 KD 2Għ c5 ρD 8G 2 ħ

= 2.3 ë 10−35 m

= 5.2 ë 10−70 m2

= 1.2 ë 10−104 m3 = 1.9 ë 1069 ~m2

= 6.5 ë 1095 kg~m3

(748) (749) (750)

Dvipsbugw

(751) (752)

Copyright © Christoph Schiller November 1997–May 2006

Of course, speed, action, angular momentum, power and force are also limited, as has already been stated. Except for a small numerical factor, for every physical observable these limits correspond to the Planck value. (The limit values are deduced from the commonly used Planck values simply by substituting 4G for G and ħ~2 for ħ.) These values are the true natural units of nature. In fact, the most aesthetically pleasing solution is to redefine the usual Planck values for every observable to these extremal values by absorbing the numerical factors into the respective definitions. In the following, we call the redefined limits the (corrected) Planck limits and assume that the factors have been properly included. In other words, every natural unit or (corrected) Planck unit is at the same time the limit value of the corresponding physical observable. Most of these limit statements are found throughout the literature, though the numerical factors are often different. Each limit has a string of publications attached to it. The existence of a smallest measurable distance and time interval of the order of the Planck values is discussed in quantum gravity and string theory. The largest curvature has been studied in quantum gravity; it has important consequences for the ‘beginning’ of the universe, where it excludes any infinitely large or small observable. The maximum mass density appears regularly in discussions on the energy of vacuum. With the present deduction of the limits, two results are achieved. First of all, the various arguments used in the literature are reduced to three generally accepted principles. Second, the confusion about the numerical factors is solved. During the history of Planck units, the numerical factors have varied greatly. For example, the fathers of quantum theory forgot the 1/2 in the definition of the quantum of action. Similarly, the specialists of relativity did not emphasize the factor 4. With the present framework, the issue of the correct factors in the Planck units can be considered as settled. We also note that the dimensional independence of the three limits in nature also means that quantum effects cannot be used to overcome the force limit; similarly, the power limit cannot be used to overcome the speed limit. The same is valid for any other combination of limits: they are independent and consistent at the same time.

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Ref. 1101, Ref. 1102

dE

¾

Dvipsbugw


1078

xi general rel ativity versus quantum mechanics

Limits to space and time

Ref. 1104

Ref. 1106 Ref. 1107

Ref. 1102

Mass plays a special role in all these arguments. The set of limits (743) does not make it possible to extract a limit statement on the mass of physical systems. To find one, the aim has to be restricted. The Planck limits mentioned so far apply for all physical systems, whether they are composed or elementary. Additional limits can only be found for elementary systems. In quantum theory, the distance limit is a size limit only for composed systems. A particle is

Copyright © Christoph Schiller November 1997–May 2006

Mass and energy limits

Dvipsbugw

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Ref. 1102, Ref. 1105

The three limits of nature (743) result in a minimum distance and a minimum time interval. These minimum intervals directly result from the unification of quantum theory and relativity. They do not appear if the theories are kept separate. In short, unification implies that there is a smallest length in nature. The result is important: the formulation of physics as a set of limit statements shows that the continuum description of space and time is not correct. Continuity and manifolds are only approximations; they are valid for large values of action, low speeds and low values of force. The reformulation of general relativity and quantum theory with limit statements makes this result especially clear. The result is thus a direct consequence of the unification of quantum theory and general relativity. No other assumptions are needed. In fact, the connection between minimum length and gravity is not new. In 1969, Andrei Sakharov pointed out that a minimum length implies gravity. He showed that regularizing quantum field theory on curved space with a cut-off will induce counter-terms that include to lowest order the cosmological constant and then the Einstein Hilbert action. The existence of limit values for the length observable (and all others) has numerous consequences discussed in detail elsewhere. In particular, the existence of a smallest length – and a corresponding shortest time interval – implies that no surface is physical if any part of it requires a localization in space-time to dimensions smaller that the minimum length. (In addition, a physical surface must not cross any horizon.) Only by stipulation of this condition can unphysical examples that contravene the force and power limits be eliminated. For example, this condition has been overlooked in Bousso’s earlier discussion of Bekenstein’s entropy bound – though not in his more recent ones. The corrected value of the Planck length should also be the expression that appears in the so-called theories of ‘doubly special relativity’. These then try to expand special relativity in such a way that an invariant length appears in the theory. A force limit in nature implies that no physical system can be smaller than a Schwarzschild black hole of the same mass. The force limit thus implies that point particles do not exist. So far, this prediction has not been contradicted by observations, as the predicted sizes are so small that they are outside experimental reach. If quantum theory is taken into account, this bound is sharpened. Because of the minimum length, elementary particles are now predicted to be larger than the corrected Planck length. Detecting the sizes of elementary particles would thus make it possible to check the force limit directly, for example with future electric dipole measurements.

Dvipsbugw


units and limit values for all physical observables

1079

elementary if the system size l is smaller than any conceivable dimension: for elementary particles:

lD

ħ . 2mc

(753)

Ref. 1108

These single-particle limits, corresponding to the corrected Planck mass, energy and momentum, were already discussed in 1968 by Andrei Sakharov, though again with different numerical factors. They are regularly cited in elementary particle theory. Obviously, all known measurements comply with the limits. It is also known that the unification of the electromagnetic and the two nuclear interactions takes place at an energy near, but still clearly below the maximum particle energy. Virtual particles – a new definition

Page 315 Page 759

In fact, there are physical systems that exceed all three limits. Nature does contain systems that move faster than light, that show action values below half the quantum of action and that experience forces larger than the force limit. The systems in question are called virtual particles. We know from special relativity that the virtual particles exchanged in collisions move faster than light. We know from quantum theory that virtual particle exchange implies action values below the minimum action. Virtual particles also imply an instantaneous change of momentum; they thus exceed the force limit. Virtual particles are thus those particles that exceed all the limits that hold for (real) physical systems.

Thermodynamics can also be summarized in a single statement on motion: There is a smallest entropy in nature. k SE . (755) 2

Ref. 1109

The entropy S is limited by half the Boltzmann constant k. The result is almost 100 years old; it was stated most clearly by Leo Szilard, though with a different numerical factor. In the same way as in the other fields of physics, this result can also be phrased as a

Copyright © Christoph Schiller November 1997–May 2006

Limits in thermodynamics

Dvipsbugw

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By using this new limit, valid only for elementary particles, the well-known mass, energy and momentum limits are found: ¾ ħc = 7.7 ë 10−9 kg = 0.42 ë 1019 GeV~c2 for elementary particles: m D 8G ¾ ħc 5 for elementary particles: E D = 6.9 ë 108 J = 0.42 ë 1019 GeV 8G ¾ ħc 3 for elementary particles: p D = 2.3 kg m~s = 0.42 ë 1019 GeV~c (754) 8G

Dvipsbugw


xi general rel ativity versus quantum mechanics

1080

indeterminacy relation: ∆ Ref. 1110

k 1 ∆U E . T 2

(756)

This relation was given by Bohr and discussed by Heisenberg and many others (though with k instead of k~2). It is mentioned here in order to complete the list of indeterminacy relations and fundamental constants. With the single-particle limits, the entropy limit leads to an upper limit for temperature: TD

¾

Dvipsbugw 5

ħc = 1.0 ë 1032 K . 2Gk 2

(757)

Electromagnetic limits and units The discussion of limits can be extended to include electromagnetism. Using the (lowenergy) electromagnetic coupling constant α, one gets the following limits for physical systems interacting electromagnetically: electric charge: electric field: magnetic field: voltage: inductance:

¼

(758) (759) (760) (761) (762)

With the additional assumption that in nature at most one particle can occupy one Planck volume, one gets

charge density: capacitance:

¾ πε o α c 5 c9 = e = 1.3 ë 1085 C~m3 ρe D 2G 3 ħ 8G 3 ħ 3 ¾ ¾ 2ħG 8G 2 =e = 1.0 ë 10−46 F C E 8πε 0 α 3 c c5 ħ ½

(763) (764)

Copyright © Christoph Schiller November 1997–May 2006

4πε 0 αcħ = e = 0.16 aC ¾ c7 c4 = ED = 2.4 ë 1061 V~m 64πε o αħG 2 4Ge ¾ c3 c5 = = 7.9 ë 1052 T BD 64πε 0 αħG 2 4Ge ¾ ¾ c4 ħc 5 1 UD = = 1.5 ë 1027 V 32πε 0 αG e 8G ¾ ¾ 2ħG ħ3 G 1 1 = = 4.4 ë 10−40 H LE 8πε o α c7 e2 2c 5 qE

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This corresponds to the temperature at which the energy per degree of freedom is given by the (corrected) Planck energy. A more realistic value would have to take into account the number of degrees of freedom of a particle at Planck energy. This would change the numerical factor.

Dvipsbugw


units and limit values for all physical observables

1081

For the case of a single conduction channel, one gets electric resistance: electric conductivity: electric current:

Ref. 1111

(765) (766) (767)

Many of these limits have been studied already. The magnetic field limit plays a role in the discussion of extreme stars and black holes. The maximum electric field plays a role in the theory of gamma ray bursters. The studies of limit values for current, conductivity and resistance in single channels are well known; the values and their effects have been studied extensively in the 1980s and 1990s. They will probably win a Nobel prize in the future. The observation of quarks and of collective excitations in semiconductors with charge e~3 does not necessarily invalidate the charge limit for physical systems. In neither case is there is a physical system – defined as localized mass–energy interacting incoherently with the environment – with charge e~3. Vacuum and mass–energy – two sides of the same coin

Page 996 Page 1032

Page 349

Copyright © Christoph Schiller November 1997–May 2006

Ref. 1102, Ref. 1105

In this discussion we have found that there is a fixed limit to every physical observable. Many consequences have been discussed already in previous sections. There we saw that the existence of a limit to all observables implies that at Planck scales no physical observable can be described by real numbers and that no low-energy symmetry is valid. One conclusion is especially important for the rest of our adventure. We saw that there is a limit for the precision of length measurements in nature. The limit is valid both for the length measurements of empty space and for the length measurements of matter (or radiation). Now let us recall what we do when we measure the length of a table with a ruler. To find the ends of the table, we must be able to distinguish the table from the surrounding air. In more precise terms, we must be able to distinguish matter from vacuum. But we have no way to perform this distinction at Planck energy. In these domains, the intrinsic measurement limitations of nature do not allow us to say whether we are measuring vacuum or matter. There is no way, at Planck scales, to distinguish the two. We have explored this conclusion in detail above and have shown that it is the only consistent description at Planck scales. The limitations in length measurement precision, in mass measurement precision and in the precision of any other observable do not allow to tell whether a box at Planck scale is full or empty. You can pick any observable you want to distinguish vacuum from matter. Use colour, mass, size, charge, speed, angular momentum, or anything you want. At Planck scales, the limits to observables also lead to limits in measurement precision. At Planck scales, the measurement limits are of the same size as the observable to be measured. As a result, it is impossible to distinguish matter and vacuum at Planck scales. We put the conclusion in the sharpest terms possible: Vacuum and matter do not differ

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Ref. 1112

ħ 1 = 2 = 2.1 kΩ 8πε 0 αc 2e 2e 2 G D 8πε 0 αc = = 0.49 mS ħ ¾ ¾ 2πε 0 αc 6 c5 ID =e = 7.4 ë 1023 A G 2ħG RE

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1082

Ref. 1113

xi general rel ativity versus quantum mechanics

at Planck scales. This counter-intuitive result is one of the charms of theoretical high energy physics. This result alone inspired many researchers in the field and induced them to write best-sellers. Brian Greene was especially successful in presenting this side of quantum geometry to the wider public. However, at this point of our adventure, most issues are still open. The precise manner in which a minimum distance leads to a homogeneous and isotropic vacuum is unclear. The way to describe matter and vacuum with the same concepts has to be found. And of course, the list of open questions in physics, given above, still waits. However, the conceptual results give hope; there are interesting issues awaiting us.

Dvipsbugw

Curiosities and fun challenges about Planck limits

** The minimum angular momentum may surprise at first, especially when one thinks about particles with spin zero. However, the angular momentum of the statement is total angular momentum, including the orbital part with respect to the observer. The total angular momentum is never smaller than ħ~2. ** If any interaction is stronger than gravity, how can the maximum force be determined by gravity alone, which is the weakest interaction? It turns out that in situations near the maximum force, the other interactions are negligible. This is the reason that gravity must be included in a unified description of nature. **

Challenge 1488 ny

At first sight, it seems that electric charge can be used in such a way that the acceleration of a charged body towards a charged black hole is increased to a value exceeding the force limit. However, the changes in the horizon for charged black holes prevent this. **

Ref. 1114

where e is the positron charge, C capacity and U potential difference, and one between electric current I and time t ∆I ∆t E e (769) and both relations are found in the literature.

Copyright © Christoph Schiller November 1997–May 2006

The general connection that to every limit value in nature there is a corresponding indeterminacy relation is also valid for electricity. Indeed, there is an indeterminacy relation for capacitors of the form ∆C ∆U E e (768)

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The (corrected) Planck limits are statements about properties of nature. There is no way to measure values exceeding these limits, whatever experiment is performed. As can be expected, such a claim provokes the search for counter-examples and leads to many paradoxes.

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units and limit values for all physical observables

1083

** The gravitational attraction between two masses never yields force values sufficiently high to exceed the force limit. Why? First of all, masses m and M cannot come closer than the sum of their horizon radii. Using F = GmM~r 2 with the distance r given by the (naive) sum of the two black hole radii as r = 2G(M + m)~c 2 , one gets FD

Mm c4 , 4G (M + m)2

(770)

Dvipsbugw

which is never larger than the force limit. Even two attracting black holes thus do not exceed the force limit – in the inverse square approximation of universal gravity. The minimum size of masses does not allow to exceed the a maximum force. **

F=

GMm . ¼ M d 2 1 − 2G d c2

(771)

** An absolute power limit implies a limit on the energy that can be transported per time unit through any imaginable surface. At first sight, it may seem that the combined power emitted by two radiation sources that each emit 3/4 of the maximum value should give 3/2 times the upper value. However, the combination forms a black hole or at least prevents part of the radiation to be emitted by swallowing some of it between the sources. **

Copyright © Christoph Schiller November 1997–May 2006

The expression diverges at d = 0, the location of the horizon. However, even a test mass cannot be smaller than its own gravitational radius. If we want to reach the horizon with a realistic test mass, we need to chose a small test mass m; only a small – and thus light – mass can get near the horizon. For vanishingly small masses however, the resulting force tends to zero. Indeed, letting the distance tend to the smallest possible value by letting d = 2G(m + M)~c 2 d = 2GM~c 2 requires m 0, which makes the force F(m, d) vanish. If on the other hand, we remain away from the horizon and look for the maximum force by using a mass as large as can possibly fit into the available distance (the calculation is straightforward algebra) again the force limit is never exceeded. In other words, for realistic test masses, expression (771) is never larger than c 4 ~4G. Taking into account the minimal size of test masses thus prevents that the maximum force is exceeded in gravitational systems.

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Ref. 1115

It is well known that gravity bends space. To be fully convincing, the calculation needs to be repeated taking space curvature into account. The simplest way is to study the force generated by a black hole on a test mass hanging from a wire that is lowered towards a black hole horizon. For an unrealistic point mass, the force would diverge on the horizon. Indeed, for a point mass m lowered towards a black hole of mass M at (conventionally defined radial) distance d, the force would be

Dvipsbugw


1084

xi general rel ativity versus quantum mechanics

One possible system that actually achieves the power limit is the final stage of black hole evaporation. But even in this case the power limit is not exceeded. ** Ref. 1095

The maximum force limit states that the stress–energy tensor, when integrated over any physical surface, does not exceed the limit value. No such integral, over any physical surface whatsoever, of any tensor component in any coordinate system, can exceed the force limit, provided that it is measured by a nearby observer or a test body with a realistic proper size. The maximum force limit thus applies to any component of any force vector, as well as to its magnitude. It applies to gravitational, electromagnetic, and nuclear forces. It applies to all realistic observers. Whether the forces are real or fictitious is not important. It also plays no role whether we discuss 3-forces of Galilean physics or 4-forces of special relativity. Indeed, the force limit applied to the 0-th component of the 4-force is the power limit.

Translated to mass flows, the power limit implies that flow of water through a tube is limited in throughput. Also this limit seems unknown in the literature. ** The force limit cannot be overcome with Lorentz boosts. A Lorentz boost of any nonvanishing force value seems to allow exceeding the force limit at high speeds. However, such a transformation would create a horizon that makes any point with a potentially higher force value inaccessible. ** The power limits is of interest if applied to the universe as a whole. Indeed, it can be used to explain Olber’s paradox. The sky is dark at night because the combined luminosity of all light sources in the universe cannot be brighter than the maximum value. **

One notes that the negative energy volume density −Λc 4 ~4πG introduced by the positive cosmological constant Λ corresponds to a negative pressure (both quantities have the same dimensions). When multiplied with the minimum area it yields a force value Λħc = 4.8 ë 10−79 N . 2π

(772)

This is also the gravitational force between two corrected Planck masses located at the » cosmological distance π~4Λ . If we make the (wishful) assumption that this is the smallest possible force in nature (the numerical prefactor is not finalized yet), we get the fascinating conjecture that the full theory of general relativity, including the cosmological constant, is defined by the combination of a maximum and a minimum force in nature. Another consequence of the limits merits a dedicated section.

Copyright © Christoph Schiller November 1997–May 2006

F=

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**

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upper and lower limits to observables

1085

Upper and lower limits to observables

Page 349

Page 1068

Size and energy dependence Looking for additional limits in nature, we directly note a fundamental property. Any upper limit for angular momentum or any lower limit for power must be system dependent. Such limits will not be absolute, but will depend on properties of the system. Now, any physical system is a part of nature characterized by a boundary and its content.** The simplest properties all systems share are thus their size (characterized in the following by the diameter) L and energy E. With these characteristics we can enjoy deducing systemdependent limits for every physical observable. The general method is straightforward. We take the known inequalities for speed, action, power, charge and entropy and then extract a limit for any observable, by inserting length and energy as required. We then have to select the strictest of the limits we find. Angular momentum, action and speed It only takes a moment to note that the ratio of angular momentum D to mass times length has the dimension of speed. Since speeds are limited by the speed of light, we get

Ref. 1116

Page 800

1 LE . c

(773)

Indeed, there do not seem to be any exceptions to this limit in nature. No known system has a larger angular momentum value, from atoms to molecules, from ice skaters to galaxies. For example, the most violently rotating objects, the so-called extremal black holes, are also limited in angular momentum by D D LE~c. (In fact, this limit is correct only if the energy is taken as the irreducible mass times c 2 ; if the usual mass is used, the limit is too large by a factor 4.) One remarks that the limit deduced from general relativity, given * This section was added in June 2004. ** We mention here that quantum theory narrows down this definition as a part of nature that in addition interacts incoherently with its environment. We assume that this condition is realized in the following.

Copyright © Christoph Schiller November 1997–May 2006

DD

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In our quest to understand motion, we have focused our attention to the limitations it is subjected to. Special relativity poses a limit to speed, namely the speed of light c. General relativity limits force and power respectively by c 4~4G and c 5 ~4G, and quantum theory introduces a smallest value ħ~2 for angular momentum or action. We saw that nature poses a limit on entropy and a limit on electrical charge. Together, all these limits provide an extreme value for every physical observable. The extremum is given by the corresponding (corrected) Planck value. We have explored the list of all Planck limits in the previous section. The question may now arise whether nature provides a limit for physical observables also on the opposite end of the measurement scale. For example, there is a highest force and a highest power in nature; is there also a lowest force and a lowest power? Is there also a lowest speed? We show in the following that there indeed are such limits, for all observables. We give the general method to generate such bounds and explore several examples.* The exploration will lead us along an interesting scan across modern physics.

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1086

Challenge 1489 ny

xi general rel ativity versus quantum mechanics

by D D L 2 c 3 ~4G is not stricter than the one just given. In addition, no system-dependent lower limit for angular momentum can be deduced. The maximum angular momentum value is also interesting when it is seen as action limit. Action is the time integral of the difference between kinetic and potential energy. In fact, since nature always minimizes action W, we are not used to search for systems which maximize its value. You might check by yourself that the action limit W D LE~c is not exceeded in any physical process. Similarly, speed times mass times length is an action. Since action values in nature are limited from below by ħ~2, we get vE

Challenge 1491 e

(774)

This relation is a rewritten form of the indeterminacy relation of quantum theory and is no news. No system of energy E and diameter L has a smaller speed than this limit. Even the slowest imaginable processes show this speed value. For example, the extremely slow radius change of a black hole by evaporation just realizes this minimal speed. Continuing with the method just outlined, one also finds that the limit deduced from general relativity, v D (c 2 ~4G)(L~E), gives no new information. Therefore, no system-dependent upper speed limit exists. Incidentally, the limits are not unique. Additional limits can be found in a systematic way. Upper limits can be multiplied, for example, by factors of (L~E)(c 4 ~4G) or (LE)(2~ħc) yielding additional, but less strict upper limits. A similar rule can be given for lower limits.* With the same approach we can now systematically deduce all size and energy dependent limits for physical observables. We have a tour of the most important ones. Force, power and luminosity We saw that force and power are central to general relativity. Due to the connection W = FLT between action W, force, distance and time, we can deduce ħ 1 . 2c T 2

(775)

Experiments do not reach this limit. The smallest forces measured in nature are those in atomic force microscopes, where values as small as 1 aN are observed. However, the values are all above the lower force limit. The power P emitted by a system of size L and mass M is limited by c3

M M E P E 2ħG 3 L L

(776)

The left, upper limit gives the upper limit for any engine or lamp deduced from relativity; not even the universe exceeds it. The right, lower limit gives the minimum power emit* The strictest upper limits are thus those with the smallest possible exponent for length, and the strictest lower limits are those with the largest sensible exponent of length.

Copyright © Christoph Schiller November 1997–May 2006

FE

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Challenge 1490 ny

ħc 2 1 . 2 LE

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upper and lower limits to observables

1087

ted by any system due to quantum gravity effects. Indeed, no system is completely tight. Even black holes, the systems with the best ability in nature to keep components inside their enclosure, nevertheless radiate. The power radiated by black holes should just saturate this limit, provided the length L is taken to be the circumference of the black hole. (However, present literature values of the numerical factors in the black hole power are not yet consistent). The claim of the quantum gravity limit is thus that the power emitted by a black hole is the smallest power that is emitted by any composed system of the same surface gravity.

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Acceleration When the acceleration of a system is measured by a nearby inertial observer, the acceleration a of a system of size L and mass M is limited by

Ref. 1117

1 M E a E 4G 2 L L

The lower, right limit gives the acceleration due to universal gravity. Indeed, the relative acceleration between a system and an observer has at least this value. The left, upper limit to acceleration is the value due to special relativity. No exception to either of these limits has ever been observed. Using the limit to the size of masses, the upper limit can be transformed into the equivalent acceleration limit 2c 3 MEa ħ

Challenge 1492 ny

(777)

(778)

which has never been approached either, despite many attempts. The upper limit to acceleration is thus a quantum limit, the lower one a gravitational limit. The acceleration of the radius of a black hole due to evaporation can be much slower than the limit a E 4GM~L 2 . Why is this not a counter-example? Momentum

The momentum p of a system of size L is limited by (779)

The lower limit is obviously due to quantum theory; experiments confirmed it for all radiation and matter. The upper limit for momentum is due to general relativity. It has never been exceeded.

Copyright © Christoph Schiller November 1997–May 2006

ħ 1 c3 LEpE 4G 2 L

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c2

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1088

xi general rel ativity versus quantum mechanics

Lifetime, distance and curvature Time is something special. What are the limits to time measurements? Like before, we find that any measured time interval t in a system in thermal equilibrium is limited by 4G 2 ML 2 E t E 3 M ħ c

(780)

2c 4G ML 2 E d E 2 M ħ c

(781)

Since curvature is an inverse square distance, curvature of space-time is also limited. Mass change The mass change dM~dt of a system of size L and mass M is limited by c 5 L dM ħ 1 E E 16G 2 M dt 2 L2

(782)

The limits apply to systems in thermal equilibrium. The left, upper limit is due to general relativity; it is never exceeded. The right, lower limit is due to quantum theory. Again, all experiments are consistent with the limit values.

Limits for rest mass make only sense if the system is characterized by a size L only. We then have ħ 1 c2 LEME (783) 4G 2c L

The upper limit for mass was discussed in general relativity. Adding mass or energy to a black hole always increases its size. No system can show higher values than this value, and indeed, no such system is known or even imaginable. The lower limit on mass is obviously due to quantum theory; it follows directly from the quantum of action.

Copyright © Christoph Schiller November 1997–May 2006

Mass and density

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The lower time limit is the gravitational time. No clock can measure a smaller time than this value. Similarly, no system can produce signals shorter than this duration. Black holes, for example, emit radiation with a frequency given by this minimum value. The upper time limit is expected to be the exact lifetime of a black hole. (There is no consensus in the literature on the numerical factor yet.) The upper limit to time measurements is due to quantum theory. It leads to a question: What happens to a single atom in space after the limit time has passed by? Obviously, an atom is not a composed system comparable with a black hole. The lifetime expression assumes that decay can take place in the most tiny energy steps. As long as there is no decay mechanism, the life-time formula does not apply. The expression (780) thus does not apply to atoms. Distance limits are straightforward.

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upper and lower limits to observables

1089

The mass density a system of size L is limited by ħ 1 c2 1 EρE 4G L 2 2c L 4

Page 1086

(784)

The upper limit for mass density, due to general relativity, is only achieved for black holes. The lower limit is the smallest density of a system due to quantum theory. It also applies to the vacuum, if a piece of vacuum of site L is taken as a physical system. We note again that many equivalent (but less strict) limits can be formulated by using the transformations rules mentioned above.

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The strange charm of the entropy bound Ref. 1118

S S limit which gives SD

Ref. 1119 Page 245

A

(785)

A limit

kc 3 A, 4Għ

(786)

where A is the surface of the system. Equality is realized only for black holes. The old question of the origin of the factor 4 in the entropy of black holes is thus answered here in he following way: it is due to the factor 4 in the force or power bound in nature (provided that the factors from the Planck entropy and the Planck action cancel). The future will tell whether this explanation will stand the winds of time. Stay tuned. We can also derive a more general relation if we use a mysterious assumption that we discuss afterwards. We assume that the limits for vacuum are opposite to those for matter. We can then write c 2 ~4G D M~L for the vacuum. This gives

S S corr. Planck

SD

D

πkc 2πkc ML = MR . ħ ħ M

A

M corr. Planck A corr. Planck

L corr. Planck . L

(787)

(788)

Expression (787) is called Bekenstein’s entropy bound. Up to today, no exception has been found or constructed, despite many attempts. Again, the limit value itself is only realized for black holes. We still need to explain the strange assumption used above. We are exploring the entropy of a horizon. Horizons are not matter, but limits to empty space. The entropy of

Copyright © Christoph Schiller November 1997–May 2006

In other words, we used

Ref. 1106

D

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In 1973, Bekenstein discovered a famous limit that connects the entropy S of a physical system with its size and mass. No system has a larger entropy than one bounded by a horizon. The larger the horizon surface, the larger the entropy. We write

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1090

Ref. 1120

xi general rel ativity versus quantum mechanics

horizons is due to the large amount of virtual particles found there. In order to deduce the maximum entropy of expression (788) one therefore has to use the properties of the vacuum. In other words, either (1) we use a mass to length ratio for vacuum above the Planck limit or (2) we use the Planck entropy as maximum value for vacuum. Other, equivalent limits for entropy can be found if other variables are introduced. For example, since the ratio of shear viscosity η to the volume density of entropy (times k) has the dimensions of an action, we can directly write SD

Challenge 1494 e

Dvipsbugw (789)

Again, equality is only reached in the case of black holes. With time, the list of similar bounds will grow longer and longer. Is there also a smallest entropy limit? So far, there does not seem to be a systemdependent minimum value for entropy; the approach gives no expression that is larger than k. The entropy limit is an important step in making the description of motion consistent. If space-time can move, as general relativity maintains, it also has an entropy. How could entropy be limited if space-time is continuous? Clearly, due to the minimum distance and a minimum time in nature, space-time is not continuous, but has a finite number of degrees of freedom. The number of degrees of freedom and thus the entropy of spacetime is thus finite. In addition, the Bekenstein limit also allows some interesting speculations. Let us speculate that the universe itself, being surrounded by a horizon, saturates the Bekenstein bound. The entropy bound gives a bound to all degrees of freedom inside a system; it tells us that the number N d.o.f . of degrees of freedom of the universe is roughly N d.o.f .  10132 .

(790)

This compares with the number N Pl. vol. of Planck volumes in the universe N Pl. vol.  10183

(791)

and with the number N part. of particles in the universe (792)

In other words, particles are only a tiny fraction of what moves around. Most motion must be that of space-time. At the same time, space-time moves much less than naively expected. Finding out how all this happens is the challenge of the unified description of motion. Temperature A lower limit for the temperature of a thermal system can be found using the idea that the number of degrees of freedom of a system is limited by its surface, more precisely, by

Copyright © Christoph Schiller November 1997–May 2006

N part.  1091 .

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Challenge 1493 ny

k ηV ħ

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upper and lower limits to observables

1091

the ratio between the surface and the Planck surface. One gets the limit TE

Page 310

Challenge 1496 ny

(793)

Alternatively, using the method given above, one can use the limit on the thermal energy kT~2 E ħc~(2πL) (the thermal wavelength must be smaller than the size of the system) together with the limit on mass c 2 ~4G E M~L and deduce the same result. We know the limit already: when the system is a black hole, it gives the temperature of the emitted radiation. In other words, the temperature of black holes is the lower limit for all physical systems for which a temperature can be defined, provided they share the same boundary gravity. As a criterion, boundary gravity makes sense: boundary gravity is accessible from the outside and describes the full physical system, since it makes use both of its boundary and its content. So far, no exception to this claim is known. All systems from everyday life comply with it, as do all stars. Even the coldest known systems in the universe, namely Bose–Einstein condensates and other cold matter in various laboratories, are much hotter than the limit, and thus much hotter than black holes of the same surface gravity. (Since a consistent Lorentz transformation for temperature is not possible, as we saw earlier, the limit of minimum temperature is only valid for an observer at the same gravitational potential and at zero relative speed to the system under consideration.) There seems to be no consistent way to define an upper limit for a system-dependent temperature. However, limits for other thermodynamic quantities can be found, but are not discussed here. Electromagnetic observables When electromagnetism plays a role, the involved system also needs to be characterized by a charge Q. The method used so far then gives the following lower limit for the electric field E: M2 (794) E E 4Ge 2 2 Q L

BE

Challenge 1498 ny

4Ge M 2 c Q 2 L2

(795)

Again, this limit is satisfied all known systems in nature. Similar limits can be found for the other electromagnetic observables. In fact, several of the earlier limits are modified when electrical charge is included. Can you show how the size limit changes when electric charge is taken into account? In fact, a dedicated research field is concerned only with the deduction of the most general limits valid in

Copyright © Christoph Schiller November 1997–May 2006

Challenge 1497 ny

We write the limitºusing the elementary charge e, though writing it using the fine structure constant via e = 4πε 0 αħc would be more appropriate. Experimentally, this limit is not exceeded in any system in nature. Can you show whether it is achieved by maximally charged black holes? For the magnetic field we get

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Challenge 1495 n

4Għ M πkc L 2

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1092

xi general rel ativity versus quantum mechanics

nature. Curiosities and fun challenges about limits to observables The limits yield a plethora of interesting paradoxes that can be discussed in lectures and student exercises. All paradoxes can be solved by carefully taking into account the combination of effects from general relativity and quantum theory. All apparent violations only appear when one of the two aspects is somehow forgotten. We study a few examples.

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**

**

Challenge 1499 r

The content of a system is not only characterized by its mass and charge, but also by its strangeness, isospin, colour charge, charge and parity. Can you deduce the limits for these quantities? **

Challenge 1500 n

In our discussion of black hole limits, we silently assumed that they interact, like any thermal system, in an incoherent way with the environment. What changes in the results of this section when this condition is dropped? Which limits can be overcome? **

Challenge 1501 e

Can you find a general method to deduce all limits? **

Copyright © Christoph Schiller November 1997–May 2006

Page 1068

Brushing some important details aside, we can take the following summary of our study of nature. Galilean physics is that description for which the difference between composed and elementary systems does not exist. Quantum theory is the description of nature with no (really large) composed systems; general relativity is the description of nature with no elementary systems. This distinction leads to the following interesting conclusion. A unified theory of nature has to unify quantum theory and general relativity. Since the first theory affirms that no (really large) composed systems exist, while the other that no elementary systems exist, a unified theory should state that no systems exist. This strange result indeed seems to be one way to look at the issue. The conclusion is corroborated by the result that in the unified description of nature, the observables time, space and mass cannot be distinguished clearly from each other, which implies that systems cannot be clearly distinguished from their surroundings. To be precise, systems thus do not really exist at unification energy.

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Several black hole limits are of importance to the universe itself. For example, the observed average mass density of the universe is not far from the corresponding limit. Also the lifetime limit is obviously valid for the universe and provides an upper limit for its age. However, the age of the universe is far from that limit by a large factor. In fact, since the universe’s size and age still increase, the age limit is pushed further into the future with every second that passes. The universe evolves in a way to escape its own decay.

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limits to measurement precision and their challenge to thought

1093

L imits to measurement precision and their challenge to thought

Challenge 1502 r

Ref. 1102

Measurement precision and the existence of sets

Copyright Š Christoph Schiller November 1997â&#x20AC;&#x201C;May 2006

Ref. 1102, Ref. 1105

The impossibility of completely eliminating measurement errors has an additional and important consequence. In physics, it is assumed that nature is a set of components. These components are assumed to be separable from each other. This tacit assumption is introduced in three main situations: it is assumed that matter consists of separable particles, that space-time consists of separable events or points, and that the set of states consists of separable initial conditions. So far, all of physics has thus built its complete description of nature on the concept of set. A fundamentally limited measurement precision implies that nature is not a set of such separable elements. A limited measurement precision implies that distinguishing physical entities is possible only approximately. The approximate distinction is only possible at energies much lower than the Planck energy. As humans we do live at such smaller energies; thus we can safely make the approximation. Indeed, the approximation is excel-

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Ref. 1102

We now know that in nature, every physical measurement has a lower and an upper bound. One of the bounds is size-dependent, the other is absolute. So far, no violation of these claims is known. The smallest relative measurement error possible in nature is thus the ratio of the two bounds. In short, a smallest length, a highest force and a smallest action, when taken together, imply that all measurements are limited in precision. A fundamental limit to measurement precision is not a new result any more. But it raises many issues. If the mass and the size of any system are themselves imprecise, can you imagine or deduce what happens then to the limit formulae given above? Due to the fundamental limits to measurement precision, the measured values of physical observables do not require the full set of real numbers. In fact, limited precision implies that no observable can be described by real numbers. We thus recover again a result that appears whenever quantum theory and gravity are brought together. In addition, we found that measurement errors increase when the characteristic measurement energy approaches the Planck energy. In that domain, the measurement errors of any observable are comparable with the measurement values themselves. Limited measurement precision thus implies that at the Planck energy it is impossible to speak about points, instants, events or dimensionality. Limited measurement precision also implies that at the Planck length it is impossible to distinguish positive and negative time values: particle and anti-particles are thus not clearly distinguished at Planck scales. A smallest length in nature thus implies that there is no way to define the exact boundaries of objects or elementary particles. However, a boundary is what separates matter from vacuum. In short, a minimum measurement error means that, at Planck scales, it is impossible to distinguish objects from vacuum with complete precision. To put it bluntly, at Planck scales, time and space do not exist. The mentioned conclusions are the same as those that are drawn by modern research on unified theories. The force limit, together with the other limits, makes it possible to reach the same conceptual results found by string theory and the various quantum gravity approaches. To show the power of the maximum force limit, we now explore a few conclusions which go beyond present approaches.

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1094

Ref. 1102

Ref. 1105

lent; we do not notice any error when performing it. But the discussion of the situation at Planck energies shows that a perfect separation is impossible in principle. In particular, at the cosmic horizon, at the big bang, and at Planck scales any precise distinction between two events or two particles becomes impossible. Another way to reach this result is the following. Separation of two entities requires different measurement results, such as different positions, different masses, different velocities, etc. Whatever observable is chosen, at the Planck energy the distinction becomes impossible, due to the large measurements errors. Only at everyday energies is a distinction approximately possible. Any distinction between two physical systems, such as between a toothpick and a mountain, is thus possible only approximately; at Planck scales, a boundary cannot be drawn. A third argument is the following. In order to count any entities in nature – a set of particles, a discrete set of points, or any other discrete set of physical observables – the entities have to be separable. The inevitable measurement errors, however, contradict separability. At the Planck energy, it is thus impossible to count physical objects with precision. Nature has no parts. In short, at Planck energies a perfect separation is impossible in principle. We cannot distinguish observations at Planck energies. In short, at Planck scale it is impossible to split nature into separate entities. There are no mathematical elements of any kind – or of any set – in nature. Elements of sets cannot be defined. As a result, in nature, neither discrete nor continuous sets can be constructed. Nature does not contain sets or elements. Since sets and elements are only approximations, the concept of ‘set’, which assumes separable elements, is already too specialized to describe nature. Nature cannot be described at Planck scales – i.e., with full precision – if any of the concepts used for its description presupposes sets. However, all concepts used in the past twenty-five centuries to describe nature – space, time, particles, phase space, Hilbert space, Fock space, particle space, loop space or moduli space – are based on sets. They all must be abandoned at Planck energy. No approach used so far in theoretical physics, not even string theory or the various quantum gravity approaches, satisfies the requirement to abandon sets. Nature is one and has no parts. Nature must be described by a mathematical concept that does not contain any set. This requirement must guide us in the future search for the unification of relativity and quantum theory.

Why are observers needed?

Certain papers on quantum theory give the impression that observers are indispensable for quantum physics. We have debunked this belief already, showing that the observer in quantum theory is mainly a bath with a definite interaction. Often, humans are observers. At Planck scales, observers also play a role. At Planck scales, quantum theory is mandatory. In these domains, observers must realize an additional requirement: they must * ‘It is almost impossible to carry the torch of truth through a crowd without scorching somebody’s beard.’

Copyright © Christoph Schiller November 1997–May 2006

Es ist fast unmöglich, die Fackel der Wahrheit durch ein Gedränge zu tragen, ohne jemandem den Bart zu sengen.* Georg Christoph Lichtenberg (1742–1799)

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Ref. 1102

xi general rel ativity versus quantum mechanics

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limits to measurement precision and their challenge to thought

1095

function at low energies. Only at low energy, an observer can introduce sets for the description of nature. Introducing observers is thus the same as introducing sets. To put it in another way, the limits of human observers is that they cannot avoid using sets. However, human observers share this limitation with video recorders, cameras, computers and pencil and paper. Nothing singles out humans in this aspect. In simple terms, observers are needed to describe nature at Planck scales only in so far as they are and use sets. We should not get too bloated about our own importance.

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A solution to Hilbert’s sixth problem Ref. 1121

Physics can be summarized in a few limit statements. They imply that in nature every physical observable is limited by a value near the Planck value. The speed limit is equivalent to special relativity, the force limit to general relativity, and the action limit to quantum theory. Even though this summary could have been made (or at least conjectured) by Planck, Einstein or the fathers of quantum theory, it is much more recent. The numerical factors for most limit values are new. The limits provoke interesting Gedanken experiments, none of which leads to violations of the limits. On the other hand, the force limit is not yet within direct experimental reach. The existence of limit values to all observables implies that the description of spacetime with a continuous manifold is not correct at Planck scales; it is only an approx-

Copyright © Christoph Schiller November 1997–May 2006

Outlook

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Ref. 1105

In the year 1900, David Hilbert gave a well-known lecture in which he listed twentythree of the great challenges facing mathematics in the twentieth century. Most problems provided challenges to many mathematicians for decades afterwards. Of the still unsolved ones, Hilbert’s sixth problem challenges mathematicians and physicists to find an axiomatic treatment of physics. The challenge has remained in the minds of many physicists since that time. Since nature does not contain sets, we can deduce that such an axiomatic description of nature does not exist! The reasoning is simple; we only have to look at the axiomatic systems found in mathematics. Axiomatic systems define mathematical structures. These structures are of three main types: algebraic systems, order systems or topological systems. Most mathematical structures – such as symmetry groups, vector spaces, manifolds or fields – are combinations of all three. But all mathematical structures contain sets. Mathematics does not provide axiomatic systems that do not contain sets. The underlying reason is that every mathematical concept contains at least one set. Furthermore, all physical concepts used so far in physics contain sets. For humans, it is difficult even simply to think without first defining a set of possibilities. However, nature is different; nature does not contain sets. Therefore, an axiomatic formulation of physics is impossible. Of course, this conclusion does not rule out unification in itself; however, it does rule out an axiomatic version of it. The result surprises, as separate axiomatic treatments of quantum theory or general relativity (see above) are possible. Indeed, only their unification, not the separate theories, must be approached without an axiomatic systems. Axiomatic systems in physics are always approximate. The requirement to abandon axiomatic systems is one of the reasons for the difficulties in reaching the unified description of nature.

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imation. For the same reason, is predicted that elementary particles are not point-like. Nature’s limits also imply the non-distinguishability of matter and vacuum. As a result, the structure of particles and of space-time remains to be clarified. So far, we can conclude that nature can be described by sets only approximately. The limit statements show that Hilbert’s sixth problem cannot be solved and that unification requires fresh approaches, taking unbeaten paths into unexplored territory. We saw that at Planck scales there is no time, no space, and there are no particles. Motion is a low energy phenomenon. Motion only appears for low-energy observers. These are observers who use sets. The (inaccurate) citation of Zeno at the beginning of our walk, stating that motion is an illusion, turns out to be correct! Therefore, we now need to find out how motion actually arises. The discussion so far hints that motion appears as soon as sets are introduced. To check this hypothesis, we need a description of nature without sets. The only way to avoid the use of sets seems a description of empty space-time, radiation and matter as being made of the same underlying entity. The inclusion of space-time dualities and of interaction dualities is most probably a necessary step. Indeed, both string theory and modern quantum gravity attempt this, though possibly not yet with the full radicalism necessary. Realizing this unification is our next task.

Copyright © Christoph Schiller November 1997–May 2006

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Biblio graphy 1093 Bohr explained the indivisibilty of the quantum of action in his famous Como lecture. N.

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B ohr, Atomtheorie und Naturbeschreibung, Springer, Berlin, 1931. More statements about the indivisibility of the quantum of action can be found in N. B ohr, Atomic Physics and Human Knowledge, Science Editions, New York, 1961. For summaries of Bohr’s ideas by others see Max Jammer, The Philosophy of Quantum Mechanics, Wiley, first edition, 1974, pp. 90–91, and John Honner, The Description of Nature – Niels Bohr and the Philosophy of Quantum Physics, Clarendon Press, 1987, p. 104. Cited on page 1069. For an overview of the quantum of action as basis of quantum theory, see the introduction to quantum theory in this textbook, available as by C. Schiller, An appetizer – quantum theory for poets and lawyers, http://www.motionmountain.net/C-5-QEDA.pdf. Cited on page 1070. The first published statements of the principle of maximum force were in the present textbook and independently in the paper by G.W. Gibbons, The maximum tension principle in general relativity, Foundations of Physics 32, pp. 1891–1901, 2002, or http://www.arxiv. org/abs/hep-th/0210109. Cited on pages 1071 and 1084. Maximal luminosity of often mentioned in studies around gravitational wave detection; nevertheless, the general maximum is never mentioned. See for example L. Ju, D.G. Bl air & C. Zhao, Detection of gravitational waves, Reports on Progress in Physics 63, pp. 1317–1427, 2000. See also C. Misner, K. Thorne & J.A. Wheeler, Gravitation, Freeman, 1973, page 980. Cited on page 1072. See for example Wolf gang R indler, Relativity – Special, General and Cosmological, Oxford University Press, 2001, p. 70 ff, or R ay d’Inverno Introducing Einstein’s Relativity, Clarendon Press, 1992, p. 36 ff. Cited on page 1073. T. Jacobson, Thermodynamics of spacetime: the Einstein equation of state, Physical Review Letters 75, pp. 1260–1263, 1995, or preprint http://www.arxiv.org/abs/gr-qc/9504004 Cited on page 1073. C. Schiller, Maximum force: a simple principle encompassing general relativity, part of this text and downloadable as http://www.motionmountain.net/C-2-MAXF.pdf. Cited on page 1075. Indeterminacy relations in general relativity are discussed in C.A. Mead, Possible connection between gravitation and fundamental length, Physical Review B 135, pp. 849–862, 1964. See also P.K. Townsend, Small-scale structure of space-time as the origin of the gravitational constant, Physical Review D 15, pp. 2795–2801, 1977, or the paper by M.T. Jaekel & S. R enaud, Gravitational quantum limit for length measurement, Physics Letters A 185, pp. 143–148, 1994. Cited on page 1075. Minimal length and minimal time intervals are discussed for example by G. Amelino– Camelia, Limits on the measurability of space-time distances in (the semiclassical approximation of) quantum gravity, Modern Physics Letters A 9, pp. 3415–3422, 1994, and by Y.J. Ng & H. Van Dam, Limit to space-time measurement, Modern Physics Letters A 9, pp. 335–340, 1994. Many other authors have explored the topic. Cited on page 1077. A pedagogical summary is C. Schiller, http://www.motionmountain.net/C10-QMGR. pdf or C. Schiller, Le vide diffère-t-il de la matière? in E. Gunzig & S. Diner, editeurs, Le vide – Univers du tout et du rien – Des physiciens et des philosophes s’interrogent, Les Editions de l’Université de Bruxelles, 1998. An older English version is also available as C. Schiller, Does matter differ from vacuum? http://www.arxiv.org/abs/gr-qc/9610066.

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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006

Cited on pages 1077, 1078, 1081, 1093, and 1094. 1103 Maximal curvature, as well as area and volume quantization are discussed by A. Ashtekar, Quantum geometry and gravity: recent advances, http://www.arxiv.org/ abs/gr-qc/0112038 and by A. Ashtekar, Quantum geometry in action: big bang and black holes, preprint downloadable at http://www.arxiv.org/abs/math-ph/0202008. Cited on page 1077. 1104 This was kindly pointed out by Achim Kempf. The story is told in A.D. Sakharov, General Relativity and Gravitation 32, p. 365, 2000, a reprint of a paper of 1969. Cited on page 1078. 1105 About set definition and Hilbert’s sixth problem see C. Schiller, Nature at large scales – is the universe something or nothing? http://www.motionmountain.net/C11-LGSM.pdf. Cited on pages 1078, 1081, 1093, 1094, and 1095. 1106 Several incorrect counterclaims to the entropy limit were made in R. B ousso, The holographic principle, Review of Modern Physics 74, pp. 825–874, 2002, also available as downloadable preprint at http://www.arxiv.org/abs/hep-th/0203101. Bousso has changed position in the meantime. Cited on pages 1078 and 1089. 1107 G. Amelino–Camelia, Doubly-special relativity: first results and key open problems, International Journal of Modern Physics 11, pp. 1643–1669, 2002, electronic preprint available as http://www.arxiv.org/abs/gr-qc/0210063. See also J. Magueijo & L. Smolin, Lorentz invariance with an invariant energy scale, Physical Review Letters 88, p. 190403, 2002, or http://www.arxiv.org/abs/hep-th/0112090. Cited on page 1078. 1108 Maximons, elementary particles of Planck mass, are discussed by A.D. Sakharov, Vacuum quantum fluctuations in curved space and the theory of gravitation, Soviet Physics – Doklady, 12, pp. 1040–1041, 1968. Cited on page 1079. 1109 Minimal entropy is discussed by L. Szil ard, Über die Entropieverminderung in einem thermodynamischen System bei Eingriffen intelligenter Wesen, Zeitschrift für Physik 53, pp. 840–856, 1929. This classic paper can also be found in English translation in his collected works. Cited on page 1079. 1110 See for example A.E. Shalyt-Margolin & A.Ya. Tregubovich, Generalized uncertainty relation in thermodynamics, http://www.arxiv.org/abs/gr-qc/0307018, or J. Uffink & J. van Lith-van Dis, Thermodynamic uncertainty relations, Foundations of Physics 29, pp. 655–692, 1999. Cited on page 1080. 1111 Gamma ray bursts are discussed by G. Preparata, R. Ruffini & S.-S. Xue, The dyadosphere of black holes and gamma-ray bursts, Astronomy and Astrophysics 338, pp. L87– L90, 1998, and C.L. Bianco, R. Ruffini & S.-S. Xue, The elementary spike produced by a pure e+ e− pair-electromagnetic pulse from a black hole: the PEM pulse, Astronomy and Astrophysics 368, pp. 377–390, 2001. Cited on page 1081. 1112 See for example the review in C.W.J. Beenakker & al., Quantum transport in semiconductor nanostructures, pp. 1–228, in H. Ehrenreich & D. Turnbull, eds., Solid State Physics, 44, Academic Press, Cambridge, 1991. The prediction of a future Nobel Prize was made to the authors by C. Schiller in 1992. Cited on page 1081. 1113 Brian Greene, The Elegant Universe - Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory, Vintage, 2000, Cited on page 1082. 1114 A discussion of a different electric uncertainty relation, namely between current and charge, can be found in Y-Q. Li & B. Chen, Quantum theory for mesoscopic electronic circuits and its applications, http://www.arxiv.org/abs/cond-mat/9907171. Cited on page 1082.

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1115 Wolf gang R indler, Relativity – Special, General and Cosmological, Oxford University 1116 1117

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1119 1120

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Press, 2001, p. 230. Cited on page 1083. Hans C. Ohanian & R emo Ruffini, Gravitation and Spacetime, W.W. Norton & Co., 1994. Cited on page 1085. A maximal acceleration for microscopic systems was stated by E.R. Caianiello, Lettere al Nuovo Cimento 41, p. 370, 1984. It is discussed by G. Papini, Shadows of a maximal acceleration, http://www.arxiv.org/abs/gr-qc/0211011. Cited on page 1087. The entropy limit for black holes is discussed by J.D. Bekenstein, Entropy bounds and black hole remnants, Physical Review D 49, pp. 1912–1921, 1994. See also J.D. Bekenstein, Universal upper bound on the entropy-to-energy ration for bounded systems, Physical Review D 23, pp. 287–298, 1981. Cited on page 1089. Private communications with Jos Uffink and Bernard Lavenda. Cited on page 1089. P. Kovtun, D.T. Son & A.O. Starinets, A viscosity bound conjecture, preprint found at http://www.arxiv.org/abs/hep-th/0405231. Cited on page 1090. H. Wussing & P.S. Alexandrov, (eds.) Die Hilbertschen Probleme, Akademische Verlagsgesellschaft Geest & Portig, 1983, or Ben H. Yandell, The Honours Class: Hilbert’s Problems and their Solvers, A.K. Peters, 2002. Cited on page 1095.

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the shape of points – extension in nature

37.

1101

the shape of p oints – extension in nature

Nil tam difficile est, quin quaerendo investigari possiet. Terence*

Copyright © Christoph Schiller November 1997–May 2006

* ‘Nothing is so difficult that it could not be investigated.’ Terence is Publius Terentius Afer (c. 190–159 bce ), important roman poet. He writes this in his play Heauton Timorumenos, verse 675. ** This section describes a research topic and as such is not a compendium of generally accepted results (yet). It was written between December 2001 and May 2002.

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The expressions for the Compton wavelength λ = h~mc and for the Schwarzschild radius r s = 2Gm~c 2 imply a number of arguments which lead to the conclusion that at Planck energies, space-time points and point particles must be described, in contrast to their name, by extended entities. These arguments point towards a connection between microscopic and macroscopic scales, confirming the present results of string theory and quantum gravity. At the same time, they provide a pedagogical summary of this aspect of present day theoretical physics.** 1 – It is shown that any experiment trying to measure the size or the shape of an elementary particle with high precision inevitably leads to the result that at least one dimension of the particle is of macroscopic size. 2 – There is no data showing that empty space-time is continuous, but enough data showing that it is not. It is then argued that in order to build up an entity such as the vacuum, that is extended in three dimensions, one necessarily needs extended building blocks 3 – The existence of minimum measurable distances and time intervals is shown to imply the existence of space-time duality, i.e. a symmetry between very large and very small distances. Space-time duality in turn implies that the fundamental entities that make up vacuum and matter are extended. 4 – It is argued that the constituents of the universe and thus of space-time, matter and radiation cannot form a set. As a consequence any precise description of nature must use extended entities. 5 – The Bekenstein–Hawking expression for the entropy of black holes, in particular its surface dependence, confirms that both space-time and particles are composed of extended entities. 6 – Trying to extend statistical properties to Planck scales shows that both particles and space-time points behave as braids at high energies, a fact which also requires extended entities. 7 – The Dirac construction for spin provides a model for fermions, without contradiction with experiments, that points to extended entities. An overview of other arguments in favour of extended entities provided by present research efforts is also given. To complete the discussion, experimental and theoretical checks for the idea of extended building blocks of nature are presented.

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1102 xi general rel ativity versus quantum mechanics • the shape of points

Introduction – vacuum and particles Our greatest pretenses are built up not to hide the evil and the ugly in us, but our emptyness. The hardest thing to hide is something that is not there. Eric Hoffer, The Passionate State of Mind.

“ Ref. 1122, Ref. 1123

Copyright © Christoph Schiller November 1997–May 2006

The existence of a minimal length and space interval in nature implies that points in space, time or space-time have no experimental backing and that we are forced to part from the traditional idea of continuity. Even though, properly speaking, points do not exist, and thus space points, events or point particles do not exist either, we can still ask what happens when we study these entities in detail. The results provide many fascinating surprises. Using a simple Gedanken experiment, we have found above that particles and spacetime cannot be distinguished from each other at Planck scales. The argument was the following. The largest mass that can be put in a box of size R is a black hole with a Schwarzschild radius of the same value. That is also the largest possible mass measurement error. But any piece of vacuum also has a smallest mass measurement error. The issue of a smallest mass measurement error is so important that it merits special attention. Mass measurement errors always prevent humans to state that a region of space has zero mass. In exactly the same way, also nature cannot ‘know’ that the mass of a region is zero, provided that this error is due to quantum indeterminacy. Otherwise nature would circumvent quantum theory itself. Energy and mass are always unsharp in quantum theory. We are not used to apply this fact to vacuum itself, but at Planck scales we have to. We remember from quantum field theory that the vacuum, like any other system, has a mass; of course, its value is zero for long time averages. For finite measuring times, the mass value will be uncertain and thus different from zero. Not only limitations in time, but also limitations in space lead to mass indeterminacy for the vacuum. These indeterminacies in turn lead to a minimum mass errors for vacuum regions of finite size. Quantum theory implies that nobody, not even nature, knows the exact mass value of a system or of a region of empty space. A box is empty if it does not contain anything. But emptiness is not well defined for photons with wavelength of the size R of the box or larger. Thus the mass measurement error for an ‘empty’ box – corresponding to what we call vacuum – is due to the indeterm-

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Page 1013

Only the separation of the description of nature into general relativity and quantum theory allows us to use continuous space-time and point particles as basic entities. When the two theories are united, what we use to call ‘point’ turns out to have quite counterintuitive properties. We explore a few of them in the following. Above, we have given a standard argument showing that points do not exist in nature. The Compton wavelength and the Schwarzschild radius together determine a minimal length and a minimal time interval in nature. They are given (within a factor of order one, usually omitted in the following) by the Planck length and the Planck time, with the values » l Pl = »ħG~c 3 = 1.6 ë 10−35 m (796) t Pl = ħG~c 5 = 5.4 ë 10−44 s .

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1103

inacy relation and is given by that mass whose Compton wavelength matches the size of the box. As shown earlier on, the same mass value is found by every other Gedanken experiment: trying to determine the gravitational mass by weighing the ‘piece’ of vacuum or by measuring its kinetic energy gives the same result. Another, but in the end equivalent way to show that a region of vacuum has a finite mass error is to study how the vacuum energy depends on the position indeterminacy of the border of the region. Any region is defined through its border. The position indeterminacy of the border will induce a mass error for the contents of the box, in the same way that a time limit does. Again, the resulting mass error value for a region of vacuum is the one for which the box size is the Compton wavelength. Summarizing, for a box of size R, nature allows only mass values and mass measurement error value m between two limits: (full box)

(797)

We see directly that for sizes R of the order of the Planck length, the two limits coincide; they both give the Planck mass ħ M Pl = = c l Pl

¾

ħc  10−8 kg  1019 GeV~c2 . G

(798)

In other words, for boxes of Planck size, we cannot distinguish a full box from an empty one. This means that there is no difference between vacuum and matter at Planck scales. Of course, a similar statement holds for the distinction between vacuum and radiation. At Planck scales, vacuum and particles cannot be distinguished.

where the differences are taken between the values before and after the collision. The error δM of such a measurement is simply given by δM δ∆v δm δ∆V = + + . M ∆v m ∆V

(800)

At Planck scales we have δ∆v~∆v  1, because the velocity error is always, like the velocities themselves, of the order of the speed of light. In other words, at Planck scales the mass measurement error is so large that we cannot determine whether a mass is different

Copyright © Christoph Schiller November 1997–May 2006

How else can we show that matter and vacuum cannot be distinguished? The impossibility to distinguish vacuum from particles is a strong statement. A strong statement needs additional proof. Mass can also be measured by probing its inertial aspect, i.e. by colliding the unknown mass M with known velocity V with a known probe particle of mass m and momentum p = mv. We then have ∆p , (799) M= ∆V

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ħ c2 R EmE (empty box) . G cR

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1104 xi general rel ativity versus quantum mechanics • the shape of points

from zero: vacuum is indistinguishable from matter. The same conclusion arises if we take light with a wavelength λ as the probe particle. In this case, expression (799) leads to a mass error δM δ∆λ δ∆V = + . M ∆λ ∆V

Challenge 1503 ny

In order that photon scattering can probe Planck dimensions, we need a wavelength of the order of the Planck value; but in this case the first term is approximately unity. Again we find that at Planck scales the energy indeterminacy is always of the same size as the energy value to be measured. Measurements cannot distinguish between vanishing and non-vanishing mass M at Planck scales. In fact, this result appears for all methods of mass measurement that can be imagined. At Planck scales, matter cannot be distinguished from vacuum. Incidentally, the same arguments are valid if instead of the mass of matter we measure the energy of radiation. In other words, it is also impossible to distinguish radiation from vacuum at high energies. In short, no type of particle differs from vacuum at high energies. The indistinguishability of particles and vacuum, together with the existence of minimum space-time intervals, suggest that space, time, radiation and matter are macroscopic approximations of an underlying, common and discrete structure. This structure is often called quantum geometry. How do the common constituents of the two aspects of nature look like? We will provide several arguments showing that these constituents are extended and fluctuating. Also, die Aufgabe ist nicht zu sehen, was noch nie jemand gesehen hat, sondern über dasjenige was jeder schon gesehen hat zu denken was noch nie jemand gedacht hat.* Erwin Schrödinger

Argument 1: The size and shape of elementary particles

* ‘Our task is not to see what nobody has ever seen, but to think what nobody has ever thought about that which everybody has seen already.’

Copyright © Christoph Schiller November 1997–May 2006

Size is the length of vacuum taken by an object. This definition comes natural in everyday life, quantum theory and relativity. However, approaching Planck energy, vacuum and matter cannot be distinguished: it is impossible to define the boundary between the two, and thus it is impossible to define the size of an object. As a consequence, every object becomes as extended as the vacuum. Care is therefore required. What happens if Planck energy is approached, advancing step by step to higher energy? Every measurement requires comparison with a standard. A standard is made of matter and comparison is performed using radiation (see Figure 387). Thus any measurement requires to distinguish between matter, radiation and space-time. However, the distinction between matter and radiation is possible only up to the (grand) unification energy, which is about an 800th of the Planck energy. Measurements do not allow us to prove that particles are point-like. Let us take a step back and check whether measurements allow us to say whether particles can at least be contained inside small spheres.

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Ref. 1124

(801)

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1105

object

comparison

standard F I G U R E 387 Measurement requires matter

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and radiation

The first and simplest way to determine the size of a compact particle such as a sphere or something akin to it, is to measure the size of a box it fits in. To be sure that the particle fits inside, we first of all must be sure that the box is tight. This is done by checking whether something, such as matter or radiation, can leave the box. However, nature does not provide a way to ensure that a box has no holes. Potential hills cannot get higher than the maximum energy, namely the Planck energy. The tunnelling effect cannot be ruled out. In short, there is no way to make fully tight boxes. In addition, already at the unification energy there is no way to distinguish between the box and the object enclosed in it, as all particles can be transformed from any one type into any other. Let us cross-check this result. In everyday life, we call particles ‘small’ because they can be enclosed. Enclosure is possible because in daily life walls are impenetrable. However, walls are impenetrable for matter particles only up to roughly 10 MeV and for photons only up to 10 keV. In fact, boxes do not even exist at medium energies. We thus cannot extend the idea of ‘box’ to high energies at all. In summary, we cannot state that particles are compact or of finite size using boxes. We need to try other methods. Can the Greeks help? – The limits of knives

Copyright © Christoph Schiller November 1997–May 2006

The Greeks deduced the existence of atoms by noting that division of matter must end. In contrast, whenever we think of space (or space-time) as made of points, we assume that it can be subdivided without end. Zeno noted this already long time ago and strongly criticized this assumption. He was right: at Planck energy, infinite subdivision is impossible. Any attempt to divide space stops at Planck dimensions at the latest. The process of cutting is the insertion of a wall. Knifes are limited in the same ways that walls are. The limits of walls imply limits to size determination. In particular, the limits to walls and knives imply that at Planck energies, a cut does not necessarily lead to two separate parts. One cannot state that the two parts have been really separated; a thin connection between can never be excluded. In short, cutting objects at Planck scales does not prove compactness.

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Do boxes exist?

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1106 xi general rel ativity versus quantum mechanics • the shape of points

Are cross-sections finite?

Ref. 1125

Can one take a photograph of a point?

Ref. 1126 Ref. 1122, Ref. 1127

check the greek

Pittacus.

Humans or any other types of observers can only observe a part of the world with finite resolution in time and in space. In this, humans resemble a film camera. The highest possible resolution has (almost) been discovered in 1899: the Planck time and the Planck length. No human, no film camera and no measurement apparatus can measure space or time intervals smaller than the Planck values. But what would happen if we took photographs with shutter times approaching the Planck time? Imagine that you have the world’s best shutter and that you are taking photographs at increasingly shorter times. Table 79 gives a rough overview of the possibilities. For * ‘Recognize the right moment.’ also rendered as: ‘Recognize thine opportunity.’ Pittacus (Πιττακος) of Mitylene, (c. 650–570 BCE ) was the Lesbian tyrant that was also one of the ancient seven sages.

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Καιρὸν γνῶθι.*

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Particles are not point like. Particles are not compact. Are particles are at least of finite size? To determine particle size, we can take try to determine their departure from pointlikeness. At high energy, detecting this departure requires scattering. For example, we can suspend the particle in some trap and then shoot some probe at it. What happens in a scattering experiment at high energies? The question has been studied already by Leonard Susskind and his coworkers. When shooting at the particle with a high energy probe, the scattering process is characterized by an interaction time. Extremely short interaction times imply sensitivity to the size and shape fluctuations due to the quantum of action. An extremely short interaction time provides a cut-off for high energy shape and size fluctuations and thus determines the measured size. As a result, the size measured for any microscopic, but extended object increases when the probe energy is increased towards the Planck value. In summary, even though at experimentally achievable energies the size is always smaller than measurable, when approaching the Planck energy, size increases above all bounds. As a result, at high energies we cannot give a limit to sizes of particles! In other words, since particles are not point-like at everyday energy, at Planck energies they are enormous: particles are extended. That is quite a deduction. Right at the start of our mountain ascent, we distinguished objects from their environment. Objects are by definition localized, bounded and compact. All objects have a boundary, i.e. a surface which itself does not have a boundary. Objects are also bounded in abstract ways; boundedness is also a property of the symmetries of any object, such as its gauge group. In contrast, the environment is not localized, but extended and unbounded. All these basic assumptions disappear at Planck scales. At Planck energy, it is impossible to determine whether something is bounded or compact. Compactness and localisation are only approximate properties; they are not correct at high energies. The idea of a point particle is a low energy, approximated concept. Particles at Planck scales are as extended as the vacuum. Let us perform another check of this conclusion.

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argument 1: the size and shape of elementary particles

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TA B L E 79 Effects of various camera shutter times on photographs

O b s e r va t i o n p o s s i b i l i t i e s

1h 1s 20 ms 10 ms

high high lower lower

0.25 ms

lower

1 µs c. 10 ps

very low lowest

10 fs 100 zs

higher high

shorter times

very high

10−41 s

highest

Ability to see faint quasars at night if motion is compensated Everyday motion is completely blurred Interruption by eyelids; impossibility to see small changes Effective eye/brain shutter time; impossibility to see tennis ball when hitting it Shortest commercial photographic camera shutter time; allows to photograph fast cars Ability to photograph flying bullets; requires strong flashlight Study of molecular processes; ability to photograph flying light pulses; requires laser light to get sufficient illumination Light photography becomes impossible due to wave effects X-ray photography becomes impossible; only γ-ray imaging is left over Photographs get darker as illumination gets dimmer; gravitational effects start playing a role imaging makes no sense

Copyright © Christoph Schiller November 1997–May 2006

shorter and shorter shutter times, photographs get darker and darker. Once the shutter time reaches the oscillation time of light, strange things happen: light has no chance to pass undisturbed; signal and noise become impossible to distinguish; in addition, the moving shutter will produce colour shifts. In contrast to our intuition, the picture would get blurred at extremely short shutter times. Photography is not only impossible at long but also at short shutter times. The difficulty of taking photographs is independent of the used wavelength. The limits move, but do not disappear. A short shutter time τ does not allow photons of energy lower than ħ~τ to pass undisturbed. The blur is small when shutter times are those of everyday life, but increases when shutter times are shortened towards Planck times. As a result, there is no way to detect or confirm the existence of point objects by taking pictures. Points in space, as well as instants of time, are imagined concepts; they do not allow a precise description of nature. At Planck shutter times, only signals with Planck energy can pass through the shutter. Since at these energies matter cannot be distinguished from radiation or from empty space, all objects, light and vacuum look the same. As a result, it becomes impossible to say how nature looks at shortest times. But the situation is much worse: a Planck shutter does not exist at all, as it would need to be as small as a Planck length. A camera using it could not be built, as lenses do not work at this energy. Not even a camera obscura – without any lens – would work, as diffraction effects would make image production impossible. In other words, the idea that at short shutter times, a photograph of nature shows a frozen version of everyday life, like a stopped film, is completely wrong! Zeno criticized this image already in ancient Greece, in his discussions about motion, though not so clearly as we can do now. Indeed,

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D u r at i o n B lu r

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1108 xi general rel ativity versus quantum mechanics • the shape of points

at a single instant of time nature is not frozen at all.* At short times, nature is blurred and fuzzy. This is also the case for point particles. In summary, whatever the intrinsic shape of what we call a ‘point’ might be, we know that being always blurred, it is first of all a cloud. Whatever method to photograph of a point particle is used, it always shows an extended entity. Let us study its shape in more detail. What is the shape of an electron?

Copyright © Christoph Schiller November 1997–May 2006

* In fact, a shutter does not exist even at medium energy, as shutters, like walls, stop existing at around 10 MeV.

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Ref. 1122

Given that particles are not point-like, they have a shape. How can we determine it? Everyday object shape determination is performed by touching the object from all sides. This works with plants, people or machines. It works with molecules, such as water molecules. We can put them (almost) at rest, e.g. in ice, and then scatter small particles off them. Scattering is just a higher energy version of touching. However, scattering cannot determine shapes smaller than the wavelength of the used probes. To determine a size as small as that of an electron, we need the highest energies available. But we already know what happens when approaching Planck scales: the shape of a particle becomes the shape of all the space surrounding it. Shape cannot be determined in this way. Another method to determine the shape is to build a tight box filled of wax around the system under investigation. We let the wax cool and and observe the hollow part. However, near Planck energies boxes do not exist. We are unable to determine the shape in this way. A third way to measure shapes is cutting something into pieces and then study the pieces. But cutting is just a low-energy version of a scattering process. It does not work at high energies. Since the term ‘atom’ means ‘uncuttable’ or ‘indivisible’, we have just found out that neither atoms nor indivisible particles can exist. Indeed, there is no way to prove this property. Our everyday intuition leads us completely astray at Planck energies. A fourth way to measure shapes could appear by distinguishing transverse and longitudinal shape, with respect to the direction of motion. However, for transverse shape we get the same issues as for scattering; transverse shape diverges for high energy. To determine longitudinal shape, we need at least two infinitely high potential walls. Again, we already know that this is impossible. A further, indirect way of measuring shapes is the measurement of the moment of inertia. A finite moment of inertia means a compact, finite shape. However, when the measurement energy is increased, rotation, linear motion and exchange become mixed up. We do not get meaningful results. Still another way to determine shapes is to measure the entropy of a collection of particles we want to study. This allows to determine the dimensionality and the number of internal degrees of freedom. At high energies, a collection of electrons would become a black hole. We study the issue separately below, but again we find no new information. Are these arguments water-tight? We assumed three dimensions at all scales, and assumed that the shape of the particle itself is fixed. Maybe these assumptions are not valid at Planck scales. Let us check the alternatives. We have already shown above that due

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argument 1: the size and shape of elementary particles

Ref. 1122

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to the fundamental measurement limits, the dimensionality of space-time cannot be determined at Planck scales. Even if we could build perfect three-dimensional boxes, holes could remain in other dimensions. It does not take long to see that all the arguments against compactness work even if space-time has additional dimensions. Is the shape of an electron fixed?

Ref. 1129

* Examples are the neutron, positronium, or the atoms. Note that the argument does not change when the elementary particle itself is unstable, such as the muon. Note also that the possibility that all components be heavier than the composite, which would avoid this argument, does not seem to lead to satisfying physical properties; e.g. it leads to intrinsically unstable composites. ** Thus at Planck scales there is no quantum Zeno effect any more.

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Ref. 1128

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Page 761

Only an object composed of localized entities, such as a house or a molecule, can have a fixed shape. The smaller a system gets, the more quantum fluctuations play a role. Any entity with a finite size, thus also an elementary particle, cannot have a fixed shape. Every Gedanken experiment leading to finite shape also implies that the shape itself fluctuates. But we can say more. The distinction between particles and environment resides in the idea that particles have intrinsic properties. In fact, all intrinsic properties, such as spin, mass, charge and parity, are localized. But we saw that no intrinsic property is measurable or definable at Planck scales. It is impossible to distinguish particles from the environment, as we know already. In addition, at Planck energy particles have all properties the environment also has. In particular, particles are extended. In short, we cannot prove by experiments that at Planck energy elementary particles are finite in size in all directions. In fact, all experiments one can think of are compatible with extended particles, with ‘infinite’ size. We can also say that particles have tails. More precisely, a particle always reaches the borders of the region of space-time under exploration. Not only are particles extended; in addition, their shape cannot be determined by the methods just explored. The only possibility left over is also suggested by quantum theory: The shape of particles is fluctuating. We note that for radiation particles we reach the same conclusions. The box argument shows that also radiation particles are extended and fluctuating. In our enthusiasm we have also settled an important detail about elementary particles. We saw above that any particle which is smaller than its own Compton wavelength must be elementary. If it were composite, there would be a lighter component inside it; this lighter particle would have a larger Compton wavelength than the composite particle. This is impossible, since the size of a composite particle must be larger than the Compton wavelength of its components.* However, an elementary particle can have constituents, provided that they are not compact. The difficulties of compact constituents were already described by Sakharov in the 1960s. But if the constituents are extended, they do not fall under the argument, as extended entities have no localized mass. As a result, a flying arrow, Zeno’s famous example, cannot be said to be at a given position at a given time, if it is made of extended entities. Shortening the observation time towards the Planck time makes an arrow disappear in the same cloud that also makes up space-time.**

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1110 xi general rel ativity versus quantum mechanics • the shape of points

Ref. 1122

Challenge 1504 e

In summary, only the idea of points leads to problems at Planck scales. If space-time and matter are imagined to be made, at Planck scales, of extended and fluctuating entities, all problems disappear. We note directly that for extended entities the requirement of a non-local description is realized. Similarly, the entities being fluctuating, the requirement of a statistical description of vacuum is realized. Finally, the argument forbidding composition of elementary particles is circumvented, as extended entities have no clearly defined mass. Thus the concept of Compton wavelength cannot be defined or applied. Elementary particles can thus have constituents if they are extended. But if the components are extended, how can compact ‘point’ particles be formed with them? A few options will be studied shortly.

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Argument 2: The shape of points in vacuum

Ref. 1130

Ref. 1131

* Imagining the vacuum as a collection of entities with Planck size in all directions, such as spheres, would

Copyright © Christoph Schiller November 1997–May 2006

Ref. 1122

We are used to think that empty space is made of spatial points. Let us check whether this is true at high energy. At Planck scales no measurement can give zero length, zero mass, zero area or zero volume. There is no way to state that something in nature is a point without contradicting experimental results. In addition, the idea of a point is an extrapolation of what is found in small empty boxes getting smaller and smaller. However, we just saw that at high energies small boxes cannot be said to be empty. In fact, boxes do not exist at all, as they are never tight and do not have impenetrable walls at high energies. Also the idea of a point as a continuous subdivision of empty space is untenable. At small distances, space cannot be subdivided, as division requires some sort of dividing wall, which does not exist. Even the idea of repeatedly putting a point between two others cannot be applied. At high energy, it is impossible to say whether a point is exactly on the line connecting the outer two points; and near Planck energy, there is no way to find a point between them at all. In fact, the term ‘in between’ makes no sense at Planck scales. We thus find that space points do not exist, in the same way that point particles do not exist. But there is more; space cannot be made of points for additional reasons. Common sense tells us that points need to be kept apart somehow, in order to form space. Indeed, mathematicians have a strong argument stating why physical space cannot be made of mathematical points: the properties of mathematical spaces described by the Banach– Tarski paradox are quite different from that of the physical vacuum. The Banach–Tarski paradox states states that a sphere made of mathematical points can be cut into 5 pieces which can be reassembled into two spheres each of the same volume as the original sphere. Mathematically, volume makes no sense. Physically speaking, we can say that the concept of volume does not exist for continuous space; it is only definable if an intrinsic length exists. This is the case for matter; it must also be the case for vacuum. But any concept with an intrinsic length, also the vacuum, must be described by one or several extended components.* In summary, we need extended entities to build up space-time!

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Thus, since there is an impossibility that [finite] quantities are built from contacts and points, it is necessary that there be indivisible material elements and [finite] quantities. Aristotle, Of Generation and Corruption.

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argument 2: the shape of points in vacuum

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Measuring the void

What is the maximum number of particles that fits inside a piece of vacuum? The maximum mass that fits into a piece of vacuum is a black hole. But also in this case, the maximum mass depends only on the surface of the given vacuum piece. The maximum

Challenge 1505 n

avoid the Banach–Tarski paradox, but would not allow to deduce the numbers of dimensions of space and time. It would also contradict all other results of this section. We therefore do not explore it further.

Copyright © Christoph Schiller November 1997–May 2006

To check whether space-time constituents are extended, let us perform a few additional Gedanken experiments. First, let us measure the size of a point of space-time. The clearest definition of size is through the cross-section. How can we determine the cross-section of a point? We can determine the cross section of a piece of vacuum and determine the number of points inside it. From the two determinations we can deduce the cross-section of a single point. At Planck energies however, we get a simple result: the cross-section of a volume of empty space is depth independent. At Planck energies, vacuum has a surface, but no depth. In other words, at Planck energy we can only state that a Planck layer covers the surface of a volume. We cannot say anything about its interior. One way to picture the result is to say that space points are long tubes. Another way to determine the size of a point is to count the points found in a given volume of space-time. One approach is to count the possible positions of a point particle in a volume. However, point particles are extended at Planck energies and indistinguishable from vacuum. At Planck energy, the number of points is given by surface area of the volume divided by the Planck area. Again, the surface dependence suggests that particles are long tubes. Another approach to count the number of points in a volume is to fill a piece of vacuum with point particles.

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Not only is it impossible to generate a volume with mathematical points; it is also impossible to generate exactly three physical dimensions with mathematical points. Mathematics shows that any compact one-dimensional set has as many points as any compact three-dimensional set. And the same is true for any other pair of dimension values. To build up the physical three-dimensional vacuum we need entities which organize their neighbourhood. This cannot be done with purely mathematical points. The fundamental entities must possess some sort of bond forming ability. Bonds are needed to construct or fill three dimensions instead of any other number. Bonds require extended entities. But also a collection of tangled entities extending to the maximum scale of the region under consideration would work perfectly. Of course the precise shape of the fundamental entities is not known at this point in time. In any case we again find that any constituents of physical three-dimensional space must be extended. In summary, we need extension to define dimensionality and to define volume. We are not surprised. Above we deduced that the constituents of particles are extended. Since vacuum is not distinguishable from matter, we expect the constituents of vacuum to be extended as well. Stated simply, if elementary particles are not point-like, then space-time points cannot be either.

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Page 223 Ref. 1132

mass increases less rapidly than the volume. In other words, the number of points in a volume is only proportional to the surface area of that volume. There is only one solution: vacuum must be made of extended entities crossing the whole volume, independently of the shape of the volume. Two thousand years ago, the Greek argued that matter must be made of particles because salt can be dissolved in water and because fish can swim through water. Now that we know more about Planck scales, we have to reconsider the argument. Like fish through water, particles can move through vacuum; but since vacuum has no bounds and since it cannot be distinguished from matter, vacuum cannot be made of particles. However, there is another possibility that allows for motion of particles through vacuum: both vacuum and particles can be made of a web of extended entities. Let us study this option in more detail.

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Argument 3: The l arge, the small and their connection

* There is also an S-duality, which connects large and small coupling constants, and a U-duality, which is the combination of S- and T-duality.

Copyright © Christoph Schiller November 1997–May 2006

Ref. 1135

If two observables cannot be distinguished, there is a symmetry transformation connecting them. For example, when switching observation frame, an electric field may change into a magnetic one. A symmetry transformation means that we can change the viewpoint (i.e. the frame of observation) with the consequence that the same observation is described by one quantity from one viewpoint and by the other quantity from the other viewpoint. When measuring a length at Planck scales it is impossible to say whether we are measuring the length of a piece of vacuum, the Compton wavelength of a body, or the Schwarzschild diameter of a body. For example, the maximum size for an elementary object is its Compton wavelength. The minimum size for an elementary object is its Schwarzschild radius. The actual size of an elementary object is somewhere in between. If we want to measure the size precisely, we have to go to Planck energy: but then all these quantities are the same. In other words, at Planck scales, there is a symmetry transformation between Compton wavelength and Schwarzschild radius. In short, at Planck scales there is a symmetry between mass and inverse mass. As a further consequence, at Planck scales there is a symmetry between size and inverse size. Matter–vacuum indistinguishability means that there is a symmetry between length and inverse length at Planck energies. This symmetry is called space-time duality or T-duality in the literature of superstrings.* Space-time duality is a symmetry between situations at scale n l Pl and at scale f l Pl ~n, or, in other words, between R and ( f l Pl )2 ~R, where the experimental number f has a value somewhere between 1 and 1000. Duality is a genuine non-perturbative effect. It does not exist at low energy, since dual2 ity automatically also relates energy E and energy E Pl ~E = ħc 3 ~GE, i.e. it relates energies below and above Planck scale. Duality is a quantum symmetry. It does not exist in everyday life, as Planck’s constant appears in its definition. In addition, it is a general relativistic

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I could be bounded in a nutshell and count myself a king of infinite space, were it not that I have bad dreams. William Shakespeare, Hamlet.

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effect, as it includes the gravitational constant and the speed of light. Let us study duality in more detail. Small is large?

“ Ref. 1133

[Zeno of Elea maintained:] If the existing are many, it is necessary that they are at the same time small and large, so small to have no size, and so large to be without limits. Simplicius, Commentary on the Physics of Aristotle, 140, 34.

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Unification and total symmetry

* A symmetry between size and Schwarzschild radius, i.e. a symmetry between length and mass, will lead to general relativity. Additionally, at Planck energy there is a symmetry between size and Compton wavelength. In other words, there is a symmetry between length and 1/mass. It means that there is a symmetry between coordinates and wave functions. Note that this is a symmetry between states and observables. It leads to quantum theory.

Copyright © Christoph Schiller November 1997–May 2006

So far, we have shown that at Planck energy, time and length cannot be distinguished. Duality has shown that mass and inverse mass cannot be distinguished. As a consequence, length, time and mass cannot be distinguished from each other. Since every observable is a combination of length, mass and time, space-time duality means that there is a symmetry between all observables. We call it the total symmetry.*

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To explore the consequences of duality, we can compare it to the 2π rotation symmetry in everyday life. Every object in daily life is symmetrical under a full rotation. For the rotation of an observer, angles make sense only as long as they are smaller than 2π. If a rotating observer would insist on distinguishing angles of 0, 2π, 4π etc., he would get a new copy of the universe at each full turn. 2 Similarly, in nature, scales R and l Pl ~R cannot be distinguished. Lengths make no sense when they are smaller than l Pl . If however, we insist on using even smaller values and insist on distinguishing them from large ones, we get a new copy of the universe at those small scales. Such an insistence is part of the standard continuum description of motion, where it is assumed that space and time are described by the real numbers, which are defined for arbitrary small intervals. Whenever the (approximate) continuum description 2 with infinite extension is used, the R  l Pl ~R symmetry pops up. Duality implies that diffeomorphism invariance is only valid at medium scales, not at extremal ones. At extremal scales, quantum theory has to be taken into account in the proper manner. We do not know yet how to do this. Space-time duality means that introducing lengths smaller than the Planck length (like when one defines space points, which have size zero) means at the same time introducing things with very large (‘infinite’) value. Space-time duality means that for every small enough sphere the inside equals outside. Duality means that if a system has a small dimension, it also has a large one. And vice versa. There are thus no small objects in nature. As a result, space-time duality is consistent with the idea that the basic entities are extended.

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Page 556 Ref. 1134

Ref. 1135

* Renormalization energy does connect different energies, but not in the correct way; in particular, it does not include duality.

Copyright © Christoph Schiller November 1997–May 2006

Challenge 1507 e

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Challenge 1506 d

Total symmetry implies that there are many types of specific dualities, one for each pair of quantities under investigation. Indeed, in string theory, the number of duality types discovered is increasing every year. It includes, among others, the famous electric–magnetic duality we first encountered in the chapter on electrodynamics, coupling constant duality, surface–volume duality, space-time duality and many more. All this confirms that there is an enormous symmetry at Planck scales. Similar statements are also well-known right from the beginning of string theory. Most importantly, total symmetry implies that gravity can be seen as equivalent to all other forces. Space-time duality shows that unification is possible. Physicist have always dreamt about unification. Duality tells us that this dream can indeed be realized. It may seem that total symmetry is in complete contrast with what was said in the previous section, where we argued that all symmetries are lost at Planck scales. Which result is correct? Obviously, both of them are. At Planck scales, all low energy symmetries are indeed lost. In fact, all symmetries that imply a fixed energy are lost. Duality and its generalizations however, combine both small and large dimensions, or large and small energies. Most symmetries of usual physics, such as gauge, permutation and space-time symmetries, are valid at each fixed energy separately. But nature is not made this way. The precise description of nature requires to take into consideration large and small energies at the same time. In everyday life, we do not do that. Everyday life is a low and fixed energy approximation of nature. For most of the twentieth century, physicists aimed to reach higher and higher energies. We believed that precision increases with increasing energy. But when we combine quantum theory and gravity we are forced to change this approach; to achieve high precision, we must take both high and low energy into account at the same time.* The large differences in phenomena at low and high energies are the main reason why unification is so difficult. So far, we were used to divide nature along the energy scale. We thought about high energy physics, atomic physics, chemistry, biology, etc. The differences between these sciences is the energy of the processes involved. But now we are not allowed to think in this way any more. We have to take all energies into account at the same time. That is not easy, but we do not have to despair. Important conceptual progress has been achieved in the last decade of the twentieth century. In particular, we now know that we need only one single concept for all things which can be measured. Since there is only one concept, there are many ways to study it. We can start from any (low-energy) concept in physics and explore how it looks and behaves when we approach Planck scales. In the present section, we are looking at the concept of point. Obviously, the conclusions must be the same, independently of the concept we start with, be it electric field, spin, or any other. Such studies thus provide a check for the results in this section. Unification thus implies to think using duality and using concepts which follow from it. In particular, we need to understand what exactly happens to duality when we restrict ourselves to low energy only, as we do in everyday life. This question is left for the next section.

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Argument 4: D oes nature have parts?

Ref. 1136

Another argument, independent of the ones given above, underlines the correctness of a model of nature made of extended entities. Let us take a little broader view. Any concept for which we can distinguish parts is described by a set. We usually describe nature as a set of objects, positions, instants, etc. The most famous set description of nature is the oldest known, given by Democritus: ‘The world is made of indivisible particles and void.’ This description was extremely successful in the past; there were no discrepancies with observations yet. However, after 2500 years, the conceptual difficulties of this approach are obvious. We now know that Democritus was wrong, first of all, because vacuum and matter cannot be distinguished at Planck scales. Thus the word ‘and’ in his sentence is already mistaken. Secondly, due to the existence of minimal scales, the void cannot be made of ‘points,’ as we usually assume nowadays. Thirdly, the description fails because particles are not compact objects. Fourth, the total symmetry implies that we cannot distinguish parts in nature; nothing can really be distinguished from anything else with complete precision, and thus the particles or points in space making up the naive model of void cannot exist. In summary, quantum theory and general relativity together show that in nature, all differences are only approximate. Nothing can really be distinguished from anything else with complete precision. In other words, there is no way to define a ‘part’ of nature, neither for matter, nor for space, nor for time, nor for radiation. Nature cannot be a set. The conclusion that nature is not a set does not come as a surprise. We have already encountered another reason to doubt that nature is a set. Whatever definition we use for the term ‘particle’, Democritus cannot be correct for a purely logical reason. The description he provided is not complete. Every description of nature defining nature as a set of parts necessarily misses certain aspects. Most importantly, it misses the number of these parts. In particular, the number of particles and the number of dimensions of space-time must be specified if we describe nature as made from particles and vacuum. Above we saw that it is rather dangerous to make fun of the famous statement by Arthur Eddington

In fact, practically all physicists share this belief; usually they either pretend to favour some other number, or worse, they keep the number unspecified. We have seen during * ‘Multitude should not be introduced without necessity.’ This famous principle is commonly called Occam’s razor. William of Ockham (b. 1285/1295 Ockham, d. 1349/50 München), or Occam in the common Latin spelling, was one of the great thinkers of his time. In his famous statement he expresses that only those concepts which are strictly necessary should be introduced to explain observations. It can be seen as the requirement to abandon beliefs when talking about nature. In addition, at this stage of our mountain ascent it has an even more direct interpretation.

Copyright © Christoph Schiller November 1997–May 2006

I believe there are 15,747,724,136,275,002,577,605,653,961,181,555,468,044,717,914,527, 116,709,366,231,425,076,185,631,031,296 protons in the universe and the same number of electrons.

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Page 681

Pluralitas non est ponenda sine necessitate.* William of Occam

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1116 xi general rel ativity versus quantum mechanics • the shape of points

* As a curiosity, practically the same discussion can already be found, in Plato’s Parmenides, written in the fourth century bce. There, Plato musically ponders different arguments on whether nature is or can be a unity or a multiplicity, i.e. a set. It seems that the text is based on the real visit by Parmenides and Zeno in Athens, where they had arrived from their home city Elea, which lies near Naples. Plato does not reach a conclusion. Modern physics however, does.

Copyright © Christoph Schiller November 1997–May 2006

Ref. 1137

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Ref. 1138

our walk that in modern physics many specialized sets are used to describe nature. We have used vector spaces, linear spaces, topological spaces and Hilbert spaces. But very consistently we refrained, like all physicists, from asking about the origin of their sizes (mathematically speaking of their dimensionality or their cardinality). In fact, it is equally unsatisfying to say that the universe contains some specific number of atoms as it is to say that space-time is made of point-like events arranged in 3+1 dimensions. Both statements are about set sizes in the widest sense. In a complete, i.e. in a unified description of nature the number of smallest particles and the number of space-time points must not be added to the description, but must result from the description. Only in this case is unification achieved. Requiring a complete explanation of nature leads to a simple consequence. Any part of nature is by definition smaller than the whole of nature and different from other parts. As a result, any description of nature by a set cannot possibly yield the number of particles nor space-time dimensionality. As long as we insist in using space-time or Hilbert spaces for the description of nature, we cannot understand the number of dimensions or the number of particles. Well, that is not too bad, as we know already that nature is not made of parts. We know that parts are only approximate concepts. In short, if nature were made of parts, it could not be a unity, or a ‘one.’ If however, nature is a unity, a one, it cannot have parts.* Nature cannot be separable exactly. It cannot be made of particles. To sum up, nature cannot be a set. Sets are lists of distinguishable elements. When general relativity and quantum theory are unified, nature shows no elements: nature stops being a set at Planck scales. The result confirms and clarifies a discussion we have started in classical physics. There we had discovered that matter objects were defined using space and time, and that space and time were defined using objects. Including the results of quantum theory, this implies that in modern physics particles are defined with the help of the vacuum and the vacuum with particles. That is not a good idea. We have just seen that since the two concepts are not distinguishable from each other, we cannot define them with each other. Everything is the same; in fact, there is no ‘every’ and no ‘thing.’ Since nature is not a set, the circular reasoning is dissolved. Space-time duality also implies that space is not a set. Duality implies that events cannot be distinguished from each other. They thus do not form elements of some space. Phil Gibbs has given the name event symmetry to this property of nature. This thoughtprovoking term, even though still containing the term ‘event’, underlines that it is impossible to use a set to describe space-time. In summary, nature cannot be made of vacuum and particles. That is bizarre. People propagating this idea have been persecuted for 2000 years. This happened to the atomists from Democritus to Galileo. Were their battles it all in vain? Let us continue to clarify our thoughts.

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argument 4: d oes nature have parts?

1117

Does the universe contain anything? Stating that the universe contains something implies that we are able to distinguish the universe from its contents. However, we now know that precise distinctions are impossible. If nature is not made of parts, it is wrong to say that the universe contains something. Let us go further. As nothing can be distinguished, we need a description of nature which allows to state that at Planck energies nothing can be distinguished from anything else. For example, it must be impossible to distinguish particles from each other or from the vacuum. There is only one solution: everything, or at least what we call ‘everything’ in everyday life, must be made of the same single entity. All particles are made of one same ‘piece.’ Every point in space, every event, every particle and every instant of time must be made of the same single entity.

Dvipsbugw

An amoeba

“ ”

Page 651

Copyright © Christoph Schiller November 1997–May 2006

We found that parts are approximate concepts. The parts of nature are not strictly smaller than nature itself. As a result, any ‘part’ must be extended. Let us try to extract more information about the constituents of nature. The search for a unified theory is the search for a description in which all concepts appearing are only approximately parts of the whole. Thus we need an entity Ω, describing nature, which is not a set but which can be approximated by one. This is unusual. We all are convinced very early in our life that we are a part of nature. Our senses provide us with this information. We are not used to think otherwise. But now we have to. Let us eliminate straight away a few options for Ω. One concept without parts is the empty set. Perhaps we need to construct a description of nature from the empty set? We could be inspired by the usual construction of the natural numbers from the empty set. However, the empty set makes only sense as the opposite of some full set. That is not the case here. The empty set is not a candidate for Ω. Another possibility to define approximate parts is to construct them from multiple copies of Ω. But in this way we would introduce a new set through the back door. In addition, new concepts defined in this way would not be approximate. We need to be more imaginative. How can we describe a whole which has no parts, but which has parts approximately? Let us recapitulate. The world must be described by a single entity, sharing all properties of the world, but which can be approximated into a set of parts. For example, the approximation should yield a set of space points and a set of particles. We also saw that whenever we look at any ‘part’ of nature without any approximation, we should not be able to distinguish it from the whole world. In other words, composed entities are not always larger than constituents. On the other hand, composed entities must usually appear to be larger than their constituents. For example, space ‘points’ or ‘point’ particles are tiny, even though they are only approximations. Which concept without boundaries can be at their origin? Using usual concepts the world is everywhere at the same time; if nature is to be described by a single constituent, this entity must be

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A theory of nothing describing everything is better than a theory of everything describing nothing.

Dvipsbugw


1118 xi general rel ativity versus quantum mechanics • the shape of points

Argument 5: The entropy of bl ack holes

* Is this the only method to describe nature? Is it possible to find another description, in particular if space and time are not used as background? The answers are unclear at present.

Copyright © Christoph Schiller November 1997–May 2006

Challenge 1508 ny

We are still collecting arguments to determine particle shape. For a completely different way to explore the shape of particles it is useful to study situations where they appear in large numbers. Collections of high numbers of constituents behave differently if they are point-like or extended. In particular, their entropy is different. Studying large-number entropy thus allows to determine component shape. The best approach is to study situations in which large numbers of particles are crammed in a small volume. This leads to study the entropy of black holes. A black hole is a body whose gravity is so strong that even light cannot escape. Black holes tell us a lot about the fundamental entities of nature. It is easily deduced from general relativity that any body whose mass m fits inside the so-called Schwarzschild radius r S = 2Gm~c 2 (802)

Dvipsbugw

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extended. The entity has to be a single one, but it must seem to be multiple, i.e. it has to be multiple approximately, as nature shows multiple aspects. The entity must be something folded. It must be possible to count the folds, but only approximately. (An analogy is the question of how many tracks are found on an LP or a CD; depending on the point of view, local or global, one gets different answers.) Counting folds would correspond to a length measurement. The simplest model would be the use of a single entity which is extended, fluctuating, going to infinity and allowing approximate localization, thus allowing approximate definition of parts and points.* In more vivid imagery, nature could be described by some deformable, folded and tangled up entity: a giant, knotted amoeba. An amoeba slides between the fingers whenever one tries to grab a part of it. A perfect amoeba flows around any knife trying to cut it. The only way to hold it would be to grab it in its entirety. However, for an actor himself made of amoeba strands this is impossible. He can only grab it approximately, by catching part of it and approximately blocking it, e.g. using a small hole so that the escape takes a long time. To sum up, nature is modelled by an entity which is a single unity (to eliminate distinguishability), extended (to eliminate localizability) and fluctuating (to ensure approximate continuity). A far-reaching, fluctuating fold, like an amoeba. The tangled branches of the amoeba allow a definition of length via counting of the folds. In this way, discreteness of space, time and particles could also be realized; the quantization of space-time, matter and radiation thus follows. Any flexible and deformable entity is also a perfect candidate for the realization of diffeomorphism invariance, as required by general relativity. A simple candidate for the extended fold is the image of a fluctuating, flexible tube. Counting tubes implies to determine distances or areas. The minimum possible count of one gives the minimum distance, and thus allows us to deduce quantum theory. In fact, we can use as model any object which has flexibility and a small dimension, such as a tube, a thin sheet, a ball chain or a woven collection of rings. These options give the different but probably equivalent models presently explored in simplicial quantum gravity, in Ashtekar’s variables and in superstrings.

Dvipsbugw


argument 5: the entropy of bl ack holes

Ref. 1139, Ref. 1140

is a black hole. A black hole can be formed when a whole star collapses under its own weight. A black hole is thus a macroscopic body with a large number of constituents. For black holes, like for every macroscopic body, an entropy can be defined. The entropy S of a macroscopic black hole was determined by Bekenstein and Hawking and is given by S=

Ref. 1141

Ref. 1143

Ref. 1145 Ref. 1146

k A 2 4 l Pl

or

S=k

4πGm 2 ħc

(803)

where k is the Boltzmann constant and A = 4πr S2 is the surface of the black hole horizon. This important result has been derived in many different ways. The various derivations also confirm that space-time and matter are equivalent, by showing that the entropy value can be seen both as an entropy of matter and as one of space-time. In the present context, the two main points of interest are that the entropy is finite, and that it is proportional to the area of the black hole horizon. In view of the existence of minimum lengths and times, the finiteness of entropy is not surprising any more. A finite black hole entropy confirms the idea that matter is made of a finite number of discrete entities per volume. The existence of an entropy also shows that these entities behave statistically; they fluctuate. In fact, quantum gravity leads to a finite entropy for any object, not only for black holes; Bekenstein has shown that the entropy of any object is always smaller than the entropy of a (certain type of) black hole of the same mass. The entropy of a black hole is also proportional to its horizon area. Why is this the case? This question is the topic of a stream of publications up to this day.* A simple way to understand the entropy–surface proportionality is to look for other systems in nature with the property that entropy is proportional to system surface instead of system volume. In general, the entropy of any collection of one-dimensional flexible objects, such as polymer chains, shows this property. Indeed, the expression for the entropy of a polymer chain made of N monomers, each of length a, whose ends are kept a distance r apart, is given by º 3r 2 S(r) = k for Na Q Na Q r . (804) 2Na 2

Ref. 1144

* The result can be derived from quantum statistics alone. However, this derivation does not yield the proportionality coefficient.

Copyright © Christoph Schiller November 1997–May 2006

The formula is derived in a few lines from the properties of a random walk on a lattice, using only two assumptions: the chains are extended, and they have a characteristic internal length a given by the smallest straight segment. Expression (804) is only valid if the polymers are effectively infinite, i.e. if the length Na of the chain and their effective º average size, the elongation a N , are much larger than the radius r of the region of interest; if the chain length is comparable or smaller than the region of interest, one gets the usual extensive entropy, fulfilling S  r 3 . Thus only flexible extended entities yield a S  r 2 dependence. However, there is º a difficulty. From º the entropy expression of a black hole we deduce that the elongation a N is given by a N  l Pl ; thus it is much smaller than the radius of a general, macroscopic black hole which can have diameters of several kilometres. On

Dvipsbugw

Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

Ref. 1142

1119

Dvipsbugw


1120 xi general rel ativity versus quantum mechanics • the shape of points

Ref. 1125

Argument 6: Exchanging space points or particles at Pl anck scales We are still collecting arguments for the extension of fundamental entities in nature. Let us focus on their exchange behaviour. We saw above that points in space have to be eliminated in favour of continuous, fluctuating entities common to space, time and matter. Is such a space ‘point’ or space entity a boson or a fermion? If we exchange two points of empty space in everyday life, nothing happens. Indeed, quantum field theory is based – among others – on the relation [x, y] = x y − yx = 0

(805)

Ref. 1122

Ref. 1147 Ref. 1147

This means that ‘points’ are neither bosons nor fermions.* They have more complex exchange properties. In fact, the term on the right hand side will be energy dependent, with an effect increasing towards Planck scales. In particular, we saw that gravity implies that double exchange does not lead back to the original situation at Planck scales. Entities following this or similar relations have been studied in mathematics for many decades: braids. In summary, at Planck scales space-time is not made of points, but of braids or some of their generalizations. Thus quantum theory and general relativity taken together again show that vacuum must be made of extended entities. * The same reasoning destroys the fermionic or Grassmann coordinates used in supersymmetry.

Copyright © Christoph Schiller November 1997–May 2006

between any two points with coordinates x and y, making them bosons. But at Planck scale, due to the existence of minimal distances and areas, this relation is at least changed to 2 [x, y] = l Pl + ... . (806)

Dvipsbugw

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the other hand, the formula for long entities is only valid when the chains are longer than the distance r between the end points. This difficulty disappears once we remember that space near a black hole is strongly curved. All lengths have to be measured in the same coordinate system. It is well known that for an outside observer, any object of finite size falling into a black hole seems to cover the complete horizon for long times (whereas it falls into the hole in its original size for an observer attached to the object). In short, an extended entity can have a proper length of Planck size but still, when seen by an outside observer, be as long as the horizon of the black hole in question. We thus find that black holes are made of extended entities. Another viewpoint can confirm the result. Entropy is (proportional to) the number of yes/no questions needed to know the exact state of the system. This view of black holes has been introduced by Gerard ’t Hooft. But if a system is defined by its surface, like a black hole is, its components must be extended. Finally, imagining black holes as made of extended entities is also consistent with the so-called no-hair theorem: black holes’ properties do not depend on what material falls into them, as all matter and radiation particles are made of the same extended components. The final state only depends on the number of entities and on nothing else. In short, the entropy of a black hole is consistent with the idea that it is made of a big tangle of extended entities, fluctuating in shape.

Dvipsbugw


argument 7: the meaning of spin

1121

Ref. 1122

Ref. 1147

At Planck energies this cannot be correct. At those energies, quantum gravity effects appear and modify the right hand side; they add an energy dependent term that is negligible at experimentally accessible energies, but which becomes important at Planck energy. We know from our experience with Planck scales that exchanging particles twice cannot lead back to the original situation, in contrast to everyday life. It is impossible that a double exchange at Planck energy has no effect, because at planck energy such statements are impossible. The simplest extension of the commutation relation (807) satisfying the requirement that the right side does not vanish is again braid symmetry. Thus Planck scales suggest that particles are also made of extended entities.

Argument 7: The meaning of spin

* With a flat (or other) background, it is possible to define a local energy–momentum tensor. Thus particles can be defined. Without background, this is not possible, and only global quantities can be defined. Without background, even particles cannot be defined! Therefore, we assume that we have a slowly varying spacetime background in this section.

Copyright © Christoph Schiller November 1997–May 2006

In the last argument, we will show that even at everyday energy, the extension of particles makes sense. Any particle is a part of the universe. A part is something which is different from anything else. Being ‘different’ means that exchange has some effect. Distinction means possibility of exchange. In other words, any part of the universe is described by its exchange behaviour. Everyday life tells us that exchange can be seen as composed of rotation. In short, distinguishing parts are described by their rotation behaviour. For this reason, for microscopic particles, exchange behaviour is specified by spin. Spin distinguishes particles from vacuum.* We note that volume does not distinguish vacuum from particles; neither does rest mass or charge: there are particles without rest mass or without charge, such as photons. The only candidate observables to distinguish particles from vacuum are spin and momentum. In fact, linear momentum is only a limiting case of angular momentum. We again find that rotation behaviour is the basic aspect distinguishing particles from vacuum. If spin is the central property distinguishing particles from vacuum, finding a model for spin is of central importance. But we do not have to search for long. An well-known model for spin 1/2 is part of physics folklore. Any belt provides an example, as we dis-

Dvipsbugw

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Particles behave in a similar way. We know that at low, everyday energies, particles of the same type are identical. Experiments sensitive to quantum effects show that there is no way to distinguish them: any system of several identical particles obeys permutation symmetry. On the other hand we know that at Planck energy all low-energy symmetries disappear. We also know that, at Planck energy, permutation cannot be carried out, as it implies exchanging positions of two particles. At Planck energy, nothing can be distinguished from vacuum; thus no two entities can be shown to have identical properties. Indeed, no two particles can be shown to be indistinguishable, as they cannot even be shown to be separate. What happens when we slowly approach Planck energy? At everyday energies, permutation symmetry is defined by commutation or anticommutation relations of any two particle creation operators a† b†  b† a† = 0 . (807)

Dvipsbugw


1122 xi general rel ativity versus quantum mechanics • the shape of points

flexible bands in unspecified number reaching the border of space

particle position

Dvipsbugw J=1/2 F I G U R E 388 A possible model for a spin 1/2

particle

Ref. 1148

cussed in detail in chapter VI on permutation symmetry. Any localized structure with any number of long tails attached to it – and reaching the border of the region of space under consideration – has the same properties as a spin 1/2 particle. It is a famous exercise to show that such a model, shown in Figure 388, is indeed invariant under 4π rotation but not under 2π rotations, that two such particles get entangled when exchanged, but get untangled when exchanged twice. The model has all properties of spin 1/2 particles, independently of the precise structure of the central region, which remains unknown at this point. The tail model even has the same problems with highly curved space as real spin 1~2 particles have. We explore the idea in detail shortly. The tail model thus confirms that rotation is partial exchange. More interestingly, it shows that rotation implies connection with the border of space-time. Extended particles can be rotating. Particles can have spin 1~2 provided that they have tails going to the border of space-time. If the tails do not reach the border, the model does not work. Spin 1~2 thus even seems to require extension. We again reach the conclusion that extended entities are a good description for particles.

Present research

The Greek deduced the existence of atoms because fish can swim through water. They argued that only if water is made of atoms, can a fish find its way through it by pushing the atoms aside. We can ask a similar question when a particle flies through vacuum: why are particles able to move through vacuum at all? Vacuum cannot be a fluid or a solid of small entities, as this would not fix its dimensionality. Only one possibility remains: both vacuum and particles are made of a web of extended entities. Describing matter as composed of extended entities is an idea from the 1960s. Describing nature as composed of ‘infinitely’ extended entities is an idea from the 1980s. Indeed, in addition to the arguments presented so far, present research provides several other approaches that arrive at the same conclusion.

Copyright © Christoph Schiller November 1997–May 2006

“ Ref. 1132

To understand is to perceive patterns. Isaiah Berlin

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Page 771

Dvipsbugw


present research

1123

** Ref. 1149

Bosonization, the construction of fermions using an infinite number of bosons, is a central property of modern unification attempts. It also implies coupling duality, and thus the extension of fundamental constituents. **

Ref. 1150

String theory and in particular its generalization to membranes are explicitly based on extended entities, as the name already states. The fundamental entities are indeed assumed to reach the limits of space-time.

Dvipsbugw

** Ref. 1151

Research into quantum gravity, in particular the study of spin networks and spin foams, has shown that the vacuum must be thought as a collection of extended entities.

Ref. 1152

In the 1990s, Dirk Kreimer has shown that high-order QED diagrams are related to knot theory. He thus proved that extension appears through the back door even when electromagnetism is described using point particles. **

Ref. 1153

A recent discovery in particle physics, ‘holography’, connects the surface and volume of physical systems at high energy. Even if it were not part of string theory, it would still imply extended entities. **

Page 805

Other fundamental nonlocalities in the description of nature, such as wave function collapse, can be seen as the result of extended entities. ** The start of the twenty-first century has brought forwards a number of new approaches, such as string net condensation or knotted particle models. All these make use of extended entities. Conceptual checks of extension

— As Ed Witten likes to say, any unified model of nature must be supersymmetric and dual. The idea of extended entities would be dead as soon as it is shown not to be compatible with these requirements. — Any model of nature must be easily extendible to a model for black holes. If not, it cannot be correct. — Showing that the results on quantum gravity, such as the results on the area and volume quantization, are in contradiction with extended entities would directly invalidate the

Copyright © Christoph Schiller November 1997–May 2006

Challenge 1509 ny

Is nature really described by extended entities? The idea is taken for granted by all present approaches in theoretical physics. How can we be sure about this result? The arguments presented above provide several possible checks.

Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

**

Dvipsbugw


1124 xi general rel ativity versus quantum mechanics • the shape of points

model. — The same conclusion of extended entities must appear if one starts from any physical (low-energy) concept – not only from length measurements – and continues to study how it behaves at Planck scales. If the conclusion were not reached, the idea of extension would not be consistent and thus incorrect. — Showing that any conclusion of the idea of extension is in contrast with string theory or with M-theory would lead to strong doubts. — Showing that the measurement of length cannot be related to the counting of folds would invalidate the model. — Finding a single Gedanken experiment invalidating the extended entity idea would prove it wrong.

Dvipsbugw

Experimental falsification of models based on extended entities

Ref. 1154

Ref. 1155

— Observing a single particle in cosmic rays with energy above the Planck energy would bring this approach to a sudden stop. (The present record is a million times lower.) — Finding any property of nature not consistent with extended entities would spell the end of the idea. — Finding an elementary particle of spin 0 would invalidate the idea. In particular, finding the Higgs boson and showing that it is elementary, i.e. that its size is smaller than its own Compton wavelength, would invalidate the model. — Most proposed experimental checks of string theory can also yield information on the ideas presented. For example, Paul Mende has proposed a number of checks on the motion of extended objects in space-time. He argues that an extended object moves differently from a mass point; the differences could be noticeable in scattering or dispersion of light near masses. — In July 2002, the Italian physicist Andrea Gregori has made a surprising prediction valid for any model using extended entities that reach the border of the universe: if particles are extended in this way, their mass should depend on the size of the universe. Particle masses should thus change with time, especially around the big bang. This completely new point is still a topic of research.

Possibilities for confirmation of extension models — The best confirmation would be to find a concrete model for the electron, muon, tau and for their neutrinos. In addition, a concrete model for photons and gravitons is needed. With these models, finding a knot-based definition for the electrical charge and the lepton number would be a big step ahead. All quantum numbers should be topological quantities deduced from these models and should behave as observed. — Estimating the coupling constants and comparing them with the experimental values is of course the main dream of modern theoretical physics.

Copyright © Christoph Schiller November 1997–May 2006

Incidentally, most of these scenarios would spell the death penalty for almost all present unification attempts.

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Physics is an experimental science. What kind of data could show that extension is incorrect?

Dvipsbugw


present research

1125 Nature's energy scale

Eeveryday

EPl

F I G U R E 389 Planck effects make the energy

axis an approximation

Dvipsbugw — Proving in full detail that extended entities imply exactly three plus one space-time dimensions is still necessary. — Estimating the total number of particles in the visible universe would provide the final check of any extended entity model. Generally speaking, the only possible confirmations are those from the one-page table of unexplained properties of nature given in Chapter X. No other confirmations are possible. The ones mentioned here are the main ones. Curiosities and fun challenges about extension

No problem is so formidable that you can’t walk away from it. Charles Schultz

Even though this section already provided sufficient food for thought, here is some more. ** Challenge 1510 ny

If measurements become impossible near Planck energy, we cannot even draw a diagram with an energy axis reaching that value. (See Figure 389) Is this conclusion valid in all cases? **

Challenge 1511 n

Quantum theory implies that even if tight walls would exist, the lid of such a box can never be tightly shut. Can you provide the argument? ** Is it correct that a detector able to detect Planck mass particles would be of infinite size? What about a detector to detect a particle moving with Planck energy? **

Challenge 1513 e

Can you provide an argument against the idea of extended entities?* **

Challenge 1514 ny

Does duality imply that the cosmic background fluctuations (at the origin of galaxies and clusters) are the same as vacuum fluctuations?

* If so, please email it to the author

Copyright © Christoph Schiller November 1997–May 2006

Challenge 1512 n

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Page 959

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1126 xi general rel ativity versus quantum mechanics • the shape of points

** Challenge 1515 ny

Does duality imply that a system with two small masses colliding is the same as one with two large masses gravitating? **

Challenge 1516 d

It seems that in all arguments so far we assumed and used continuous time, even though we know it is not. Does this change the conclusions so far?

Dvipsbugw

** Duality also implies that large and small masses are equivalent in some sense. A mass 2 2 m in a radius r is equivalent to a mass m Pl ~m in a radius l Pl ~r. In other words, duality 2 transforms mass density from ρ to ρ Pl ~ρ. Vacuum and maximum density are equivalent. Vacuum is thus dual to black holes. Duality implies that initial conditions for the big bang make no sense. Duality again shows the uselessness of the idea, as minimal distance did before. As duality implies a symmetry between large and small energies, the big bang itself becomes an unclearly defined concept. ** Challenge 1517 n

The total symmetry, as well as space-time duality, imply that there is a symmetry between all values an observable can take. Do you agree? **

Challenge 1518 n

Can supersymmetry be an aspect or special case of total symmetry or is it something else? **

Challenge 1519 n

Any description is a mapping from nature to mathematics, i.e. from observed differences (and relations) to thought differences (and relations). How can we do this accurately, if differences are only approximate? Is this the end of physics? ** What do extended entities imply for the big-bang? **

Challenge 1521 d

Can you show that going to high energies or selecting a Planck size region of space-time is equivalent to visiting the big-bang? **

Ref. 1155, Ref. 1156 Challenge 1522 ny

Additionally, one needs a description for the expansion of the universe in terms of extended entities. First approaches are being explored; no final conclusions can be drawn yet. Can you speculate about the solution?

Copyright © Christoph Schiller November 1997–May 2006

Challenge 1520 d

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**

Dvipsbugw


present research

1127

An intermediate status report

Many efforts for unification advance by digging deeper and deeper into details of quantum field theory and general relativity. Here we took the opposite approach: we took a step back and looked at the general picture. Guided by this idea we found several arguments, all leading to the same conclusion: space-time points and point particles are made of extended entities. Somehow it seems that the universe is best described by a fluctuating, multi-branched entity, a crossing between a giant amoeba and a heap of worms. Another analogy is a big pot of boiling and branched spaghetti. Such an extended model of quantum geometry is beautiful and simple, and these two criteria are often taken as indication, before any experimental tests, of the correctness of a description. We toured topics such as the existence of Planck limits, 3-dimensionality, curvature, renormalization, spin, bosonization, the cosmological constant problem, as well as the search for a background free description of nature. We will study and test specific details of the model in the next section. All these tests concern one of only three possible topics: the construction of space-time, the construction of particles and the construction of the universe. These are the only issues remaining on our mountain ascent of Motion Mountain. We will discuss them in the next section. Before we do so, we enjoy two small thoughts. Sexual preferences in physics

Page 223

Ref. 1158

A physical aphorism Ref. 1159, Ref. 1160

To ‘show’ that we are not far from the top of Motion Mountain, we give a less serious argument as final curiosity. Salecker and Wigner and then Zimmerman formulated the fundamental» limit for the measurement precision τ attainable by a clock of mass M. It is given by τ = ħT~Mc 2 , where T is the time to be measured. We can then ask what time * ‘We must know, we will know.’ This was Hilbert’s famous personal credo.

Copyright © Christoph Schiller November 1997–May 2006

Fluctuating entities can be seen to answer an old and not so serious question. When nature was defined as made of tiny balls moving in vacuum, we described this as a typically male idea. Suggesting that it is male implies that the female part is missing. Which part would that be? From the present point of view, the female part of physics might be the quantum description of the vacuum. The unravelling of the structure of the vacuum, as extended container of localized balls, could be seen as the female half of physics. If women had developed physics, the order of the discoveries would surely have been different. Instead of studying matter, as men did, women might have studied the vacuum first. In any case, the female and the male approaches, taken together, lead us to the description of nature by extended entities. Extended entities, which show that particles are not balls and that the vacuum is not a container, transcend the sexist approaches and lead to the unified description. In a sense, extended entities are thus the politically correct approach to nature.

Dvipsbugw

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Ref. 1157

Wir müssen wissen, wir werden wissen.* David Hilbert, 1930.

Dvipsbugw


1128

xi general rel ativity versus quantum mechanics • string theory

T can be measured with a precision of a Planck time t Pl , given a clock of the mass of the whole universe. We get a maximum time of T=

Challenge 1523 e

2 2 t Pl c M. ħ

(808)

Inserting numbers, we find rather precisely that the time T is the present age of the universe. With the right dose of humour we can see the result as an omen for the belief that time is now ripe, after so much waiting, to understand the universe down to the Planck scale. We are getting nearer to the top of Motion Mountain. Be prepared for even more fun.

38.

string theory – a web of dualities

By incorporating these four ideas, string theory acquires a number of appealing properties. First of all, the theory is unique. It has no adjustable parameters. Furthermore, as expected from any theory with extended entities,string theory contains gravity. In addition, string theory describes interactions. It does describe gauge fields such as the electromagnetic field. The theory is thus an extension of quantum field theory. All essential points of quantum field theory are retained.

Copyright © Christoph Schiller November 1997–May 2006

— String theory describes particles as extended entities. In a further generalization, Mtheory describes particles as higher-dimensional membranes. — String theory uses supersymmetry. Supersymmetry is the symmetry that relates matter to radiation, or equivalently, fermions to bosons. Supersymmetry is the most general local interaction symmetry that is known. — String theory uses higher dimensions to introduce extended entities and to unify interactions. A number of dimensions higher than 3 + 1 seems to be necessary for a consistent quantum field theory. At present however, the topology and size of the dimensions above 4 is still unclear. (In fact, to be honest, the topology of the first 4 dimensions is also unknown.) — String theory makes heavy use of dualities. Dualities, in the context of high energy physics, are symmetries that map large to small values of physical observables. Important examples are space-time duality and coupling constant duality. Dualities are global interaction and space-time symmetries. They are essential to include gravitation in the description of nature. With dualities, string theory integrates the equivalence between space-time and matter-radiation.

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String theory, superstring theory or membrane theory – all are names for related approaches – can be characterized as the most investigated theory ever. String theory explores the idea of particles as extended entities. String theory contains a maximum speed, a minimum action and a maximum force. It thus incorporates special relativity, quantum theory and general relativity. In its attempt to achieve a unified description, string theory uses four ideas that go beyond usual general relativity and quantum theory: extension, supersymmetry, higher dimensions and dualities.

Ref. 1161

Dvipsbugw

Dvipsbugw


strings and membranes – why string theory is so difficult

1129

String theory has many large symmetries, a consequence of its many dualities. These symmetries connect many wildly different situations; that makes the theory fascinating but also difficult to picture. Finally, string theory shows special cancellations of anomalies and inconsistencies. Historically, the first example was the Green-Schwarz anomaly cancellation; however, many other mathematical problems of quantum field theory are solved. In short, string theory research has shown that general relativity and quantum field theory can be unified using extended entities.

Dvipsbugw

Strings and membranes – why string theory is so difficult

– CS – more to be added in the future – CS –

Matrix models and M-theory

– CS – the rest of the chapter will be made available in the future – CS – In other words, M-theory complies with all conceptual requirements that a unified theory must fulfil. Let us now turn to experiment. * The history of string theory was characterized by short periods of excitement followed by long periods of disappointment. The main reason for this ineffective evolution was that most researchers only studied topics that everybody else was also studying. Due to the fear of unemployment of young researchers and out of the fear of missing out something, people were afraid to research topics that nobody else was looking at. Thus string theory had an extremely difficult birth.

Copyright © Christoph Schiller November 1997–May 2006

One of the most beautiful results of string theory research is the fact that string theory can be deduced from an extremely small number of fundamental ideas. The approach is to take a few basic ideas and to deduce M-theory form them. M-theory is the most recent formulation of string theory. M-theory assumes that the basic entities of nature are membranes of different dimensions.M-theory incorporates dualities and holography. The existence of interactions can been deduced. Supersymmetry and, in particular, super Yang–Mills theory can be deduced. Also the entropy of black holes follows from M-theory. Finally, supergravity has been deduced from M-theory. This means that the existence of space-time follows from M-theory. M-theory is thus a promising candidate for a theory of motion.

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What is string theory? There are two answers. The first answer is to follow its historical development. Unfortunately, that is almost the hardest of all possible ways to learn and understand string theory.* Thus we only give a short overview of this approach here. The full description of string theory along historical lines starts from classical strings, then proceeds to string field theory, incorporating dualities, higher dimensions and supersymmetry on the way. Each step can be described by a Lagrangian. Let us have a look at them. The first idea of string theory, as the name says, is the use of strings to describe particles. Strings replace the idea of point particles. The simplest form of string theory can be given if the existence of a background space-time is already taken for granted.

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1130

xi general rel ativity versus quantum mechanics • string theory

Masses and couplings Ref. 1166

One of the main results of QCD, the theory of strong interactions, is the explanation of mass relations such as m proton  e−k~α unif m Pl

k = 11~2π , α unif = 1~25 .

(809)

Here, the value of the coupling constant α unif is taken at the unifying energy. This energy is known to be a factor of about 800 below the Planck energy. In other words, a general understanding of masses of bound states of the strong interaction, such as the proton, requires almost only a knowledge of the unification energy and the coupling constant at that energy. The approximate value α unif = 1~25 is an extrapolation from the low energy value, using experimental data. Any unified theory must allow one to calculate the coupling constants as function of energy, including the value α unif at the unification energy itself. At present, researchers state that the main difficulty of string theory and of M-theory is the search for the vacuum state. Without the vacuum state, no calculations of the masses and coupling constants are possible. The vacuum state of string theory or M-theory is expected by researchers to be one of an extremely involved set of topologically distinct manifolds. At present, is it estimated that there are around 10750250 candidate vacuum states. One notes that the error alone is as large as the number of Planck 4-volumes in the history of the universe. This poorness of the estimate reflects the poor understanding of the theory. On can describe the situation also in the following way. String and M-theory predict states with Planck mass and with zero mass. The zero mass particles are then thought to get a tiny mass (compared to the Planck mass) due to the Higgs mechanism. However, the Higgs mechanism, with all its coupling constants, has not yet been deduced from string theory. In other words, so far, string theory predicts no masses for elementary particles. This disappointing situation is the reason that several scientists are dismissing string theory altogether. Future will show. Outlook

Copyright © Christoph Schiller November 1997–May 2006

It is estimated that 10 000 man-years have been invested in the search for string theory; compare this with about 5 for Maxwell’s electrodynamics, 10 for general relativity and a similar number for the foundation of quantum theory. Historically, the community of string theorists took over 10 years to understand that strings were not the basic entities of string theory. The fundamental entities are membranes. Then string theorists took another 10 years and more to understand that membranes are not the most practical fundamental entities for calculation. However, the search for the most practical entities is still ongoing. It is probable that knotted membranes must be taken into consideration. Why is M-theory so hard? Doing calculations on knotted membranes is difficult. In fact, it is extremely difficult. Mathematicians and physicists all over the world are still struggling to find simple ways for such calculations. In short, a new approach for calculations with extended entities is needed. We then can ask the following question. M-theory is based on several basic ideas: extension, higher di-

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Ref. 1167

and

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1131

mensions, supersymmetry and duality. Which of these basic assumptions is confusing? Dualities seem to be related to extension, for which experimental and theoretical evidence exists. However, supersymmetry and higher dimensions are completely speculative. We thus leave them aside, and continue our adventure.

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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006

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1132

Biblio graphy 1122 C. Schiller, Does matter differ from vacuum? http://www.arxiv.org/abs/gr-qc/9610066.

1123

1124

1126

1127 1128 1129 1130

1132 1133 1134 1135

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A French language version has been published as C. Schiller, Le vide diffère-t-il de la matière? in E. Gunzig & S. Diner, editeurs, Le vide – Univers du tout et du rien – Des physiciens et des philosophes s’interrogent, Les Éditions de l’Université de Bruxelles, 1998. Cited on pages 1102, 1103, 1106, 1108, 1109, 1110, 1120, and 1121. Most arguments for minimal temporal and spatial intervals are due to M. Schön, Operative time definition and principal uncertainty, preprint http://www.arxiv.org/abs/gr-qc/ 9304024, and to T. Padmanabhan, Limitations on the operational definition of spacetime events and quantum gravity, Classical and Quantum Gravity 4, pp. L107–L113, 1987; see also his earlier papers referenced there. Cited on page 1102. A similar point of view, often called monism, was proposed by Baruch Spinoza, Ethics Demonstrated in Geometrical Order, 1677, originally in Latin; an affordable French edition is B. Spinoza, L’Ethique, Folio-Gallimard, 1954. For a discussion of his ideas, especially his monism, see Don Garret, ed., The Cambridge Companion to Spinoza, Cambridge University Press, 1996, or any general text on the history of philosophy. Cited on page 1104. See L. Susskind & J. Uglum, Black holes, interactions, and strings, http://www.arxiv. org/abs/hep-th/9410074, or L. Susskind, String theory and the principle of black hole complementarity, Physical Review Letters 71, pp. 2367–2368, 1993, and M. Karliner, I. Klebanov & L. Susskind, Size and shape of strings, International Journal of Modern Physics A 3, pp. 1981–1996, 1988, as well as L. Susskind, Structure of hadrons implied by duality, Physical Review D 1, pp. 1182–1186, 1970. Cited on pages 1106 and 1120. M. Pl anck, Über irreversible Strahlungsvorgänge, Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften zu Berlin, pp. 440–480, 1899. Today it is commonplace to use ħ = h~2π instead of Planck’s h, which he originally called b in his paper. Cited on page 1106. C.A. Mead, Possible connection between gravitation and fundamental length, Physical Review B 135, pp. 849–862, 1964. Cited on page 1106. A.D. Sakharov, Vacuum quantum fluctuations in curved space and the theory of gravitation, Soviet Physics - Doklady 12, pp. 1040–1041, 1968. Cited on page 1109. P. Facchi & S. Pascazio, Quantum Zeno and inverse quantum Zeno effects, pp. 147– 217, in E. Wolf, editor, Progress in Optics, 42, 2001. Cited on page 1109. Aristotle, Of Generation and Corruption, I, II, 316 a 13. See Jean-Paul Dumont, Les écoles présocratiques, Folio Essais, Gallimard, p. 427, 1991. Cited on page 1110. See for example the speculative model of vacuum as composed of Planck-size spheres proposed by F. Winterberg, Zeitschrift für Naturforschung 52a, p. 183, 1997. Cited on page 1110. The Greek salt and fish arguments are given by Lucrece, or Titus Lucretius Carus, De natura rerum, around 50 bce. Cited on pages 1112 and 1122. Cited in Jean-Paul Dumont, Les écoles présocratiques, Folio Essais, Gallimard, p. 379, 1991. Cited on page 1113. D. Olive & C. Montonen, Magnetic monopoles as gauge particles, Physics Letters 72B, pp. 117–120, 1977. Cited on page 1114. J.H. Schwarz, The second superstring revolution, http://www.arxiv.org/abs/hep-th/ 9607067. More on dualities in ... Cited on pages 1112 and 1114.

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bibliography

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1136 The famous quote is found at the beginning of chapter XI, ‘The Physical Universe’, in Ar-

thur Eddington, The Philosophy of Physical Science, Cambridge, 1939. Cited on page 1115. 1137 Pl ato, Parmenides, c. 370 bce. It has been translated into most languages. Reading it aloud, like a song, is a beautiful experience. A pale reflection of these ideas is Bohm’s concept of ‘unbroken wholeness’. Cited on page 1116. 1138 P. Gibbs, Event-symmetric physics, http://www.arxiv.org/abs/hep-th/9505089; see also 1139 1140

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1143 1144 1145 1146 1147

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his http://www.weburbia.com/pg/contents.htm website. Cited on page 1116. J.D. Bekenstein, Black holes and entropy, Physical Review D 7, pp. 2333–2346, 1973. Cited on page 1119. S.W. Hawking, Particle creation by black holes, Communications in Mathematical Physics 43, pp. 199–220, 1975; see also S.W. Hawking, Black hole thermodynamics, Physical Review D13, pp. 191–197, 1976. Cited on page 1119. J.B. Hartle & S.W. Hawking, Path integral derivation of black hole radiance, Physical Review D 13, pp. 2188–2203, 1976. See also A. Strominger & C. Vafa, Microscopic origin of Bekenstein–Hawking entropy, Physics Letters B379, pp. 99–104, 1996, or the downloadable version found at http://www.arxiv.org/abs/hep-th/9601029. For another derivation of black hole entropy, see Gary T. Horowitz & J. Polchinski, A correspondence principle for black holes and strings, Physical Review D 55, pp. 6189–6197, 1997, or http://www.arxiv.org/abs/hep-th/9612146. Cited on page 1119. J.D. Bekenstein, Entropy bounds and black hole remnants, Physical Review D 49, pp. 1912–1921, 1994. Cited on page 1119. J. Madd ox, When entropy does not seem extensive, Nature 365, p. 103, 1993. Other references on the proportionality of entropy and surface are ... Cited on page 1119. L. B ombelli, R.K. Koul, J. Lee & R.D. Sorkin, Quantum source of entropy of black holes, Physical Review D 34, pp. 373–383, 1986. Cited on page 1119. The analogy between polymers and black holes is by G. Weber, Thermodynamics at boundaries, Nature 365, p. 792, 1993. Cited on page 1119. P.-G. de Gennes, Scaling Concepts in Polymer Physics, Cornell University Press, 1979. Cited on page 1119. See for example S. Majid, Introduction to braided geometry and q-Minkowski space, preprint found at http://www.arxiv.org/abs/hep-th/9410241 or S. Majid, Duality principle and braided geometry, preprint http://www.arxiv.org/abs/gr-qc/940957. Cited on pages 1120 and 1121. The relation between spin and statistics has been studied recently by M.V. Berry & J.M. Robbins, Quantum indistinguishability: spin–statistics without relativity or field theory?, in R.C. Hilborn & G.M. Tino, editors, Spin–Statistics Connection and Commutation Relations, American Institute of Physics CP545 2000. Cited on page 1122. An example is given by A.A. Sl avnov, Fermi–Bose duality via extra dimension, preprint http://www.arxiv.org/abs/hep-th/9512101. See also the standard work by Michael Stone, editor, Bosonization, World Scientific, 1994. Cited on page 1123. A popular account making the point is the bestseller by the US-american string theorist Brian Greene, The Elegant Universe – Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory, Jonathan Cape 1999. Cited on page 1123. The weave model of space-time appears in certain approaches of quantum gravity, as told by A. Ashtekar, Quantum geometry and gravity: recent advances, December 2001,

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1157 1158

1159 1160 1161

1163 1164

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preprint found at http://www.arxiv.org/abs/gr-qc/0112038. See also A. Ashtekar, Quantum geometry in action: big bang and black holes, http://www.arxiv.org/abs/ math-ph/0202008. On a slightly different topic, see also S.A. Major, A spin network primer, http://www.arxiv.org/abs/gr-qc/9905020. Cited on page 1123. A good introduction into his work is the paper D. Kreimer, New mathematical structures in renormalisable quantum field theories, http://www.arxiv.org/abs/hep-th/0211136 to appear in Annals of Physics Cited on page 1123. Introductions to holography are E. Alvarez, J. Conde & L. Hernandez, Rudiments of Holography, http://www.arxiv.org/abs/hep-th/0205075 or R. B ousso, The holographic principle, Review of Modern Physics 74, pp. 825–874, 2002, also at http://www.arxiv. org/abs/hep-th/0203101. (However, this otherwise good review has some argumentation errors, as explained on page 871.) The importance of holography in theoretical high energy physics was underlined by the discovery of J. Maldacena, The large N limit of superconformal field theories and supergravity, http://www.arxiv.org/abs/hep-th/9711200. Cited on page 1123. P.F. Mende, String theory at short distance and the principle of equivalence, preprint available at http://www.arxiv.org/abs/hep-th/9210001. Cited on page 1124. A. Gregori, Entropy, string theory and our world, http://www.arxiv.org/abs/hep-th/ 0207195 preprint. Cited on pages 1124 and 1126. String cosmology is a passtime for many. Examples are N.E. Mavromatos, String cosmology, http://www.arxiv.org/abs/hep-th/0111275 and N.G. Sanchez, New developments in string gravity and string cosmology - a summary report, http://www.arxiv.org/ abs/hep-th/0209016. Cited on page 1126. Searches for background free approaches are described by E. Witten, .... Cited on page 1127. Not even the most combative feminists have ever said how women-centered physics should look like. This lack of alternatives pervades also the agressive text by Margaret Wertheim, Pythagoras’ Trousers – God, Physics and the Gender Wars, Fourth Estate, 1997, and thus makes its author look rather silly. Cited on page 1127. H. Salecker & E.P. Wigner, Quantum limitations of the measurement of space-time distances, Physical Review 109, pp. 571–577, 1958. Cited on page 1127. E.J. Zimmerman, The macroscopic nature of space-time, American Journal of Physics 30, pp. 97–105, 1962. Cited on page 1127. See A. Sen, An introduction to duality symmetries in string theory, in Les Houches Summer School: Unity from Duality: Gravity, Gauge Theory and Strings, (Les Houches, France, 2001), Springer Verlag, 76, pp. 241–322, 2002. Cited on page 1128. See for example C. Rovelli, Notes for a brief history of quantum gravity, electronic preprint available at http://www.arxiv.org/abs/gr-qc/0006061. No citations. See the http://www.nuclecu.unam.mx/~alberto/physics/stringrev.html website for an upto date list of string theory and M theory reviews. No citations. W.G. Unruh, Notes on black hole evaporation, Physical Review D 14, pp. 870–875, 1976. W.G. Unruh & R.M. Wald, What happens when an accelerating observer detects a Rindler particle, Physical Review D 29, pp. 1047–1056, 1984. No citations. The most prominent proposer of the idea that particles might be knots was William Thomson. See W. Thomson, On vortex motion, Transactions of the Royal Society in Edinburgh pp. 217–260, 1868. This paper stimulated work on knot theory, as it proposed the idea that different particles might be differently knotted vortices in the aether. No citations.

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1166 F. Wilczek, Getting its from bits, Nature 397, pp. 303–306, 1999. Cited on page 1130. 1167 See the well-known blog by Peter Woit, Not even wrong, at http://www.math.columbia.

edu/~woit/blog. Cited on page 1130.

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C h a p t e r XII

UNI FIC ATION (NOT YET AVAI L ABLE)

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– CS – this chapter will appear in the near future – CS –

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C h a p t e r XIII

THE TOP OF THE MOUNTAI N (NOT YET AVAI L ABLE)

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– CS – this chapter will appear in the near future – CS –

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Fourth Part

Appendices

Where necessary reference information for mountain ascents is provided, preparing the reader for this and any other future adventure.


Appendix A

NOTATION AND C ONV ENTIONS

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N

The L atin alphabet What is written without effort is in general read without pleasure. Samuel Johnson

Copyright © Christoph Schiller November 1997–May 2006

Books are collections of symbols. Writing was probably invented between 3400 and 3300 bce by the Sumerians in Mesopotamia (though other possibilities are also discussed). It then took over a thousand years before people started using symbols to represent sounds instead of concepts: this is the way in which the first alphabet was created. This happened between 2000 and 1600 bce (possibly in Egypt) and led to the Semitic alphabet. The use of an alphabet had so many advantages that it was quickly adopted in all neighbouring cultures, though in different forms. As a result, the Semitic alphabet is the forefather of all alphabets used in the world. This text is written using the Latin alphabet. At first sight, this seems to imply that its pronunciation cannot be explained in print, in contrast to the pronunciation of other alphabets or of the International Phonetic Alphabet (IPA). (They can be explained using the alphabet of the main text.) However, it is in principle possible to write a text that describes exactly how to move lips, mouth and tongue for each letter, using physical concepts where necessary. The descriptions of pronunciations found in dictionaries make indirect use of this method: they refer to the memory of pronounced words or sounds found in nature. Historically, the Latin alphabet was derived from the Etruscan, which itself was a derivation of the Greek alphabet. There are two main forms.

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Ref. 1168

ewly introduced and defined concepts in this text are indicated by italic typeface. ew definitions are also referred to in the index by italic page numbers. We naturally use SI units throughout the text; these are defined in Appendix B. Experimental results are cited with limited precision, usually only two digits, as this is almost always sufficient for our purposes. High-precision reference values can be found in Appendices B and C. In relativity we use the time convention, where the metric has the signature (+ − − − ). This is used in about 70 % of the literature worldwide. We use indices i, j, k for 3-vectors and indices a, b, c, etc. for 4-vectors. Other conventions specific to general relativity are explained as they arise in the text.

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The ancient Latin alphabet, used from the sixth century bce onwards: A B C D E F Z

H I K L M N O P Q R S T V X

The classical Latin alphabet, used from the second century bce until the eleventh century: A B C D E F G H I K L M N O P Q R S T V X Y Z

The Latin alphabet was derived from the Etruscan; the Etruscan from the Greek. The Greek alphabet was itself derived from the Phoenician or a similar northern Semitic al-

Ref. 1170

* To meet Latin speakers and writers, go to http://www.alcuinus.net/. ** The Runic script, also called Futhark or futhorc, a type of alphabet used in the Middle Ages in Germanic, Anglo–Saxon and Nordic countries, probably also derives from the Etruscan alphabet. The name derives from the first six letters: f, u, th, a (or o), r, k (or c). The third letter is the letter thorn mentioned above; it is often written ‘Y’ in Old English, as in ‘Ye Olde Shoppe.’ From the runic alphabet Old English also took the letter wyn to represent the ‘w’ sound, and the already mentioned eth. (The other letters used in Old English – not from futhorc – were the yogh, an ancient variant of g, and the ligatures æ or Æ, called ash, and œ or Œ, called ethel.)

Copyright © Christoph Schiller November 1997–May 2006

The Greek alphabet

Outside a dog, a book is a man’s best friend. Inside a dog, it’s too dark to read. Groucho Marx

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Ref. 1169

The letter G was added in the third century bce by the first Roman to run a fee-paying school, Spurius Carvilius Ruga. He added a horizontal bar to the letter C and substituted the letter Z, which was not used in Latin any more, for this new letter. In the second century bce, after the conquest of Greece, the Romans included the letters Y and Z from the Greek alphabet at the end of their own (therefore effectively reintroducing the Z) in order to be able to write Greek words. This classical Latin alphabet was stable for the next thousand years.* The classical Latin alphabet was spread around Europe, Africa and Asia by the Romans during their conquests; due to its simplicity it began to be used for writing in numerous other languages. Most modern ‘Latin’ alphabets include a few other letters. The letter W was introduced in the eleventh century in French and was then adopted in most European languages. The letters J and U were introduced in the sixteenth century in Italy, to distinguish certain sounds which had previously been represented by I and V. In other languages, these letters are used for other sounds. The contractions æ and œ date from the Middle Ages. Other Latin alphabets include more letters, such as the German sharp s, written ß, a contraction of ‘ss’ or ‘sz’, the and the Nordic letters thorn, written Þ or þ, and eth, written Ð or ð, taken from the futhorc,** and other signs. Lower-case letters were not used in classical Latin; they date only from the Middle Ages, from the time of Charlemagne. Like most accents, such as ê, ç or ä, which were also first used in the Middle Ages, lower-case letters were introduced to save the then expensive paper surface by shortening written words.

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TA B L E 80 The ancient and classical Greek alphabets, and the correspondence with Latin and Indian

digits

Α Β Γ ∆ Ε F ϝ, Ϛ

Α Β Γ ∆ Ε

Ζ Η Θ Ι Κ Λ Μ

Ζ Η Θ Ι Κ Λ Μ

α β γ δ ε

ζ η θ ι κ λ µ

alpha beta gamma delta epsilon digamma, stigma2 zeta eta theta iota kappa lambda mu

Corresp.

Anc.

Cl ass. Name

a b g, n1 d e w

1 2 3 4 5 6

Ν ν Ξ ξ Ο ο Π π

z e th i, j k l m

7 8 9 10 20 30 40

Ν Ξ Ο Π P Ϟ, Ρ Σ Τ

Λ Ϡ v

Cl ass. Name

Ρ Σ Τ Υ Φ Χ Ψ Ω

ρ σ, ς τ υ φ χ ψ ω

nu xi omicron pi qoppa3 rho sigma4 tau upsilon phi chi psi omega sampi6

Corresp. n x o p q r, rh s t y, u5 ph, f ch ps o s

50 60 70 80 90 100 200 300 400 500 600 700 800 900

The regional archaic letters yot, sha and san are not included in the table. The letter san was the ancestor of sampi. 1. Only if before velars, i.e. before kappa, gamma, xi and chi. 2. ‘Digamma’ is the name used for the F-shaped form. It was mainly used as a letter (but also sometimes, in its lower-case form, as a number), whereas the shape and name ‘stigma’ is used only for the number. Both names were derived from the respective shapes; in fact, the stigma is a medieval, uncial version of the digamma. The name ‘stigma’ is derived from the fact that the letter looks like a sigma with a tau attached under it – though unfortunately not in all modern fonts. The original letter name, also giving its pronunciation, was ‘waw’. 3. The version of qoppa that looks like a reversed and rotated z is still in occasional use in modern Greek. Unicode calls this version ‘koppa’. 4. The second variant of sigma is used only at the end of words. 5. Uspilon corresponds to ‘u’ only as the second letter in diphthongs. 6. In older times, the letter sampi was positioned between pi and qoppa.

phabet in the tenth century bce. The Greek alphabet, for the first time, included letters also for vowels, which the Semitic alphabets lacked (and often still lack). In the Phoenician alphabet and in many of its derivatives, such as the Greek alphabet, each letter has a proper name. This is in contrast to the Etruscan and Latin alphabets. The first two Greek letter names are, of course, the origin of the term alphabet itself. In the tenth century bce, the Ionian or ancient (eastern) Greek alphabet consisted of the upper-case letters only. In the sixth century bce several letters were dropped, while a few new ones and the lower-case versions were added, giving the classical Greek alphabet. Still later, accents, subscripts and breathings were introduced. Table 80 also gives the values signified by the letters took when they were used as numbers. For this special use, the obsolete ancient letters were kept during the classical period; thus they also acquired

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Ref. 1171

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Anc.

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TA B L E 81 The beginning of the Hebrew abjad

Letter

Name

Correspondence

ℵ ° ‫ג‬ ² etc.

aleph beth gimel daleth

a b g d

1 2 3 4

Copyright © Christoph Schiller November 1997–May 2006

* The Greek alphabet is also the origin of the Gothic alphabet, which was defined in the fourth century by Wulfila for the Gothic language, using also a few signs from the Latin and futhorc scripts. The Gothic alphabet is not to be confused with the so-called Gothic letters, a style of the Latin alphabet used all over Europe from the eleventh century onwards. In Latin countries, Gothic letters were replaced in the sixteenth century by the Antiqua, the ancestor of the type in which this text is set. In other countries, Gothic letters remained in use for much longer. They were used in type and handwriting in Germany until 1941, when the National Socialist government suddenly abolished them, in order to comply with popular demand. They remain in sporadic use across Europe. In many physics and mathematics books, Gothic letters are used to denote vector quantities.

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lower-case forms. The Latin correspondence in the table is the standard classical one, used for writing Greek words. The question of the correct pronunciation of Greek has been hotly debated in specialist circles; the traditional Erasmian pronunciation does not correspond either to the results of linguistic research, or to modern Greek. In classical Greek, the sound that sheep make was βη–βη. (Erasmian pronunciation wrongly insists on a narrow η; modern Greek pronunciation is different for β, which is now pronounced ‘v’, and for η, which is now pronounced as ‘i’ – a long ‘i’.) Obviously, the pronunciation of Greek varied from region to region and over time. For Attic Greek, the main dialect spoken in the classical period, the question is now settled. Linguistic research has shown that chi, phi and theta were less aspirated than usually pronounced in English and sounded more like the initial sounds of ‘cat’, ‘perfect’ and ‘tin’; moreover, the zeta seems to have been pronounced more like ‘zd’ as in ‘buzzed’. As for the vowels, contrary to tradition, epsilon is closed and short whereas eta is open and long; omicron is closed and short whereas omega is wide and long, and upsilon is really a ‘u’ sound as in ‘boot’, not like a French ‘u’ or German ‘ü.’ The Greek vowels can have rough or smooth breathings, subscripts, and acute, grave, circumflex or diaeresis accents. Breathings – used also on ρ – determine whether the letter is aspirated. Accents, which were interpreted as stresses in the Erasmian pronunciation, actually represented pitches. Classical Greek could have up to three of these added signs per letter; modern Greek never has more than one. Another descendant of the Greek alphabet* is the Cyrillic alphabet, which is used with slight variations, in many Slavic languages, such as Russian and Bulgarian. However, there is no standard transcription from Cyrillic to Latin, so that often the same Russian name is spelled differently in different countries or even in the same country on different occasions.

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1146

The Hebrew alphabet and other scripts Ref. 1172 Page 649

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Novem figure indorum he sunt 9 8 7 6 5 4 3 2 1. Cum his itaque novem fig-

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* A well-designed website on the topic is http://www.omniglot.com. The main present and past writing systems are encoded in the Unicode standard, which at present contains 52 writing systems. See http:// www.unicode.org. ** The story of the development of the numbers is told most interestingly by G. Ifrah, Histoire universelle des chiffres, Seghers, 1981, which has been translated into several languages. He sums up the genealogy in ten beautiful tables, one for each digit, at the end of the book. However, the book contains many factual errors, as explained in the http://www.ams.org/notices/200201/rev-dauben.pdf and http://www.ams.org/notices/ 200202/rev-dauben.pdf review. It is not correct to call the digits 0 to 9 Arabic. Both the actual Arabic digits and the digits used in Latin texts such as this one derive from the Indian digits. Only the digits 0, 2, 3 and 7 resemble those used in Arabic writing, and then only if they are turned clockwise by 90°. *** Leonardo di Pisa, called Fibonacci (b. c. 1175 Pisa, d. 1250 Pisa), Italian mathematician, and the most important mathematician of his time.

Copyright © Christoph Schiller November 1997–May 2006

Ref. 1173

The phoenician alphabet is also the origin of the Hebrew consonant alphabet or abjad. Its first letters are given in Table 81. Only the letter aleph is commonly used in mathematics, though others have been proposed. Around one hundred writing systems are in use throughout the world. Experts classify them into five groups. Phonemic alphabets, such as Latin or Greek, have a sign for each consonant and vowel. Abjads or consonant alphabets, such as Hebrew or Arabic, have a sign for each consonant (sometimes including some vowels, such as aleph), and do not write (most) vowels; most abjads are written from right to left. Abugidas, also called syllabic alphabets or alphasyllabaries, such as Balinese, Burmese, Devanagari, Tagalog, Thai, Tibetan or Lao, write consonants and vowels; each consonant has an inherent vowel which can be changed into the others by diacritics. Syllabaries, such as Hiragana or Ethiopic, have a sign for each syllable of the language. Finally, complex scripts, such as Chinese, Mayan or the Egyptian hieroglyphs, use signs which have both sound and meaning. Writing systems can have text flowing from right to left, from bottom to top, and can count book pages in the opposite sense to this book. Even though there are about 7000 languages on Earth, there are only about one hundred writing systems used today. About fifty other writing systems have fallen out of use. * For physical and mathematical formulae, though, the sign system used in this text, based on Latin and Greek letters, written from left to right and from top to bottom, is a standard the world over. It is used independently of the writing system of the text containing it. Digits and numbers Both the digits and the method used in this text to write numbers originated in India. They were brought to the Mediterranean by Arabic mathematicians in the Middle Ages. The number system used in this text is thus much younger than the alphabet.** The Indian numbers were made popular in Europe by Leonardo of Pisa, called Fibonacci,*** in his book Liber Abaci or ‘Book of Calculation’, which he published in 1202. That book revolutionized mathematics. Anybody with paper and a pen (the pencil had not yet been invented) was now able to calculate and write down numbers as large as reason allowed, or even larger, and to perform calculations with them. Fibonacci’s book started:

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uris, et cum hoc signo 0, quod arabice zephirum appellatur, scribitur quilibet numerus, ut inferius demonstratur.*

The symbols used in the text To avoide the tediouse repetition of these woordes: is equalle to: I will sette as I doe often in woorke use, a paire of paralleles, or Gemowe lines of one lengthe, thus: = , bicause noe .2. thynges, can be moare equalle. Robert Recorde***

Ref. 1172

Ref. 1174

* ‘The nine figures of the Indians are: 9 8 7 6 5 4 3 2 1. With these nine figures, and with this sign 0 which in Arabic is called zephirum, any number can be written, as will be demonstrated below.’ ** Currently, the shortest time for finding the thirteenth (integer) root of a hundred-digit number, a result with 8 digits, is 11.8 seconds. For more about the stories and the methods of calculating prodigies, see the fascinating book by Steven B. Smith, The Great Mental Calculators – The Psychology, Methods and Lives of the Calculating Prodigies, Columbia University Press, 1983. The book also presents the techniques that they use, and that anybody else can use to emulate them. *** Robert Recorde (c. 1510–1558), English mathematician and physician; he died in prison, though not for his false pretence to be the inventor of the ‘equal’ sign, which he took over from his Italian colleagues, but for a smaller crime, namely debt. The quotation is from his The Whetstone of Witte, 1557. An image showing the quote in manuscript can be found at the http://members.aol.com/jeff94100/witte.jpg website.

Copyright © Christoph Schiller November 1997–May 2006

Ref. 1174

Besides text and numbers, physics books contain other symbols. Most symbols have been developed over hundreds of years, so that only the clearest and simplest are now in use. In this mountain ascent, the symbols used as abbreviations for physical quantities are all taken from the Latin or Greek alphabets and are always defined in the context where they are used. The symbols designating units, constants and particles are defined in Appendices B and C. There is an international standard for them (ISO 31), but it is virtually inaccessible; the symbols used in this text are those in common use. The mathematical symbols used in this text, in particular those for operations and relations, are given in the following list, together with their origins. The details of their history have been extensively studied in the literature.

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The Indian method of writing numbers consists of a large innovation, the positional system, and a small one, the digit zero. The positional system, as described by Fibonacci, was so much more efficient that it completely replaced the previous Roman number system, which writes 1996 as IVMM or MCMIVC or MCMXCVI, as well as the Greek number system, in which the Greek letters were used for numbers in the way shown in Table 80, thus writing 1996 as ͵αϠϞϚʹ. Compared to these systems, the Indian numbers are a much better technology. Indeed, the Indian system proved so practical that calculations done on paper completely eliminated the need for the abacus, which therefore fell into disuse. The abacus is still in use only in those countries which do not use a positional system to write numbers. (The Indian system also eliminated the need fir systems to represent numbers with fingers. Such systems, which could show numbers up to 10 000 and more, have left only one trace: the term ‘digit’ itself, which derives from the Latin word for finger.) Similarly, only the positional number system allows mental calculations and made – and still makes – calculating prodigies possible.**

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Symbol

Meaning

+, −

plus, minus

Origin

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J. Regiomontanus 1456; the plus sign is derived from Latin ‘et’ º read as ‘square root’ used by C. Rudolff in 1525; the sign stems from a deformation of the letter ‘r’, initial of the Latin radix = equal to Italian mathematicians, early sixteenth century, then brought to England by R. Recorde { }, [ ], ( ) grouping symbols use starts in the sixteenth century A, < larger than, smaller than T. Harriot 1631  multiplied with, times W. Oughtred 1631  divided by G. Leibniz 1684 ë multiplied with, times G. Leibniz 1698 an power R. Descartes 1637 x, y, z coordinates, unknowns R. Descartes 1637 ax +by + c = 0 constants and equations for unknowns R. Descartes 1637 d~dx, d2 x, derivative, differential, integral G. Leibniz 1675 ∫ y dx φx function of x J. Bernoulli 1718 f x, f (x) function of x L. Euler 1734 ∆x, P difference, sum L. Euler 1755 x is different from L. Euler eighteenth century ∂~∂x partial derivative, read like ‘d~dx’ it was derived from a cursive form of ‘d’ or of the letter ‘dey’ of the Cyrillic alphabet by A. Legendre in 1786 ∆ Laplace operator R. Murphy 1833 SxS absolute value K. Weierstrass 1841 ∇ read as ‘nabla’ (or ‘del’) introduced by William Hamilton in 1853 and P.G. Tait in 1867, named after the shape of an old Egyptian musical instrument [x] the measurement unit of a quantity x twentieth century ª infinity J. Wallis 1655 π 4 arctan 1 H. Jones 1706 ª 1 n L. Euler 1736 e Pn=0 n! = lim n ª (1 + 1~n) º L. Euler 1777 i + −1 8, 9 set union and intersection G. Peano 1888 > element of G. Peano 1888 g empty set André Weil as member of the N. Bourbaki group in the early twentieth century `ψS, Sψe bra and ket state vectors Paul Dirac 1930 a dyadic product or tensor product or unknown outer product

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Ref. 1175

Ref. 1176 Ref. 1177

Ref. 1179

Other signs used here have more complicated origins. The & sign is a contraction of Latin et meaning ‘and’, as is often more clearly visible in its variations, such as &, the common italic form. Each of the punctuation signs used in sentences with modern Latin alphabets, such as , . ; : ! ? ‘ ’ » « – ( ) ... has its own history. Many are from ancient Greece, but the question mark is from the court of Charlemagne, and exclamation marks appear first in the sixteenth century.* The @ or at-sign probably stems from a medieval abbreviation of Latin ad, meaning ‘at’, similarly to how the & sign evolved from Latin et. In recent years, the smiley :-) and its variations have become popular. The smiley is in fact a new version of the ‘point of irony’ which had been formerly proposed, without success, by A. de Brahm (1868–1942). The section sign § dates from the thirteenth century in northern Italy, as was shown by the German palaeographer Paul Lehmann. It was derived from ornamental versions of the capital letter C for capitulum, i.e. ‘little head’ or ‘chapter.’ The sign appeared first in legal texts, where it is still used today, and then spread into other domains. The paragraph sign ¶ was derived from a simpler ancient form looking like the Greek letter Γ, a sign which was used in manuscripts from ancient Greece until well into the Middle Ages to mark the start of a new text paragraph. In the Middle Ages it took the modern form, probably because a letter c for caput was added in front of it. One of the most important signs of all, the white space separating words, was due to Celtic and Germanic influences when these people started using the Latin alphabet. It became commonplace between the ninth and the thirteenth century, depending on the language in question. Calendars

* On the parenthesis see the beautiful book by J. L ennard, But I Digress, Oxford University Press, 1991.

Copyright © Christoph Schiller November 1997–May 2006

The many ways to keep track of time differ greatly from civilization to civilization. The most common calendar, and the one used in this text, is also one of the most absurd, as it is a compromise between various political forces who tried to shape it. In ancient times, independent localized entities, such as tribes or cities, preferred lunar calendars, because lunar timekeeping is easily organized locally. This led to the use of the month as a calendar unit. Centralized states imposed solar calendars, based on the year. Solar calendars require astronomers, and thus a central authority to finance them. For various reasons, farmers, politicians, tax collectors, astronomers, and some, but not all, religious groups wanted the calendar to follow the solar year as precisely as possible. The compromises necessary between days and years are the origin of leap days. The compromises necessary between months and year led to the varying lengths are different in different calendars. The most commonly used year–month structure was organized over 2000 years ago by Gaius Julius Ceasar, and is thus called the Julian calendar. The system was destroyed only a few years later: August was lengthened to 31 days when it was named after Augustus. Originally, the month was only 30 days long; but in order to show that Augustus was as important as Caesar, after whom July is named, all

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Ref. 1178

1149

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Copyright © Christoph Schiller November 1997–May 2006

* Remembering the intermediate result for the current year can simplify things even more, especially since the dates 4.4, 6.6, 8.8, 10.10, 12.12, 9.5, 5.9, 7.11, 11.7 and the last day of February all fall on the same day of the week, namely on the year’s intermediate result plus 4. ** The present counting of years was defined in the Middle Ages by setting the date for the foundation of Rome to the year 753 bce, or 753 before the Common Era, and then counting backwards, so that the bce years behave almost like negative numbers. However, the year 1 follows directly after the year 1 bce: there was no year 0. Some other standards set by the Roman Empire explain several abbreviations used in the text: - c. is a Latin abbreviation for circa and means ‘roughly’; - i.e. is a Latin abbreviation for id est and means ‘that is’; - e.g. is a Latin abbreviation for exempli gratia and means ‘for the sake of example’; - ibid. is a Latin abbreviation for ibidem and means ‘at that same place’; - inf. is a Latin abbreviation for infra and means ‘(see) below’; - op. cit. is a Latin abbreviation for opus citatum and means ‘the cited work’; - et al. is a Latin abbreviation for et alii and means ‘and others’. By the way, idem means ‘the same’ and passim means ‘here and there’ or ‘throughout’. Many terms used in physics, like frequency, acceleration, velocity, mass, force, momentum, inertia, gravitation and temperature, are derived from Latin. In fact, it is arguable that the language of science has been Latin for over two thousand years. In Roman times it was Latin vocabulary with Latin grammar, in modern times it switched to Latin vocabulary with French grammar, then for a short time to Latin vocabulary with German grammar, after

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month lengths in the second half of the year were changed, and February was shortened by an additional day. The week is an invention of Babylonia, from where it was spread through the world by various religious groups. (The way astrology and astronomy cooperated to determine the order of the weekdays is explained in the section on gravitation.) Although it is about three thousand years old, the week was fully included into the Julian calendar only around the year 300, towards the end of the Western Roman Empire. The final change in the Julian calendar took place between 1582 and 1917 (depending on the country), when more precise measurements of the solar year were used to set a new method to determine leap days, a method still in use today. Together with a reset of the date and the fixation of the week rhythm, this standard is called the Gregorian calendar or simply the modern calendar. It is used by a majority of the world’s population. Despite its complexity, the modern calendar does allow you to determine the day of the week of a given date in your head. Just execute the following six steps: 1. take the last two digits of the year, and divide by 4, discarding any fraction; 2. add the last two digits of the year; 3. subtract 1 for January or February of a leap year; 4. add 6 for 2000s or 1600s, 4 for 1700s or 2100s, 2 for 1800s and 2200s, and 0 for 1900s or 1500s; 5. add the day of the month; 6. add the month key value, namely 144 025 036 146 for JFM AMJ JAS OND. The remainder after division by 7 gives the day of the week, with the correspondence 1-23-4-5-6-0 meaning Sunday-Monday-Tuesday-Wednesday-Thursday-Friday-Saturday.* When to start counting the years is a matter of choice. The oldest method not attached to political power structures was that used in ancient Greece, when years were counted from the first Olympic games. People used to say, for example, that they were born in the first year of the twenty-third Olympiad. Later, political powers always imposed the counting of years from some important event onwards.** Maybe reintroducing the Olympic

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counting is worth considering? Abbreviations and eponyms or concepts? Sentences like the following are the scourge of modern physics: The EPR paradox in the Bohm formulation can perhaps be resolved using the GRW approach, using the WKB approximation of the Schrödinger equation.

Copyright © Christoph Schiller November 1997–May 2006

Ref. 1180

which it changed to Latin vocabulary with British/American grammar. Many units of measurement also date from Roman times, as explained in the next appendix. Even the infatuation with Greek technical terms, as shown in coinages such as ‘gyroscope’, ‘entropy’ or ‘proton’, dates from Roman times.

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Ref. 1181

Using such vocabulary is the best way to make language unintelligible to outsiders. First of all, it uses abbreviations, which is a shame. On top of this, the sentence uses people’s names to characterize concepts, i.e. it uses eponyms. Originally, eponyms were intended as tributes to outstanding achievements. Today, when formulating radical new laws or variables has become nearly impossible, the spread of eponyms intelligible to a steadily decreasing number of people simply reflects an increasingly ineffective drive to fame. Eponyms are a proof of scientist’s lack of imagination. We avoid them as much as possible in our walk and give common names to mathematical equations or entities wherever possible. People’s names are then used as appositions to these names. For example, ‘Newton’s equation of motion’ is never called ‘Newton’s equation’; ‘Einstein’s field equations’ is used instead of ‘Einstein’s equations’; and ‘Heisenberg’s equation of motion’ is used instead of ‘Heisenberg’s equation’. However, some exceptions are inevitable: certain terms used in modern physics have no real alternatives. The Boltzmann constant, the Planck scale, the Compton wavelength, the Casimir effect, Lie groups and the Virasoro algebra are examples. In compensation, the text makes sure that you can look up the definitions of these concepts using the index. In addition, it tries to provide pleasurable reading.

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Biblio graphy 1168 For a clear overview of the various sign conventions in general relativity, see the front cover

of Charles W. Misner, Kip S. Thorne & John A. Wheeler, Gravitation, Freeman, 1973. We use the gravitational sign conventions of Hans C. Ohanian & R emo Ruffini, Gravitazione e spazio-tempo, Zanichelli, 1997. Cited on page 1142. 1169 For more information about the letters thorn and eth, have a look at the extensive report

to be found on the website http://www.everytype.com/standards/wynnyogh/thorn.html. Cited on page 1143.

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1170 For a modern history of the English language, see David Crystal, The Stories of English,

Allen Lane, 2004. Cited on page 1143. 1171 Hans Jensen, Die Schrift, Berlin, 1969, translated into English as Sign, Symbol and Script:

an Account of Man’s Efforts to Write, Putnam’s Sons, 1970. Cited on page 1144. 1172 David R. Lide, editor, CRC Handbook of Chemistry and Physics, 78th edition, CRC Press,

1173 See the mighty text by Peter T. Daniels & William Bright, The World’s Writing

Systems, Oxford University Press, 1996. Cited on page 1146. 1174 See for example the article ‘Mathematical notation’ in the Encyclopedia of Mathematics,

10 volumes, Kluwer Academic Publishers, 1988–1993. But first all, have a look at the informative and beautiful http://members.aol.com/jeff570/mathsym.html website. The main source for all these results is the classic and extensive research by Florian Cajori, A History of Mathematical Notations, 2 volumes, The Open Court Publishing Co., 1928–1929. The square root sign is used in Christoff Rud olff, Die Coss, Vuolfius Cephaleus Joanni Jung: Argentorati, 1525. (The full title was Behend vnnd Hubsch Rechnung durch die kunstreichen regeln Algebre so gemeinlicklich die Coss genent werden. Darinnen alles so treülich an tag gegeben, das auch allein auss vleissigem lesen on allen mündtliche vnterricht mag begriffen werden, etc.) Cited on page 1147. 1175 J. Tschichold, Formenwamdlungen der et-Zeichen, Stempel AG, 1953. Cited on page 1176 Malcolm B. Parkes, Pause and Effect: An Introduction to the History of Punctuation

in the West, University of California Press, 1993. Cited on page 1149. 1177 This is explained by Berthold Louis Ullman, Ancient Writing and its Influence, 1932.

Cited on page 1149. 1178 Paul Lehmann, Erforschung des Mittelalters – Ausgewählte Abhandlungen und Aufsätze,

Anton Hiersemann, 1961, pp. 4–21. Cited on page 1149. 1179 Bernard Bischoff, Paläographie des römischen Altertums und des abendländischen

Mittelalters, Erich Schmidt Verlag, 1979, pp. 215–219. Cited on page 1149. 1180 The connections between Greek roots and many French words – and thus many English

ones – can be used to rapidly build up a vocabulary of ancient Greek without much study, as shown by the practical collection by J. Chaineux, Quelques racines grecques, Wetteren

Copyright © Christoph Schiller November 1997–May 2006

1149.

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1997. This classic reference work appears in a new edition every year. The full Hebrew alphabet is given on page 2-90. The list of abbreviations of physical quantities for use in formulae approved by ISO, IUPAP and IUPAC can also be found there. However, the ISO 31 standard, which defines these abbreviations, costs around a thousand euro, is not available on the internet, and therefore can safely be ignored, like any standard that is supposed to be used in teaching but is kept inaccessible to teachers. Cited on pages 1146 and 1147.

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– De Meester, 1929. See also Donald M. Ayers, English Words from Latin and Greek Elements, University of Arizona Press, 1986. Cited on page 1151. 1181 To write well, read William Strunk & E.B. White, The Elements of Style, Macmillan, 1935, 1979, or Wolf Schneider, Deutsch für Kenner – Die neue Stilkunde, Gruner und Jahr, 1987. Cited on page 1151.

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Appendix B

UNI T S, MEA SUR EMENT S AND C ONSTANT S

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M

Copyright © Christoph Schiller November 1997–May 2006

* The respective symbols are s, m, kg, A, K, mol and cd. The international prototype of the kilogram is a

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Ref. 1182

easurements are comparisons with standards. Standards are based on a unit. any different systems of units have been used throughout the world. ost standards confer power to the organization in charge of them. Such power can be misused; this is the case today, for example in the computer industry, and was so in the distant past. The solution is the same in both cases: organize an independent and global standard. For units, this happened in the eighteenth century: to avoid misuse by authoritarian institutions, to eliminate problems with differing, changing and irreproducible standards, and – this is not a joke – to simplify tax collection, a group of scientists, politicians and economists agreed on a set of units. It is called the Système International d’Unités, abbreviated SI, and is defined by an international treaty, the ‘Convention du Mètre’. The units are maintained by an international organization, the ‘Conférence Générale des Poids et Mesures’, and its daughter organizations, the ‘Commission Internationale des Poids et Mesures’ and the ‘Bureau International des Poids et Mesures’ (BIPM), which all originated in the times just before the French revolution. All SI units are built from seven base units, whose official definitions, translated from French into English, are given below, together with the dates of their formulation: ‘The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom.’ (1967)* ‘The metre is the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second.’ (1983) ‘The kilogram is the unit of mass; it is equal to the mass of the international prototype of the kilogram.’ (1901)* ‘The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 metre apart in vacuum, would produce between these conductors a force equal to 2 ë 10−7 newton per metre of length.’ (1948) ‘The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water.’ (1967)* ‘The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12.’ (1971)* ‘The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 ë 1012 hertz and has a radiant intensity in that direction of (1/683) watt per steradian.’ (1979)*

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platinum–iridium cylinder kept at the BIPM in Sèvres, in France. For more details on the levels of the caesium atom, consult a book on atomic physics. The Celsius scale of temperature θ is defined as: θ~°C = T~K−273.15; note the small difference with the number appearing in the definition of the kelvin. SI also states: ‘When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles.’ In the definition of the mole, it is understood that the carbon 12 atoms are unbound, at rest and in their ground state. In the definition of the candela, the frequency of the light corresponds to 555.5 nm, i.e. green colour, around the wavelength to which the eye is most sensitive. * Jacques Babinet (1794–1874), French physicist who published important work in optics.

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Ref. 1183

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Note that both time and length units are defined as certain properties of a standard example of motion, namely light. This is an additional example making the point that the observation of motion as the fundamental type of change is a prerequisite for the definition and construction of time and space. By the way, the use of light in the definitions had been proposed already in 1827 by Jacques Babinet.* From these basic units, all other units are defined by multiplication and division. Thus, all SI units have the following properties: SI units form a system with state-of-the-art precision: all units are defined with a precision that is higher than the precision of commonly used measurements. Moreover, the precision of the definitions is regularly being improved. The present relative uncertainty of the definition of the second is around 10−14 , for the metre about 10−10 , for the kilogram about 10−9 , for the ampere 10−7 , for the mole less than 10−6 , for the kelvin 10−6 and for the candela 10−3 . SI units form an absolute system: all units are defined in such a way that they can be reproduced in every suitably equipped laboratory, independently, and with high precision. This avoids as much as possible any misuse by the standard-setting organization. (The kilogram, still defined with help of an artefact, is the last exception to this requirement; extensive research is under way to eliminate this artefact from the definition – an international race that will take a few more years. There are two approaches: counting particles, or fixing ħ. The former can be achieved in crystals, the latter using any formula where ħ appears, such as the formula for the de Broglie wavelength or that of the Josephson effect.) SI units form a practical system: the base units are quantities of everyday magnitude. Frequently used units have standard names and abbreviations. The complete list includes the seven base units, the supplementary units, the derived units and the admitted units. The supplementary SI units are two: the unit for (plane) angle, defined as the ratio of arc length to radius, is the radian (rad). For solid angle, defined as the ratio of the subtended area to the square of the radius, the unit is the steradian (sr). The derived units with special names, in their official English spelling, i.e. without capital letters and accents, are:

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Name hertz pascal watt volt ohm weber henry lumen becquerel sievert

Hz = 1~s Pa = N~m2 = kg~m s2 W = kg m2 ~s3 V = kg m2 ~As3 Ω = V~A = kg m2 ~A2 s3 Wb = Vs = kg m2 ~As2 H = Vs~A = kg m2 ~A2 s2 lm = cd sr Bq = 1~s Sv = J~kg = m2 ~s2

Name

A b b r e v i at i o n N = kg m~s2 J = Nm = kg m2 ~s2 C = As F = As~V = A2 s4 ~kg m2 S = 1~Ω T = Wb~m2 = kg~As2 = kg~Cs °C (see definition of kelvin) lx = lm~m2 = cd sr~m2 Gy = J~kg = m2 ~s2 kat = mol~s

newton joule coulomb farad siemens tesla degree Celsius lux gray katal

We note that in all definitions of units, the kilogram only appears to the powers of 1, 0 and -1. The final explanation for this fact appeared only recently. Can you try to formulate the reason? The admitted non-SI units are minute, hour, day (for time), degree 1X = π~180 rad, minute 1′ = π~10 800 rad, second 1′′ = π~648 000 rad (for angles), litre and tonne. All other units are to be avoided. All SI units are made more practical by the introduction of standard names and abbreviations for the powers of ten, the so-called prefixes:* Power Name Power Name

Power Name

101 102 103 106 109 1012 1015

1018 Exa 1021 Zetta 1024 Yotta unofficial: 1027 Xenta 1030 Wekta 1033 Vendekta 36 10 Udekta

deca da hecto h kilo k Mega M Giga G Tera T Peta P

10−1 10−2 10−3 10−6 10−9 10−12 10−15

deci d centi c milli m micro µ nano n pico p femto f

Power Name E Z Y

10−18 10−21 10−24

atto zepto yocto

a z y

xenno weko vendeko udeko

x w v u

Ref. 1184

X W V U

10−27 10−30 10−33 10−36

Challenge 1525 e

* Some of these names are invented (yocto to sound similar to Latin octo ‘eight’, zepto to sound similar to Latin septem, yotta and zetta to resemble them, exa and peta to sound like the Greek words for six and five, the unofficial ones to sound similar to the Greek words for nine, ten, eleven and twelve); some are from Danish/Norwegian (atto from atten ‘eighteen’, femto from femten ‘fifteen’); some are from Latin (from mille ‘thousand’, from centum ‘hundred’, from decem ‘ten’, from nanus ‘dwarf ’); some are from Italian (from piccolo ‘small’); some are Greek (micro is from µικρός ‘small’, deca/deka from δέκα ‘ten’, hecto from ἑκατόν ‘hundred’, kilo from χίλιοι ‘thousand’, mega from µέγας ‘large’, giga from γίγας ‘giant’, tera from τέρας ‘monster’). Translate: I was caught in such a traffic jam that I needed a microcentury for a picoparsec and that my car’s fuel consumption was two tenths of a square millimetre.

Copyright © Christoph Schiller November 1997–May 2006

SI units form a complete system: they cover in a systematic way the complete set of observables of physics. Moreover, they fix the units of measurement for all other sciences as well.

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Challenge 1524 ny

A b b r e v i at i o n

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b units, measurements and constants

Page 1104

1157

SI units form a universal system: they can be used in trade, in industry, in commerce, at home, in education and in research. They could even be used by extraterrestrial civilizations, if they existed. SI units form a coherent system: the product or quotient of two SI units is also an SI unit. This means that in principle, the same abbreviation, e.g. ‘SI’, could be used for every unit. The SI units are not the only possible set that could fulfil all these requirements, but they are the only existing system that does so.* Remember that since every measurement is a comparison with a standard, any measurement requires matter to realize the standard (yes, even for the speed standard), and radiation to achieve the comparison. The concept of measurement thus assumes that matter and radiation exist and can be clearly separated from each other.

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Planck’s natural units

TA B L E 83 Planck’s (uncorrected) natural units

Name

Definition

Basic units the Planck length

l Pl

the Planck time

t Pl

the Planck mass

m Pl

the Planck current

I Pl

the Planck temperature

TPl

=

=

=

=

¼

¼

¼

¼

ħG~c 3 ħG~c 5 ħc~G

4πε 0 c ~G 6

ħc 5 ~Gk 2

= = = = =

1.616 0(12) ë 10−35 m 5.390 6(40) ë 10−44 s 21.767(16) µg 3.479 3(22) ë 1025 A 1.417 1(91) ë 1032 K

* Most non-SI units still in use in the world are of Roman origin. The mile comes from milia passum, which used to be one thousand (double) strides of about 1480 mm each; today a nautical mile, once defined as minute of arc on the Earth’s surface, is exactly 1852 m). The inch comes from uncia/onzia (a twelfth – now of a foot). The pound (from pondere ‘to weigh’) is used as a translation of libra – balance – which is the origin of its abbreviation lb. Even the habit of counting in dozens instead of tens is Roman in origin. These and all other similarly funny units – like the system in which all units start with ‘f ’, and which uses furlong/fortnight as its unit of velocity – are now officially defined as multiples of SI units. ** The natural units x Pl given here are those commonly used today, i.e. those defined using the constant ħ, and not, as Planck originally did, by using the constant h = 2πħ. The electromagnetic units can also be defined with other factors than 4πε 0 in the expressions: for example, using 4πε 0 α, with the fine structure constant α, gives q Pl = e. For the explanation of the numbers between brackets, the standard deviations, see page 1164.

Copyright © Christoph Schiller November 1997–May 2006

=

¼

Va l u e

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Challenge 1526 e

Since the exact form of many equations depends on the system of units used, theoretical physicists often use unit systems optimized for producing simple equations. The chosen units and the values of the constants of nature are related. In microscopic physics, the system of Planck’s natural units is frequently used. They are defined by setting c = 1, ħ = 1, G = 1, k = 1, ε 0 = 1~4π and µ 0 = 4π. Planck units are thus defined from combinations of fundamental constants; those corresponding to the fundamental SI units are given in Table 83.** The table is also useful for converting equations written in natural units back to SI units: just substitute every quantity X by X~X Pl .

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b units, measurements and constants

1158

Name

Definition

Trivial units the Planck velocity the Planck angular momentum the Planck action the Planck entropy

v Pl L Pl S aPl S ePl

Composed units the Planck energy

E Pl

the Planck momentum

p Pl

the Planck force the Planck power

FPl PPl

the Planck acceleration

a Pl

the Planck frequency

f Pl

the Planck electric charge

q Pl

the Planck voltage

U Pl

the Planck resistance

R Pl

the Planck capacitance

C Pl

the Planck inductance

L Pl

the Planck electric field

E Pl

the Planck magnetic flux density

B Pl

=

=

=

= =

=

=

=

=

=

=

=

=

=

c 5 ~G 2 ħ ¼ ħc 5 ~G ¼ ħc 3 ~G

c 4 ~G c 5 ~G ¼ ¼

¼

¼

c 7 ~ħG c 5 ~ħG

4πε 0 cħ

c 4 ~4πε 0 G

1~4πε¼ 0c 4πε 0 ħG~c 3 ¼ (1~4πε 0 ) ħG~c 7 ¼ c 7 ~4πε 0 ħG 2 ¼ c 5 ~4πε 0 ħG 2

= = = =

=

= =

= =

= = = =

=

= = = =

0.3 Gm~s 1.1 ë 10−34 Js 1.1 ë 10−34 Js 13.8 yJ~K

Dvipsbugw 5.2 ë 1096 kg~m3

2.0 GJ = 1.2 ë 1028 eV 6.5 Ns

1.2 ë 1044 N 3.6 ë 1052 W

5.6 ë 1051 m~s2 1.9 ë 1043 Hz

1.9 aC = 11.7 e 1.0 ë 1027 V 30.0 Ω

1.8 ë 10−45 F 1.6 ë 10−42 H 6.5 ë 1061 V~m 2.2 ë 1053 T

The natural units are important for another reason: whenever a quantity is sloppily called ‘infinitely small (or large)’, the correct expression is ‘as small (or as large) as the corresponding corrected Planck unit’. As explained throughout the text, and especially in the third part, this substitution is possible because almost all Planck units provide, within a correction factor of order 1, the extremal value for the corresponding observable – some an upper and some a lower limit. Unfortunately, these correction factors are not yet widely known. The exact extremal value for each observable in nature is obtained when G is substituted by 4G, ħ by ħ~2, k by k~2 and 4πε 0 by 8πε 0 α in all Planck quantities. These extremal values, or corrected Planck units, are the true natural units. To exceeding the extremal values is possible only for some extensive quantities. (Can you find out which ones?) Other unit systems A central aim of research in high-energy physics is the calculation of the strengths of all interactions; therefore it is not practical to set the gravitational constant G to unity, as in the Planck system of units. For this reason, high-energy physicists often only set

Copyright © Christoph Schiller November 1997–May 2006

Challenge 1527 n

ρ Pl

c ħ ħ k

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Page 1076

the Planck mass density

= = = =

Va l u e

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b units, measurements and constants

1159

c = ħ = k = 1 and µ 0 = 1~ε 0 = 4π,* leaving only the gravitational constant G in the equations. In this system, only one fundamental unit exists, but its choice is free. Often a standard length is chosen as the fundamental unit, length being the archetype of a measured quantity. The most important physical observables are then related by

1 eV 

Challenge 1529 e

Ref. 1187

Ref. 1186 Challenge 1528 n

aJ

(811)

which is easily remembered. The simplification c = ħ = 1 yields G = 6.9 ë 10−57 eV−2 and allows one to use the unit eV also for mass, momentum, temperature, frequency, time and length, with the respective correspondences 1 eV  1.8 ë 10−36 kg  5.4 ë 10−28 Ns  242 THz  11.6 kK and 1 eV−1  4.1 fs  1.2 µm. To get some feeling for the unit eV, the following relations are useful. Room temperature, usually taken as 20°C or 293 K, corresponds to a kinetic energy per particle of 0.025 eV or 4.0 zJ. The highest particle energy measured so far belongs to a cosmic ray with an energy of 3 ë 1020 eV or 48 J. Down here on the Earth, an accelerator able to produce an energy of about 105 GeV or 17 nJ for electrons and antielectrons has been built, * Other definitions for the proportionality constants in electrodynamics lead to the Gaussian unit system often used in theoretical calculations, the Heaviside–Lorentz unit system, the electrostatic unit system, and the electromagnetic unit system, among others. ** In the list, l is length, E energy, F force, E electric the electric and B the magnetic field, m mass, p momentum, a acceleration, f frequency, I electric current, U voltage, T temperature, v speed, q charge, R resistance, P power, G the gravitational constant. The web page http://www.chemie.fu-berlin.de/chemistry/general/units_en.html provides a tool to convert various units into each other. Researchers in general relativity often use another system, in which the Schwarzschild radius rs = 2Gm~c 2 is used to measure masses, by setting c = G = 1. In this case, mass and length have the same dimension, and ħ has the dimension of an area. Already in the nineteenth century, George Stoney had proposed to use as length, time and mass units » 2 4 » 2 6 the Ge ~(c 4πε 0 ) = 1.4 ë 10−36 m, t S = Ge ~(c 4πε 0 ) = 4.6 ë 10−45 s and m S = » 2quantities l S = e ~(G4πε 0 ) = 1.9 µg. How are these units related to the Planck units?

Copyright © Christoph Schiller November 1997–May 2006

Ref. 1185

1 6

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1~[l 2 ] = [E]2 = [F] = [B] = [E electric ] , 1~[l] = [E] = [m] = [p] = [a] = [ f ] = [I] = [U] = [T] , 1 = [v] = [q] = [e] = [R] = [S action ] = [S entropy ] = ħ = c = k = [α] , [l] = 1~[E] = [t] = [C] = [L] and [l]2 =1~[E]2 = [G] = [P] (810) where we write [x] for the unit of quantity x. Using the same unit for time, capacitance and inductance is not to everybody’s taste, however, and therefore electricians do not use this system.** Often, in order to get an impression of the energies needed to observe an effect under study, a standard energy is chosen as fundamental unit. In particle physics the most common energy unit is the electronvolt (eV), defined as the kinetic energy acquired by an electron when accelerated by an electrical potential difference of 1 volt (‘protonvolt’ would be a better name). Therefore one has 1 eV = 1.6 ë 10−19 J, or roughly

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1160

Ref. 1188

b units, measurements and constants

and one able to produce an energy of 10 TeV or 1.6 µJ for protons will be finished soon. Both are owned by CERN in Geneva and have a circumference of 27 km. The lowest temperature measured up to now is 280 pK, in a system of rhodium nuclei held inside a special cooling system. The interior of that cryostat may even be the coolest point in the whole universe. The kinetic energy per particle corresponding to that temperature is also the smallest ever measured: it corresponds to 24 feV or 3.8 vJ = 3.8 ë 10−33 J. For isolated particles, the record seems to be for neutrons: kinetic energies as low as 10−7 eV have been achieved, corresponding to de Broglie wavelengths of 60 nm.

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Curiosities and fun challenges about units Here are a few facts making the concept of physical unit more vivid. **

**

Ref. 1189

A gray is the amount of radioactivity that deposits 1 J on 1 kg of matter. A sievert is the unit of radioactivity adjusted to humans by weighting each type of human tissue with a factor representing the impact of radiation deposition on it. Four to five sievert are a lethal dose to humans. In comparison, the natural radioactivity present inside human bodies leads to a dose of 0.2 mSv per year. An average X-ray image implies an irradiation of 1 mSv; a CAT scan 8 mSv. **

* This story revived an old (and false) urban legend that states that only three countries in the world do not use SI units: Liberia, the USA and Myanmar.

Copyright © Christoph Schiller November 1997–May 2006

Challenge 1530 e

Are you confused by the candela? The definition simply says that 683 cd = 683 lm~sr corresponds to 1 W~sr. The candela is thus a unit for light power per (solid) angle, except that it is corrected for the eye’s sensitivity: the candela measures only visible power per angle. Similarly, 683 lm = 683 cd sr corresponds to 1 W. So both the lumen and the watt measure power, or energy flux, but the lumen measures only the visible part of the power. This difference is expressed by inserting ‘radiant’ or ‘luminous’: thus, the Watt measures radiant flux, whereas the lumen measures luminous flux. The factor 683 is historical. An ordinary candle emits a luminous intensity of about a candela. Therefore, at night, a candle can be seen up to a distance of 10 or 20 kilometres. A 100 W incandescent light bulb produces 1700 lm, and the brightest light emitting diodes about 5 lm. Cinema projectors produce around 2 Mlm, and the brightest flashes, like lightning, 100 Mlm. The irradiance of sunlight is about 1300 W~m2 on a sunny day; on the other hand, the illuminance is only 120 klm~m2 = 120 klux or 170 W~m2 . (A cloud-covered summer day or a clear winter day produces about 10 klux.) These numbers show that most of the energy from the Sun that reaches the Earth is outside the visible spectrum.

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Not using SI units can be expensive. In 1999, NASA lost a satellite on Mars because some software programmers had used imperial units instead of SI units in part of the code. As a result, the Mars Climate Orbiter crashed into the planet, instead of orbiting it; the loss was around 100 million euro.*

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b units, measurements and constants

Ref. 1190

1161

On a glacier, near the sea shore, on the top of a mountain, or in particular weather condition the brightness can reach 150 klux. The brightest lamps, those used during surgical operations, produce 120 klux. Humans need about 30 lux for comfortable reading. Museums are often kept dark because water-based paintings are degraded by light above 100 lux, and oil paintings by light above 200 lux. The full moon produces 0.1 lux, and the sky on a dark moonless night about 1 mlux. The eyes lose their ability to distinguish colours somewhere between 0.1 lux and 0.01 lux; the eye stops to work below 1 nlux. Technical devices to produce images in the dark, such as night goggles, start to work at 1 µlux. By the way, the human body itself shines with about 1 plux, a value too small to be detected with the eye, but easily measured with specialized apparatus. The origin of this emission is still a topic of research.

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**

** The luminous density is a quantity often used by light technicians. Its unit is 1 cd~m2 , unofficially called 1 Nit and abbreviated 1 nt. Eyes see purely with rods from 0.1 µcd~m2 to 1 mcd~m2 ; they see purely with cones above 5 cd~m2 ; they see best between 100 and 50 000 cd~m2 ; and they get completely overloaded above 10 Mcd~m2 : a total range of 15 orders of magnitude. **

Ref. 1192

The Planck length is roughly the de Broglie wavelength λ B = h~mv of a man walking comfortably (m = 80 kg, v = 0.5 m~s); this motion is therefore aptly called the ‘Planck stroll.’ ** The Planck mass is equal to the mass of about 1019 protons. This is roughly the mass of a human embryo at about ten days of age.

The second does not correspond to 1/86 400th of the day any more, though it did in the year 1900; the Earth now takes about 86 400.002 s for a rotation, so that the International Earth Rotation Service must regularly introduce a leap second to ensure that the Sun is at the highest point in the sky at 12 o’clock sharp.* The time so defined is called Universal Time Coordinate. The speed of rotation of the Earth also changes irregularly from day to day due to the weather; the average rotation speed even changes from winter to summer because of the changes in the polar ice caps; and in addition that average decreases over * Their website at http://hpiers.obspm.fr gives more information on the details of these insertions, as does http://maia.usno.navy.mil, one of the few useful military websites. See also http://www.bipm.fr, the site of the BIPM.

Copyright © Christoph Schiller November 1997–May 2006

**

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Ref. 1191

The highest achieved light intensities are in excess of 1018 W~m2 , more than 15 orders of magnitude higher than the intensity of sunlight. They are produced by tight focusing of pulsed lasers. The electric field in such light pulses is of the same order as the field inside atoms; such a beam therefore ionizes all matter it encounters.

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1162

b units, measurements and constants

time, because of the friction produced by the tides. The rate of insertion of leap seconds is therefore higher than once every 500 days, and not constant in time. ** The most precisely measured quantities in nature are the frequencies of certain millisecond pulsars,* the frequency of certain narrow atomic transitions, and the Rydberg constant of atomic hydrogen, which can all be measured as precisely as the second is defined. The caesium transition that defines the second has a finite linewidth that limits the achievable precision: the limit is about 14 digits.

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** Ref. 1193 Ref. 1194

** Ref. 1195 Challenge 1531 n

The shortest times measured are the lifetimes of certain ‘elementary’ particles. In particular, the lifetime of certain D mesons have been measured at less than 10−23 s. Such times are measured using a bubble chamber, where the track is photographed. Can you estimate how long the track is? (This is a trick question – if your length cannot be observed with an optical microscope, you have made a mistake in your calculation.) **

Ref. 1196

The longest times encountered in nature are the lifetimes of certain radioisotopes, over 1015 years, and the lower limit of certain proton decays, over 1032 years. These times are thus much larger than the age of the universe, estimated to be fourteen thousand million years. **

**

Challenge 1532 n

The precision of mass measurements of solids is limited by such simple effects as the adsorption of water. Can you estimate the mass of a monolayer of water – a layer with thickness of one molecule – on a metal weight of 1 kg?

* An overview of this fascinating work is given by J.H. Taylor, Pulsar timing and relativistic gravity, Philosophical Transactions of the Royal Society, London A 341, pp. 117–134, 1992.

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Page 1167

The least precisely measured of the fundamental constants of physics are the gravitational constant G and the strong coupling constant α s . Even less precisely known are the age of the universe and its density (see Table 86).

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The most precise clock ever built, using microwaves, had a stability of 10−16 during a running time of 500 s. For longer time periods, the record in 1997 was about 10−15 ; but values around 10−17 seem within technological reach. The precision of clocks is limited for short measuring times by noise, and for long measuring times by drifts, i.e. by systematic effects. The region of highest stability depends on the clock type; it usually lies between 1 ms for optical clocks and 5000 s for masers. Pulsars are the only type of clock for which this region is not known yet; it certainly lies at more than 20 years, the time elapsed at the time of writing since their discovery.

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b units, measurements and constants

1163

**

Ref. 1197

Ref. 1198

Variations of quantities are often much easier to measure than their values. For example, in gravitational wave detectors, the sensitivity achieved in 1992 was ∆l~l = 3 ë 10−19 for lengths of the order of 1 m. In other words, for a block of about a cubic metre of metal it is possible to measure length changes about 3000 times smaller than a proton radius. These set-ups are now being superseded by ring interferometers. Ring interferometers measuring frequency differences of 10−21 have already been built; and they are still being improved.

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**

Ref. 1199

** In the previous millennium, thermal energy used to be measured using the unit calorie, written as cal. 1 cal is the energy needed to heat 1 g of water by 1 K. To confuse matters, 1 kcal was often written 1 Cal. (One also spoke of a large and a small calorie.) The value of 1 kcal is 4.1868 kJ. ** SI units are adapted to humans: the values of heartbeat, human size, human weight, human temperature and human substance are no more than a couple of orders of magnitude near the unit value. SI units thus (roughly) confirm what Protagoras said 25 centuries ago: ‘Man is the measure of all things.’ **

Challenge 1534 n

The table of SI prefixes covers 72 orders of magnitude. How many additional prefixes will be needed? Even an extended list will include only a small part of the infinite range of possibilities. Will the Conférence Générale des Poids et Mesures have to go on forever, defining an infinite number of SI prefixes? Why?

The French philosopher Voltaire, after meeting Newton, publicized the now famous story that the connection between the fall of objects and the motion of the Moon was discovered by Newton when he saw an apple falling from a tree. More than a century later, just before the French Revolution, a committee of scientists decided to take as the unit of force precisely the force exerted by gravity on a standard apple, and to name it after the English scientist. After extensive study, it was found that the mass of the standard apple was 101.9716 g; its weight was called 1 newton. Since then, visitors to the museum in Sèvres near Paris have been able to admire the standard metre, the standard kilogram and the standard apple.* * To be clear, this is a joke; no standard apple exists. It is not a joke however, that owners of several apple

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**

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Challenge 1533 n

The Swedish astronomer Anders Celsius (1701–1744) originally set the freezing point of water at 100 degrees and the boiling point at 0 degrees. Later the scale was reversed. However, this is not the whole story. With the official definition of the kelvin and the degree Celsius, at the standard pressure of 1013.25 Pa, water boils at 99.974°C. Can you explain why it is not 100°C any more?

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b units, measurements and constants

1164 N number of measurements

standard deviation full width at half maximum (FWHM)

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limit curve for a large number of measurements

x average value

x measured values

Precision and accuracy of measurements As explained on page 264, precision means how well a result is reproduced when the measurement is repeated; accuracy is the degree to which a measurement corresponds to the actual value. Lack of precision is due to accidental or random errors; they are best measured by the standard deviation, usually abbreviated σ; it is defined through σ2 =

Challenge 1535 n

1 n 2 Q(x i − x¯ ) , n − 1 i=1

where x¯ is the average of the measurements x i . (Can you imagine why n − 1 is used in the formula instead of n?) For most experiments, the distribution of measurement values tends towards a normal distribution, also called Gaussian distribution, whenever the number of measurements is increased. The distribution, shown in Figure 390, is described by the expression

Ref. 1201

Ref. 1200

¯ 2 (x− x) 2σ 2

.

(813)

The square σ 2 of the standard deviation is also called the variance. For a Gaussian distribution of measurement values, 2.35σ is the full width at half maximum. Lack of accuracy is due to systematic errors; usually these can only be estimated. This estimate is often added to the random errors to produce a total experimental error, sometimes also called total uncertainty. The following tables give the values of the most important physical constants and particle properties in SI units and in a few other common units, as published in the standtrees in Britain and in the US claim descent, by rerooting, from the original tree under which Newton had his insight. DNA tests have even been performed to decide if all these derive from the same tree. The result was, unsurprisingly, that the tree at MIT, in contrast to the British ones, is a fake.

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N(x)  e−

Challenge 1536 e

(812)

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F I G U R E 390 A precision experiment and its measurement distribution

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b units, measurements and constants

Ref. 1202

Challenge 1537 ny Page 1035

1165

Basic physical constants Ref. 1202

In principle, all quantitative properties of matter can be calculated with quantum theory. For example, colour, density and elastic properties, can be predicted using the values of the following constants using the equations of the standard model of high-energy physics. TA B L E 84 Basic physical constants

U n c e r t. a

number of space-time dimensions

3+1

0b

vacuum speed of light c vacuum permeabilityc

299 792 458 m~s 4π ë 10−7 H~m = 1.256 637 061 435 ... µH~m 8.854 187 817 620 ... pF~m 6.626 068 76(52) ë 10−34 Js 1.054 571 596(82) ë 10−34 Js 0.160 217 646 2(63) aC 1.380 650 3(24) ë 10−23 J~K 6.673(10) ë 10−11 Nm2 ~kg2 2.076(3) ë 10−43 s2 ~kg m 1~137.035 999 76(50)

0 0 0 0 7.8 ë 10−8 7.8 ë 10−8 3.9 ë 10−8 1.7 ë 10−6 1.5 ë 10−3 1.5 ë 10−3 3.7 ë 10−9

vacuum permittivityc original Planck constant reduced Planck constant positron charge Boltzmann constant gravitational constant gravitational coupling constant fine structure constant,d e.m. coupling constant

Symbol

c µ0 ε 0 = 1~µ 0 c 2 h ħ e k G κ = 8πG~c 4 2 α = 4πεe 0 ħc

= дem (m e2 c 2 )

= 0.007 297 352 533(27)

3.7 ë 10−9

* Some of the stories can be found in the text by N.W. Wise, The Values of Precision, Princeton University Press, 1994. The field of high-precision measurements, from which the results on these pages stem, is a world on its own. A beautiful introduction to it is J.D. Fairbanks, B.S. Deaver, C.W. Everitt & P.F. Michaelson, eds., Near Zero: Frontiers of Physics, Freeman, 1988.

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Va l u e i n S I u n i t s

Q ua n t i t y

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ard references. The values are the world averages of the best measurements made up to the present. As usual, experimental errors, including both random and estimated systematic errors, are expressed by giving the standard deviation in the last digits; e.g. 0.31(6) means – roughly speaking – 0.31 0.06. In fact, behind each of the numbers in the following tables there is a long story which is worth telling, but for which there is not enough room here.* What are the limits to accuracy and precision? There is no way, even in principle, to measure a length x to a precision higher than about 61 digits, because ∆x~x A l Pl ~d horizon = 10−61 . (Is this valid also for force or for volume?) In the third part of our text, studies of clocks and metre bars strengthen this theoretical limit. But it is not difficult to deduce more stringent practical limits. No imaginable machine can measure quantities with a higher precision than measuring the diameter of the Earth within the smallest length ever measured, about 10−19 m; that is about 26 digits of precision. Using a more realistic limit of a 1000 m sized machine implies a limit of 22 digits. If, as predicted above, time measurements really achieve 17 digits of precision, then they are nearing the practical limit, because apart from size, there is an additional practical restriction: cost. Indeed, an additional digit in measurement precision often means an additional digit in equipment cost.

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b units, measurements and constants

1166

Q ua n t i t y Fermi coupling constant,d weak coupling constant weak mixing angle weak mixing angle strong coupling constantd

Page 860

Symbol G F ~(ħc)3 α w (M Z ) = дw2 ~4π sin2 θ W (MS) sin2 θ W (on shell) = 1 − (m W ~m Z )2 α s (M Z ) = дs2 ~4π

Va l u e i n S I u n i t s

U n c e r t. a

1.166 39(1) ë 10−5 GeV−2 1~30.1(3) 0.231 24(24) 0.2224(19)

8.6 ë 10−6 1 ë 10−2 1.0 ë 10−3 8.7 ë 10−3

0.118(3)

25 ë 10−3

Why do all these constants have the values they have? The answer is different in each case. For any constant having a dimension, such as the quantum of action ħ, the numerical value has only historical meaning. It is 1.054 ë 10−34 Js because of the SI definition of the joule and the second. The question why the value of a dimensional constant is not larger or smaller always requires one to understand the origin of some dimensionless number giving the ratio between the constant and the corresponding natural unit. Understanding the sizes of atoms, people, trees and stars, the duration of molecular and atomic processes, or the mass of nuclei and mountains, implies understanding the ratios between these values and the corresponding natural units. The key to understanding nature is thus the understanding of all ratios, and thus of all dimensionless constants. The story of the most important ratios is told in the third part of the adventure. The basic constants yield the following useful high-precision observations. TA B L E 85 Derived physical constants

Q ua n t i t y

Va l u e i n S I u n i t s

U n c e r t.

376.730 313 461 77... Ω 6.022 141 99(47) ë 1023 10 973 731.568 549(83) m−1 77.480 916 96(28) µS 2.067 833 636(81) pWb 483.597 898(19) THz~V 25 812.807 572(95) Ω 9.274 008 99(37) yJ~T 27.992 4925(11) GHz~T

0 7.9 ë 10−8 7.6 ë 10−12 3.7 ë 10−9 3.9 ë 10−8 3.9 ë 10−8 3.7 ë 10−9 4.0 ë 10−8 4.0 ë 10−8

r e = e 2 ~4πε 0 m e c 2 2.817 940 285(31) fm λ c = h~m e c 2.426 310 215(18) pm

1.1 ë 10−8 7.3 ë 10−9

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Vacuum wave resistance Avogadro’s number Rydberg constant a conductance quantum magnetic flux quantum Josephson frequency ratio von Klitzing constant Bohr magneton cyclotron frequency of the electron classical electron radius Compton wavelength

Symbol » Z 0 = µ 0 ~ε 0 NA Rª = m e cα 2 ~2h G 0 = 2e 2 ~h φ 0 = h~2e 2e~h h~e 2 = µ 0 c~2α µ B = eħ~2m e f c ~B = e~2πm e

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a. Uncertainty: standard deviation of measurement errors. b. Only down to 10−19 m and up to 1026 m. c. Defining constant. d. All coupling constants depend on the 4-momentum transfer, as explained in the section on renormalization. Fine structure constant is the traditional name for the electromagnetic coupling constant дem in the case of a 4-momentum transfer of Q 2 = m e2 c 2 , which is the smallest one possible. At higher 2 2 momentum transfers it has larger values, e.g. дem (Q 2 = M W c )  1~128. The strong coupling constant has higher values at lower momentum transfers; e.g., α s (34 GeV) = 0.14(2).

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b units, measurements and constants

Q ua n t i t y of the electron Bohr radius a nuclear magneton proton–electron mass ratio Stefan–Boltzmann constant Wien displacement law constant bits to entropy conversion const. TNT energy content

Symbol λ c = ħ~m e c = r e ~α aª = r e ~α 2 µ N = eħ~2m p m p ~m e σ = π 2 k 4 ~60ħ 3 c 2 b = λ max T

1167

Va l u e i n S I u n i t s

U n c e r t.

0.386 159 264 2(28) pm 52.917 720 83(19) pm 5.050 783 17(20) ë 10−27 J~T 1 836.152 667 5(39) 56.704 00(40) nW~m2 K4 2.897 768 6(51) mmK 1023 bit = 0.956 994 5(17) J~K 3.7 to 4.0 MJ~kg

7.3 ë 10−9 3.7 ë 10−9 4.0 ë 10−8 2.1 ë 10−9 7.0 ë 10−6 1.7 ë 10−6 1.7 ë 10−6 4 ë 10−2

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a. For infinite mass of the nucleus.

Some properties of nature at large are listed in the following table. (If you want a challenge, can you determine whether any property of the universe itself is listed?) TA B L E 86 Astrophysical constants

Q ua n t i t y

Symbol

Va l u e

Copyright © Christoph Schiller November 1997–May 2006

gravitational constant G 6.672 59(85) ë 10−11 m3 ~kg s2 cosmological constant Λ c. 1 ë 10−52 m−2 tropical year 1900 a a 31 556 925.974 7 s tropical year 1994 a 31 556 925.2 s mean sidereal day d 23 h 56′ 4.090 53′′ light year al 9.460 528 173 ... Pm b astronomical unit AU 149 597 870.691(30) km parsec pc 30.856 775 806 Pm = 3.261 634 al c age of the universe t0 4.333(53) ë 1017 s = 13.73(0.17) ë 109 a (determined from space-time, via expansion, using general relativity) age of the universe c t0 over 3.5(4) ë 1017 s = 11.5(1.5) ë 109 a (determined from matter, via galaxies and stars, using quantum theory) Hubble parameter c H0 2.3(2) ë 10−18 s−1 = 0.73(4) ë 10−10 a−1 H 0 = h 0 ë 100 km~sMpc = h 0 ë 1.0227 ë 10−10 a−1 reduced Hubble parameter c h0 0.71(4) ¨ 0 ~H 02 −0.66(10) deceleration parameter q 0 = −( a~a) universe’s horizon distance c d 0 = 3ct 0 40.0(6) ë 1026 m = 13.0(2) Gpc universe’s topology unknown number of space dimensions 3, for distances up to 1026 m 2 h 02 ë 1.878 82(24) ë 10−26 kg~m3 critical density ρ c = 3H 0 ~8πG of the universe = 0.95(12) ë 10−26 kg~m3 c (total) density parameter Ω 0 = ρ 0 ~ρ c 1.02(2) c baryon density parameter Ω B0 = ρ B0 ~ρ c 0.044(4) cold dark matter density parameter c Ω CDM0 = ρ CDM0 ~ρ c 0.23(4) neutrino density parameter c Ω ν0 = ρ ν0 ~ρ c 0.001 to 0.05

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Challenge 1538 n

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b units, measurements and constants

1168

Q ua n t i t y

Symbol Ω X0 = ρ X0 ~ρ c w = p X ~ρ X

dark energy density parameter c dark energy state parameter baryon mass baryon number density luminous matter density stars in the universe baryons in the universe

mb

ns nb

Planck length Planck time Planck mass instants in history c space-time points inside the horizon c mass inside horizon

º S º T σ8 n dn~d ln k ¼ l Pl = ħG~c 3 ¼ t Pl = ħG~c 5 » m Pl = ħc~G t 0 ~t Pl N 0 = (R 0 ~l Pl )3 ë (t 0 ~t Pl ) M

1.67 ë 10−27 kg 0.25(1) ~m3 3.8(2) ë 10−28 kg~m3 10221 10811 2.725(1) K 1089 4.6 ë 10−31 kg~m3 410.89 ~cm3 or 400 ~cm3 (T0 ~2.7 K)3 5.6(1.5) ë 10−6 º < 0.71 S 0.84(4) 0.93(3) -0.03(2) 1.62 ë 10−35 m 5.39 ë 10−44 s 21.8 µg 8.7(2.8) ë 1060 102441 10541 kg

a. Defining constant, from vernal equinox to vernal equinox; it was once used to define the second. (Remember: π seconds is about a nanocentury.) The value for 1990 is about 0.7 s less, corresponding to a slowdown of roughly 0.2 ms~a. (Watch out: why?) There is even an empirical formula for the change of the length of the year over time. b. Average distance Earth–Sun. The truly amazing precision of 30 m results from time averages of signals sent from Viking orbiters and Mars landers taken over a period of over twenty years. c. The index 0 indicates present-day values. d. The radiation originated when the universe was 380 000 years old and had a temperature of about 3000 K; the fluctuations ∆T0 which led to galaxy formation are today about 16  4 µK = 6(2) ë 10−6 T0 .

Warning: in the third part of this text it is shown that many of the constants in Table 86 are not physically sensible quantities. They have to be taken with many grains of salt. The more specific constants given in the following table are all sensible, though. TA B L E 87 Astronomical constants

Q ua n t i t y

Symbol

Va l u e

Earth’s mass

M

5.972 23(8) ë 1024 kg

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Copyright © Christoph Schiller November 1997–May 2006

Page 451

T0 nγ ρ γ = π 2 k 4 ~15T04

0.73(4) −1.0(2)

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microwave background temperature d photons in the universe photon energy density photon number density density perturbation amplitude gravity wave amplitude mass fluctuations on 8 Mpc scalar index running of scalar index

Challenge 1539 n Ref. 1203

Va l u e

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b units, measurements and constants

Q ua n t i t y Earth’s gravitational length Earth radius, equatorial a Earth’s radius, polar a Equator–pole distance a Earth’s flattening a Earth’s av. density Earth’s age Moon’s radius Moon’s radius Moon’s mass Moon’s mean distance b Moon’s distance at perigee b

Moon’s angular size c Moon’s average density Sun’s mass Sun’s gravitational length Sun’s luminosity solar radius, equatorial Sun’s angular size

Va l u e

l = 2GM~c 2 R eq Rp

8.870(1) mm 6378.1367(1) km 6356.7517(1) km 10 001.966 km (average) e 1~298.25231(1) ρ 5.5 Mg~m3 4.55 Ga = 143 Ps R mv 1738 km in direction of Earth R mh 1737.4 km in other two directions Mm 7.35 ë 1022 kg dm 384 401 km typically 363 Mm, hist. minimum 359 861 km typically 404 Mm, hist. maximum 406 720 km average 0.5181X = 31.08′ , minimum 0.49X , maximum - shortens line 0.55X ρ 3.3 Mg~m3 Mb 1.988 43(3) ë 1030 kg 2 lb = 2GMb ~c 2.953 250 08 km Lb 384.6 YW Rb 695.98(7) Mm 0.53X average; minimum on fourth of July (aphelion) 1888′′ , maximum on fourth of January (perihelion) 1952′′ ρ 1.4 Mg~m3 AU 149 597 870.691(30) km 4.6 Ga v bg 220(20) km~s 370.6(5) km~s 8.0(5) kpc = 26.1(1.6) kal 13.6 Ga c. 1021 m or 100 kal 1012 solar masses, c. 2 ë 1042 kg 1.90 ë 1027 kg 71.398 Mm 67.1(1) Mm 778 412 020 km 13.2 ë 109 al = 1.25 ë 1026 m, red-shift 10

Copyright © Christoph Schiller November 1997–May 2006

Sun’s average density Sun’s average distance Sun’s age solar velocity around centre of galaxy solar velocity v bb against cosmic background distance to galaxy centre Milky Way’s age Milky Way’s size Milky Way’s mass Jupiter’s mass M Jupiter’s radius, equatorial R Jupiter’s radius, polar R Jupiter’s average distance from Sun D most distant galaxy known 1835 IR1916

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Moon’s distance at apogee b

Symbol

1169

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b units, measurements and constants

1170

a. The shape of the Earth is described most precisely with the World Geodetic System. The last edition dates from 1984. For an extensive presentation of its background and its details, see the http://www.wgs84.com website. The International Geodesic Union refined the data in 2000. The radii and the flattening given here are those for the ‘mean tide system’. They differ from those of the ‘zero tide system’ and other systems by about 0.7 m. The details constitute a science in itself. b. Measured centre to centre. To find the precise position of the Moon at a given date, see the http://www. fourmilab.ch/earthview/moon_ap_per.html page. For the planets, see the page http://www.fourmilab.ch/ solar/solar.html and the other pages on the same site. c. Angles are defined as follows: 1 degree = 1X = π~180 rad, 1 (first) minute = 1′ = 1X ~60, 1 second (minute) = 1′′ = 1′ ~60. The ancient units ‘third minute’ and ‘fourth minute’, each 1~60th of the preceding, are not in use any more. (‘Minute’ originally means ‘very small’, as it still does in modern English.)

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Useful numbers

Challenge 1540 n

3.14159 26535 89793 23846 26433 83279 50288 41971 69399 375105 2.71828 18284 59045 23536 02874 71352 66249 77572 47093 699959 0.57721 56649 01532 86060 65120 90082 40243 10421 59335 939923 0.69314 71805 59945 30941 72321 21458 17656 80755 00134 360255 2.30258 50929 94045 68401 79914 54684 36420 76011 01488 628772 3.16227 76601 68379 33199 88935 44432 71853 37195 55139 325216

If the number π is normal, i.e. if all digits and digit combinations in its decimal expansion appear with the same limiting frequency, then every text ever written or yet to be written, as well as every word ever spoken or yet to be spoken, can be found coded in its sequence. The property of normality has not yet been proven, although it is suspected to hold. Does this mean that all wisdom is encoded in the simple circle? No. The property is nothing special: it also applies to the number 0.123456789101112131415161718192021... and many others. Can you specify a few examples? By the way, in the graph of the exponential function ex , the point (0, 1) is the only point with two rational coordinates. If you imagine painting in blue all points on the plane with two rational coordinates, the plane would look quite bluish. Nevertheless, the graph goes through only one of these points and manages to avoid all the others.

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Ref. 1204

π e γ ln 2 ln 10 º 10

Copyright © Christoph Schiller November 1997–May 2006

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bibliography

1171

Biblio graphy 1182 Le Système International d’Unités, Bureau International des Poids et Mesures, Pavillon de

Breteuil, Parc de Saint Cloud, 92310 Sèvres, France. All new developments concerning SI units are published in the journal Metrologia, edited by the same body. Showing the slow pace of an old institution, the BIPM launched a website only in 1998; it is now reachable at http://www.bipm.fr. See also the http://www.utc.fr/~tthomass/Themes/Unites/index.html website; this includes the biographies of people who gave their names to various units. The site of its British equivalent, http://www.npl.co.uk/npl/reference, is much better; it provides many details as well as the English-language version of the SI unit definitions. Cited on page 1154.

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1183 The bible in the field of time measurement is the two-volume work by J. Vanier & C.

1184 The unofficial prefixes were first proposed in the 1990s by Jeff K. Aronson of University of

Oxford, and might come into general usage in the future. Cited on page 1156. 1185 For more details on electromagnetic unit systems, see the standard text by John David

Jackson, Classical Electrodynamics, 3rd edition, Wiley, 1998. Cited on page 1159. 1186 G.J. Stoney, On the physical units of nature, Philosophical Magazine 11, pp. 381–391,

1881. Cited on page 1159. 1187 D.J. Bird & al., Evidence for correlated changes in the spectrum and composition of

cosmic rays at extremely high energies, Physical Review Letters 71, pp. 3401–3404, 1993. Cited on page 1159. 1188 P.J. Hakonen, R.T. Vuorinen & J.E. Martikainen, Nuclear antiferromagnetism

in rhodium metal at positive and negative nanokelvin temperatures, Physical Review Letters 70, pp. 2818–2821, 1993. See also his article in Scientific American, January 1994. Cited on page 1160. 1189 G. Charpak & R.L. Garwin, The DARI, Europhysics News 33, pp. 14–17, Janu-

ary/February 2002. Cited on page 1160. 1190 Measured values and ranges for physical quantities are collected in Horst Völz &

1191 See e.g. K. Codling & L.J. Frasinski, Coulomb explosion of simple molecules in in-

tense laser fields, Contemporary Physics 35, pp. 243–255, 1994. Cited on page 1161. 1192 A. Zeilinger, The Planck stroll, American Journal of Physics 58, p. 103, 1990. Can you Challenge 1541 e

find another similar example? Cited on page 1161. 1193 The most precise clock built in 2004, a caesium fountain clock, had a precision of one

part in 1015 . Higher precision has been predicted to be possible soon, among others by M. Takamoto, F.-L. Hong, R. Higashi & H. Katori, An optical lattice clock, Nature 435, pp. 321–324, 2005. Cited on page 1162. 1194 J. Bergquist, ed., Proceedings of the Fifth Symposium on Frequency Standards and Met-

rology, World Scientific, 1997. Cited on page 1162.

Copyright © Christoph Schiller November 1997–May 2006

Peter Ackermann, Die Welt in Zahlen, Spektrum Akademischer Verlag, 1996. Cited on page 1161.

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Aud oin, The Quantum Physics of Atomic Frequency Standards, Adam Hilge, 1989. A popular account is Tony Jones, Splitting the Second, Institute of Physics Publishing, 2000. The site http://opdaf1.obspm.fr/www/lexique.html gives a glossary of terms used in the field. On length measurements, see ... On electric current measurements, see ... On mass and atomic mass measurements, see page 314. On high-precision temperature measurements, see page 270. Cited on page 1155.

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b units, measurements and constants

1172

1195 See the information on Ds mesons from the particle data group at http://pdg.web.cern.ch/

pdg. Cited on page 1162.

1196 About the long life of tantalum 180, see D. Belic & al., Photoactivation of 180 Tam and its

implications for the nucleosynthesis of nature’s rarest naturally occurring isotope, Physical Review Letters 83, pp. 5242–5245, 20 December 1999. Cited on page 1162. 1197 See the review by L. Ju, D.G. Bl air & C. Zhao, The detection of gravitational waves,

Reports on Progress in Physics 63, pp. 1317–1427, 2000. Cited on page 1163. 1198 See the clear and extensive paper by G.E. Stedman, Ring laser tests of fundamental

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physics and geophysics, Reports on Progress in Physics 60, pp. 615–688, 1997. Cited on page 1163. 1199 Following a private communication by Richard Rusby, this is the value of 1997, whereas it

1200 J. Short, Newton’s apples fall from grace, New Scientist, 2098, p. 5, 6 September 1997.

More details can be found in R.G. Keesing, The history of Newton’s apple tree, Contemporary Physics 39, pp. 377–391, 1998. Cited on page 1164. 1201 The various concepts are even the topic of a separate international standard, ISO 5725, with

the title Accuracy and precision of measurement methods and results. A good introduction is John R. Taylor, An Introduction to Error Analysis: the Study of Uncertainties in Physical Measurements, 2nd edition, University Science Books, Sausalito, 1997. Cited on page 1164. 1202 P.J. Mohr & B.N. Taylor, CODATA recommended values of the fundamental physical

constants: 1998, Reviews of Modern Physics 59, p. 351, 2000. This is the set of constants resulting from an international adjustment and recommended for international use by the Committee on Data for Science and Technology (CODATA), a body in the International Council of Scientific Unions, which brings together the International Union of Pure and Applied Physics (IUPAP), the International Union of Pure and Applied Chemistry (IUPAC) and other organizations. The website of IUPAC is http://www.iupac.org. Cited on page 1165. 1203 The details are given in the well-known astronomical reference, P. Kenneth Seidel-

mann, Explanatory Supplement to the Astronomical Almanac, 1992. Cited on page 1168. 1204 For information about the number π, as well as about other constants, the website http://

or the beautiful formula discovered in 1996 by Bailey, Borwein and Plouffe 1 4 2 1 1 ( − − − ). n 8n + 1 16 8n + 4 8n + 5 8n +6 n=0

π=Q ª

(815)

The site also explains the newly discovered methods for calculating specific binary digits of π without having to calculate all the preceding ones. By the way, the number of (consecutive) digits known in 1999 was over 1.2 million million, as told in Science News 162, p. 255, 14 December 2002. They pass all tests of randomness, as the http://mathworld.wolfram.com/ PiDigits.html website explains. However, this property, called normality, has never been

Copyright © Christoph Schiller November 1997–May 2006

oldweb.cecm.sfu.ca/pi/pi.html provides lots of data and references. It also has a link to the overview at http://mathworld.wolfram.com/Pi.html and to many other sites on the topic. Simple formulae for π are ª n 2n (814) π + 3 = Q 2n n=1 ‰ n Ž

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was estimated as 99.975°C in 1989, as reported by Gareth Jones & R ichard Rusby, Official: water boils at 99.975°C, Physics World 2, pp. 23–24, September 1989, and R.L. Rusby, Ironing out the standard scale, Nature 338, p. 1169, March 1989. For more details on temperature measurements, see page 270. Cited on page 1163.

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bibliography

Challenge 1542 r Challenge 1543 n

1173

proven; it is the biggest open question about π. It is possible that the theory of chaotic dynamics will lead to a solution of this puzzle in the coming years. Another method to calculate π and other constants was discovered and published by D.V. Chudnovsky & G.V. Chudnovsky, The computation of classical constants, Proceedings of the National Academy of Sciences (USA) 86, pp. 8178–8182, 1989. The Chudnowsky brothers have built a supercomputer in Gregory’s apartment for about 70 000 euros, and for many years held the record for calculating the largest number of digits of π. They have battled for decades with Kanada Yasumasa, who held the record in 2000, calculated on an industrial supercomputer. New formulae to calculate π are still occasionally discovered. For the calculation of Euler’s constant γ see also D.W. DeTemple, A quicker convergence to Euler’s constant, The Mathematical Intelligencer, pp. 468–470, May 1993. Note that little is known about the basic properties of some numbers; for example, it is still not known whether π + e is a rational number or not! (It is believed that it is not.) Do you want to become a mathematician? Cited on page 1170.

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Appendix C

PARTIC LE PROPERTI ES

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T

he following table lists the known and predicted elementary particles. he list has not changed since the mid-1970s, mainly because of the inefficient use hat was made of the relevant budgets across the world.

Radiation

electromagnetic interaction

weak interaction

strong interaction

γ

W+ , W−

д1 ... д8

Z0 photon

intermediate vector bosons

gluons

Radiation particles are bosons with spin 1. W − is the antiparticle of W + ; all others are their own antiparticles. Matter

generation 1

generation 2

generation 3

Leptons

e

µ

τ

νe

νµ

ντ

Quarks

d

s

t

(each in three colours)

u

c

b

Hypothetical matter and radiation Higgs boson Supersymmetric partners

H se ...

predicted to have spin 0 and to be elementary

h

one partner for each of the above

Copyright © Christoph Schiller November 1997–May 2006

Matter particles are fermions with spin 1/2; all have a corresponding antiparticle.

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TA B L E 88 The elementary particles

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c particle properties

1175

The following table lists all properties for the elementary particles (for reasons of space, the colour quantity is not given explicitely).* The table and its header thus allow us, in principle, to deduce a complete characterization of the intrinsic properties of any composed moving entity, be it an object or an image. TA B L E 89 Elementary particle properties

Particle

Mass m

a

photon γ

0 (<10−52 kg)

W

80.425(38) GeV~c 2

Z

91.1876(21) GeV~c 2

gluon

0

Isospin I, spin J, c parity P, charge parity C

I(J PC ) = 0, 1(1−− ) 2.124(41) GeV J = 1 67.96(35)% hadrons, 32.04(36)% l + ν J=1 2.4952(43) GeV~c 2 69.91(6)% hadrons 10.0974(69)% l+l− stable I(J P ) = 0(1− ) stable

Charge, isospin, strangeness, c charm, beau ty, topness: QISCBT

Lepton & baryon e numbers L B, Rparity

000000

0, 0, 1

100000

0, 0, 1

000000

0, 0, 1

000000

0, 0, 1

Elementary matter (fermions): leptons electron e

−100000 2.197 03(4) µs J = 21 99 % e − ν¯e ν µ = 105.658 3568(52) MeV~c 2 = 0.113 428 9168(34) u gyromagnetic ratio µ µ ~(eħ~2m µ ) = −1.001 165 916 02(64)

0.188 353 109(16) yg

1, 0, 1

1, 0, 1

* The official reference for all these data, worth a look for every physicist, is the massive collection of information compiled by the Particle Data Group, with the website http://pdg.web.cern.ch/pdg containing the most recent information. A printed review is published about every two years, with updated data, in one of the major journals on elementary particle physics. See for example S. Eidelman & al., Review of Particle Physics, Physics Letters B 592, p. 1, 2004. For many measured properties of these particles, the official reference is the set of CODATA values. The most recent list was published by P.J. Mohr & B.N. Taylor, Reviews of Modern Physics 59, p. 351, 2000.

Copyright © Christoph Schiller November 1997–May 2006

muon µ

−100 000 9.109 381 88(72) ë A 13 ë 1030 s J = 21 −31 2 10 kg = 81.871 0414(64) pJ~c = 0.510 998 902(21) MeV~c 2 = 0.000 548 579 9110(12) u gyromagnetic ratio µ e ~µ B = −1.001 159 652 1883(42) electric dipole moment d = (−0.3  0.8) ë 10−29 e m f

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Elementary radiation (bosons)

Lifetime τ or energy width, b main decay modes

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c particle properties

1176

Particle

tau τ

Mass m

a

Lifetime τ or energy width, b main decay modes

electric dipole moment d = (3.7  3.4) ë 10−22 e m 1.777 05(29) GeV~c 2

290.0(1.2) fs

J=

J=

Elementary matter (fermions): quarks д 1.5 to 5 MeV~c 2 3 to 9 MeV~c 2 60 to 170 MeV~c 2 1.25(15) GeV~c 2 4.25(15) GeV~c 2 173.8(5.2) GeV~c 2

J=

see proton see proton τ = 1.33(11) ps

Hypothetical, maybe elementary (boson)

−100000

1, 0, 1

1 2

1, 0, 1

1 2

1, 0, 1

I(J P ) = 21 ( 21 ) 23 21 0000 + I(J P ) = 21 ( 21 ) − 31 − 21 0000 + I(J P ) = 0( 21 ) − 31 0−1000 + I(J P ) = 0( 21 ) 23 00+100 + I(J P ) = 0( 21 ) − 31 000−10 + I(J P ) = 0( 21 ) 23 0000+1 +

J J J J J

J J J J J

=0 =0 =0 =0 =0 = = = = =

1 2 1 2 1 2 1 2 1 2

0 31 1 0 31 1 0 31 1 0 31 1 0 31 1 0 31 1

R = −1 R = −1 R = −1 R = −1 R = −1 R = −1 R = −1 R = −1 R = −1 R = −1

Notes: a. See also the table of SI prefixes on page 1155. About the eV~c 2 mass unit, see page 1159. b. The energy width Γ of a particle is related to its lifetime τ by the indeterminacy relation Γτ = ħ. There

Copyright © Christoph Schiller November 1997–May 2006

Selectron h Smuon h Stauon h Sneutrinos h Squark h Hypothetical elementary matter (fermions)

1, 0, 1

1 2

Hypothetical elementary radiation (bosons)

Higgsino(s) h Wino h (a chargino) Zino h (a neutralino) Photino h Gluino h

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1 2

J=0

Higgs h H

A 79.3 GeV~c 2

J=

Lepton & baryon e numbers L B, Rparity

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el. neutrino < 3 eV~c 2 νe muon < 0.19 MeV~c 2 neutrino ν µ tau neutrino < 18.2 MeV~c 2 ντ up u down d strange s charm c bottom b top t

Charge, isospin, strangeness, c charm, beau ty, topness: QISCBT

Isospin I, spin J, c parity P, charge parity C

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c particle properties

Challenge 1544 n

is a difference between the half-life t 1~2 and the lifetime τ of a particle: they are related by t 1~2 = τ ln 2, where ln 2  0.693 147 18; the half-life is thus shorter than the lifetime. The unified atomic mass unit u is defined as 1/12 of the mass of a carbon 12 atom at rest and in its ground state. One has 1 u = 121 m(12 C) = 1.660 5402(10) yg. c. To keep the table short, the header does not explicitly mention colour, the charge of the strong interactions. This has to be added to the list of basic object properties. Quantum numbers containing the word ‘parity’ are multiplicative; all others are additive. Time parity T (not to be confused with topness T), better called motion inversion parity, is equal to CP. The isospin I (or I Z ) is defined only for up and down quarks and their composites, such as the proton and the neutron. In the literature one also sees references to the so-called G-parity, defined as G = (−1)IC . d. ‘Beauty’ is now commonly called bottomness; similarly, ‘truth’ is now commonly called topness. The signs of the quantum numbers S, I, C, B, T can be defined in different ways. In the standard assignment shown here, the sign of each of the non-vanishing quantum numbers is given by the sign of the charge of the corresponding quark. e. R-parity is a quantum number important in supersymmetric theories; it is related to the lepton number L, the baryon number B and the spin J through the definition R = (−1)3B+L+2J . All particles from the standard model are R-even, whereas their superpartners are odd. f . The electron radius is less than 10−22 m. It is possible to store single electrons in traps for many months. д. See page 933 for the precise definition and meaning of the quark masses. h. Currently a hypothetical particle.

Using the table of elementary particle properties, together with the standard model and the fundamental constants, in principle all properties of composite matter and radiation can be deduced, including all those encountered in everyday life. (Can you explain how the size of an object follows from them?) In a sense, this table contains all our knowledge of matter and radiation, including materials science, chemistry and biology. The most important examples of composites are grouped in the following table. TA B L E 90 Properties of selected composites

C o m p o si t e

M a ss m, q ua n t u m L i f e t i m e τ , m a i n numbers d e c ay m o d e s

mesons (hadrons, bosons) (selected from over 130 known types) º ¯ pion π 0 (u u¯ − d d)~ 2 134.976 4(6) MeV~c 2 84(6) as, 2γ 98.798(32)% ¯ pion π+ (u d)

kaon K S0 kaon K L0 ¯ kaon K (u¯s , us) 

PC

−+

m K S0 + 3.491(9) µeV~c 2

493.677(16) MeV~c

2

51.7(4) ns

12.386(24) ns, µ + ν µ 63.51(18)% π+ π 0 21.16(14)% n.a.

kaon K 0 (d¯s) (50 % K S , 50 % 497.672(31) MeV~c 2 KL ) all kaons K  , K 0 , K S0 , K L0 I(J P ) = 21 (0− ), S = 1, B = C = 0

baryons (hadrons, fermions) (selected from over 100 known types)

 1 fm

 1 fm  1 fm

 1 fm

 1 fm

 1 fm

Copyright © Christoph Schiller November 1997–May 2006

I (J ) = 1 (0 ), S = C = B = 0 139.569 95(35) MeV~c 2 26.030(5) ns, µ + ν µ 99.987 7(4)% G P − − I (J ) = 1 (0 ), S = C = B = 0 m K S0 89.27(9) ps G

Size (diam.)

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Ref. 1205, Ref. 1206

1177

Dvipsbugw


c particle properties

1178

C o m p o si t e proton p or N + (uud)

neutron n or N 0 (udd)

1.672 621 58(13) yg τ total A 1.6 ë 1025 a, = 1.007 276 466 88(13) u τ(p e + π 0 ) A5.5 ë 1032 a = 938.271 998(38) MeV~c 2 + I(J P ) = 21 ( 21 ), S = 0 gyromagnetic ratio µ p ~µ N = 2.792 847 337(29) electric dipole moment d = (−4  6) ë 10−26 e m electric polarizability α e = 12.1(0.9) ë 10−4 fm3 magnetic polarizability α m = 2.1(0.9) ë 10−4 fm3

1.674 927 16(13) yg 887.0(2.0) s, pe − ν¯e 100 % = 1.008 664 915 78(55) u = 939.565 330(38) MeV~c 2 + I(J P ) = 21 ( 21 ), S = 0 gyromagnetic ratio µ n ~µ N = −1.913 042 72(45) electric dipole moment d n = (−3.3  4.3) ë 10−28 e m electric polarizability α = 0.98(23) ë 10−3 fm3 1672.43(32) MeV~c 2 82.2(1.2) ps, ΛK − 67.8(7)%, Ξ 0 π− 23.6(7)% gyromagnetic ratio µ Ω ~µ N = −1.94(22)

composite radiation: glueballs glueball f 0 (1500)

1503(11) MeV I G (J PC ) = 0+ (0++ )

full width 120(19) MeV

atoms (selected from 114 known elements with over 2000 known nuclides) 1

1.007 825 032(1) u = 1.6735 yg 1.007 u = 1.67 yg 4.002 603250(1) u = 6.6465 yg 12 u = 19.926 482(12) yg 209 u 0.1 ps 180 u

 1 fm

 1 fm

 1 fm

Ref. 1208

Ref. 1209

223 u 289 u

22 min 0.6 ms

A 1025 a A 1025 a

Dvipsbugw

Ref. 1210

1.9(2)1019 a

 2u  18 u

Ref. 1207

2 ë 30 pm 2 ë 30 pm 2 ë 32 pm 2 ë 77 pm

209 u

molecules (selected from over 107 known types) hydrogen (H2 ) water (H2 O)

A 1015 a

0.89(1) fm

Ref. 1210

2 ë 0.28 nm

Copyright © Christoph Schiller November 1997–May 2006

hydrogen ( H) [lightest] antihydrogen helium (4 He) [smallest] carbon (12 C) bismuth (209 Bi‡ ) [shortest living and rarest] tantalum (180 Ta) [second longest living radioactive] bismuth (209 Bi) [longest living radioactive] francium (223 Fr) [largest] atom 116 (289 Uuh) [heaviest]

Size (diam.)

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omega Ω− (sss)

M a ss m, q ua n t u m L i f e t i m e τ , m a i n numbers d e c ay m o d e s

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c particle properties

1179

C o m p o si t e

M a ss m, q ua n t u m L i f e t i m e τ , m a i n numbers d e c ay m o d e s

ATP

507 u

(adenosinetriphosphate) human Y chromosome

70 ë 106 base pairs

other composites

 50 a 7 plus 120 days not fecundated:  5 d

cell (ovule)

1 µg

cell (E. coli) adult human

1 pg 35 kg < m < 350 kg

fecundated: over 4000 million years 4000 million years body: 2 µm 9 τ  2.5 ë 10 s Ref. 1211  1.7 m  600 million breaths  2 500 million heartbeats < 122 a, 60 % H2 O and 40 % dust A 130 a A 4 km

Notes (see also those of the previous table): G-parity is defined only for mesons and given by G = (−1) L+S+I = (−1) I C. In 2003, experiments provided candidates for tetraquarks, namely the X(3872), Ds(2317), f(980) and Ds(2460), and for pentaquarks, namely the Θ+ (1500) particle. Time will tell whether these interpretations are correct. Neutrons bound in nuclei have a lifetime of at least 1020 years. The f 0 (1500) resonance is a candidate for the glueball ground state and thus for a radiation composite. The Y (3940) resonance is a candidate for a hybrid meson, a composite of a gluon and a quark– antiquark pair. This prediction of 1980 seems to have been confirmed in 2005. In 2002, the first evidence for the existence of tetra-neutrons was published by a French group. However, more recent investigations seem to have refuted the claim. The number of existing molecules is several orders of magnitude larger than the number of molecules that have been analysed and named. Some nuclei have not yet been observed; in 2006 the known nuclei ranged from 1 to 116, but 113 and 115 were still missing. The first anti-atoms, made of antielectrons and antiprotons, were made in January 1996 at CERN in Geneva. All properties of antimatter checked so far are consistent with theoretical predictions. The charge parity C is defined only for certain neutral particles, namely those that are different from their antiparticles. For neutral mesons, the charge parity is given by C = (−1) L+S , where L

Dvipsbugw

Copyright © Christoph Schiller November 1997–May 2006

Ref. 1214

See the table on page 185.

20 m  10 µm length 60 µm, head 3 µm5 µm  120 µm

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Ref. 1213

c. 50 mm (uncoiled)

 1 kg 0.1 ng 10 pg

larger composites

Ref. 1212

A 106 a

c. 3 nm

blue whale nerve cell cell (red blood) cell (sperm)

heaviest living thing: colony 6.6 ë 106 kg of aspen trees

Page 1176

A 1010 a

Size (diam.)

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1180

c particle properties

is the orbital angular momentum. P is the parity under space inversion r −r. For mesons, it is related to the orbital angular momentum L through P = (−1) L+1 . The electric polarizability, defined on page 594, is predicted to vanish for all elementary particles.

Copyright © Christoph Schiller November 1997–May 2006

* The ‘average formula’ of life is approximately C5 H40 O18 N.

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Ref. 1217

The most important matter composites are the atoms. Their size, structure and interactions determine the properties and colour of everyday objects. Atom types, also called elements in chemistry, are most usefully set out in the so-called periodic table, which groups together atoms with similar properties in rows and columns. It is given in Table 91 and results from the various ways in which protons, neutrons and electrons can combine to form aggregates. Comparable to the periodic table of the atoms, there are tables for the mesons (made of two quarks) and the baryons (made of three quarks). Neither the meson nor the baryon table is included here; they can both be found in the cited Review of Particle Physics at http://pdg.web.cern.ch/pdg. In fact, the baryon table still has a number of vacant spots. However, the missing particles are extremely heavy and short-lived (which means expensive to make and detect), and their discovery is not expected to yield deep new insights. The atomic number gives the number of protons (and electrons) found in an atom of a given element. This number determines the chemical behaviour of an element. Most – but not all – elements up to 92 are found on Earth; the others can be produced in laboratories. The highest element discovered is element 116. (In a famous case of research fraud, a scientist in the 1990s tricked two whole research groups into claiming to have made and observed elements 116 and 118. Element 116 was independently made and observed by another group later on.) Nowadays, extensive physical and chemical data are available for every element. Elements in the same group behave similarly in chemical reactions. The periods define the repetition of these similarities. More elaborate periodic tables can be found on the http://chemlab.pc.maricopa.edu/periodic website. The most beautiful of them all can be found on page 838 of this text. Group 1 are the alkali metals (though hydrogen is a gas), group 2 the Earth-alkali metals. Actinoids, lanthanoids and groups 3 to 13 are metals; in particular, groups 3 to 12 are transition or heavy metals. The elements of group 16 are called chalkogens, i.e. oreformers; group 17 are the halogens, i.e. the salt-formers, and group 18 are the inert noble gases, which form (almost) no chemical compounds. The groups 13, 14 and 15 contain metals, semimetals, a liquid and gases; they have no special name. Groups 1 and 13 to 17 are central for the chemistry of life; in fact, 96 % of living matter is made of C, O, N, H;* almost 4 % of P, S, Ca, K, Na, Cl; trace elements such as Mg, V, Cr, Mn, Fe, Co, Ni, Cu, Zn, Cd, Pb, Sn, Li, Mo, Se, Si, I, F, As, B form the rest. Over 30 elements are known to be essential for animal life. The full list is not yet known; candidate elements to extend this list are Al, Br, Ge and W. Many elements exist in versions with different numbers of neutrons in their nucleus, and thus with different mass; these various isotopes – so called because they are found at the same place in the periodic table – behave identically in chemical reactions. There are

Dvipsbugw


c particle properties

1181

TA B L E 91 The periodic table of the elements known in 2006, with their atomic numbers

Group 1 I

2 II

3 4 5 6 7 IIIa IVa Va VIa VIIa

8

9 10 VIIIa

11 Ia

12 IIa

13 III

14 IV

15 V

16 VI

17 18 VII VIII

Period 1

2

4

5

6

7

2

H

He

3

4

5

6

7

8

9

10

Li

Be

B

C

N

O

F

Ne

11

12

13

14

15

16

17

18

Al

Si

P

S

Cl

Ar

31

32

33

Na Mg 19

20

21

22

23

24

K

Ca

Sc

Ti

V

Cr Mn Fe

37

38

39

40

41

42

Rb

Sr

Y

Zr

55

56

72

73

74

75

76

77

78

Cs

Ba

‡

Hf

Ta

W

Re

Os

Ir

Pt

104

105

106

107

108

109

110

Rf

Db Sg Bh Hs Mt Ds Uuu Uub Uut Uuq Uup Uuh

87

88

Fr

Ra

‡‡

Lanthanoids ‡ Actinoids

43

Nb Mo Tc

57

58

59

La

Ce

Pr

60

89

90

91

92

Ac

Th

Pa

U

26

44

27

28

Co

Ni

45

46

29

30

Cu Zn Ga Ge As 47

48

62

63

94

95

36

Kr

50

51

52

53

54

In

Sn

Sb

Te

I

Xe

81

82

83

84

85

86

Au Hg

Tl

Pb

Bi

Po

At

Rn

111

113

114

115

116

117

118

79

64

80

112

65

66

67

Nd Pm Sm Eu Gd Tb Dy Ho 93

35

Br

49

Ru Rh Pd Ag Cd

61

34

Se

96

97

Np Pu Am Cm Bk

68

Er 100

69

70

Tm Yb 101

71

Lu

98

99

102

103

Cf

Es Fm Md No

Lr

over 2000 of them. TA B L E 92 The elements, with their atomic number, average mass, atomic radius and main properties

Name

Sym- At. bol n.

Aver. mass a in u (error), longest lifetime

Ato- Main properties, (naming) h dismic e covery date and use radius in pm

Actiniumb

Ac

(227.0277(1)) 21.77(2) a

(188)

89

highly radioactive metallic rare Earth (Greek aktis ray) 1899, used as alphaemitting source

Copyright © Christoph Schiller November 1997–May 2006

Ref. 1208, Ref. 1215

‡‡

25

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3

1

Dvipsbugw


c particle properties

1182

Aver. mass a in u (error), longest lifetime

Ato- Main properties, (naming) h dismic e covery date and use radius in pm

Aluminium

Al

13

26.981 538 (8) stable

118c, 143m

Americiumb

Am

95

(184)

Antimony

Sb

51

(243.0614(1)) 7.37(2) ka 121.760(1) f stable

Argon

Ar

18

39.948(1) f stable

(71n)

Arsenic

As

33

74.921 60(2) stable

120c, 185v

Astatineb

At

85

(140)

Barium

Ba

56

(209.9871(1)) 8.1(4) h 137.327(7) stable

Berkeliumb

Bk

97

Beryllium

Be

4

Bismuth

Bi

Bohriumb

137c, 159m, 205v

224m

(247.0703(1)) 1.4(3) ka 9.012 182(3) stable

n.a. 106c, 113m

83

208.980 40(1) stable

170m, 215v

Bh

107

(264.12(1)) 0.44 s д

n.a.

Boron

B

5

10.811(7) f stable

83c

Bromine

Br

35

79.904(1) stable

120c, 185v

light metal (Latin alumen alum) 1827, used in machine construction and living beings radioactive metal (Italian America from Amerigo) 1945, used in smoke detectors toxic semimetal (via Arabic from Latin stibium, itself from Greek, Egyptian for one of its minerals) antiquity, colours rubber, used in medicines, constituent of enzymes noble gas (Greek argos inactive, from anergos without energy) 1894, third component of air, used for welding and in lasers poisonous semimetal (Greek arsenikon tamer of males) antiquity, for poisoning pigeons and doping semiconductors radioactive halogen (Greek astatos unstable) 1940, no use Earth-alkali metal (Greek bary heavy) 1808, used in vacuum tubes, paint, oil industry, pyrotechnics and X-ray diagnosis made in lab, probably metallic (Berkeley, US town) 1949, no use because rare toxic Earth-alkali metal (Greek beryllos, a mineral) 1797, used in light alloys, in nuclear industry as moderator diamagnetic metal (Latin via German weisse Masse white mass) 1753, used in magnets, alloys, fire safety, cosmetics, as catalyst, nuclear industry made in lab, probably metallic (after Niels Bohr) 1981, found in nuclear reactions, no use semimetal, semiconductor (Latin borax, from Arabic and Persian for brilliant) 1808, used in glass, bleach, pyrotechnics, rocket fuel, medicine red-brown liquid (Greek bromos strong odour) 1826, fumigants, photography, water purification, dyes, medicines

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Sym- At. bol n.

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Name

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c particle properties

1183

Name

Sym- At. bol n.

Aver. mass a in u (error), longest lifetime

Ato- Main properties, (naming) h dismic e covery date and use radius in pm

Cadmium

Cd

112.411(8) f stable

157m

48

55

132.905 4519(2) 273m stable

Calcium

Ca

20

40.078(4) f stable

197m

Californiumb

Cf

98

(251.0796(1)) 0.90(5) ka

n.a.

Carbon

C

6

12.0107(8) f stable

77c

Cerium

Ce

58

140.116(1) f stable

183m

Chlorine

Cl

17

35.453(2) f stable

102c, 175v

Chromium

Cr

24

51.9961(6) stable

128m

Cobalt

Co

27

58.933 195(5) stable

125m

Copper

Cu

29

63.546(3) f stable

128m

Curiumb

Cm

96

(247.0704(1)) 15.6(5) Ma

n.a.

Copyright © Christoph Schiller November 1997–May 2006

Cs

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Caesium

heavy metal, cuttable and screaming (Greek kadmeia, a zinc carbonate mineral where it was discovered) 1817, electroplating, solder, batteries, TV phosphors, dyes alkali metal (Latin caesius sky blue) 1860, getter in vacuum tubes, photoelectric cells, ion propulsion, atomic clocks Earth-alkali metal (Latin calcis chalk) antiquity, pure in 1880, found in stones and bones, reducing agent, alloying made in lab, probably metallic, strong neutron emitter (Latin calor heat and fornicare have sex, the land of hot sex :-) 1950, used as neutron source, for well logging makes up coal and diamond (Latin carbo coal) antiquity, used to build most life forms rare Earth metal (after asteroid Ceres, Roman goddess) 1803, cigarette lighters, incandescent gas mantles, glass manufacturing, self-cleaning ovens, carbon-arc lighting in the motion picture industry, catalyst, metallurgy green gas (Greek chloros yellow-green) 1774, drinking water, polymers, paper, dyes, textiles, medicines, insecticides, solvents, paints, rubber transition metal (Greek chromos colour) 1797, hardens steel, makes steel stainless, alloys, electroplating, green glass dye, catalyst ferromagnetic transition metal (German Kobold goblin) 1694, part of vitamin B12 , magnetic alloys, heavy-duty alloys, enamel dyes, ink, animal nutrition red metal (Latin cuprum from Cyprus island) antiquity, part of many enzymes, electrical conductors, bronze, brass and other alloys, algicides, etc. highly radioactive, silver-coloured (after Pierre and Marie Curie) 1944, used as radioactivity source

Dvipsbugw


c particle properties

1184

Name

Sym- At. bol n.

Darmstadtiumb Ds Dubniumb Db

110 105

Aver. mass a in u (error), longest lifetime

Ato- Main properties, (naming) h dismic e covery date and use radius in pm

(271) 1.6 min д n.a. (262.1141(1)) n.a. 34(5) s

66

162.500(1) f stable

177m

Einsteiniumb

Es

99

n.a.

Erbium

Er

68

Europium

Eu

63

(252.0830(1)) 472(2) d 167.259(3) f stable 151.964(1) f stable

Fermiumb

Fm

100

Fluorine

F

9

Franciumb

Fr

87

Gadolinium

Gd

64

Gallium

Ga

31

Germanium

Ge

Gold

176m 204m

(257.0901(1)) n.a. 100.5(2) d 18.998 4032(5) 62c, stable 147v (223.0197(1)) 22.0(1) min 157.25(3) f stable 69.723(1) stable

125c, 141m

32

72.64(1) stable

122c, 195v

Au

79

196.966 569(4) 144m stable

Hafnium

Hf

72

Hassiumb

Hs

108

178.49(2) c 158m stable (277) 16.5 min д n.a.

Helium

He

2

4.002 602(2) f stable

(278) 180m

(31n)

gaseous halogen (from fluorine, a mineral, from Greek fluo flow) 1886, used in polymers and toothpaste radioactive metal (from France) 1939, no use (after Johan Gadolin) 1880, used in lasers and phosphors almost liquid metal (Latin for both the discoverer’s name and his nation, France) 1875, used in optoelectronics semiconductor (from Germania, as opposed to gallium) 1886, used in electronics heavy noble metal (Sanskrit jval to shine, Latin aurum) antiquity, electronics, jewels metal (Latin for Copenhagen) 1923, alloys, incandescent wire radioactive element (Latin form of German state Hessen) 1984, no use noble gas (Greek helios Sun) where it was discovered 1895, used in balloons, stars, diver’s gas and cryogenics

Copyright © Christoph Schiller November 1997–May 2006

Dy

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Dysprosium

(after the German city) 1994, no use made in lab in small quantities, radioactive (Dubna, Russian city) 1967, no use (once known as Hahnium) rare Earth metal (Greek dysprositos difficult to obtain) 1886, used in laser materials, as infrared source material, and in nuclear industry made in lab, radioactive (after Albert Einstein) 1952, no use rare Earth metal (Ytterby, Swedish town) 1843, used in metallurgy and optical fibres rare Earth metal (named after the continent) 1901, used in red screen phosphor for TV tubes (after Enrico Fermi) 1952, no use

Dvipsbugw


c particle properties

1185

Aver. mass a in u (error), longest lifetime

Ato- Main properties, (naming) h dismic e covery date and use radius in pm

Holmium

Ho

67

177m

Hydrogen

H

1

Indium

In

49

Iodine

I

53

Iridium

Ir

77

Iron

Fe

26

Krypton

Kr

36

Lanthanum

La

57

164.930 32(2) stable 1.007 94(7) f stable 114.818(3) stable 126.904 47(3) stable 192.217(3) stable 55.845(2) stable 83.798(2) f stable 138.905 47(7) c, f

Lawrenciumb

Lr

103

Lead

Pb

82

Lithium

Li

3

6.941(2) f stable

156m

Lutetium

Lu

71

173m

Magnesium

Mg

12

174.967(1) f stable 24.3050(6) stable

Manganese

Mn

25

54.938 045(5) stable

126m

Meitneriumb

Mt

109

n.a.

Mendeleviumb Md

101

(268.1388(1)) 0.070 s д (258.0984(1)) 51.5(3) d

30c 141c, 166m 140c, 198v 136m 127m (88n) 188m

stable (262.110 97(1)) n.a. 3.6(3) h 207.2(1) c, f 175m stable

160m

n.a.

metal (Stockholm, Swedish capital) 1878, alloys reactive gas (Greek for water-former) 1766, used in building stars and universe soft metal (Greek indikon indigo) 1863, used in solders and photocells blue-black solid (Greek iodes violet) 1811, used in photography precious metal (Greek iris rainbow) 1804, electrical contact layers metal (Indo-European ayos metal, Latin ferrum) antiquity, used in metallurgy noble gas (Greek kryptos hidden) 1898, used in lasers reactive rare Earth metal (Greek lanthanein to be hidden) 1839, used in lamps and in special glasses appears in reactions (after Ernest Lawrence) 1961, no use poisonous, malleable heavy metal (Latin plumbum) antiquity, used in car batteries, radioactivity shields, paints light alkali metal with high specific heat (Greek lithos stone) 1817, used in batteries, anti-depressants, alloys and many chemicals rare Earth metal (Latin Lutetia for Paris) 1907, used as catalyst light common alkaline Earth metal (from Magnesia, a Greek district in Thessalia) 1755, used in alloys, pyrotechnics, chemical synthesis and medicine, found in chlorophyll brittle metal (Italian manganese, a mineral) 1774, used in alloys, colours amethyst and permanganate appears in nuclear reactions (after Lise Meitner) 1982, no use appears in nuclear reactions (after Dimitri Ivanovitch Mendeleiev) 1955, no use

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Copyright © Christoph Schiller November 1997–May 2006

Sym- At. bol n.

Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

Name

Dvipsbugw


c particle properties

1186

Name

Sym- At. bol n.

Aver. mass a in u (error), longest lifetime

Ato- Main properties, (naming) h dismic e covery date and use radius in pm

Mercury

Hg

200.59(2) stable

157m

80

42

95.94(2) f stable

140m

Neodymium

Nd

60

182m

Neon

Ne

10

Neptuniumb

Np

93

144.242(3) c, f stable 20.1797(6) f stable (237.0482(1)) 2.14(1) Ma

Nickel

Ni

28

58.6934(2) stable

125m

Niobium

Nb

41

92.906 38(2) stable

147m

Nitrogen

N

7

14.0067(2) f stable

70c, 155v

Nobeliumb

No

102

n.a.

Osmium

Os

76

(259.1010(1)) 58(5) min 190.23(3) f stable

Oxygen

O

8

15.9994(3) f stable

66c, 152v

Palladium

Pd

46

106.42(1) f stable

138m

Phosphorus

P

15

30.973 762(2) stable

109c, 180v

(36n) n.a.

135m

noble gas (Greek neos new) 1898, used in lamps, lasers and cryogenics radioactive metal (planet Neptune, after Uranus in the solar system) 1940, appears in nuclear reactors, used in neutron detection and by the military metal (German Nickel goblin) 1751, used in coins, stainless steels, batteries, as catalyst ductile metal (Greek Niobe, mythical daughter of Tantalos) 1801, used in arc welding, alloys, jewellery, superconductors diatomic gas (Greek for nitre-former) 1772, found in air, in living organisms, Viagra, fertilizers, explosives (after Alfred Nobel) 1958, no use heavy metal (from Greek osme odour) 1804, used for fingerprint detection and in very hard alloys transparent, diatomic gas (formed from Greek to mean ‘acid former’) 1774, used for combustion, blood regeneration, to make most rocks and stones, in countless compounds, colours auroras red heavy metal (from asteroid Pallas, after the Greek goddess) 1802, used in alloys, white gold, catalysts, for hydride storage poisonous, waxy, white solid (Greek phosphoros light bearer) 1669, fertilizers, glasses, porcelain, steels and alloys, living organisms, bones

Copyright © Christoph Schiller November 1997–May 2006

Mo

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Molybdenum

liquid heavy metal (Latin god Mercurius, Greek hydrargyrum liquid silver) antiquity, used in switches, batteries, lamps, amalgam alloys metal (Greek molybdos lead) 1788, used in alloys, as catalyst, in enzymes and lubricants (Greek neos and didymos new twin) 1885

Dvipsbugw


c particle properties

1187

Name

Sym- At. bol n.

Aver. mass a in u (error), longest lifetime

Ato- Main properties, (naming) h dismic e covery date and use radius in pm

Platinum

Pt

195.084(9) stable

139m

78

94

(244.0642(1)) 80.0(9) Ma

n.a.

Polonium

Po

84

(208.9824(1)) 102(5) a

(140)

Potassium

K

19

39.0983(1) stable

238m

Praeseodymium Pr

59

140.907 65(2) stable

183m

Promethiumb

Pm

61

(144.9127(1)) 17.7(4) a

181m

Protactinium

Pa

91

(231.035 88(2)) n.a. 32.5(1) ka

Copyright © Christoph Schiller November 1997–May 2006

Pu

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Plutonium

silvery-white, ductile, noble heavy metal (Spanish platina little silver) pre-Columbian, again in 1735, used in corrosion-resistant alloys, magnets, furnaces, catalysts, fuel cells, cathodic protection systems for large ships and pipelines; being a catalyst, a fine platinum wire glows red hot when placed in vapour of methyl alcohol, an effect used in hand warmers extremely toxic alpha-emitting metal (after the planet) synthesized 1940, found in nature 1971, used as nuclear explosive, and to power space equipment, such as satellites and the measurement equipment brought to the Moon by the Apollo missions alpha-emitting, volatile metal (from Poland) 1898, used as thermoelectric power source in space satellites, as neutron source when mixed with beryllium; used in the past to eliminate static charges in factories, and on brushes for removing dust from photographic films reactive, cuttable light metal (German Pottasche, Latin kalium from Arabic quilyi, a plant used to produce potash) 1807, part of many salts and rocks, essential for life, used in fertilizers, essential to chemical industry white, malleable rare Earth metal (Greek praesos didymos green twin) 1885, used in cigarette lighters, material for carbon arcs used by the motion picture industry for studio lighting and projection, glass and enamel dye, darkens welder’s goggles radioactive rare Earth metal (from the Greek mythical figure of Prometheus) 1945, used as beta source and to excite phosphors radioactive metal (Greek protos first, as it decays into actinium) 1917, found in nature, no use

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c particle properties

1188

Name

Sym- At. bol n.

Aver. mass a in u (error), longest lifetime

Ato- Main properties, (naming) h dismic e covery date and use radius in pm

Radium

Ra

(226.0254(1)) 1599(4) a

(223)

88

Rn

86

(222.0176(1)) 3.823(4) d

Rhenium

Re

75

186.207(1) c stable

Rhodium

Rh

45

102.905 50(2) stable

Roentgeniumb Rg

111

Rubidium

Rb

37

(272.1535(1)) 1.5 ms д 85.4678(3) f stable

Ruthenium

Ru

44

101.107(2) f stable

134m

Rutherfordiumb Rf

104

n.a.

Samarium

Sm

62

(261.1088(1)) 1.3 min д 150.36(2) c, f stable

Scandium

Sc

21

44.955 912(6) stable

164m

Seaborgiumb

Sg

106

n.a.

Selenium

Se

34

266.1219(1) 21 s д 78.96(3) f stable

255m

180m

120c, 190v

silvery-white, reactive alkali metal (Latin rubidus red) 1861, used in photocells, optical glasses, solid electrolytes white metal (Latin Rhuthenia for Russia) 1844, used in platinum and palladium alloys, superconductors, as catalyst; the tetroxide is toxic and explosive radioactive transactinide (after Ernest Rutherford) 1964, no use silver-white rare Earth metal (from the mineral samarskite, after Wassily Samarski) 1879, used in magnets, optical glasses, as laser dopant, in phosphors, in high-power light sources silver-white metal (from Latin Scansia Sweden) 1879, the oxide is used in highintensity mercury vapour lamps, a radioactive isotope is used as tracer radioactive transurane (after Glenn Seaborg) 1974, no use red or black or grey semiconductor (Greek selene Moon) 1818, used in xerography, glass production, photographic toners, as enamel dye

Copyright © Christoph Schiller November 1997–May 2006

Radon

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highly radioactive metal (Latin radius ray) 1898, no use any more; once used in luminous paints and as radioactive source and in medicine (130n) radioactive noble gas (from its old name ‘radium emanation’) 1900, no use (any more), found in soil, produces lung cancer 138m (Latin rhenus for Rhine river) 1925, used in filaments for mass spectrographs and ion gauges, superconductors, thermocouples, flash lamps, and as catalyst 135m white metal (Greek rhodon rose) 1803, used to harden platinum and palladium alloys, for electroplating, and as catalyst n.a. 1994, no use

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c particle properties

1189

Name

Sym- At. bol n.

Aver. mass a in u (error), longest lifetime

Ato- Main properties, (naming) h dismic e covery date and use radius in pm

Silicon

Si

28.0855(3) f stable

105c, 210v

14

Silver

Ag

47

107.8682(2) f stable

Sodium

Na

11

22.989 769 28(2)191m

145m

Sr

38

87.62(1) f stable

215m

Sulphur

S

16

32.065(5) f stable

105c, 180v

Tantalum

Ta

73

180.947 88(2) stable

147m

Technetiumb

Tc

43

(97.9072(1)) 6.6(10) Ma

136m

Tellurium

Te

52

127.60(3) f stable

139c, 206v

Terbium

Tb

65

158.925 35(2) stable

178m

Thallium

Tl

81

204.3833(2) stable

172m

Thorium

Th

90

232.038 06(2)d , f 180m 14.0(1) Ga

Copyright © Christoph Schiller November 1997–May 2006

Strontium

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stable

grey, shiny semiconductor (Latin silex pebble) 1823, Earth’s crust, electronics, sand, concrete, bricks, glass, polymers, solar cells, essential for life white metal with highest thermal and electrical conductivity (Latin argentum, Greek argyros) antiquity, used in photography, alloys, to make rain light, reactive metal (Arabic souwad soda, Egyptian and Arabic natrium) component of many salts, soap, paper, soda, salpeter, borax, and essential for life silvery, spontaneously igniting light metal (Strontian, Scottish town) 1790, used in TV tube glass, in magnets, and in optical materials yellow solid (Latin) antiquity, used in gunpowder, in sulphuric acid, rubber vulcanization, as fungicide in wine production, and is essential for life; some bacteria use sulphur instead of oxygen in their chemistry heavy metal (Greek Tantalos, a mythical figure) 1802, used for alloys, surgical instruments, capacitors, vacuum furnaces, glasses radioactive (Greek technetos artificial) 1939, used as radioactive tracer and in nuclear technology brittle, garlic-smelling semiconductor (Latin tellus Earth) 1783, used in alloys and as glass component malleable rare Earth metal (Ytterby, Swedish town) 1843, used as dopant in optical material soft, poisonous heavy metal (Greek thallos branch) 1861, used as poison and for infrared detection radioactive (Nordic god Thor, as in ‘Thursday’) 1828, found in nature, heats Earth, used as oxide, in alloys and as coating

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c particle properties

1190

Aver. mass a in u (error), longest lifetime

Ato- Main properties, (naming) h dismic e covery date and use radius in pm

Thulium

Tm

69

175m

Tin

Sn

50

168.934 21(2) stable 118.710(7) f stable

Titanium

Ti

22

Tungsten

W

74

Ununbiumb Ununtrium Ununquadiumb Ununpentium Ununhexiumb Ununseptium Ununoctium Uranium

Uub Uut Uuq Uup Uuh Uus Uuo U

112 113 114 115 116 117 118 92

Vanadium

V

23

Xenon

Xe

54

Ytterbium

Yb

70

Yttrium

Y

39

Zinc

Zn

30

Zirconium

Zr

40

rare Earth metal (Thule, mythical name for Scandinavia) 1879, found in monazite 139c, grey metal that, when bent, allows one 210v, to hear the ‘tin cry’ (Latin stannum) an162m tiquity, used in paint, bronze and superconductors 47.867(1) 146m metal (Greek hero Titanos) 1791, alloys, stable fake diamonds 183.84(1) 141m heavy, highest-melting metal (Swedish stable tung sten heavy stone, German name Wolfram) 1783, lightbulbs (285) 15.4 min д n.a. 1996, no use n.a. 2004, no use (289) 30.4 s д n.a. 1999, no use n.a. 2004, no use (289) 0.6 ms д n.a. 2000 (earlier claim was false), no use n.a. not yet observed n.a. not yet observed, but false claim in 1999 238.028 91(3)d , f 156m radioactive and of high density (planet Ur4.468(3) ë 109 a anus, after the Greek sky god) 1789, found in pechblende and other minerals, used for nuclear energy 50.9415(1) 135m metal (Vanadis, scandinavian goddess of stable beauty) 1830, used in steel 131.293(6) f (103n) noble gas (Greek xenos foreign) 1898, stable 200v used in lamps and lasers f 173.04(3) 174m malleable heavy metal (Ytterby, Swedish stable town) 1878, used in superconductors 88.905 85(2) 180m malleable light metal (Ytterby, Swedish stable town) 1794, used in lasers 65.409(4) 139m heavy metal (German Zinke protuberstable ance) antiquity, iron rust protection 91.224(2) f 160m heavy metal (from the mineral zircon, stable after Arabic zargum golden colour) 1789, chemical and surgical instruments, nuclear industry

a. The atomic mass unit is defined as 1 u = 121 m(12 C), making 1 u = 1.660 5402(10) yg. For elements found on Earth, the average atomic mass for the naturally occurring isotope mixture is given, with the error in the last digit in brackets. For elements not found on Earth, the mass of the longest living isotope is given; as it is

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Sym- At. bol n.

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Ref. 1215

Name

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c particle properties

Ref. 1216 Ref. 1216

Challenge 1545 n

not an average, it is written in brackets, as is customary in this domain. b. The element is not found on Earth because of its short lifetime. c. The element has at least one radioactive isotope. d. The element has no stable isotopes. e. Strictly speaking, the atomic radius does not exist. Because atoms are clouds, they have no boundary. Several approximate definitions of the ‘size’ of atoms are possible. Usually, the radius is defined in such a way as to be useful for the estimation of distances between atoms. This distance is different for different bond types. In the table, radii for metallic bonds are labelled m, radii for (single) covalent bonds with carbon c, and Van der Waals radii v. Noble gas radii are labelled n. Note that values found in the literature vary by about 10 %; values in brackets lack literature references. The covalent radius can be up to 0.1 nm smaller than the metallic radius for elements on the (lower) left of the periodic table; on the (whole) right side it is essentially equal to the metallic radius. In between, the difference between the two decreases towards the right. Can you explain why? By the way, ionic radii differ considerably from atomic ones, and depend both on the ionic charge and the element itself. All these values are for atoms in their ground state. Excited atoms can be hundreds of times larger than atoms in the ground state; however, excited atoms do not form solids or chemical compounds. f . The isotopic composition, and thus the average atomic mass, of the element varies depending on the place where it was mined or on subsequent human treatment, and can lie outside the values given. For example, the atomic mass of commercial lithium ranges between 6.939 and 6.996 u. The masses of isotopes are known in atomic mass units to nine or more significant digits, and usually with one or two fewer digits in kilograms. The errors in the atomic mass are thus mainly due to the variations in isotopic composition. д. The lifetime errors are asymmetric or not well known. h. Extensive details on element names can be found on http://elements.vanderkrogt.net.

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Ref. 1208 Ref. 1215

1191

Copyright © Christoph Schiller November 1997–May 2006

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c particle properties

1192

Biblio graphy 1205 The electron radius limit is deduced from the д − 2 measurements, as explained in the

1206

1207

1209

1210

1211

1212

1214 1215

Copyright © Christoph Schiller November 1997–May 2006

1213

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1208

Nobel Prize talk by Hans Dehmelt, Experiments with an isolated subatomic particle at rest, Reviews of Modern Physics 62, pp. 525–530, 1990, or in Hans Dehmelt, Is the electron a composite particle?, Hyperfine Interactions 81, pp. 1–3, 1993. Cited on page 1177. G. Gabrielse, H. Dehmelt & W. Kells, Observation of a relativistic, bistable hysteresis in the cyclotron motion of a single electron, Physical Review Letters 54, pp. 537–540, 1985. Cited on page 1177. The proton charge radius was determined by measuring the frequency of light emitted by hydrogen atoms to high precision by T. Udem, A. Huber, B. Gross, J. R eichert, M. Prevedelli, M. Weitz & T.W. Hausch, Phase-coherent measurement of the hydrogen 1S–2S transition frequency with an optical frequency interval divider chain, Physical Review Letters 79, pp. 2646–2649, 1997. Cited on page 1178. For a full list of isotopes, see R.B. Firestone, Table of Isotopes, Eighth Edition, 1999 Update, with CD-ROM, John Wiley & Sons, 1999. For a list of isotopes on the web, see the Korean website by J. Chang, http://atom.kaeri.re.kr/. For a list of precise isotope masses, see the http://csnwww.in2p3.fr website. Cited on pages 1178, 1181, and 1191. For information on the long life of tantalum 180, see D. Belic & al., Photoactivation of 180 m Ta and its implications for the nucleosynthesis of nature’s rarest naturally occurring isotope, Physical Review Letters 83, pp. 5242–5245, 20 December 1999. Cited on page 1178. The ground state of bismuth 209 was thought to be stable until early 2003. It was then discovered that it was radioactive, though with a record lifetime, as reported by P. de Marcill ac, N. Coron, G. Dambier, J. Lebl anc & J.-P. Moalic, Experimental detection of α-particles from the radioactive decay of natural bismuth, Nature 422, pp. 876– 878, 2003. By coincidence, the excited state 83 MeV above the ground state of the same bismuth 209 nucleus is the shortest known radioactive nuclear state. Cited on page 1178. Stephen J. Gould, The Panda’s thumb, W.W. Norton & Co., 1980. This is one of several interesting and informative books on evolutionary biology by the best writer in the field. Cited on page 1179. A tetraquark is thought to be the best explanation for the f0 (980) resonance at 980 MeV. However, this assignment is still under investigation. Pentaquarks were first predicted by Maxim Polyakov, Dmitri Diakonov, and Victor Petrov in 1997. Two experimental groups in 2003 claimed to confirm their existence, with a mass of 1540 MeV; see K. Hicks, An experimental review of the Θ+ pentaquark, http://www.arxiv.org/abs/hep-ex/0412048. Results from 2005, however, seem to rule out that the 1540 MeV particle is a pentaquark. Cited on page 1179. F. Marques & al., Detection of neutron clusters, Physical Review C 65, p. 044006, 2002. Opposite results have been obtained by B.M. Sherrill & C.A. Bertul ani, Proton-tetraneutron elastic scattering, Physical Review C 69, p. 027601, 2004, and D.V. Aleksandrov & al., Search for resonances in the three- and four-neutron systems in the 7Li(7Li, 11C)3n and 7Li(7Li, 10C)4n reactions, JETP Letters 81, p. 43, 2005. Cited on page 1179. For a good review, see the article by P.T. Greenl and, Antimatter, Contemporary Physics 38, pp. 181–203, 1997. Cited on page 1179. The atomic masses, as given by IUPAC, can be found in Pure and Applied Chemistry 73, pp. 667–683, 2001, or on the http://www.iupac.org website. For an isotope mass list, see

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bibliography

1193

the http://csnwww.in2p3.fr website. Cited on pages 1181, 1190, and 1191. 1216 The metallic, covalent and Van der Waals radii are from Nathaniel W. Alcock, Bonding and Structure, Ellis Horwood, 1999. This text also explains in detail how the radii are defined and measured. Cited on page 1191. 1217 Almost everything known about each element and its chemistry can be found in the encyclopaedic Gmelin, Handbuch der anorganischen Chemie, published from 1817 onwards. There are over 500 volumes, now all published in English under the title Handbook of Inorganic and Organometallic Chemistry, with at least one volume dedicated to each chemical element. On the same topic, an incredibly expensive book with an equally bad layout is Per Enhag, Encyclopedia of the Elements, Wiley–VCH, 2004. Cited on page 1180.

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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006

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Appendix D

NUMBER S AND SPAC ES

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M

athematical concepts can all be expressed in terms of ‘sets’ and ‘relations.’ any fundamental concepts were presented in the first Intermezzo. Why does athematics, given this simple basis, grow into a passion for certain people? The following pages present a few more advanced concepts as simply* and vividly as possible, for all those who want to smell the passion for mathematics. In particular, in this appendix we shall expand the range of algebraic and of topological structures; the third basic type of mathematical structures, order structures, are not so important in physics. Mathematicians are concerned not only with the exploration of concepts, but also with their classification. Whenever a new mathematical concept is introduced, mathematicians try to classify all the possible cases and types. This has been achieved most spectacularly for the different types of numbers, for finite simple groups, and for many types of spaces and manifolds.

Numbers as mathematical structures Challenge 1546 ny

A person who can solve x 2 − 92y 2 = 1 in less than a year is a mathematician. Brahmagupta (b. 598 Sindh, d. 668) (implied: solve in integers)

— R is a commutative group with respect to addition, i.e. a + b > R, a + b = b + a, a + 0 = a, a + (−a) = a − a = 0 and a + (b + c) = (a + b) + c; — R is closed under multiplication, i.e. ab > R; * The opposite approach can have the same effect: it is taken in the delightful text by Carl E. L inderholm, Mathematics Made Difficult, Wolfe Publishing, 1971.

Copyright © Christoph Schiller November 1997–May 2006

We start with a short introduction to the vocabulary. Any mathematical system with the same basic properties as the natural numbers is called a semi-ring. Any mathematical system with the same basic properties as the integers is called a ring. (The term is due to David Hilbert. Both structures can also be finite rather than infinite.) More precisely, a ring (R, +, ë) is a set R of elements with two binary operations, called addition and multiplication, usually written + and ë (the latter may simply be understood without notation), for which the following properties hold for all elements a, b, c > R:

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Ref. 1218

A mathematician is a machine that transforms coffee into theorems. Paul Erdős (b. 1913 Budapest, d. 1996 Warsaw)

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numbers as mathematical structures

1195

— multiplication is associative, i.e. a(bc) = (ab)c; — distributivity holds, i.e. a(b + c) = ab + ac and (b + c)a = ba + ca.

Defining properties such as these are called axioms. Note that axioms are not basic beliefs, as is often stated; axioms are the basic properties used in the definition of a concept: in this case, of a ring. A semi-ring is a set satisfying all the axioms of a ring, except that the existence of neutral and negative elements for addition is replaced by the weaker requirement that if a + c = b + c then a = b. To incorporate division and define the rational numbers, we need another concept. A field K is a ring with

Ref. 1220

A ring or field is said to be commutative if the multiplication is commutative. A noncommutative field is also called a skew field. Fields can be finite or infinite. (A field or a ring is characterized by its characteristic p. This is the smallest number of times one has to add 1 to itself to give zero. If there is no such number the characteristic is set to 0. p is always a prime number or zero.) All finite fields are commutative. In a field, all equations of the type cx = b and xc = b (c x 0) have solutions for x; there is a unique solution if b x 0. To sum up sloppily by focusing on the most important property, a field is a set of elements for which, together with addition, subtraction and multiplication, a division (by non-zero elements) is also defined. The rational numbers are the simplest field that incorporates the integers. The system of the real numbers is the minimal extension of the rationals which is complete and totally ordered.* However, the concept of ‘number’ is not limited to these examples. It can be generalized in several ways. The simplest generalization is achieved by extending the real numbers to manifolds of more than one dimension.

— if a D b and b D c, then a D c; — if a D b and b D a, then a = b; — a D b or b D a holds. In summary, a set is totally ordered if there is a binary relation that allows to say about any two elements which one is the predecessor of the other in a consistent way.

Copyright © Christoph Schiller November 1997–May 2006

* A set is mathematically complete if physicists call it continuous. More precisely, a set of numbers is complete if every non-empty subset that is bounded above has a lest upper bound. A set is totally ordered if there exists a binary relation D between pairs of elements such that for all elements a and b

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— an identity 1, such that all elements a obey 1a = a; — at least one element different from zero; and most importantly — a (multiplicative) inverse a −1 for every element a =~ 0.

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d numbers and spaces

1196

Complex numbers A complex number is defined by z = a + ib, where a and b are real numbers, and i is a new symbol. Under multiplication, the generators of the complex numbers, 1 and i, obey ë 1 i

1 i 1 i i −1

(816)

(a 12 + a 22 )(b 12 + b 22 ) = (a 1 b 1 − a 2 b 2 )2 + (a 1 b 2 + a 2 b 1 )2

Challenge 1547 e

valid for all real numbers a i , b i . It was already known, in its version for integers, to Diophantus of Alexandria. Complex numbers can also be written as ordered pairs (a, A) of real numbers, with their addition defined as (a, A) + (b, B) = (a + b, A + ic B) and their multiplication defined as (a, A) ë (b, B) = (ab − AB, aB + bA). This notation allows us to identify the complex numbers with the points on a plane. Translating the definition of multiplication into geometrical language allows ih = − i ab c us to rapidly prove certain geometrical theorems, such as the one of Figure 391. Complex numbers a + ib can also be repres0 a b ented as 2  2 matrices

Page 1204 Challenge 1549 ny

witha, b > R .

F I G U R E 391 A property of triangles easily

(818)

provable with complex numbers

Matrix addition and multiplication then correspond to complex addition and multiplication. In this way, complex numbers can be represented by a special type of real matrix. What is SzS in matrix language? The set C of complex numbers with addition and multiplication as defined above forms both a commutative two-dimensional field and a vector space over R. In the field of complex numbers, quadratic equations az 2 + bz + c = 0 for an unknown z always have two solutions (for a x 0 and counting multiplicity). Complex numbers can be used to describe the points of a plane. A rotation around the origin can be described by multiplication by a complex number of unit length. Other twodimensional quantities can also be described with complex numbers. Electrical engineers

Copyright © Christoph Schiller November 1997–May 2006

a b ‘ Œ −b a

Challenge 1548 n

(817)

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º often summarized as i = + −1 . The complex conjugate z ‡ , also written z¯, of a complex number z = a + ib isºdefined as z ‡ = aº− ib. The absolute value SzS of a complex number is defined as SzS = zz ‡ = º z ‡ z = a 2 + b 2 . It defines a norm on the vector space of the complex numbers. From SwzS = SwS SzS follows the two-squares theorem

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numbers as mathematical structures

Challenge 1550 e

Challenge 1551 n

use complex numbers to describe quantities with phases, such as alternating currents or electrical fields in space. Writing complex numbers of unit length as cos θ + i sin θ is a useful method for remembering angle addition formulae. Since one has cos nθ + i sin nθ = (cos θ + i sin θ)n , one can easily deduce formulae cos 2θ = cos2 θ − sin2 θ and sin 2θ = 2 sin θ cos θ. By the way, there are exactly as many complex numbers as there are real numbers. Can you show this?

Quaternions

The positions of the points on a line can be described by real numbers. Complex numbers can be used to describe the positions of the points of a plane. It is natural to try to generalize the idea of a number to higher-dimensional spaces. However, it turns out that no useful number system can be defined for three-dimensional space. A new number system, the quaternions, can be constructed which corresponds the points of four-dimensional space, but only if the commutativity of multiplication is sacrificed. No useful number system can be defined for dimensions other than 1, 2 and 4. The quaternions were discovered by several mathematicians in the nineteenth century, among them Hamilton,* who studied them for much of his life. In fact, Maxwell’s theory of electrodynamics was formulated in terms of quaternions before three-dimensional vectors were used. Under multiplication, the quaternions H form a 4-dimensional algebra over the reals with a basis 1, i, j, k satisfying ë 1 i j k

1 i j k 1 i j k i −1 k − j . j −k −1 i k j −i −1

(819)

These relations are also often written i 2 = j2 = k 2 = −1, i j = − ji = k, jk = −k j = i, ki = −ik = j. The quaternions 1, i, j, k are also called basic units or generators. The lack of symmetry across the diagonal of the table shows the non-commutativity of quaternionic multiplication. With the quaternions, the idea of a non-commutative product appeared for the first time in mathematics. However, the multiplication of quaternions is associative. As a consequence of non-commutativity, polynomial equations in quaternions have many more solutions than in complex numbers: just search for all solutions of the equation X 2 + 1 = 0 to convince yourself of it. Every quaternion X can be written in the form X = x 0 + x 1 i + x 2 j + x 3 k = x 0 + v = (x 0 , x 1 , x 2 , x 3 ) = (x 0 , v) ,

(820)

* William Rowan Hamilton (b. 1805 Dublin, d. 1865 Dunsink), Irish child prodigy and famous mathematician, named the quaternions after an expression from the Vulgate (Acts. 12: 4).

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Copyright © Christoph Schiller November 1997–May 2006

Challenge 1552 ny

Love is complex: it has real and imaginary parts.

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Ref. 1221 Page 1207

1197

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d numbers and spaces

1198

where x 0 is called the scalar part and v the vector part. The multiplication is thus defined as (x, v)(y, w) = (x y−vëw, xw+yv+vw). The multiplication of two general quaternions can be written as (a 1 , b 1 , c 1 , d 1 )(a 2 , b 2 , c 2 , d 2 ) = (a 1 a 2 − b 1 b 2 − c 1 c 2 − d 1 d 2 , a 1 b 2 + b 1 a 2 + c 1 d 2 − d 1 c 2 , a1 c2 − b1 d2 + c1 a2 + d1 b2 , a1 d2 + b1 c2 − c1 b2 + d1 a2 ) . (821)

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The conjugate quaternion X is defined as X = x 0 − v, so that XY = Y X. The norm SXS of a quaternion X is defined as SXS2 = X X = X X = x 02 + x 12 + x 22 + x 32 = x 02 + v 2 . The norm is multiplicative, i.e. SXYS = SXS SYS. Unlike complex numbers, every quaternion is related to its complex conjugate by

(822)

No relation of this type exists for complex numbers. In the language of physics, a complex number and its conjugate are independent variables; for quaternions, this is not the case. As a result, functions of quaternions are less useful in physics than functions of complex variables. The relation SXYS = SXS SYS implies the four-squares theorem (a 12 + a 22 + a 32 + a 42 )(b 12 + b 22 + b 32 + b 42 )

= (a 1 b 1 − a 2 b 2 − a 3 b 3 − a 4 b 4 )2 + (a 1 b 2 + a 2 b 1 + a 3 b 4 − a 4 b 3 )2

+(a 1 b 3 + a 3 b 1 + a 4 b 2 − a 2 b 4 )2 + (a 1 b 4 + a 4 b 1 + a 2 b 3 − a 3 b 2 )2

valid for all real numbers a i and b i , and thus also for any set of eight integers. It was discovered in 1748 by Leonhard Euler (1707–1783) when trying to prove that each integer is the sum of four squares. (That fact was proved only in 1770, by Joseph Lagrange.) Hamilton thought that a quaternion with zero scalar part, which he simply called a vector (a term which he invented), could be identified with an ordinary three-dimensional translation vector; but this is wrong. Such a quaternion is now called a pure, or homogeneous, or imaginary quaternion. The product of two pure quaternions V = (0, v) and W = (0, w) is given by V W = (−v ë w, v  w), where ë denotes the scalar product and  denotes the vector product. Note that any quaternion can be written as the ratio of two pure quaternions. In reality, a pure quaternion (0, v) does not behave like a translation vector under coordinate transformations; in fact, a pure quaternion represents a rotation by the angle π or 180° around the axis defined by the direction v = (v x , v y , v z ). It turns out that in three-dimensional space, a general rotation about the origin can be described by a unit quaternion Q, also called a normed quaternion, for which SQS = 1. Such a quaternion can be written as (cos θ~2, n sin θ~2), where n = (n x , n y , n z ) is the normed vector describing the direction of the rotation axis and θ is the rotation angle. Such a unit quaternion Q = (cos θ~2, n sin θ~2) rotates a pure quaternion V = (0, v) into

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Challenge 1553 ny

(823)

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X = − 21 (X + iXi + jX j + kXk) .

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numbers as mathematical structures

1199

k

i j

1

k

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i

j

back of right hand

palm of right hand

another pure quaternion W = (0, w) given by

W = QV Q ‡ .

(824)

Thus, if we use pure quaternions such as V or W to describe positions, we can use unit quaternions to describe rotations and to calculate coordinate changes. The concatenation of two rotations is then given by the product of the corresponding unit quaternions. Indeed, a rotation by an angle α about the axis l followed by a rotation by an angle β about the axis m gives a rotation by an angle γ about the axis n, with the values determined by (cos γ~2, sin γ~2n) = (cos α~2, sin α~2l)(cos β~2, sin β~2m) .

Challenge 1554 e

Challenge 1555 n

Page 783

One way to show the result graphically is given in Figure 392. By drawing a triangle on a unit sphere, and taking care to remember the factor 1~2 in the angles, the combination of two rotations can be simply determined. The interpretation of quaternions as rotations is also illustrated, in a somewhat different way, in the motion of any hand. To see this, take a green marker and write the letters 1, i, j and k on your hand as shown in Figure 393. Defining the three possible 90° rotations axes as shown in the figure and taking concatenation as multiplication, the motion of the right hand follows the same ‘laws’ as those of pure unit quaternions. (One still needs to distinguish +i and −i, and the same for the other units, by the sense of the arm twist. And the result of a multiplication is that letter that can be read by a person facing you.) You can show that i 2 = j2 = k 2 = −1, that i 4 = 1, and all other quaternion relations.) The model also shows that the rotation angle of the arm is half the rotation angle of the corresponding quaternion. In other words, quaternions can be used to describe the belt trick, if the multiplication V W of two quaternions is taken to mean that rotation V is performed after rotation W. Quaternions, or the human hand, thus behaves like a spin

Copyright © Christoph Schiller November 1997–May 2006

Ref. 1219

(825)

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F I G U R E 393 The hand and the quaternions

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1200

Page 1218

Ref. 1222

Œ

A B ‘ with A, B > C, ‡ −B A‡

c d “ ’ a b – −b a −d c — — with a, b, c, d > R, (826) or as – – −c d a −b — — – ” −d −c b a •

where A = a + ib, B = c + id and the quaternion X is X = A + B j = a + ib + jc + kd; matrix addition and multiplication then corresponds to quaternionic addition and multiplication.

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Copyright © Christoph Schiller November 1997–May 2006

1/2 particle. Quaternions and spinors are isomorphic. The reason for the half-angle behaviour of rotations can be specified more precisely using mathematical language. The rotations in three α~2 dimensions around a point form the ‘special orthogonal group’ in three dimensions, which l π − γ~2 is called SO(3). But the motions of a hand attached to a shoulder via an arm form a different group, isomorphic to the Lie group SU(2). n The difference is due to the appearance of half angles in the parametrization of rotations; inβ~2 deed, the above parametrizations imply that a m rotation by 2π corresponds to a multiplication by −1. Only in the twentieth century was it real- F I G U R E 392 Combinations of rotations ized that there exist fundamental physical observables that behaves like hands attached to arms: they are called spinors. More on spinors can be found in the section on permutation symmetry, where belts are used as an analogy as well as arms. In short, the group SU(2) of the quaternions is the double cover of the rotation group SO(3). The simple representation of rotations and positions with quaternions is used in by computer programmes in robotics, in astronomy and in flight simulation. In the software used to create three-dimensional images and animations, visualization software, quaternions are often used to calculate the path taken by repeatedly reflected light rays and thus give surfaces a realistic appearance. The algebra of the quaternions is the only associative, non-commutative. finite-dimensional normed algebra with an identity over the field of real numbers. Quaternions form a non-commutative field, i.e. a skew field, in which the inverse of a quaternion X is X~SXS. We can therefore define division of quaternions (while being careful to distinguish XY −1 and Y −1 X). Therefore quaternions are said to form a division algebra. In fact, the quaternions H, the complex numbers C and the reals R are the only three finite-dimensional associative division algebras. In other words, the skew-field of quaternions is the only finite-dimensional real associative non-commutative algebra without divisors of zero. The centre of the quaternions, i.e. the set of quaternions that commute with all other quaternions, is just the set of real numbers. Quaternions can be represented as matrices of the form

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Page 783

d numbers and spaces

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numbers as mathematical structures

1201

The generators of the quaternions can be realized as 1  σ0

,

i  −iσ1

,

j  −iσ2

,

k  −iσ3

(827)

where the σn are the Pauli spin matrices.* Real 4  4 representations are not unique, as the alternative representation

Challenge 1556 ny

Ref. 1221 Challenge 1557 n

Dvipsbugw (829)

shows; however, no representation by 3  3 matrices is possible. These matrices contain real and complex elements, which pose no special problems. In contrast, when matrices with quaternionic elements are constructed, care has to be taken, because quaternionic multiplication is not commutative, so that simple relations such as trAB = trBA are not generally valid. What can we learn from quaternions about the description of nature? First of all, we see that binary rotations are similar to positions, and thus to translations: all are represented by 3-vectors. Are rotations the basic operations of nature? Is it possible that translations are only ‘shadows’ of rotations? The connection between translations and rotations s is investigated in the third part of our mountain ascent. When Maxwell wrote down his equations of electrodynamics, he used quaternion notation. (The now usual 3-vector notation was introduced later by Hertz and Heaviside.) The equations can be written in variousº ways using quaternions. The simplest is achieved when one keeps a distinction between −1 and the units i, j, k of the quaternions. One then can write all of electrodynamics in a single equation: dF = −

Q ε0

(830)

where F is the generalized electromagnetic field and Q the generalized charge. These are

σ0 = 1 =

Page 1209

’ 1 ” 0

0 “ 1 •

,

σ1 =

’ 0 ” 1

1 “ 0 •

,

σ2 =

’ 0 ” i

−i “ 0 •

,

σ3 =

’ 1 ” 0

0 “ −1 •

(828)

all of whose eigenvalues are 1; they satisfy the relations [σ i , σ k ]+ = 2 δ i k and [σ i , σ k ] = 2i ε i k l σ l . The linear combinations σ = 21 (σ1  σ2 ) are also frequently used. By the way, another possible representation of the quaternions is i  iσ3 , j  iσ2 , k  iσ1 .

Copyright © Christoph Schiller November 1997–May 2006

* The Pauli spin matrices are the complex Hermitean matrices

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Page 545

b −d −c “ ’ a – −b a −c d — — – — – c a b — – d ” c −d −b a •

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d numbers and spaces

1202

defined by

º F = E + −1 cB E = iE x + jE y + kE z B = iB x + jB y + kB z º d = δ + −1 ∂ t ~c δ = i∂ x + j∂ y + k∂ z º Q = ρ + −1 J~c

(831)

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Octonions In the same way that quaternions are constructed from complex numbers, octonions can be constructed from quaternions. They were first investigated by Arthur Cayley (1821–1895). Under multiplication, octonions (or octaves) are the elements of an eightdimensional algebra over the reals with the generators 1, i n with n = 1 . . . 7 satisfying ë 1 i1 i2 i3 i4 i5 i6 i7

1 1

i1 i2 i3 i4 i5 i6 i7

i1 i2 i3 i4 i5 i6 i7 i1 i2 i3 i4 i5 i6 i7 −1 i 3 −i 2 i 5 −i 4 i 7 −i 6 −i 3 −1 i 1 −i 6 i7 i 4 −i 5 i 2 −i 1 −1 i7 i 6 −i 5 −i 4 −i 5 i 6 −i 7 −1 i 1 −i 2 i3 i 4 −i 7 −i 6 −i 1 −1 i3 i2 −i 7 −i 4 i5 i 2 −i 3 −1 i1 i6 i5 i 4 −i 3 −i 2 −i 1 −1

(832)

Copyright © Christoph Schiller November 1997–May 2006

Nineteen other, equivalent multiplication tables are also possible. This algebra is called the Cayley algebra; it has an identity and a unique division. The algebra is non-commutative, and also non-associative. It is, however, alternative, meaning that for all elements x and y, one has x(x y) = x 2 y and (x y)y = x y 2 : a property somewhat weaker than associativity. It is the only 8-dimensional real alternative algebra without zero divisors. Because it is not associative, the set Ω of all octonions does not form a field, nor even a ring, so that the old designation of ‘Cayley numbers’ has been abandoned. The octonions are the most general hypercomplex ‘numbers’ whose norm is multiplicative. Its generators obey since (i n i m )i l = i n (i m i l ), where the minus sign, which shows the non-associativity, is valid for combinations of indices, such as 1-2-4, which are not quaternionic.

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where the fields E and B and the charge distributions ρ and J have the usual meanings. The content of equation 830 for the electromagnetic field is exactly the same as the usual formulation. Despite their charm, quaternions do not seem to be ready for the reformulation of special relativity; the main reason for this is the sign in the expression for their norm. Therefore, relativity and space-time are usually described using real numbers.

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numbers as mathematical structures

1203

Octonions can be represented as matrices of the form Œ

A B ‘ where A, B > H , or as real 8  8 matrices. −B¯ A¯

(833)

Matrix multiplication then gives the same result as octonionic multiplication. The relation SwzS = SwS SzS allows one to deduce the impressive eight-squares theorem (a 12 + a 22 + a 32 + a 42 + a 52 + a 62 + a 72 + a 82 )(b 12 + b 22 + b 32 + b 42 + b 52 + b 62 + b 72 + b 82 )

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= (a 1 b 1 − a 2 b 2 − a 3 b 3 − a 4 b 4 − a 5 b 5 − a 6 b 6 − a 7 b 7 − a 8 b 8 )2

+ (a 1 b 2 + a 2 b 1 + a 3 b 4 − a 4 b 3 + a 5 b 6 − a 6 b 5 + a 7 b 8 − a 8 b 7 )2

+ (a 1 b 4 + a 2 b 3 − a 3 b 2 + a 4 b 1 + a 5 b 8 + a 6 b 7 − a 7 b 6 − a 8 b 5 )2

+ (a 1 b 5 − a 2 b 6 + a 3 b 7 − a 4 b 8 + a 5 b 1 + a 6 b 2 − a 7 b 3 + a 8 b 4 )2

+ (a 1 b 6 + a 2 b 5 − a 3 b 8 − a 4 b 7 − a 5 b 2 + a 6 b 1 + a 7 b 4 + a 8 b 3 )2

+ (a 1 b 7 − a 2 b 8 − a 3 b 5 + a 4 b 6 + a 5 b 3 − a 6 b 4 + a 7 b 1 + a 8 b 2 )2

+ (a 1 b 8 + a 2 b 7 + a 3 b 6 + a 4 b 5 − a 5 b 4 − a 6 b 3 − a 7 b 2 + a 8 b 1 )2

(834)

valid for all real numbers a i and b i and thus in particular also for all integers. (There are many variations of this expression, with different possible sign combinations.) The theorem was discovered in 1818 by Carl Ferdinand Degen (1766–1825), and then rediscovered in 1844 by John Graves and in 1845 by Arthur Cayley. There is no generalization to higher numbers of squares, a fact proved by Adolf Hurwitz (1859–1919) in 1898. The octonions can be used to show that a vector product can be defined in more than three dimensions. A vector product or cross product is an operation  satisfying

Using the definition

anticommutativity exchange rule.

1 X  Y = (XY − Y X) , 2

(835)

(836)

the -products of imaginary quaternions, i.e. of quaternions of the type (0, u), are again imaginary, and correspond to the usual vector product, thus fulfilling (835). Interestingly, it is possible to use definition (836) for octonions as well. In that case, the product of imaginary octonions is also imaginary, and (835) is again satisfied. In fact, this is the only other non-trivial example of a vector product. Thus a vector product exists only in three and in seven dimensions.

Copyright © Christoph Schiller November 1997–May 2006

Ref. 1220 Challenge 1558 e

u  v = −v  u (u  v)w = u(v  w)

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+ (a 1 b 3 − a 2 b 4 + a 3 b 1 + a 4 b 2 − a 5 b 7 + a 6 b 8 + a 7 b 5 − a 8 b 6 )2

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d numbers and spaces

1204

Other types of numbers

Ref. 1223 Challenge 1559 n

Challenge 1560 ny Ref. 1220

Grassmann numbers

Challenge 1561 ny

θ2 = 0

and

θi θ j + θ jθi = 0 .

(837)

You may want to look for a matrix representation of these numbers.

Vector spaces Vector spaces, also called linear spaces, are mathematical generalizations of certain aspects of the intuitive three-dimensional space. A set of elements any two of which can * Hermann Günther Grassmann (1809–1877), schoolteacher in Stettin, and one of the most profound mathematical thinkers of the nineteenth century.

Copyright © Christoph Schiller November 1997–May 2006

With the discovery of supersymmetry, another system of numbers became important, called the Grassmann numbers.* They are in fact a special type of hypercomplex numbers. In supersymmetric Lagrangians, fields depend on two types of coordinates: on the usual real space-time coordinates and additionally on the Grassmann coordinates. Grassmann numbers, also called fermionic coordinates, θ have the defining properties

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Ref. 1224 Page 653

The process of constructing new systems of hypercomplex ‘numbers’ or real algebras by ‘doubling’ a given one can be continued ad infinitum. However, octonions, sedenions and all the following doublings are neither rings nor fields, but only non-associative algebras with unity. Other finite-dimensional algebras with unit element over the reals, once called hypercomplex ‘numbers’, can also be defined: they include the so-called such as ‘dual numbers’, ‘double numbers’, ‘Clifford–Lifshitz numbers’ etc. They play no special role in physics. Mathematicians have also defined number fields which have ‘one and a bit’ dimensions, such as algebraic number fields. There is also a generalization of the concept of integers to the complex domain: the Gaussian integers, defined as n+im, where n and m are ordinary integers. Gauss even defined what are now known as Gaussian primes. (Can you find out how?) They are not used in the description of nature, but are important in number theory. Physicists used to call quantum-mechanical operators ‘q-numbers.’ But this term has now fallen out of fashion. Another way in which the natural numbers can be extended is to include numbers larger infinite numbers. The most important such classes of transfinite number are the ordinals, the cardinals and the surreals (which were defined in the first intermezzo). The ordinals are essentially an extension of the integers beyond infinity, whereas the surreals are a continuous extension of the reals, also beyond infinity. Loosely speaking, among the transfinites, the ordinals have a similar role as the integers have among the reals; the surreals fill in all the gaps between the ordinals, like the reals do for integers. Interestingly, many series that diverge in R converge n the surreals. Can you find one example? The surreals include infinitely small numbers, as do the numbers of nonstandard analysis, also called hyperreals. In both the number systems, in contrast to real numbers, the numbers 1 and 0.999 999 9 (with an infinite string of nines) do not coincide, and indeed are separated by infinitely many other numbers.

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vector spaces

1205

be added together and any one of which can be multiplied by a number is called a vector space, if the result is again in the set and the usual rules of calculation hold. More precisely, a vector space over a number field K is a set of elements, called vectors, for which a vector addition and a scalar multiplication is defined, such that for all vectors a, b, c and for all numbers s and r from K one has (a + b) + c = a + (b + c) = a + b + c n+a=a (−a) + a = n 1a = a (s + r)(a + b) = sa + sb + ra + rb

If the field K, whose elements are called scalars in this context, is taken to be the real (or complex, or quaternionic) numbers, one speaks of a real (or complex, or quaternionic) vector space. Vector spaces are also called linear vector spaces or simply linear spaces. The complex numbers, the set of all real functions defined on the real line, the set of all polynomials, the set of matrices with a given number of rows and columns, all form vector spaces. In mathematics, a vector is thus a more general concept than in physics. (What is the simplest possible mathematical vector space?) In physics, the term ‘vector’ is reserved for elements of a more specialized type of vector space, namely normed inner product spaces. To define these, we first need the concept of a metric space. A metric space is a set with a metric, i.e. a way to define distances between elements. A real function d(a, b) between elements is called a metric if d(a, b) E 0 d(a, b) + d(b, c) E d(a, c) d(a, b) = 0 if and only if a = b

Challenge 1563 n

(839)

A non-trivial example is the following. We define a special distance d between cities. If the two cities lie on a line going through Paris, we use the usual distance. In all other cases, we define the distance d by the shortest distance from one to the other travelling via Paris. This strange method defines a metric between all cities in France. A normed vector space is a linear space with a norm, or ‘length’, associated to each a vector. A norm is a non-negative number SSaSS defined for each vector a with the properties SSraSS = SrS SSaSS linearity of norm SSa + bSS D SSaSS + SSbSS triangle inequality SSaSS = 0 only if a = 0 regularity

(840)

Usually there are many ways to define a norm for a given space. Note that a norm can always be used to define a metric by setting d(a, b) = SSa − bSS

(841)

Copyright © Christoph Schiller November 1997–May 2006

Challenge 1564 ny

positivity of metric triangle inequality regularity of metric

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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net

Challenge 1562 ny

associativity of vector addition existence of null vector existence of negative vector (838) regularity of scalar multiplication complete distributivity of scalar multiplication

Dvipsbugw


d numbers and spaces

1206

so that all normed spaces are also metric spaces. This is the natural distance definition (in contrast to unnatural ones like that between French cities). The norm is often defined with the help of an inner product. Indeed, the most special class of linear spaces are the inner product spaces. These are vector spaces with an inner product, also called scalar product ë (not to be confused with the scalar multiplication!) which associates a number to each pair of vectors. An inner product space over R satisfies commutativity of scalar product bilinearity of scalar product left distributivity of scalar product right distributivity of scalar product positivity of scalar product regularity of scalar product

Dvipsbugw

(842)

for all vectors a, b, c and all scalars r, s. A real inner product space of finite dimension is also called a Euclidean vector space. The set of all velocities, the set of all positions, or the set of all possible momenta form such spaces. An inner product space over C satisfies* aëb =bëa =bëa (ra) ë (sb) = rs(a ë b) (a + b) ë c = a ë c + b ë d a ë (b + c) = a ë b + a ë c aëaE0 a ë a = 0 if and only if a = 0

Page 1195

Challenge 1566 ny

(843)

for all vectors a, b, c and all scalars r, s. A complex inner product space (of finite dimension) is also called a unitary or Hermitean vector space. If the inner product space is complete, it is called, especially in the infinite-dimensional complex case, a Hilbert space. The space of all possible states of a quantum system forms a Hilbert space. All inner product spaces are also metric spaces, and thus normed spaces, if the metric is defined by » d(a, b) = (a − b) ë (a − b) . (844) Only in the context of an inner product spaces we can speak about angles (or phase differences) between vectors, as we are used to in physics. Of course, like in normed spaces, inner product spaces also allows us to speak about the length of vectors and to define a basis, the mathematical concept necessary to define a coordinate system. The dimension of a vector space is the number of linearly independent basis vectors. Can you define these terms precisely? Which vector spaces are of importance in physics? * Two inequivalent forms of the sequilinearity axiom exist. The other is (ra) ë (sb) = rs(a ë b). The term sesquilinear is derived from Latin and means for ‘one-and-a-half-linear’.

Copyright © Christoph Schiller November 1997–May 2006

Challenge 1565 ny

Hermitean property sesquilinearity of scalar product left distributivity of scalar product right distributivity of scalar product positivity of scalar product regularity of scalar product

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aëb =bëa (ra) ë (sb) = rs(a ë b) (a + b) ë c = a ë c + b ë c a ë (b + c) = a ë b + a ë c aëaE0 a ë a = 0 if and only if a = 0

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algebras

1207

Algebras

Page 1194

The term algebra is used in mathematics with three different, but loosely related, meanings. First, it denotes a part of mathematics, as in ‘I hated algebra at school’. Secondly, it denotes a set of formal rules that are obeyed by abstract objects, as in the expression ‘tensor algebra’. Finally – and this is the only meaning used here – an algebra denotes a specific type of mathematical structure. Intuitively, an algebra is a set of vectors with a vector multiplication defined on it. More precisely, an algebra is a vector space (over a field K) that is also a ring. (The concept is due to Benjamin Peirce (1809–1880), father of Charles Sanders Peirce.) A ring is a set for which an addition and a multiplication is defined – like the integers. Thus, in an algebra, there are (often) three types of multiplications:

A precise definition of an algebra thus only needs to define properties of the (main) multiplication and to specify the number field K. An algebra is defined by the following axioms x(y + z) = x y + xz

, (x + y)z = xz + yz c(x y) = (cx)y = x(c y)

distributivity of multiplication bilinearity (845)

for all vectors x, y, z and all scalars c > K. To stress their properties, algebras are also called linear algebras. For example, the set of all linear transformations of an n-dimensional linear space (such as the translations on a plane, in space or in time) is a linear algebra, if the composition is taken as multiplication. So is the set of observables of a quantum mechanical system.* An associative algebra is an algebra whose multiplication has the additional property

Challenge 1567 n

T(e) = λe

Page 744

Page 805

(846)

where λ is a scalar, then the vector e is called an eigenvector of T, and λ is associated eigenvalue. The set of all eigenvalues of a transformation T is called the spectrum of T. Physicists did not pay much attention to these mathematical concepts until they discovered quantum theory. Quantum theory showed that observables are transformations in Hilbert space, because any measurement interacts with a system and thus transforms it. Quantum-mechanical experiments also showed that a measurement result for an observable must be an eigenvalue of the corresponding transformation. The state of the system after the measurement is given by the eigenvector corresponding to the measured eigenvalue. Therefore every expert on motion must know what an eigenvalue is.

Copyright © Christoph Schiller November 1997–May 2006

* Linear transformations are mappings from the vector space to itself, with the property that sums and scalar multiples of vectors are transformed into the corresponding sums and scalar multiples of the transformed vectors. Can you specify the set of all linear transformations of the plane? And of three-dimensional space? And of Minkowski space? All linear transformations transform some special vectors, called eigenvectors (from the German word eigen meaning ‘self ’) into multiples of themselves. In other words, if T is a transformation, e a vector, and

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— scalar multiplication: the c-fold multiple of a vector x is another vector y = cx; — the (main) algebraic multiplication: the product of two vectors x and y is another vector z = x y; — if the vector space is a inner product space, the scalar product: the scalar product of two algebra elements (vectors) x and y is a scalar c = x ë y;

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d numbers and spaces

1208

that

Challenge 1568 e

Challenge 1570 n

associativity .

(847)

Most algebras that arise in physics are associative.* Therefore, in mathematical physics, a linear associative algebra is often simply called an algebra. The set of multiples of the unit 1 of the algebra is called the field of scalars scal(A) of the algebra A. The field of scalars is also a subalgebra of A. The field of scalars and the scalars themselves behave in the same way. We explore a few examples. The set of all polynomials in one variable (or in several variables) forms an algebra. It is commutative and infinite-dimensional. The constant polynomials form the field of scalars. The set of n  n matrices, with the usual operations, also forms an algebra. It is n 2 dimensional. Those diagonal matrices (matrices with all off-diagonal elements equal to zero) whose diagonal elements all have the same value form the field of scalars. How is the scalar product of two matrices defined? The set of all real-valued functions over a set also forms an algebra. Can you specify the multiplication? The constant functions form the field of scalars. A star algebra, also written ‡-algebra, is an algebra over the complex numbers for which there is a mapping ‡  A A, x ( x ‡ , called an involution, with the properties (x ‡ )‡ = x (x + y)‡ = x ‡ + y‡ (cx)‡ = cx ‡ for all (x y)‡ = y‡ x ‡

c>C

(848)

valid for all elements x, y of the algebra A. The element x ‡ is called the adjoint of x. Star algebras are the main type of algebra used in quantum mechanics, since quantummechanical observables form a ‡-algebra. A C‡-algebra is a Banach algebra over the complex numbers with an involution ‡ (a function that is its own inverse) such that the norm YxY of an element x satisfies

Challenge 1571 n

(849)

(A Banach algebra is a complete normed algebra; an algebra is complete if all Cauchy sequences converge.) In short, C‡-algebra is a nicely behaved algebra whose elements form a continuous set and a complex vector space. The name C comes from ‘continuous functions’. Indeed, the bounded continuous functions form such an algebra, with a properly defined norm. Can you find it? Every C‡-algebra contains a space of Hermitean elements (which have a real spectrum), a set of normal elements, a multiplicative group of unitary elements and a set of positive elements (with non-negative spectrum). We should mention one important type of algebra used in mathematics. A division algebra is an algebra for which the equations ax = b and ya = b are uniquely solvable in * Note that a non-associative algebra does not possess a matrix representation.

Copyright © Christoph Schiller November 1997–May 2006

YxY2 = x ‡ x .

Dvipsbugw

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Challenge 1569 ny

x(yz) = (x y)z

Dvipsbugw


algebras

1209

Lie algebras A Lie algebra is special type of algebra (and thus of vector space). Lie algebras are the most important type of non-associative algebra. A vector space L over the field R (or C) with an additional binary operation [ , ], called Lie multiplication or the commutator, is called a real (or complex) Lie algebra if this operation satisfies

Challenge 1572 e

Challenge 1573 ny

antisymmetry (left-)linearity Jacobi identity

(850)

for all elements X, Y , Z > L and for all a, b > R (or C). (Lie algebras are named after Sophus Lie.) The first two conditions together imply bilinearity. A Lie algebra is called commutative if [X, Y] = 0 for all elements X and Y. The dimension of the Lie algebra is the dimension of the vector space. A subspace N of a Lie algebra L is called an ideal* if [L, N] ⊂ N; any ideal is also a subalgebra. A maximal ideal M which satisfies [L, M] = 0 is called the centre of L. A Lie algebra is called a linear Lie algebra if its elements are linear transformations of another vector space V (intuitively, if they are ‘matrices’). It turns out that every finitedimensional Lie algebra is isomorphic to a linear Lie algebra. Therefore, there is no loss * Can you explain the notation [L, N]? Can you define what a maximal ideal is and prove that there is only one?

Copyright © Christoph Schiller November 1997–May 2006

[X, Y] = −[Y , X] [aX + bY , Z] = a[X, Z] + b[Y , Z] [X, [Y ,Z]] + [Y , [Z, X]] + [Z, [X, Y]] = 0

Dvipsbugw

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x or y for all b and all a x 0. Obviously, all type of continuous numbers must be division algebras. Division algebras are thus one way to generalize the concept of a number. One of the important results of modern mathematics states that (finite-dimensional) division algebras can only have dimension 1, like the reals, dimension 2, like the complex numbers, dimension 4, like the quaternions, or dimension 8, like the octonions. There is thus no way to generalize the concept of (continuous) ‘number’ to other dimensions. And now for some fun. Imagine a ring A which contains a number field K as a subring (or ‘field of scalars’). If the ring multiplication is defined in such a way that a general ring element multiplied with an element of K is the same as the scalar multiplication, then A is a vector space, and thus an algebra – provided that every element of K commutes with every element of A. (In other words, the subring K must be central.) For example, the quaternions H are a four-dimensional real division algebra, but although H is a two-dimensional complex vector space, it is not a complex algebra, because i does not commute with j (one has i j = − ji = k). In fact, there are no finite-dimensional complex division algebras, and the only finite-dimensional real associative division algebras are R, C and H. Now, if you are not afraid of getting a headache, think about this remark: every Kalgebra is also an algebra over its field of scalars. For this reason, some mathematicians prefer to define an (associative) K-algebra simply as a ring which contains K as a central subfield. In physics, it is the algebras related to symmetries which play the most important role. We study them next.

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Page 1220

of generality in picturing the elements of finite-dimensional Lie algebras as matrices. The name ‘Lie algebra’ was chosen because the generators, i.e. the infinitesimal elements of every Lie group, form a Lie algebra. Since all important symmetries in nature form Lie groups, Lie algebras appear very frequently in physics. In mathematics, Lie algebras arise frequently because from any associative finite-dimensional algebra (in which the symbol ë stands for its multiplication) a Lie algebra appears when we define the commutator by [X, Y] = X ë Y − Y ë X . (851)

k

Page 1201

The numbers c ikj are called the structure constants of the Lie algebra. They depend on the choice of basis. The structure constants determine the Lie algebra completely. For example, the algebra of the Lie group SU(2), with the three generators defined by Ta = σ a ~2i, where the σ a are the Pauli spin matrices, has the structure constants C abc = ε abc .* Classification of Lie algebras

Finite-dimensional Lie algebras are classified as follows. Every finite-dimensional Lie algebra is the (semidirect) sum of a semisimple and a solvable Lie algebra. A Lie algebra is called solvable if, well, if it is not semisimple. Solvable Lie algebras have not yet been classified completely. They are not important in physics. A semisimple Lie algebra is a Lie algebra which has no non-zero solvable ideal. Other

c

The definition implies that ad(a i ) jk = c ikj , where c ikj are the structure constants of the Lie algebra. For a real Lie algebra, all elements of ad(a) are real for all a > L. Note that for any Lie algebra, a scalar product can be defined by setting X ë Y = Tr( adX ë adY ) .

(854)

This scalar product is symmetric and bilinear. (Can you show that it is independent of the representation?) The corresponding bilinear form is also called the Killing form, after the German mathematician Wilhelm Killing (1847–1923), the discoverer of the ‘exceptional’ Lie groups. The Killing form is invariant under the action of any automorphism of the Lie algebra L. In a given basis, one has X ë Y = Tr( (adX) ë (adY)) = c li k c ski x l y s = дl s x l y s

where дl s = c li k c ski is called the Cartan metric tensor of L.

(855)

Copyright © Christoph Schiller November 1997–May 2006

* Like groups, Lie algebras can be represented by matrices, i.e. by linear operators. Representations of Lie algebras are important in physics because many continuous symmetry groups are Lie groups. The adjoint representation of a Lie algebra with basis a 1 ...a n is the set of matrices ad(a) defined for each element a by (853) [a, a j ] = Q ad(a)c j a c .

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(This fact gave the commutator its name.) Lie algebras are non-associative in general; but the above definition of the commutator shows how to build one from an associative algebra. Since Lie algebras are vector spaces, the elements Ti of a basis of the Lie algebra always obey a relation of the form: [Ti , T j ] = Q c ikj Tk . (852)

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algebras

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equivalent definitions are possible, depending on your taste: — a semisimple Lie algebra does not contain non-zero abelian ideals; — its Killing form is non-singular, i.e. non-degenerate; — it splits into the direct sum of non-abelian simple ideals (this decomposition is unique); — every finite-dimensional linear representation is completely reducible; — the one-dimensional cohomology of д with values in an arbitrary finite-dimensional д-module is trivial.

Lie superalgebras

[x, y] = −(−1)SxSSyS [y, x]

0 = (−1)SzSSxS [x, [y, z]] + (−1)SxSSyS [y, [z, x]] + (−1)SySSzS [z, [x, y]]

(856)

Copyright © Christoph Schiller November 1997–May 2006

Lie superalgebras arise when the concept of Lie algebra is extended to the case of supersymmetry. A Lie superalgebra contains even and odd elements; the even elements correspond to bosons and the odd elements to fermions. Supersymmetry applies to systems with anticommuting coordinates. Lie superalgebras are a natural generalization of Lie algebras to supersymmetry, and simply add a Z2 -grading. In detail, a Lie superalgebra is a Z2 -graded algebra over a field of characteristic 0 – usually R or C – with a product [., .], called the (Lie) superbracket or supercommutator, that has the properties

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Ref. 1225

Finite-dimensional semisimple Lie algebras have been completely classified. They decompose uniquely into a direct sum of simple Lie algebras. Simple Lie algebras can be complex or real. The simple finite-dimensional complex Lie algebras all belong to four infinite classes and to five exceptional cases. The infinite classes are also called classical, and are: A n for n E 1, corresponding to the Lie groups SL(n + 1) and their compact ‘cousins’ SU(n + 1); B n for n E 1, corresponding to the Lie groups SO(2n + 1); C n for n E 1, corresponding to the Lie groups Sp(2n); and D n for n E 4, corresponding to the Lie groups SO(2n). Thus A n is the algebra of all skew-Hermitean matrices; B n and D n are the algebras of the symmetric matrices; and C n is the algebra of the traceless matrices. The exceptional Lie algebras are G 2 , F4 , E 6 , E 7 , E 8 . In all cases, the index gives the number of roots. The dimensions of these algebras are A n  n(n+2); B n and C n  n(2n+1); D n  n(2n − 1); G 2  14; F4  32; E 6  78; E 7  133; E 8  248. The simple and finite-dimensional real Lie algebras are more numerous; their classification follows from that of the complex Lie algebras. Moreover, corresponding to each complex Lie group, there is always one compact real one. Real Lie algebras are not so important in fundamental physics. The so-called superalgebras (see below) play a role in systems with supersymmetry. Of the large number of infinite-dimensional Lie algebras, only few are important in physics: among them are the Poincaré algebra, the Cartan algebra, the Virasoro algebra and a few other Kac–Moody algebras.

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where x, y and z are algebra elements that are ‘pure’ in the Z2 -grading. The expression SxS denotes the degree of the algebra element x and is either 0 or 1. The second condition is sometimes called the ‘super Jacobi identity’. Obviously, the even subalgebra of a Lie superalgebra forms a Lie algebra; in that case the superbracket becomes the usual Lie bracket. As in the case of Lie algebras, the simple Lie superalgebras have been completely classified. They fall into five infinite classes and some special cases, namely

The Virasoro algebra The Virasoro algebra is the infinite algebra of operators L n satisfying [L m , L n ] = (m − n)L m+n +

Challenge 1574 ny

c (m 3 − m)δ m,−n 12

(857)

where the number c, which may be zero, is called the central charge, and the factor 1~12 is introduced by historical convention. This rather specific algebra is important in physics because it is the algebra of conformal symmetries in two dimensions.* Can you find a representation in terms of infinite square matrices? Mathematically speaking, the Virasoro algebra is a special case of a Kac–Moody algebra. Kac–Moody algebras

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* Note that, in contrast, the conformal symmetry group in four dimensions has 15 parameters, and thus its Lie algebra is finite (15) dimensional.

Copyright © Christoph Schiller November 1997–May 2006

In physics, more general symmetries than Lie groups also appear. This happens in particular when general relativity is taken into account. Because of the symmetries of space-time, the number of generators becomes infinite. The concepts of Lie algebra and of superalgebra have to be extended to take space-time symmetry into account. The corresponding algebras are the Kac–Moody algebras. (‘Kac’ is pronounced like ‘Katz’.) Kac–Moody algebras are a particular class of infinite-parameter Lie algebras; thus they have an infinite number of generators. They used to be called associated affine algebras or affine Lie algebras; sometimes they are also called Z-graded Lie algebras. They were introduced independently in 1968 by Victor Kac and Robert Moody. We present some specific examples. Of basic physical importance are those Kac–Moody algebras associated to a symmetry (n) group G a C[t], where t is some continuous parameter. The generators M a of the corresponding algebra have two indices, one, a, that indexes the generators of the group G and another, n, that indexes the generators of the group of Laurent polynomials C[t]. The

Dvipsbugw

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— A(n, m) corresponding to the Lie supergroups SL(n + 1Sm + 1) and their compact ‘cousins’ SU(NSM); — B(n, m) corresponding to the Lie supergroups OSp(2n + 1S2m); — D(n, m) corresponding to the Lie supergroups OSp(2nS2m); — P(n); — Q(n); — the exceptional cases D α (2, 1), G(3), F(4); — and finally the Cartan superalgebras.

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generators form a Kac–Moody algebra if they obey the relations: [M (n) a , Mb

(m)

] = C abc M c(n+m)

n, m = 0, 1, 2, . . . , ª.

for

(858)

For example, if we take G as SU(2) with its generators T n = σ a ~2i, σ a being the Pauli spin matrices, then the objects M (n) = Ta a t n , a

e.g.

(n)

M3

=

1 tn 0 ‘ Œ 2i 0 −t n

Dvipsbugw

(859)

form a Kac–Moody algebra with the structure constants C abc = ε abc of SU(2).

Topolo gy – what shapes exist?

Challenge 1575 n

In a simplified view of topology that is sufficient for physicists, only one type of entity can possess shape: manifolds. Manifolds are generalized examples of pullovers: they are locally flat, can have holes, boundaries and can often be turned inside out. Pullovers are subtle entities. For example, can you turn your pullover inside out while your hands are tied together? (A friend may help you.) By the way, the same feat is also possible with your trousers, while your feet are tied together. Certain professors like to demonstrate this during topology lectures – of course with a carefully selected pair of underpants. Topological spaces

Ref. 1227

— x is in N; — if N and M are neighbourhoods, so is N 9 M; — anything containing a neighbourhood of x is itself a neighbourhood of x. The choice of the subsets is free. All the subsets Yx , for all points x, that were chosen in the definition are then called open sets. (A neighbourhood and an open set are not necessarily the same; but all open sets are also neighbourhoods.) One also calls a topological space a ‘set with a topology’. In effect, a topology specifies the systems of ‘neighbourhoods’ of every point of the set. ‘Topology’ is also the name of the branch of mathematics that studies topological spaces.

Copyright © Christoph Schiller November 1997–May 2006

The study of shapes requires a good definition of a set made of ‘points’. To be able to talk about shape, these sets must be structured in such a way as to admit a useful concept of ‘neighbourhood’ or ‘closeness’ between the elements of the set. The search for the most general type of set which allows a useful definition of neighbourhood has led to the concept of topological space. A topological space is a finite or infinite set X of elements, called points, together with a neighbourhood for each point. A neighbourhood N of a point x is a collection of subsets Yx of X with the properties that

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Topology is group theory. The Erlangen program

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Challenge 1576 e

d numbers and spaces

For example, the real numbers together with the open intervals form the usual topology of R. If one takes all subsets of Ras open sets, one speaks of the discrete topology. If one takes only the full set X and the empty set as open sets, one speaks of the trivial or indiscrete topology. The concept of topological space allows us to define continuity. A mapping from one topological space X to another topological space Y is continuous if the inverse image of every open set in Y is an open set in X. You may verify that this definition is not satisfied by a real function that makes a jump. You may also check that the term ‘inverse’ is necessary in the definition; otherwise a function with a jump would be continuous, as such a function may still map open sets to open sets.* We thus need the concept of topological space, or of neighbourhood, if we want to express the idea that there are no jumps in nature. We also need the concept of topological space in order to be able to define limits.

Of the many special kinds of topological spaces that have been studied, one type is particularly important. A Hausdorff space is a topological space in which for any two points x and y there are disjoint open sets U and V such that x is in U and y is in V . A Hausdorff space is thus a space where, no matter how ‘close’ two points are, they can always be separated by open sets. This seems like a desirable property; indeed, non-Hausdorff spaces are rather tricky mathematical objects. (At Planck energy, it seems that vacuum appears to behave like a non-Hausdorff space; however, at Planck energy, vacuum is not really a space at all. So non-Hausdorff spaces play no role in physics.) A special case of Hausdorff space is well-known: the manifold. Manifolds

Challenge 1577 ny

* The Cauchy-Weierstass definition of continuity says that a real function f (x) is continuous at a point a if (1) f is defined on a open interval containing a, (2) f (x) tends to a limit as x tends to a, and (3) the limit is f (a). In this definition, the continuity of f is defined using the intuitive idea that the real numbers form the basic model of a set that has no gaps. Can you see the connection with the general definition given above?

Copyright © Christoph Schiller November 1997–May 2006

In physics, the most important topological spaces are differential manifolds. Loosely speaking, a differential manifold is a set of points that looks like Rn under the microscope – at small distances. For example, a sphere and a torus are both two-dimensional differential manifolds, since they look locally like a plane. Not all differential manifolds are that simple, as the examples of Figure 394 show. A differential manifold is called connected if any two points can be joined by a path lying in the manifold. (The term has a more general meaning in topological spaces. But the notions of connectedness and pathwise connectedness coincide for differential manifolds.) We focus on connected manifolds in the following discussion. A manifold is called simply connected if every loop lying in the manifold can be contracted to a point. For example, a sphere is simply connected. A connected manifold which is not simply connected is called multiply connected. A torus is multiply connected. Manifolds can be non-orientable, as the well-known Möbius strip illustrates. Nonorientable manifolds have only one surface: they do not admit a distinction between front

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– CS – more on topological spaces to be added – CS –

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F I G U R E 395 Compact (left) and noncompact (right) manifolds of various dimensions

and back. If you want to have fun, cut a paper Möbius strip into two along a centre line. You can also try this with paper strips with different twist values, and investigate the regularities. In two dimensions, closed manifolds (or surfaces), i.e. surfaces that are compact and without boundary, are always of one of three types: — The simplest type are spheres with n attached handles; they are called n-tori or surfaces of genus n. They are orientable surfaces with Euler characteristic 2 − 2n. — The projective planes with n handles attached are non-orientable surfaces with Euler characteristic 1 − 2n.

Copyright © Christoph Schiller November 1997–May 2006

Challenge 1578 e

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F I G U R E 394 Examples of orientable and non-orientable manifolds of two dimensions: a disc, a Möbius strip, a sphere and a Klein bottle

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Dvipsbugw F I G U R E 396 Simply connected (left), multiply connected (centre) and

disconnected (right) manifolds of one (above) and two (below) dimensions

Therefore Euler characteristic and orientability describe compact surfaces up to homeomorphism (and if surfaces are smooth, then up to diffeomorphism). Homeomorphisms are defined below. The two-dimensional compact manifolds or surfaces with boundary are found by removing one or more discs from a surface in this list. A compact surface can be embedded in R3 if it is orientable or if it has non-empty boundary. In physics, the most important manifolds are space-time and Lie groups of observables. We study Lie groups below. Strangely enough, the topology of space-time is not known. For example, it is unclear whether or not it is simply connected. Obviously, the reason is that it is difficult to observe what happens at large distances form the Earth. However, a similar difficulty appears near Planck scales. If a manifold is imagined to consist of rubber, connectedness and similar global properties are not changed when the manifold is deformed. This fact is formalized by saying that two manifolds are homeomorphic (from the Greek words for ‘same’ and ‘shape’) if between them there is a continuous, one-to-one and onto mapping with a continuous inverse. The concept of homeomorphism is somewhat more general than that of rubber deformation, as can be seen from Figure 397. Holes, homotopy and homology

– CS – more on topology to be added – CS –

Copyright © Christoph Schiller November 1997–May 2006

Only ‘well-behaved’ manifolds play a role in physics: namely those which are orientable and connected. In addition, the manifolds associated with observables, are always compact. The main non-trivial characteristic of connected compact orientable manifolds is that they contain ‘holes’ (see Figure 398). It turns out that a proper description of the holes of manifolds allows us to distinguish between all different, i.e. non-homeomorphic, types of manifold. There are three main tools to describe holes of manifolds and the relations among them: homotopy, homology and cohomology. These tools play an important role in the study of gauge groups, because any gauge group defines a manifold.

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— The Klein bottles with n attached handles are non-orientable surfaces with Euler characteristic −2n.

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types and cl assification of groups

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Dvipsbugw F I G U R E 397 Examples of homeomorphic pairs

of manifolds

Challenge 1579 d

In other words, through homotopy and homology theory, mathematicians can classify manifolds. Given two manifolds, the properties of the holes in them thus determine whether they can be deformed into each other. Physicists are now extending these results of standard topology. Deformation is a classical idea which assumes continuous space and time, as well as arbitrarily small action. In nature, however, quantum effects cannot be neglected. It is speculated that quantum effects can transform a physical manifold into one with a different topology: for example, a torus into a sphere. Can you find out how this can be achieved? Topological changes of physical manifolds happen via objects that are generalizations of manifolds. An orbifold is a space that is locally modelled by Rn modulo a finite group. Examples are the tear-drop or the half-plane. Orbifolds were introduced by Satake Ichiro in 1956; the name was coined by William Thurston. Orbifolds are heavily studied in string theory.

Page 192

We introduced groups early on because groups play an important role in many parts of physics, from the description of solids, molecules, atoms, nuclei, elementary particles and forces up to the study of shapes, cycles and patterns in growth processes. Group theory is also one of the most importan