9780141979168

Date: 19/12/2019 Designer: c/o RG Production Controller: ISBN: 9780141988108

LEE SMOLIN

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D A I LY T E L E G R A P H

’The best explanation yet of what has yet to be explained’ G E O RG E DYS O N , AU T H O R O F T U R I N G' S C AT H E D R A L

’Full of extraordinary ideas ... a new theory of everything’ S TE PH E N BLE AC H, THE TIMES

‘An inventive and provocative thinker, as well as an engaging writer ... lucid, searingly honest, impassioned, inspiring’ PHILIP BALL, PHYSICS WORLD

EINSTEIN’S UNFINISHED REVOLUTION

How do we ‘know’ how to ’Quantum mechanics is perhaps speak? shows ever this the mostLanguage, successful theory landmark is part of our formulated.book, The only problem genetic inheritance ratheris than with it, argues Lee Smolin, a cultural creation.

Destroying that it is wrong.

He attempts to the mythsother about language examine options for a – that children learn to talk ... by theory of the atomic world copying that ideas English a fount ofparents, provocative ... defiesupbeat logic –and, Pinker reveals our lucid, finally, innate instinct to communicate. optimistic’

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Einstein’s Unfinished Revolution

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The Search for What Lies Beyond the Quantum

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’A masterful exposition on the state of quantum physics … spellbinding‘ Financial Times

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FRO NT COV E R 19/12/2019 12:25

PART 1

AN O RT H O D OXY OF THE UNREAL

ONE ONE

Nature NatureLoves LovestotoHide Hide Reality Reality is the is the business business of physics. of physics.

— ALBERT — ALBERT EINSTEIN EINSTEIN

Q Q

uantum uantum mechanics mechanics hashas been been thethe core core of our of our understandunderstandinging of nature of nature forfor nine nine decades. decades.

It isIt ubiquitous, is ubiquitous, butbut it isit is alsoalso deeply deeply mysterious. mysterious.

Little Little of modern of modern science science would would

make make sense sense without without it. But it. But experts experts have have a hard a hard time time agreeing agreeing what what it asserts it asserts about about nature. nature.

Quantum Quantum mechanics mechanics explains explains why why there there areare atoms, atoms, andand why why those those atoms atoms areare stable stable andand have have distinct distinct chemical chemical properties. properties.

QuanQuantum tum mechanics mechanics alsoalso explains explains how how atoms atoms combine combine intointo diverse diverse molmolecules. ecules.

As As a result, a result, it isitthe is the basis basis for for how how wewe understand understand thethe shapes shapes andand interactions interactions of those of those molecules. molecules.

LifeLife would would be be incomprehensiincomprehensible without ble without thethe quantum. quantum.

From From thethe behavior behavior of water of water to the to the shapes shapes of proteins of proteins to the to the fidelity fidelity andand transmittal transmittal of infor of infor mation mation by by DNA DNA andand RNA, RNA, everything everything in biology in biology depends depends on on thethe quantum. quantum.

Quantum Quantum mechanics mechanics explains explains thethe properties properties of materials, of materials, such such as what as what makes makes a metal a metal a conductor a conductor of electricity, of electricity, while while another another is is an an insulator. insulator.

It explains It explains light light andand radioactivity, radioactivity, andand is the is the basis basis of of

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nuclear nuclear physics. physics.

Without Without it we it we wouldn’t wouldn’t understand understand how how thethe stars stars shine. shine.

NorNor could could wewe have have invented invented thethe chips chips or the or the lasers lasers on on which which so so much much of of ourour technology technology is based. is based.

Quantum Quantum mechanics mechanics is the is the language language that that wewe useuse to write to write thethe standard standard model model of particle of particle physphysics,ics, which which contains contains all all wewe know know about about thethe elementary elementary particles particles andand thethe fundamental fundamental forces forces by by which which they they interact. interact.

According According to our to our best best theory theory of the of the early early universe, universe, all all matter, matter, along along with with thethe patterns patterns that that eventually eventually coalesced coalesced intointo thethe galaxies, galaxies, waswas yanked yanked intointo existence existence from from thethe quantum quantum randomness randomness of the of the vacuum vacuum of empty of empty space space by by thethe rapid rapid expansion expansion of the of the universe. universe.

I I don’t don’t expect expect thethe reader reader to understand to understand precisely precisely what what thisthis means, means, butbut perhaps perhaps thethe words words evoke evoke an an image. image.

In any In any case, case, if this if this is right, is right, then then without without quantum quantum physics physics there there would would literally literally be be nothing nothing ex-except cept empty empty spacetime. spacetime.

YetYet forfor all all its its success, success, there there is aisstubborn a stubborn puzzle puzzle at the at the heart heart of of quantum quantum mechanics. mechanics.

TheThe quantum quantum world world behaves behaves in in ways ways that that challenge challenge ourour intuition. intuition.

It isItoften is often saidsaid that that in quantum in quantum physics physics an an atom atom cancan be be in two in two places places at once, at once, butbut that that is only is only thethe start; start; thethe fullfull story story is far is far weirder weirder than than that. that.

If an If an atom atom cancan be be here here or there, or there, wewe must must speak speak of states of states in which in which it is, it somehow, is, somehow, simultaneously simultaneously both both here here andand there. there.

This This is called is called a superposition. a superposition.

If you If you areare new new to to thethe quantum quantum world, world, youyou areare undoubtedly undoubtedly wondering wondering what what it means it means forfor an an atom atom to be to be somehow somehow both both here here andand there. there.

Don’t Don’t be be discouraged discouraged if you if you find find thisthis confusing. confusing.

YouYou areare ab-absolutely solutely right right to wonder to wonder what what it means. it means.

This This is one is one of the of the central central mysteries mysteries of quantum of quantum mechanics. mechanics.

It isItenough, is enough, forfor now, now, if you if you justjust accept accept thisthis as aasmystery, a mystery, to which to which wewe attach attach thethe term term “superposi“superposition.” tion.”

Later Later wewe willwill be be ableable to demystify to demystify it. it.

Here Here is aisfirst a first step. step.

When When wewe saysay that that a quantum a quantum particle particle is is in a “superposition in a “superposition of being of being here here andand there,” there,” thisthis is related is related to to thethe

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wavelike wavelike nature nature of matter, of matter, for for a wave a wave is aisdisturbance a disturbance thatthat is spread is spread out,out, andand so it socan it can be be both both here here andand there. there.

WeWe speak speak of elementary of elementary particles, particles, butbut everything everything quantum, quantum, in- including cluding atoms atoms andand molecules, molecules, is both is both a particle a particle andand a wave. a wave.

Here Here is is a taste a taste of what of what thatthat means. means.

If we If we do do an an experiment experiment thatthat asks asks where where an an atom atom is, the is, the result result willwill be be thatthat it isitsomewhere is somewhere definite. definite.

ButBut be-between tween measurements, measurements, when when wewe areare notnot looking looking for for it, it it,turns it turns outout to to be be impossible impossible to project to project where where it might it might be.be.

It isItas is if asthe if the likelihood likelihood or propensity or propensity of finding of finding thethe particle particle spreads spreads as aaswave a wave when when wewe areare notnot looking. looking.

ButBut as soon as soon as we as we look look again, again, it isitalways is always somewhere. somewhere.

Imagine Imagine playing playing a game a game of of hidehideandandseek seek with with an an atom. atom.

WeWe open open ourour eyes, eyes, or turn or turn on on a detector, a detector, andand wewe seesee it somewhere. it somewhere.

ButBut when when wewe close close ourour eyes eyes it dissolves it dissolves intointo a wave a wave of of potentiality. potentiality.

Open Open ourour eyes eyes again again andand it isitalways is always somewhere. somewhere.

Another Another feature feature unique unique to the to the quantum quantum world world is called is called entanentanglement. glement.

If two If two particles particles interact, interact, andand then then move move apart, apart, they they remain remain intertwined intertwined in the in the sense sense that that they they seem seem to share to share properties properties which which cannot cannot be be broken broken down down to properties to properties each each enjoys enjoys individually. individually.

WeWe cancan stretch stretch ourour imagination imagination to apply to apply these these new new concepts concepts to to atoms atoms andand molecules molecules which which areare tootoo small small to see to see directly. directly.

WeWe must must study study them them indirectly, indirectly, andand to do to do that that wewe employ employ large large andand complex complex measurement measurement devices. devices.

Those Those measurement measurement devices devices areare part part of the of the everyday, everyday, familiar familiar world world of large of large objects. objects.

One One thing thing wewe cancan be be sure sure of is of that is that bigbig ev-everyday eryday things things display display none none of the of the bizarre bizarre behavior behavior quantum quantum me-mechanics chanics describes. describes.

A chair A chair is here is here or itorisitthere, is there, never never in aincombination a combination of such of such states. states.

When When wewe wake wake up up in the in the middle middle of the of the night night in ain a strange strange hotel hotel room, room, wewe may may be be unsure unsure where where thethe chair chair is, is, butbut wewe cancan be be sure sure it isit somewhere. is somewhere.

And And after after wewe collide collide with with it in it the in the dark, dark, ourour future future does does notnot become become entangled entangled with with its its future. future.

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In the In the world world as we as we experience experience it, it, catscats areare either either alive alive or dead, or dead, even even if they if they areare locked locked in ainbox. a box.

When When wewe open open thethe box, box, thethe catcat does does notnot suddenly suddenly resolve resolve from from a combination a combination of dead of dead andand alive alive to to dead. dead.

If we If we find find it dead it dead it will it will likely likely have have been been so for so for some some time, time, as as wewe willwill instantly instantly smell. smell.

Ordinary Ordinary objects objects appear appear to share to share none none of the of the quantum quantum weirdweirdnesses nesses of the of the atoms atoms of which of which they they areare made. made.

This This seems seems obvious, obvious, butbut it raises it raises a mystery. a mystery.

Quantum Quantum mechanics mechanics is the is the core core theory theory of of nature. nature.

As As such such it must it must be be universal. universal.

If itIfapplies it applies to an to an atom atom it must it must apply apply to two to two atoms, atoms, or ten or ten or ninety. or ninety.

And And wewe have have excellent excellent experiexperimental mental evidence evidence that that it does. it does.

Delicate Delicate experiments, experiments, in which in which large large molecules molecules areare putput in quantum in quantum superpositions, superpositions, show show us us that that they they areare justjust as quantum as quantum weird weird as electrons. as electrons.

ForFor oneone thing, thing, they they diffract diffract andand interfere interfere as waves. as waves.

ButBut then then quantum quantum mechanics mechanics must must apply apply to the to the vastvast collections collections of atoms of atoms that that make make up up youyou or me or me or our or our catcat or the or the chair chair on on which which sheshe is perched. is perched.

ButBut it doesn’t it doesn’t seem seem to. to.

NorNor does does quantum quantum mechanmechanics ics appear appear to to apply apply to to anyany of of thethe instruments instruments andand machines machines wewe employ employ to image to image thethe atoms atoms andand reveal reveal their their quantum quantum weirdnesses. weirdnesses.

How How cancan thisthis be?be?

In particular, In particular, when when wewe measure measure a property a property of an of an atom, atom, wewe em-employ ploy bigbig devices. devices.

TheThe atoms atoms may may be be in superpositions in superpositions of states of states andand so be so be several several places places at once, at once, butbut thethe measuring measuring instrument instrument always always indicates indicates justjust oneone outout of the of the possible possible answers answers to the to the questions questions wewe pose. pose.

Why Why is that? is that?

Why Why does does quantum quantum mechanics mechanics notnot apply apply to the to the very very devices devices wewe useuse to measure to measure quantum quantum systems? systems?

This This is called is called thethe measurement measurement problem. problem.

It has It has been been controvercontroversialsial andand unresolved unresolved since since thethe 1920s.

1920s.

TheThe factfact that, that, after after all all thisthis time, time, wewe have have found found no no agreement agreement among among experts experts means means there there is is something something basic basic about about nature nature wewe have have yetyet to understand. to understand.

So So there there is somewhere is somewhere a transition a transition between between thethe quantum quantum world, world, 6

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in which in which an an atom atom cancan be be several several places places at once, at once, andand thethe ordinary ordinary world, world, in in which which everything everything is always is always somewhere. somewhere.

If aIf molecule a molecule made made from from tenten or ninety or ninety atoms atoms cancan be be described described by by quantum quantum me-mechanics, chanics, butbut a cat a cat cannot, cannot, then then somewhere somewhere between between thethe twotwo there there is aisline a line delineating delineating where where thethe quantum quantum world world stops. stops.

AnAn answer answer to to thethe measurement measurement problem problem would would telltell us us where where that that lineline is and is and ex-explain plain thethe transition. transition.

There There areare people people who who areare sure sure they they know know thethe answer answer to to thethe measurement measurement problem. problem.

WeWe willwill meet meet some some of them of them andand their their ideas ideas later later on.on.

WeWe willwill want want to look to look outout forfor what what price price wewe have have to pay to pay to to expunge expunge thisthis quantum quantum insanity insanity from from ourour understanding understanding of of thethe world. world.

BROADLY BROADLY SPEAKING, SPEAKING, thethe people people who who aimaim to address to address thethe mysteries mysteries of of

quantum quantum mechanics mechanics fallfall intointo twotwo classes. classes.

TheThe first first group group assumes assumes that that thethe theory theory as itaswas it was formulated formulated in in thethe 1920s 1920s is essentially is essentially correct. correct.

They They believe believe thethe problem problem is not is not with with quantum quantum theory; theory; it isit instead is instead with with how how wewe understand understand it or it or speak speak about about it. it.

This This strategy strategy to mitigate to mitigate thethe strangeness strangeness of quanof quantum tum mechanics mechanics goes goes back back to some to some of the of the founders, founders, beginning beginning with with Niels Niels Bohr.

Bohr.

Niels Niels Bohr Bohr waswas a Danish a Danish physicist physicist who, who, while while stillstill in his in his twentwenties, ties, waswas thethe first first to apply to apply quantum quantum theory theory to atoms. to atoms.

As As he he grew grew older older he he became became thethe de de facto facto leader leader of of thethe quantum quantum revolurevolution, tion, partly partly duedue to to thethe attractiveness attractiveness of of hishis ideas ideas andand partly partly be-because cause he he educated educated andand mentored mentored many many of of thethe young young quantum quantum revolutionaries. revolutionaries.

TheThe second second group group hashas concluded concluded that that thethe theory theory is incomplete. is incomplete.

It can’t It can’t be be made made sense sense of because of because it isit not is not thethe whole whole story. story.

They They seek seek a completion a completion of the of the theory theory that that willwill telltell us the us the restrest of the of the story story 7

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and, and, by by doing doing so, so, resolve resolve thethe mysteries mysteries of quantum of quantum mechanics. mechanics.

This This strategy strategy goes goes back back to Albert to Albert Einstein.

Einstein.

More More than than anyone anyone else, else, Einstein Einstein waswas responsible responsible forfor initiating initiating thethe quantum quantum revolution. revolution.

HeHe waswas thethe first first to to articulate articulate thethe dual dual nature nature of light of light as aasparticle a particle andand a wave. a wave.

HeHe is by is by now now better better known known forfor hishis theory theory of relativity, of relativity, butbut hishis Nobel Nobel Prize Prize waswas forfor hishis work work on on quantum quantum theory, theory, andand he he himself himself admitted admitted that that he he spent spent much much more more time time on on quantum quantum theory theory than than on on relativity. relativity.

Yet,Yet, even even if he if he initiated initiated thethe quantum quantum revolution, revolution, Einstein Einstein diddid notnot become become oneone of of its its leaders, leaders, because because hishis realism realism required required that that he he reject reject thethe theory theory as as it was it was developed developed in the in the latelate 1920s.

1920s.

In In thethe language language introduced introduced in in thethe preface, preface, those those in in thethe first first group group areare mostly mostly antiantirealists realists or or magical magical realists. realists.

Realists Realists find find themselves themselves in the in the second second group. group.

Those Those who who argue argue forfor thethe incompleteness incompleteness of quantum of quantum mechanics mechanics point point to the to the factfact that that in most in most cases cases it can it can only only make make statistical statistical prepredictions dictions forfor thethe results results of of experiments. experiments.

Rather Rather than than saying saying what what willwill happen, happen, it gives it gives probabilities probabilities forfor what what might might happen. happen.

In aInletter a letter to his to his friend friend Max Max Born Born in 1926, in 1926, Einstein Einstein wrote: wrote: Quantum Quantum mechanics mechanics is certainly is certainly imposing. imposing.

ButBut an an inner inner voice voice tells tells meme thatthat it isit not is not yetyet thethe realreal thing. thing.

TheThe theory theory sayssays a a lot, but lot, but does does notnot really really bring bring us any us any closer closer to the to the secret secret of the of the ‘old‘old one’. one’.

I, at I, any at any rate, rate, amam convinced convinced thatthat He He is not is not playing playing 1 1 at dice. at dice.

Einstein Einstein waswas alsoalso friends friends with with Niels Niels Bohr, Bohr, andand their their divergent divergent responses responses to to quantum quantum mechanics mechanics fueled fueled a passionate a passionate debate debate be-between tween them them thatthat lasted lasted more more than than forty forty years, years, tilltill Einstein’s Einstein’s death. death.

It continues It continues between between their their intellectual intellectual descendants descendants to to thisthis day.day.

Einstein Einstein waswas thethe first first person person to to clearly clearly articulate articulate thethe need need for for a a 8

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revolutionary revolutionary new new theory theory of atoms of atoms andand radiation, radiation, butbut he he waswas unable unable to to accept accept thatthat quantum quantum mechanics mechanics waswas thatthat theory. theory.

HisHis first first re- response sponse to quantum to quantum mechanics mechanics waswas to argue to argue thatthat it was it was inconsistent. inconsistent.

When When thatthat failed, failed, he he argued argued instead instead thatthat quantum quantum mechanics mechanics gives gives an an incomplete incomplete description description of nature, of nature, which which leaves leaves outout an an essential essential part part of the of the picture. picture.

I believe I believe that that Einstein Einstein waswas unable unable to accept to accept quantum quantum mechanmechanics ics as aasdefinitive a definitive theory theory because because he he hadhad exceedingly exceedingly high high aspiraaspirations tions forfor science. science.

HeHe waswas driven driven by by thethe hope hope of of transcending transcending subjective subjective opinion opinion andand discovering discovering a true a true mirror mirror of nature of nature that that ex-exhibits hibits thethe essence essence of reality of reality in ainfew a few timeless timeless mathematical mathematical laws. laws.

ForFor him, him, science science aimed aimed to to capture capture thethe true true essence essence of the of the world, world, andand that that essence essence is independent is independent of us of us andand cancan have have nothing nothing to do to do with with what what wewe believe believe or know or know about about it. it.

Einstein, Einstein, of all of all people, people, must must have have feltfelt he he hadhad thethe right right to to de-demand mand thisthis because because he he hadhad achieved achieved it in it his in his discoveries discoveries of special of special andand general general relativity. relativity.

Having Having laidlaid thethe groundwork groundwork forfor quantum quantum physics, physics, he he sought sought to capture to capture thethe essence essence of the of the atomic atomic world world in ain a complete complete description description of atoms, of atoms, electrons, electrons, andand light. light.

Bohr Bohr replied replied that that atomic atomic physics physics required required a revolutionary a revolutionary revirevisionsion in how in how wewe understand understand what what science science is, is, as well as well as in as our in our conconception ception of the of the relationship relationship between between reality reality andand ourour knowledge knowledge of it. of it.

This This stemmed stemmed from from thethe factfact that that wewe areare a part a part of the of the world, world, so we so we must must interact interact with with thethe atoms atoms wewe seek seek to describe. to describe.

Bohr Bohr asserted asserted that that once once wewe absorbed absorbed thisthis revolutionary revolutionary change change in in ourour thinking, thinking, thethe completeness completeness of of quantum quantum mechanics mechanics would would be be unavoidable, unavoidable, because because it was it was built built intointo ourour being being participants participants in in thethe world world wewe seek seek to describe. to describe.

From From Bohr’s Bohr’s perspective, perspective, quantum quantum theory theory is complete is complete in the in the sense sense that that there there is no is no moremorecomplete complete de-description scription of the of the world world to be to be had. had.

If If wewe refuse refuse these these philosophical philosophical revolutions revolutions andand insist insist on on 9

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maintaining maintaining an an old-oldfashioned, fashioned, commonsense commonsense view view of reality of reality andand its its relation relation to our to our observations observations andand knowledge, knowledge, wewe have have to pay to pay a difa different ferent kind kind of of price. price.

WeWe have have to to contemplate contemplate that that wewe areare wrong wrong about about some some aspect aspect of nature. of nature.

WeWe have have to find to find outout which which common common assumption assumption is wrong is wrong andand replace replace it with it with a radically a radically new new physical physical hypothesis hypothesis that that opens opens thethe wayway to atonew a new theory theory that that willwill complete complete quantum quantum mechanics. mechanics.

Thanks Thanks to to a combination a combination of of theory theory andand experiment, experiment, starting starting with with a paper a paper by by Einstein Einstein andand twotwo collaborators collaborators in 1935, in 1935, wewe know know oneone aspect aspect of of thisthis completion. completion.

TheThe new new theory theory must must violate violate thethe commonplace commonplace assumption assumption that that things things interact interact only only with with other other things things that that areare near near them them in space. in space.

This This assumption assumption is called is called locality. locality.

A big A big part part of the of the story story I will I will be be telling telling in later in later chapters chapters is how is how thisthis commonsense commonsense idea idea must must be be transcended transcended in the in the theory theory which which willwill replace replace quantum quantum mechanics. mechanics.

THIS THIS BOOK BOOK HASHAS THREE THREE PURPOSES. PURPOSES. First, First, I want I want to to explain explain to to lay-lay-

people people justjust what what thethe puzzles puzzles at the at the heart heart of quantum of quantum mechanics mechanics are.are.

After After more more than than a century a century of studying of studying quantum quantum physics, physics, it isit is remarkable remarkable that that there there continues continues to this to this dayday to be to be no no agreement agreement on on thethe solution solution of these of these puzzles. puzzles.

ButBut having having explained explained thethe reasons reasons forfor thethe debate debate in ainway a way that that is is fairfair to both to both sides, sides, I will I will notnot staystay impartial. impartial.

In the In the great great debate debate about about whether whether quantum quantum mechanics mechanics is the is the lastlast word word or or not,not, I side I side with with Einstein.

Einstein.

I believe I believe that that there there is aislayer a layer of reality of reality deeper deeper than than that that described described by by Bohr, Bohr, which which cancan be be understood understood without without compromiscompromisinging old-oldfashioned fashioned notions notions of reality of reality andand ourour ability ability to comprehend to comprehend andand describe describe it. it.

Thus, Thus, mymy second second purpose purpose is to is advocate to advocate a point a point of view of view about about thethe puzzles puzzles of of quantum quantum mechanics. mechanics.

This This is that is that thethe problems problems cancan be be 10 10

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resolved resolved only only by by progress progress in science in science which which willwill uncover uncover a world a world be-beyond yond quantum quantum mechanics. mechanics.

Where Where quantum quantum mechanics mechanics is mysterious is mysterious andand confusing, confusing, thisthis deeper deeper theory theory willwill be be entirely entirely comprehensible. comprehensible.

I can I can make make thisthis claim claim because because wewe have have known known since since thethe inveninventiontion of quantum of quantum mechanics mechanics how how to present to present thethe theory theory in ainway a way that that dissolves dissolves thethe mysteries mysteries andand resolves resolves thethe puzzles. puzzles.

In this In this approach, approach, there there is no is no challenge challenge to our to our usual usual beliefs beliefs in an in an objective objective reality, reality, a a reality reality unaffected unaffected by what by what wewe know know or do or do about about it, and it, and about about which which it isit possible is possible to have to have complete complete knowledge. knowledge.

In In thisthis reality, reality, there there is is justjust oneone universe, universe, andand when when wewe observe observe something something about about it, it, it isit is because because it isittrue. is true.

This This cancan justly justly be be called called a realist a realist approach approach to the to the quantum quantum world. world.

AnAn antiantirealist realist approach approach ascribes ascribes thethe mysteries mysteries of quantum of quantum me-mechanics chanics to to subtleties subtleties having having to to do do with with how how wewe gain gain knowledge knowledge about about nature. nature.

Such Such approaches approaches have have radical radical proposals proposals to to make make about about epistemology, epistemology, which which is the is the branch branch of philosophy of philosophy concerned concerned with with how how wewe know know things. things.

Realist Realist approaches approaches assume assume wewe areare ableable to arrive to arrive sooner sooner or later or later at aattrue a true representation representation of the of the world world andand so so areare deliberately deliberately naive naive about about epistemology. epistemology.

Instead, Instead, realists realists areare in- interested terested in ontology, in ontology, which which is the is the study study of what of what exists. exists.

By By contrast, contrast, antiantirealists realists believe believe wewe cannot cannot know know what what really really exists, exists, apart apart from from ourour representation representation of the of the knowledge knowledge wewe have have of the of the world, world, which which is is gained gained only only through through interacting interacting with with it. it.

So So I will I will endeavor endeavor to reassure to reassure readers readers that that quantum quantum mechanics mechanics cancan be be understood understood completely completely within within a realist a realist perspective perspective in which in which thethe external external world world cancan be be completely completely comprehended comprehended as indepenas independent dent from from us.us.

There There is no is no mysterious mysterious effect effect of the of the observer observer on on thethe observed. observed.

Reality Reality is out is out there, there, recalcitrant recalcitrant to to ourour willwill andand thethe choices choices wewe make. make.

That That reality reality is fully is fully comprehensible. comprehensible.

And And that that reality reality consists consists of aofsingle a single world. world.

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does does notnot by by itself itself mean mean thatthat thethe philosophically philosophically more more extravagant extravagant proposals proposals areare wrong. wrong.

ButBut it does it does mean mean thatthat there there is no is no strong strong scienscientifictific reason reason to believe to believe in them, in them, because because realism realism is always is always to be to be prepreferred ferred in science, in science, when when it can it can be be achieved. achieved. then, then, is so is much so much of the of the talktalk about about quantum quantum theory theory inspired inspired Why, Why, by by thethe weirder weirder ideas ideas in which in which reality reality depends depends on on ourour knowledge knowledge of of it or it or there there areare multiple multiple realities? realities?

This This is aisproblem a problem forfor historians historians of ideas. of ideas.

One One such such historian, historian, Paul Paul Forman, Forman, hashas tiedtied thethe dominance dominance of Bohr of Bohr andand Heisenberg’s Heisenberg’s antiantirealist realist philosophy philosophy within within thethe scienscientific tific community community in the in the 1920s 1920s andand 1930s 1930s to the to the embrace embrace of chaos of chaos andand irrationality irrationality advocated advocated by by Spengler Spengler andand others others in the in the wake wake of of thethe First First World World War. War. That That history history is fascinating, is fascinating, butbut it isitfor is for scholars scholars to do to do justice justice to to it. Iit.am I am notnot a scholar, a scholar, I am I am a scientist, a scientist, andand thisthis brings brings meme to my to my third third purpose purpose in writing in writing thisthis book. book.

I have I have been been on on Einstein’s Einstein’s sideside in in thethe search search forfor a deeper a deeper butbut simpler simpler reality reality behind behind quantum quantum mechanics mechanics since since first first reading reading him him on on thethe subject subject as aashigh a high school school dropout. dropout.

MyMy journey journey in physics in physics be-begangan with with reading reading Einstein’s Einstein’s autobiographical autobiographical notes, notes, where, where, in the in the lastlast fewfew years years of his of his life,life, in the in the 1950s, 1950s, he he reflected reflected on on thethe twotwo main main tasks tasks he he feltfelt were were leftleft incomplete incomplete in physics. in physics.

These These were were to make to make sense sense of quantum of quantum physics physics and, and, after after that, that, to unify to unify thethe new new underunderstanding standing of the of the quantum quantum with with gravity, gravity, by by which which he he meant meant hishis gengeneraleral theory theory of relativity. of relativity.

I recall I recall thinking thinking that that maybe maybe I could I could trytry to to help. help.

I was I was unlikely unlikely to to succeed, succeed, butbut perhaps perhaps here here waswas something something worth worth striving striving for.for.

After, After, as as it were, it were, getting getting mymy mission mission from from reading reading Einstein’s Einstein’s autobiographical autobiographical notes, notes, I found I found that that book book by by de de Broglie, Broglie, talked talked mymy wayway intointo a good a good college, college, found found great great teachers, teachers, andand gotgot lucky lucky several several times times in my in my applications applications forfor graduate graduate school school andand beyond. beyond.

I’mI’m havhavinging a wonderful a wonderful life,life, andand as aasscientist a scientist on on thethe frontier, frontier, I’veI’ve hadhad many many 12 12

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chances chances to take to take a shot a shot on on goal, goal, aimed aimed at solving at solving Einstein’s Einstein’s twotwo bigbig questions. questions.

I haven’t I haven’t succeeded, succeeded, at least at least so far. so far.

Very Very unfortunately, unfortunately, neither neither hashas anyone anyone else. else.

At At thethe same same time, time, over over thethe past past several several decades decades there there hashas at least at least been been progress progress toward toward understanding understanding thethe problem. problem.

That That is not is not nearly nearly as good as good as it aswould it would be be to solve to solve thethe problem, problem, butbut neither neither is itisnothing. it nothing.

WeWe know know much much better better than than Einstein Einstein diddid thethe obstacles obstacles that that a theory a theory that that transcends transcends thethe limits limits of quantum of quantum me-mechanics chanics must must overcome. overcome.

And And because because of that, of that, some some very very interesting interesting proposals proposals andand hypotheses hypotheses have have been been putput forward, forward, which which may may frame frame thethe deeper deeper theory theory forfor which which wewe search.* search.* I have I have been been thinking thinking about about thethe question question of how of how to go to go beyond beyond quantum quantum mechanics mechanics since since thethe midmid1970s, 1970s, andand I’veI’ve never never been been more more excited excited andand optimistic optimistic about about thethe prospects prospects forfor success. success.

So So thisthis is is mymy third third reason reason forfor writing writing thisthis book, book, which which is to is bring to bring to atowider a wider audience audience a report a report from from thethe front front in our in our search search forfor thethe world world beyond beyond thethe quantum. quantum.

* A note * A note to my to expert my expert readers: readers: Quantum Quantum foundations foundations is presently is presently a very a very lively lively subject, subject, withwith many many exciting exciting developments developments bothboth experimental experimental and theoretical. and theoretical.

Many Many proposals proposals compete compete to resolve to resolve the the puzzles puzzles we will we will meetmeet here.here.

I should I should warnwarn the reader the reader that that our path our path through through thesethese frontiers frontiers will will be a be a narrow narrow one,one, and and therethere are many are many exciting exciting ideasideas and and results results that that I do Inot do mention not mention here.here.

HadHad I tried I tried to review to review the whole the whole field,field, or include or include all the all latest the latest supremely supremely clever clever advances, advances, the result the result would would havehave beenbeen a lessa accessible less accessible book. book.

My first My first aim aim is to isintroduce to introduce the world the world of quantum of quantum phenomena, phenomena, not the not the full full spectrum spectrum of competing of competing interpretations interpretations of those of those phenomena. phenomena.

I apologize I apologize in advance in advance to those to those experts experts whowho don’tdon’t find find theirtheir preferred preferred version version of quantum of quantum physics physics here,here, and and encourage encourage themthem to to writewrite theirtheir ownown books. books.

I alsoI also apologize apologize to the to historians. the historians.

I amInot am anot scholar, a scholar, and the and stories the stories I tellI are tell are creation creation myths, myths, handed handed down down fromfrom teacher teacher to student, to student, originating, originating, in some in some cases, cases, withwith the foundthe founders themselves. ers themselves.

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TWO TWO

Quanta Quanta

II

f we f we break break quantum quantum mechanics mechanics down down to its to its most most essential essential prinprinciple, ciple, it isitthis: is this: WeWe cancan only only know know halfhalf of what of what we we would would need need to know to know if we if we wanted wanted to completely to completely control, control, or precisely or precisely predict, predict, thethe future. future.

This This disrupts disrupts thethe basic basic ambition ambition of physics, of physics, which which is to is be to be ableable to predict to predict thethe future. future.

It was It was hoped hoped that that thisthis power power would would follow follow if if only only wewe could could givegive thethe physical physical world world a complete a complete description. description.

By By describing describing fully fully thethe motion motion of every of every particle particle andand thethe action action of every of every force, force, wewe would would be be ableable to to work work outout exactly exactly what what would would happen happen in the in the future. future.

Before Before quantum quantum mechanics mechanics waswas formulated formulated in the in the 1920s, 1920s, wewe physicists physicists were were confident confident that that if we if we could could learn learn thethe laws laws that that govern govern thethe fundamental fundamental particles, particles, wewe would would be be ableable to predict to predict andand explain explain everything everything that that happened happened in the in the world. world.

TheThe hypothesis hypothesis that that thethe future future is completely is completely determined determined by by thethe

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laws laws of of physics physics acting acting on on thethe present present configuration configuration of of thethe world world is called is called determinism. determinism.

This This is an is an extraordinarily extraordinarily powerful powerful idea, idea, whose whose influence influence cancan be be seen seen in diverse in diverse fields. fields.

If you If you appreciate appreciate thethe extent extent to which to which determinism determinism dominated dominated thought thought in the in the nineteenth nineteenth century, century, youyou cancan begin begin to understand to understand thethe revolutionary revolutionary impact impact of of quantum quantum mechanics mechanics across across all all fields, fields, because because quantum quantum mechanics mechanics precludes precludes determinism. determinism.

To To emphasize emphasize thisthis point, point, I like I like to quote to quote from from Tom Tom Stoppard’s Stoppard’s play play Arcadia, Arcadia, in which in which hishis precocious precocious heroine, heroine, Thomasina, Thomasina, explains explains to her to her tutor: tutor: If you If you could could stop stop every every atom atom in in its its position position andand direction, direction, and and if your if your mind mind could could comprehend comprehend all all thethe actions actions thus thus sus-suspended, pended, then then if you if you were were really, really, really really good good at algebra at algebra youyou could could write write thethe formula formula for for all all thethe future; future; andand although although no-nobody can body can be so be clever so clever as to as do to do it, the it, the formula formula must must exist exist justjust as as 1 1 if one if one could. could.

A complete A complete description description of nature, of nature, at aatgiven a given time, time, is called is called a a state. state.

ForFor example, example, if we if we think think of the of the world world as composed as composed of partiof particlescles whizzing whizzing around, around, thethe state state tells tells us where us where each each of them of them is, and is, and how how fastfast andand in what in what direction direction each each is moving, is moving, at that at that moment. moment.

TheThe power power of physics of physics comes comes from from its its laws, laws, which which dictate dictate how how nature nature changes changes in time. in time.

They They do do thisthis by by transforming transforming thethe state state of of thethe world world as itasisitnow is now to the to the state state at any at any future future time. time.

A law A law of physof physics ics functions functions in some in some ways ways likelike a computer a computer program: program: it reads it reads in inin inputput andand puts puts outout output. output.

TheThe input input is the is the state state at aatgiven a given time; time; thethe output output is the is the state state at some at some future future time.* time.* Along Along with with thethe computation computation comes comes an an explanation explanation of how of how thethe * The * The metaphor metaphor of the of universe the universe as a computer as a computer is helpful is helpful for illustrating for illustrating determinism, determinism, but is but onisthe on the whole whole misleading, misleading, as I will as I will argue argue below. below.

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world world changes changes in time. in time.

TheThe lawlaw acting acting on on thethe present present state state causes causes thethe future future states. states.

A successful A successful prediction prediction of the of the future future state state is taken is taken as aasvalidation a validation of that of that explanation. explanation.

TheThe prediction prediction is deterministic, is deterministic, in that in that a precise a precise input input leads leads to to a precise a precise output. output.

This This confirms confirms a a belief belief that that thethe information information that that went went intointo describing describing thethe state state is in is in factfact a complete a complete description description of the of the world world at one at one moment moment of time. of time.

This This concept concept of aoflaw a law is basic is basic to atorealist a realist conception conception of nature of nature and, and, as such, as such, transcends transcends anyany oneone theory. theory.

Newtonian Newtonian mechanics mechanics andand Einstein’s Einstein’s twotwo theories theories of relativity of relativity all all work work thethe same same way. way.

One One ap-applies plies thethe lawlaw to the to the state state at an at an initial initial time, time, andand it transforms it transforms that that state state to the to the state state at some at some future future time. time.

This This schema schema forfor explaining explaining nature nature waswas invented invented by by Newton, Newton, so so wewe callcall it the it the Newtonian Newtonian paradigm. paradigm.

It isItalso is also worth worth mentioning mentioning that that in almost in almost all all cases cases so far so far known, known, thethe laws laws areare reversible. reversible.

One One cancan input input thethe state state at some at some future future time time andand runrun thethe lawlaw backward backward to to output output thethe state state at an at an earlier earlier time. time.

(The (The issue issue of the of the reversibility reversibility of time of time andand of the of the fundamental fundamental laws laws is aiscentral a central concern concern of chapters of chapters 14 14 andand 15.)15.)

It isIt often is often thethe case case that that thethe information information needed needed to completely to completely describe describe thethe state state of aofphysical a physical system system comes comes in pairs. in pairs.

Position Position andand momentum.* momentum.* Volume Volume andand pressure. pressure.

Electric Electric field field andand magnetic magnetic field. field.

WeWe need need both both to to predict predict thethe future. future.

Quantum Quantum mechanics mechanics sayssays wewe cancan know know only only one. one.

This This means means wewe can’t can’t precisely precisely predict predict thethe future. future.

That That is just is just thethe first first of of thethe blows blows to to ourour comfortable comfortable intuitions intuitions that that wewe willwill have have to absorb to absorb from from quantum quantum theory. theory.

Which Which member member of of each each pairpair is the is the oneone that that cancan be be known? known?

Quantum Quantum mechanics mechanics sayssays youyou choose! choose!

This This is the is the basis basis of its of its chalchallenge lenge to realism. to realism. * Momentum * Momentum will will be defined be defined shortly, shortly, but roughly, but roughly, a body’s a body’s momentum momentum is proportional is proportional to both to both its its speed speed and and mass. mass.

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There There is more is more to say to say about about thethe impossibility impossibility of predicting of predicting thethe future. future.

To To getget there, there, let’slet’s take take advantage advantage of of thethe great great generality generality quantum quantum mechanics mechanics claims, claims, andand speak speak a bit a bit abstractly. abstractly.

WeWe want want to to describe describe some some physical physical system system in terms in terms of aofpair a pair of variables— of variables— wewe willwill callcall them them A A andand B. B. Quantum Quantum physics physics asserts asserts a twoa twopart part principle. principle.

1. If1.we If we knew knew both both A and A and B atB aatgiven a given time, time, we we could could precisely precisely predict predict thethe future future of the of the system. system.

2. We 2. We cancan choose choose to measure to measure A or A we or we cancan choose choose to measure to measure B; and B; and in each in each casecase we we willwill succeed. succeed.

ButBut we we can’t can’t do do bet-better. ter.

WeWe cannot cannot choose choose to to simultaneously simultaneously measure measure both both A A andand B. B.

As As I have I have stated stated it, this it, this is aisprohibition a prohibition of what of what wewe cancan measure; measure; but, but, if we if we prefer, prefer, wewe cancan express express it as it aasprohibition a prohibition of what of what wewe cancan know know about about thethe system. system.

ButBut wait, wait, why why can’t can’t youyou measure measure A and A and then, then, at aatlater a later time, time, measure measure B? B? YouYou can. can.

ButBut your your measurement measurement of B ofwill B will render render irrelirrelevant evant (for(for thethe purpose purpose of predicting of predicting thethe future) future) your your past past knowlknowledge edge of A. of A. One One wayway thisthis cancan happen happen is that is that after after thethe measurement measurement of B, of the B, the value value of A of isArandomized. is randomized.

WeWe cannot cannot measure measure B without B without disrupting disrupting thethe value value of A, of A, andand vicevice versa. versa.

Thus, Thus, if we if we measure measure A, A, then then B, then B, then A again, A again, thethe value value of A ofwe A we getget thethe second second time time willwill be be random, random, andand hence hence unrelated unrelated to the to the value value wewe gotgot thethe first first time time wewe measured measured A. A. 1. and 1. and 2. together 2. together areare called called thethe principle principle of nonof noncommutativity. commutativity.

Two Two actions actions areare saidsaid to commute to commute if itifdoesn’t it doesn’t matter matter in what in what order order wewe do do them. them.

If the If the action action that that is done is done first first matters, matters, wewe saysay they they areare nonnoncommutative. commutative.

It doesn’t It doesn’t matter matter (except (except to to a few a few fanatics) fanatics) in in what what order order youyou putput milk milk andand sugar sugar intointo coffee; coffee; they they commute. commute.

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Getting Getting dressed dressed involves involves nonnoncommutative commutative operations; operations; thethe order order in in which which youyou putput on on your your underwear underwear andand pants pants matters. matters.

ButBut it doesn’t it doesn’t matter matter which which sock sock youyou putput on on first, first, or whether or whether youyou putput your your socks socks on on first, first, partway partway through through thethe process, process, or last. or last.

So So putting putting on on socks socks commutes commutes with with everything everything except except putting putting on on shoes. shoes.

(The (The mathemathematically matically minded minded willwill understand understand thisthis as an as an application application of algebra of algebra to topology.) to topology.)

What What if we if we allow allow there there to be to be some some specified specified amount amount of uncerof uncertainty tainty in in thethe measurement measurement of of A? A? Then Then wewe cancan measure measure B, B, butbut only only up up to some to some accuracy. accuracy.

These These uncertainties uncertainties areare reciprocal—the reciprocal—the better better wewe know know A, A, thethe worse worse wewe cancan know know B, and B, and vicevice versa. versa.

ForFor example, example, let’slet’s suppose suppose that that A is A the is the position position of aofparticle. a particle.

Then Then B isBits is its momentum. momentum.

Suppose Suppose wewe do do a measurement a measurement that that tells tells its its location location to within to within a meter. a meter.

Then Then wewe cancan measure measure thethe momenmomentum tum to atocorresponding a corresponding uncertainty. uncertainty.

If we If we increase increase thethe uncertainty uncertainty in A, in A, then then wewe cancan make make thethe measurement measurement of B of more B more precise precise andand vicevice versa. versa.

This This gives gives us us a principle a principle called, called, notnot surprisingly, surprisingly, thethe uncertainty uncertainty principle.* principle.*

(Uncertainty (Uncertainty in A) in A) × (Uncertainty × (Uncertainty in B) in B) > a>constant a constant Applied Applied to position to position andand momentum, momentum, it reads it reads

(Uncertainty (Uncertainty in position) in position) × (Uncertainty × (Uncertainty in momentum) in momentum) > a>constant a constant Physics Physics is like is like a college a college campus campus where where every every building building is named is named after after someone. someone.

TheThe constant constant is named is named after after Max Max Planck, Planck, andand thethe uncertainty uncertainty principle principle is named is named after after Werner Werner Heisenberg.

Heisenberg. * > means * > means “is larger “is larger than.” than.”

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TheThe uncertainty uncertainty principle principle is quite is quite powerful, powerful, as is asshown is shown by by thisthis important important consequence. consequence.

Let’s Let’s go go back back to the to the scenario scenario in which in which youyou measure measure A, A, then then youyou measure measure B, then B, then youyou measure measure A again.

A again.

As As I I saidsaid above, above, once once youyou know know thethe result result of measuring of measuring B, B, thethe second second measurement measurement of A of isArandomized; is randomized; it isitno is no longer longer equal equal to the to the origioriginalnal value value of A. of A. ButBut suppose suppose that, that, justjust before before youyou remeasure remeasure A, A, youyou do do something something to forget to forget what what thethe value value of B ofwas.

B was.

Then Then thethe system system remembers— remembers— yes,yes, that that word word is the is the oneone wewe useuse to to describe describe thisthis situation— situation— thethe original original value value of A. of A. This This is called is called interference. interference.

It isItallowed is allowed by by thethe uncertainty uncertainty prinprinciple ciple because because once once youyou forget forget thethe measurement measurement of of B, B, B’sB’s unceruncertainty tainty is very is very large, large, so A’s so A’s uncertainty uncertainty cancan be be small. small.

ButBut how how cancan wewe undo undo a measurement? a measurement?

LetLet meme givegive a fanciful a fanciful example. example.

There There areare many many simple simple cases cases in which in which A and A and B each B each have have twotwo possible possible values. values.

LetLet thethe systems systems wewe study study be be people, people, andand let let A A be be political political identity, identity, which which wewe willwill simplify simplify to be to be a binary a binary choice: choice: either either leftleftwing wing or or right. right.

I will I will let let B be B be petpet preference, preference, catcat lovers lovers versus versus dogdog lovers. lovers.

WeWe now now play play a game a game in which in which a person a person can’t can’t have have both both a definite a definite petpet preference preference andand a political a political identity. identity.

WeWe go go to ato a party party where where everyone everyone hashas leftleftwing wing views views andand askask each each whether whether they they areare a cat a cat person person or aordog a dog person. person.

WeWe putput thethe catcat lovers lovers in the in the living living room room andand thethe dogdog lovers lovers in the in the kitchen. kitchen.

If we If we go go intointo either either room room andand inquire inquire about about their their political political views, views, then then halfhalf willwill now now be be rightrightwingers. wingers.

That That is what is what must must happen happen if political if political identity identity andand petpet preference preference don’t don’t commute. commute.

ButBut let’slet’s afterward afterward callcall everyone everyone together together intointo thethe dining dining room. room.

WeWe let let them them mingle mingle forfor a while, a while, then then wewe go go in and in and pick pick a random a random person. person.

They They could could have have come come from from either either thethe living living room room or the or the kitchen, kitchen, wewe don’t don’t know know which, which, so we’ve so we’ve lostlost track track of their of their petpet prefpreference. erence.

Then, Then, when when wewe askask them them about about politics, politics, wewe find find they they areare all all leftleftwingers wingers again. again.

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These These principles principles areare entirely entirely general. general.

A and A and B are B are often often thethe an-answers swers to yes/no to yes/no questions. questions.

ButBut in the in the original original case, case, A was A was thethe po-position sition of of an an elementary elementary particle, particle, saysay an an electron, electron, andand B was B was thethe momentum momentum of the of the particle. particle.

Momentum Momentum is one is one of those of those words words that that functions functions as aasbarrier a barrier to to comprehension, comprehension, so let’s so let’s take take a moment a moment to define to define it. it.

In physics In physics wewe often often have have to refer to refer to the to the speed speed andand thethe direction direction of motion of motion of aofparticle. a particle.

WeWe combine combine these these intointo oneone quantity quantity which which wewe callcall thethe velocity. velocity.

YouYou cancan think think of aofparticle’s a particle’s velocity velocity as an as an ar- arrowrow that that points points in the in the direction direction of its of its motion. motion.

TheThe faster faster thethe speed, speed, thethe longer longer thethe arrow. arrow.

To survive To survive a collision a collision youyou want want to experience to experience as little as little force force as possias possible.ble.

TheThe force force a truck a truck willwill impart impart on aon car a car is proportional is proportional to the to the truck’s truck’s change change of of speed. speed.

ButBut it’sit’s alsoalso proportional proportional to to thethe mass mass of of thethe truck. truck.

You’d You’d rather rather collide collide with with a Pinga PingPong Pong ballball than than a truck, a truck, even even if they if they areare traveling traveling toward toward youyou at the at the same same speed. speed.

To To express express this, this, physicists physicists define define momentum momentum as the as the product product of the of the mass mass times times thethe velocity. velocity.

This This is also is also an an arrow arrow pointing pointing in the in the direction direction of motion, of motion, only only now now thethe length length is proportional is proportional to both to both thethe speed speed andand thethe mass. mass.

Momentum Momentum is ais central a central concept concept in in physics physics because because it isit conis conserved. served.

That That means means that that in any in any processes processes at all, at all, wewe cancan addadd up up thethe momenta momenta of the of the various various particles particles involved involved at the at the beginning, beginning, and, and, no no matter matter what what happens, happens, thethe resulting resulting total total momentum momentum won’t won’t change change in time. in time.

Before, Before, during, during, andand after after a collision, a collision, thethe total total momentum momentum willwill be be thethe same. same.

What What happens happens in aincollision a collision is that is that momentum momentum is is exchanged exchanged from from oneone body body to another. to another.

This This change change of momentum of momentum is experienced is experienced as aasforce. a force.

Energy Energy is another is another conserved conserved quantity. quantity.

TheThe total total energy energy of aofsysa systemtem of of particles particles never never changes changes in in time. time.

When When particles particles interact, interact, oneone may may gain gain energy energy while while thethe restrest loselose energy. energy.

ButBut thethe total total energy energy remains remains thethe same; same; none none is created is created or destroyed. or destroyed.

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FIGURE FIGURE 1. A truck 1. A truck carries carries much much more more momentum momentum thanthan a Ping-Pong a Ping-Pong ball ball going going the same the same velocity, velocity,

because because its mass its mass is soismuch so much greater, greater, and and the momentum the momentum is the is product the product of the of mass the mass and and the the velocity. velocity.

Energy Energy andand momentum momentum areare related. related.

WeWe won’t won’t need need thethe exact exact relation, relation, butbut wewe need need to know to know that that a particle a particle that that is moving is moving freely, freely, andand hashas an an exact exact value value of momentum, of momentum, alsoalso hashas an an exact exact energy. energy.

TheThe uncertainty uncertainty principle principle then then sayssays that that wewe can’t can’t know know both both thethe position position andand momentum momentum of of a body a body at at thethe same same time. time.

This This means means wewe can’t can’t make make a precise a precise prediction prediction of its of its future, future, because because to to do do so so wewe would would need need to know to know both both where where something something is and is and how how fastfast andand in what in what direction direction it isitmoving, is moving, with with complete complete accuracy. accuracy.

If we If we want want to develop to develop an an intuition intuition about about how how quantum quantum partiparticlescles behave, behave, wewe willwill need need to be to be ableable to visualize to visualize a particle a particle with with a a definite definite position, position, but, but, because because of the of the uncertainty uncertainty principle, principle, no no def-definite inite momentum momentum or velocity. or velocity.

This This is not is not hard: hard: visualize visualize thethe particle particle being being somewhere somewhere momentarily. momentarily.

In the In the next next moment moment it will it will alsoalso be be somewhere somewhere definite, definite, justjust somewhere somewhere else. else.

Because Because its its momentum momentum is indefinite, is indefinite, it jumps it jumps around around randomly. randomly.

ButBut how how do do wewe visualize visualize a particle a particle with with a definite a definite momentum, momentum, butbut a completely a completely indefinite indefinite position? position?

This This seems seems more more challenging. challenging.

If you If you look look forfor it, you it, you have have an an equal equal chance chance of finding of finding it anywhere. it anywhere.

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So So it isitcompletely is completely spread spread out.out.

ButBut how how do do wewe visualize visualize its its definite definite momentum? momentum?

TheThe answer answer is that is that a particle a particle with with a definite a definite momentum, momentum, butbut a a completely completely indefinite indefinite position, position, cancan be be visualized visualized as aaswave. a wave.

And And notnot justjust anyany wave, wave, butbut a pure a pure wave, wave, oneone which which vibrates vibrates at aatsingle a single frequency. frequency.

A wave A wave cancan be be characterized characterized by by twotwo numbers. numbers.

One One is its is its fre-frequency; quency; thisthis is the is the number number of times of times perper second second that that it oscillates. it oscillates.

TheThe other other is the is the distance distance between between thethe peaks, peaks, which which is known is known as the as the wavelength. wavelength.

These These areare related related in the in the following following way: way: if you if you multiply multiply these these twotwo numbers numbers together, together, youyou getget thethe speed speed at which at which thethe wave wave is traveling. is traveling.

Thus Thus a wave a wave which which oscillates oscillates with with a single a single frequency frequency willwill alsoalso have have a definite a definite wavelength. wavelength.

Quantum Quantum mechanics mechanics asserts asserts that that thethe momentum momentum of the of the partiparticlecle andand thethe wavelength wavelength of the of the wave wave that that represents represents it are it are related related in in a simple a simple way, way, which which is that is that they they areare inversely inversely proportional. proportional.

That That is, is,

wavelength wavelength = h=/momentum h /momentum h ish the is the same same Planck’s Planck’s constant constant that that came came intointo thethe uncertainty uncertainty relations. relations.

LetLet us us assume assume for for a moment a moment thatthat no no force force actsacts on on ourour particle, particle, perhaps perhaps because because it isitvery is very far far from from everything everything else. else.

In the In the absence absence of of forces, forces, a particle a particle with with a definite a definite momentum momentum alsoalso hashas a definite a definite en-energy. ergy.

That That energy energy is in is turn in turn related related to the to the frequency frequency of the of the wave, wave, in in thatthat they they vary vary proportionately. proportionately.

Energy Energy = h=×hfrequency × frequency These These relations relations andand correspondences correspondences areare universal. universal.

Everything Everything in the in the quantum quantum world world cancan be be viewed viewed as both as both a wave a wave andand a particle. a particle.

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This This is aisdirect a direct consequence consequence of the of the basic basic principle principle that that wewe cancan meameasure sure thethe particle’s particle’s position position or or measure measure its its momentum, momentum, butbut wewe cannot cannot measure measure both both at the at the same same time. time.

When When wewe wish wish to measure to measure its its position, position, wewe visualize visualize it as it aaspara particle, ticle, localized, localized, butbut justjust momentarily, momentarily, at aatpoint a point in space. in space.

TheThe momomentum mentum is completely is completely uncertain, uncertain, so the so the next next moment, moment, if we if we look look again, again, wewe willwill find find it has it has randomly randomly jumped jumped somewhere somewhere else. else.

It can’t It can’t remain remain in one in one place place because, because, if itifdid, it did, it would it would have have a definite a definite value value of momentum, of momentum, namely namely zero. zero.

If, If, on on thethe other other hand, hand, wewe choose choose to to measure measure thethe particle’s particle’s momentum, momentum, wewe willwill discover discover it has it has some some definite definite value. value.

It isIt nois nowhere where in particular, in particular, so so wewe visualize visualize it as it aaswave, a wave, butbut oneone with with a a definite definite wavelength wavelength andand frequency, frequency, according according to the to the relations relations justjust mentioned. mentioned.

What What is so is crazily so crazily fabulous fabulous about about thisthis is that is that waves waves andand partiparticlescles areare quite quite different. different.

A particle A particle always always hashas a definite a definite position, position, localized localized somewhere somewhere in space. in space.

Its Its motion motion traces traces outout a path a path through through space, space, what what wewe callcall its its trajectory. trajectory.

Moreover, Moreover, according according to Newtoto Newtonian nian physics, physics, at each at each moment moment a particle a particle alsoalso hashas a definite a definite velocity velocity and, and, consequently, consequently, a definite a definite momentum. momentum.

A wave A wave is almost is almost thethe opposite. opposite.

It isItdelocalized; is delocalized; it spreads it spreads outout as itastravels, it travels, occupying occupying all all thethe space space available available to it. to it.

ButBut now now wewe areare learning learning that that waves waves andand particles particles areare different different sides sides of aofduality, a duality, that that is, different is, different ways ways of visualizing of visualizing oneone reality. reality.

A A single single reality reality with with a dual a dual nature: nature: a duality a duality of waves of waves andand particles. particles.

A quantum A quantum particle particle cancan have have a position. a position.

WeWe askask where where it is, it and is, and wewe willwill find find it somewhere. it somewhere.

ButBut a quantum a quantum particle particle never never hashas a traa trajectory, jectory, because, because, if we if we know know where where it is, it is, where where it will it will be be next next is is completely completely uncertain. uncertain.

WeWe must must getget used used to thinking to thinking of particles of particles at at definite definite positions positions which which areare notnot points points on on trajectories. trajectories.

Similarly, Similarly, if if wewe measure measure a momentum a momentum wewe willwill always always find find a value. a value.

ButBut then then it’sit’s 23 23

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a wave, a wave, spread spread outout everywhere. everywhere.

Where Where wewe willwill find find thethe particle, particle, if if wewe next next measure measure its its position, position, is completely is completely uncertain. uncertain.

This This scheme, scheme, it must it must be be admitted, admitted, hashas an an incredible incredible elegance. elegance.

ButBut what what is most is most compelling compelling is its is its universality. universality.

It applies It applies to light, to light, it it applies applies to electrons, to electrons, andand it applies it applies to all to all thethe other other elementary elementary parparticles ticles known. known.

It applies It applies to combinations to combinations of those of those particles, particles, such such as as atoms atoms andand molecules. molecules.

It has It has worked worked successfully successfully to to describe describe thethe motions motions of large of large molecules, molecules, such such as buckyballs as buckyballs andand proteins. proteins.

There There is no is no case case of of an an experiment experiment that that waswas sensitive sensitive enough enough to to reveal reveal the quantum the quantum nature nature of an of an object, object, butbut failed failed to do to do so. so.

At At least least so far, so far, sizesize andand complexity complexity provide provide no no limit. limit.

WeWe do do notnot yetyet know know if the if the wave-particle wave-particle duality duality applies applies to people to people or cats or cats or planets or planets or stars, or stars, butbut there there is no is no reason reason known known why why it definitely it definitely can’t. can’t.

In all In all these these cases cases thethe effect effect is the is the same: same: wewe cancan only only know know halfhalf of what of what wewe would would need need to know to know to precisely to precisely predict predict thethe future. future.

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How HowQuanta QuantaChange Change

II

n the n the first first lecture lecture of of hishis course course on on quantum quantum mechanics, mechanics, mymy teacher teacher Herbert Herbert Bernstein Bernstein asserted asserted that that physics physics is the is the science science of of everything. everything.

Our Our goal goal in physics in physics is to is find to find thethe most most general general laws laws of of

nature, nature, from from which which thethe multitude multitude of phenomena of phenomena exhibited exhibited by by na-nature ture may may all all be be explained. explained.

Quantum Quantum mechanics mechanics explains explains thethe widest widest variety variety of phenomena of phenomena of any of any theory theory so far. so far.

At At thethe same same time, time, it greatly it greatly restricts restricts thethe quesquestions tions that that cancan be be asked asked of of anyany particular particular phenomenon. phenomenon.

WeWe have have already already encountered encountered oneone kind kind of limitation: of limitation: that that wewe cancan know know only only halfhalf what what wewe would would need need to know to know about about a system a system to make to make precise precise predictions predictions forfor its its future. future.

As As a result, a result, wewe must must givegive up up describing describing exactly exactly what what goes goes on on in individual in individual atoms atoms in favor in favor of statistical of statistical prepredictions, dictions, which which apply apply only only to to averages averages taken taken over over many many cases. cases.

Hence, Hence, to believe to believe in quantum in quantum theory theory wewe must must givegive up up thethe ambition ambition to precisely to precisely predict predict thethe future. future.

Most Most physicists physicists have have given given up up those those ambitions ambitions in the in the faceface of the of the

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success success of of quantum quantum mechanics. mechanics.

ButBut I believe I believe that that thisthis is shortis shortsighted sighted andand there there is aisdeeper a deeper level level of reality of reality to be to be discovered, discovered, thethe mastery mastery of which of which willwill restore restore ourour ambitions ambitions forfor a complete a complete underunderstanding standing of nature. of nature.

Another Another restriction restriction limits limits thethe range range of quantum of quantum theory. theory.

WeWe cancan express express thisthis in ainprinciple a principle I call I call thethe subsystem subsystem principle: principle: Any Any system system quantum quantum mechanics mechanics applies applies to must to must be a besubsysa subsystemtem of aoflarger a larger system. system.

One One reason reason forfor thisthis is that is that quantum quantum mechanics mechanics refers refers only only to to physical physical quantities quantities which which areare measured measured by by measuring measuring instruments, instruments, andand these these must must be be outside outside thethe system system being being studied. studied.

Further, Further, thethe results results of these of these measurements measurements areare perceived perceived andand recorded recorded by by ob-observers, servers, who who areare alsoalso notnot part part of the of the system system being being studied. studied.

Most Most of us of us approach approach science science with with thethe naive naive expectation expectation that that it it willwill telltell us us what what is real. is real.

WeWe cancan follow follow John John BellBell andand callcall a real a real property property of aofsystem a system a beable: a beable: it isit part is part of what of what is. is.

BellBell coined coined thethe word word as as a contrast a contrast to to thethe term term observables, observables, which which is what is what antiantirealists realists want want outout of aoftheory. a theory.

“Observables” “Observables” andand “beables” “beables” areare loaded loaded terms, terms, whose whose useuse cancan signify signify allegiance allegiance to atoside a side of the of the debate debate between between realism realism andand antiantirealism. realism.

AnAn observable observable is aisquantity a quantity produced produced by by an an experiment experiment or or an an observation. observation.

There There is no is no commitment commitment to believe to believe it corresponded it corresponded to to something something that that exists exists apart apart from from thethe measurement measurement or or hadhad a a value value before before thethe measurement. measurement.

AntiAntirealists realists useuse thisthis term term to emto emphasize phasize that that thethe quantities quantities quantum quantum physicists physicists measure measure need need have have no no existence existence apart apart from, from, or prior or prior to, to, ourour observation observation of them. of them.

RealRealistsists useuse John John Bell’s Bell’s term term “beable” “beable” to refer to refer to the to the reality reality that that they they believe believe exists exists whether whether wewe measure measure something something or not. or not.

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Most Most scientific scientific explanations, explanations, whether whether of the of the flights flights of cannonof cannonballs balls or of or birds of birds andand bees, bees, speak speak in terms in terms of beables. of beables.

ButBut notnot quantum quantum mechanics! mechanics!

As As Heisenberg Heisenberg andand Bohr Bohr insisted, insisted, quantum quantum mechanics mechanics speaks speaks notnot in in terms terms of of what what is, is, butbut only only of of what what hashas been been observed. observed.

There There is, according is, according to them, to them, no no useful useful talktalk about about beables beables in the in the atomic atomic domain; domain; instead, instead, quantum quantum mechanics mechanics deals deals only only in observables. in observables.

To To measure measure an an atom’s atom’s observables, observables, wewe impose impose on on it aitlarge, a large, macmacroscopic roscopic instrument. instrument.

By By definition, definition, that that device device is not is not part part of the of the system system whose whose observables observables wewe areare studying. studying.

NorNor is the is the observer. observer.

Therefore, Therefore, to be to be described described in the in the language language of quantum of quantum mechanmechanics,ics, a system a system must must be part be part of aoflarger a larger system system thatthat includes includes thethe observ observ er er andand herher measuring measuring instruments. instruments.

Hence Hence ourour subsystem subsystem principle. principle.

Most Most applications applications of quantum of quantum theory theory areare to to atoms atoms andand molemolecules cules or other or other tinytiny systems; systems; in these in these cases cases thethe restriction restriction is irreleis irrelevant. vant.

ButBut some some of of us us have have thethe ambition ambition to to describe describe thethe whole whole universe. universe.

WeWe feelfeel that that is the is the ultimate ultimate goal goal of science. of science.

However, However, thethe universe universe as aaswhole a whole is not, is not, by by definition, definition, part part of aoflarger a larger system. system.

TheThe subsystem subsystem principle principle frustrates frustrates ourour hope hope to have to have a theory a theory of the of the whole whole universe. universe.

There There is aissubtle a subtle butbut keykey difference difference between between thethe ideaidea thatthat quanquantum tum mechanics mechanics is the is the theory theory of everything, of everything, andand thethe hope hope of extendof extendinging quantum quantum theory theory to include to include thethe whole whole universe. universe.

What What Professor Professor Bernstein Bernstein meant meant by by hishis claim claim is that is that physics physics is the is the root root of the of the correct correct description description of everything— of everything— each each considered considered as aassubsystem a subsystem of the of the whole. whole.

It isIt very is very different different to imagine to imagine applying applying quantum quantum theory theory to to thethe entire entire universe, universe, which which would would mean mean including including us observers us observers inside inside thethe system system being being studied, studied, andand ourour measuring measuring instru instru ments. ments.

Over Over thethe lastlast century century several several attempts attempts were were made made to to extend extend quantum quantum mechanics mechanics to to a theory a theory of of thethe whole whole universe. universe.

WeWe willwill

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meet meet oneone of these of these later later on;on; a part a part of our of our overall overall argument argument is that is that these these attempts attempts fail.fail.

ForFor oneone thing, thing, making making thethe observer observer a part a part of the of the system system being being described described raises raises tricky tricky questions questions of of selfselfreference. reference.

It is It not is not even even clear clear that that an an observer observer cancan givegive a complete a complete selfselfdescription, description, because because thethe actact of observing of observing or describing or describing yourself yourself changes changes you. you.

ButBut there there areare deeper deeper reasons reasons why why quantum quantum mechanics mechanics cannot cannot be be extended extended to atotheory a theory of the of the whole whole universe. universe.

In In several several of of mymy books books (namely (namely TheThe LifeLife of the of the Cosmos, Cosmos, Time Time Reborn, Reborn, andand TheThe Singular Singular Universe Universe andand thethe Reality Reality of Time, of Time, written written with with Roberto Roberto Mangabeira Mangabeira Unger), Unger), I investigate I investigate thethe question question of how of how physics physics may may be be extended extended to give to give a theory a theory of the of the whole whole universe. universe.

I I conclude conclude that that a theory a theory of the of the whole whole universe universe must must differ differ in several in several crucial crucial aspects aspects from from anyany of the of the physical physical theories theories so far so far developed, developed, including including quantum quantum mechanics. mechanics.

AllAll these these theories theories only only make make sense sense when when interpreted interpreted as descriptions as descriptions of aofportion a portion of the of the universe. universe.

Indeed, Indeed, thethe factfact thatthat quantum quantum mechanics mechanics only only makes makes sense sense when when read read as aastheory a theory of aofpart a part of the of the universe universe is, by is, by itself, itself, a sufficient a sufficient rea-reasonson for for regarding regarding quantum quantum mechanics mechanics as incomplete. as incomplete.

One One thing thing wewe may may askask of aoftheory a theory thatthat completes completes quantum quantum theory theory is that is that it makes it makes sense sense when when extended extended to atodescription a description of the of the universe universe as aaswhole. a whole.

However, However, thisthis is not is not thethe only only lineline of thought of thought that that leads leads to the to the conclusion conclusion that that quantum quantum mechanics mechanics is incomplete. is incomplete.

Other Other concerns concerns andand difficulties difficulties hadhad far far more more influence influence on on how how thethe subject subject hashas evolved evolved historically. historically.

ForFor thethe time time being, being, I will I will ignore ignore thethe cosmologicosmologicalcal issues issues andand focus focus on on more more immediate immediate challenges. challenges.

THETHE PROCESS PROCESS OF OF APPLYING APPLYING general general laws laws to to a specific a specific physical physical sys-sys-

temtem hashas three three steps. steps.

First, First, wewe specify specify thethe physical physical system system wewe want want to study. to study.

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TheThe second second step step is tois describe to describe thatthat system system at aatmoment a moment of time of time in in of aof list a list of properties. of properties.

If the If the system system is made is made of particles, of particles, thethe proppropterms terms erties erties willwill include include thethe positions positions andand momenta momenta of those of those particles. particles.

If itIf it is made is made of waves, of waves, then then wewe givegive their their wavelengths wavelengths andand frequencies. frequencies.

And And so on. so on.

These These listed listed properties properties make make up up thethe state state of the of the system. system.

TheThe third third step step is to is postulate to postulate a law a law to describe to describe how how thethe system system changes changes in time. in time.

Before Before quantum quantum physics, physics, physicists physicists hadhad a simple a simple butbut powerful powerful ambition ambition for for science. science.

At At thethe second second step step wewe would would be able be able to describe to describe a system a system in in terms terms thatthat were were complete, complete, in in twotwo senses. senses.

Complete Complete means, means, firstfirst of all, of all, thatthat a more a more detailed detailed description description is neither is neither needed needed nornor possible. possible.

Any Any other other property property thethe system system might might have have would would be be a a consequence consequence of those of those already already included. included.

Additionally, Additionally, thethe listlist of propof properties erties should should be be exactly exactly what what is needed is needed to give to give precise precise predictions predictions of of thethe future. future.

This This is done is done using using thethe laws. laws.

TheThe future future cancan be be deterdetermined mined precisely, precisely, given given ourour complete complete knowledge knowledge of the of the present. present.

This This is the is the second second meaning meaning of the of the description description being being complete. complete.

Between Between Newton, Newton, in the in the latelate seventeenth seventeenth century, century, andand thethe in- invention vention of quantum of quantum mechanics mechanics in the in the 1920s, 1920s, it was it was believed believed that that thethe properties properties making making up up that that complete complete description description were were thethe posipositions tions of all of all thethe particles particles andand their their momenta. momenta.

It might, It might, of course, of course, happen happen thatthat wewe don’t don’t know know thethe precise precise posipositions tions andand momenta momenta of all of all thethe particles particles making making up up a system. a system.

TheThe air air 28 atoms andand molecules, molecules, so asocoma comin this in this room room consists consists of around of around 102810atoms

plete plete listing listing of their of their positions positions is impossible. is impossible.

WeWe have have to use to use a very a very approximate approximate description description in terms in terms of density, of density, pressure, pressure, andand temperatemperature. ture.

These These refer refer to averages to averages of the of the atoms’ atoms’ positions positions andand motions. motions.

Our Our bulk bulk description description willwill have have to employ to employ probabilities, probabilities, andand thethe prepredictions dictions it makes it makes willwill then then be be to some to some degree degree uncertain. uncertain.

ButBut thethe useuse of probabilities of probabilities is just is just for for ourour convenience, convenience, andand thethe resulting resulting uncertainties uncertainties justjust express express ourour ignorance. ignorance.

Behind Behind ourour bulk bulk 29 29

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description description of aofgas a gas in terms in terms of density of density andand temperature, temperature, wewe concontinue tinue to believe to believe there there is aisprecise a precise description, description, which which includes includes listing listing where where every every lastlast atom atom is and is and how how it isitmoving. is moving.

WeWe share share a faith a faith thatthat if we if we hadhad access access to that to that description description wewe could could useuse thethe laws laws to predict to predict thethe future future precisely. precisely.

That That faith faith is based is based on on thethe belief belief in realism— in realism— thatthat there there is an is an objective objective reality, reality, which which it isitpossible is possible for for us to us know. to know.

Quantum Quantum mechanics mechanics blocks blocks thisthis complacent complacent ambition, ambition, because because its its first first principle principle asserts asserts wewe cancan know, know, at most, at most, only only halfhalf thethe inforinformation mation that that would would be be needed needed to realize to realize it. it.

THETHE COMPLETE COMPLETE INFORMATION INFORMATION NEEDED NEEDED to to precisely precisely predict predict thethe fu-fu-

ture ture is called is called a classical a classical state. state.

“Classical” “Classical” is how is how wewe refer refer to physics to physics as it aswas it was between between Newton Newton andand thethe discovery discovery of the of the quantum. quantum.

It isIt is then then natural natural to to callcall a specification a specification of of halfhalf of of that that information information a a quantum quantum state. state.

TheThe halfhalf is arbitrary; is arbitrary; it can it can be be chosen chosen to be to be only only thethe momentum, momentum, or only or only thethe position, position, or some or some mixture mixture of these, of these, as long as long as as halfhalf thethe information information needed needed to to precisely precisely predict predict thethe future future is is present, present, andand halfhalf is missing. is missing.

TheThe quantum quantum state state is aiscentral a central notion notion in in quantum quantum theory. theory.

A A realist realist willwill want want to ask: to ask: Is itIsreal? it real?

Does Does a particle’s a particle’s quantum quantum state state correspond correspond precisely precisely to the to the physical physical reality reality of that of that particle? particle?

OrOr is itis it justjust a convenient a convenient tooltool to make to make predictions? predictions?

Perhaps Perhaps thethe quantum quantum state state is aisdescription, a description, notnot of the of the particle, particle, butbut only only of the of the informainformationtion wewe have have about about thethe particle? particle?

WeWe areare notnot going going to resolve to resolve these these questions questions here. here.

Experts Experts dis-disagree agree about about them. them.

WeWe willwill soon soon enough enough have have thethe chance chance to focus to focus on on these these andand other other questions questions about about thethe meaning meaning andand correctness correctness of of quantum quantum mechanics. mechanics.

ForFor now now wewe take take a pragmatic a pragmatic viewpoint viewpoint andand regard regard thethe quantum quantum state state as aastool a tool forfor making making predictions predictions about about thethe future. future.

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A quantum A quantum state state is aisuseful a useful tooltool because because it can it can do do justjust that. that.

This This is our is our next next principle: principle: Given Given thethe quantum quantum state state of an of an isolated isolated system system at one at one time, time, there there is aislaw a law thatthat willwill predict predict thethe precise precise quantum quantum state state of of thatthat system system at any at any other other time. time.

This This lawlaw is called is called Rule Rule 1. 1.

It It is also is also sometimes sometimes called called thethe Schrödinger Schrödinger equation. equation.

TheThe principle principle that that there there is such is such a law a law is is called called unitarity. unitarity.

Thus, Thus, while while thethe relation relation between between thethe quantum quantum state state andand thethe be-behavior havior of an of an individual individual particle particle cancan be be statistical, statistical, thethe theory theory is deteris deterministic ministic when when it comes it comes to how to how thethe quantum quantum state state changes changes in time. in time.

As As wewe said, said, quantum quantum states states with with definite definite values values of energy of energy andand momentum momentum areare represented represented by by pure pure waves waves with with exact exact frequency frequency andand wavelength. wavelength.

ButBut these these quantum quantum states states areare very very special. special.

What What about about other other quantum quantum states, states, whose whose momenta momenta areare uncertain, uncertain, so that so that they they do do notnot vibrate vibrate at aatsingle a single frequency frequency andand with with a single a single wavelength? wavelength?

More More general general quantum quantum states states areare represented represented by by waves waves with with arbitrary arbitrary proprofiles. files.

These These areare sharp sharp in neither in neither position position nornor momentum, momentum, so if soeither if either of these of these quantities quantities is measured, is measured, there there willwill be be uncertainties.* uncertainties.* There There areare alsoalso states states of definite of definite position position andand completely completely indefiindefinitenite momentum; momentum; if we if we graph graph them, them, they they look look likelike spikes, spikes, which which areare zero zero everywhere everywhere except except thethe single single point point where where thethe particle particle is. is.

Other Other states states areare peaked peaked over over a region a region of space of space andand correspond correspond to to particles particles which which areare localized localized imprecisely, imprecisely, so we so we know know only only approxapproximately imately where where they they are.are.

One One wayway to make to make a general a general quantum quantum state state is by is by adding adding together together pure pure waves, waves, each each with with a different a different frequency frequency andand wavelength. wavelength. * When * When a wave a wave represents represents a quantum a quantum state, state, we sometimes we sometimes call it call a wave it a wave function. function.

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