Chapter 4 Symmetries And Conservation Laws

Chapter 4 4.1 Why Conservation And Symmetry 4.2 Angular Momentum 4.3 Flavour Symmetry Isospin 4.4 Parity and Charge Conjugation 4.5 Interactive Exercise

4.1 Why Conservation And Symmetry

Symmetry and Conservation Laws

Particle Physics

Why are some quantities conserved and why are we interested in them? In studying physics you notice that the interesting quantities are in some way the ones which are CONSERVED. In solving problems- which is what physical understanding is about-, you continually invoke CONSERVATION LAWS Energy Momentum Electric charge Flavour All these conservation laws stem from symmetries in the equations describing a system In classical/ quantum mechanics, you learn that Energy (Angular) momentum

Conservation

Arise from the fact that the laws of motion are invariant under transformations in space-time in electrodynamics Electric charge conservation arise invariance of Maxwell's equations

from

gauge

In elementary particle physics one experimentally searches for new conserved quantities (flavour, baryon no........) Dayalbagh Educational Institute

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Symmetry and Conservation Laws

Particle Physics

Knowing the conserved quantities gives one insight into the symmetry of the equations describing the new theory Gauge symmetry

Group theory

A conserved quantity is one which does not vary with time The total formal connection: Conserved quantity

Symmetry

Is most easily seen in quantum mechanics- but also true in classical mechanics Any operator which commutes with the Hamiltonian, corresponds to a conserved quantity [ F, H ] = 0 Conserved

Hamiltonian

Analog of Emily 'Nothers' theorem in classical mechanics Remember how measurable, or observable quantities appear in quantum mechanics every observable corresponds to an operator e.g. Momentum -iħ ∂/∂x In some general state Ψa Dayalbagh Educational Institute

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Symmetry and Conservation Laws

Particle Physics

If we make a measurement corresponding to the operator F there will be some result. most likely Expectation value 〈 F〉 = ∫d3x Ψ*a FΨa Could also interpret this as average result of measurement on many independent systems expectation value will obey classical equation of motion Ehrenfest's Theorem In the Schrödinger equation One can ask under what conditions is the expectation value of an operator independent of time When is F a conserved quantity or constant of motion?

Hermitian real

If Dayalbagh Educational Institute

From Schrödinger

Then 6

Symmetry and Conservation Laws

Particle Physics

If [ H, F]= 0 ; d /dt 〈 F〉 = 0 The expected value does not change with time since H, F corresponds to simultaneously measurable quantities states can be simultaneously eigenstates of H, F HΨ = EΨ FΨ = f Ψ There are two feasible ways of identifying conserved quantities not feasible to write down Hamiltonian + Commutators Usually not completely known symmetries Hamiltonian correspond to conserved quantities.

of

Experimentally: Finding new conserved quantities will identify symmetries which allow one to construct unknown Hamiltonian

Symmetry operations To investigate the symmetries of Hamiltonian we have to define symmetry operators This is an operation which corresponds to some transformation which leaves the Hamiltonian unchanged A transform operator will change the wave function of a state into another wave function Transformation

changes state of system Ψ1(x, t) = u Ψ (x, t)

New state

Transformation

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original state 7

Symmetry and Conservation Laws

Particle Physics

The particle must still be found somewhere in space Normalization unchanged

Unitary operator like eiα u is a symmetry operator if uΨ behaves just like Ψ Example

Transformed state

Mult through by u-1

But Schrödinger says

So

H = u-1 H u

For a unitary operator A+ = A-1

H = u+ H u

Since A+ A = 1 A+ = 1/A

H = u+ H u H - u+ H u = 0 u H - uu+ H u = 0 = 1 for unitary operator uH - H u = 0 [ H, u ] = 0 Dayalbagh Educational Institute

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Symmetry and Conservation Laws

Particle Physics

Symmetry operator commutes with Hamiltonian Deep connection between symmetry operators observable operators Symmetries

conserved quantum numbers [ H, u ] = 0

A symmetry operator only needs to be unitary u+ u = 1 This commutator defines u as a symmetry operator It corresponds to a set of transformations which leave the Hamiltonian unchanged. Only Hermitian observables

operators;

A+=A

correspond

to

Generally transformation operators do not correspond to observables but a symmetry transformation operator, which is an observable Hermitian operator is operator analog of real number A+ = A

R* = R

Hermitian A = A+ Unitary AA+ = 1 A+ = Ăƒ* (A.B)+ = B+ A+

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Symmetry and Conservation Laws

Particle Physics

Two distinct kinds of transformation: Even in non-mathematical terms, can see that there are two distinct kinds of transformation (and hence symmetries of the Hamiltonian) Continuous- these depend on some continuous parameter, and can differ from unit (do nothing) by arbitrarily small amount. Example → spatial translation Discrete (non-continuous)- these either happen or do not “all or nothing” Reflection ● Time reversal ● Make all charges opposing sign (particles → anti particles) Some discrete transformations do correspond to observables , spatial reflection = parity = up ●

Ψ(x) → Ψ(-x) up . up = 1 Unitary Hermitian }→ so, it must be an observable.

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Symmetry and Conservation Laws

Particle Physics

Why Conservation And Symmetry? Introduction To Invariance Symmetry is a tool to reduce the diversity of the physical world to a few fundamental formulas. A butterfly, ball, alternation of day and night...are some of the manifestations of symmetry which surround us. The notion of symmetry is inseparable from the notion of the notion of transformations and invariance: A ball invariant under notions, as the two wings of a butterfly are under mirror reflection. EMMY NOETHER IN 1917 PUBLISHED HER FAMOUS THEOREM:

Fig 1

Which implies every symmetry of nature yields a conservation law: conversely, every conservation law reflects an underlying symmetry. A simple example: the law of Physics are symmetrical with respect to translations in time. According to Noether's theorem, this invariance is related to conservation of energy. Dayalbagh Educational Institute

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Symmetry and Conservation Laws

Particle Physics

Symmetries And Conservation Laws SYMMETRY

CONSERVATION LAW

Translation in time

Energy

Translation in space

Momentum

Rotation

Angular Momentum

Gauge transformation

Charge

Table 1

Symmetry Symmetry is an operation performed on a system that leaves it invariant- i.e. transformer if into a state indistinguishable from the original one. The invariance principle means that given an experiment which obeys laws of Physics, we can create other experiments that obey the same laws by carrying out any combinations of symmetry operations. e.g. experiments performed on freely falling objects falls with a constant acceleration in Agra, translated in space, an astronaut on moon discovers the same physical law. The value of ‘g’ is different on the moon but the ‘form’ of the law is the same. The set of all symmetry operations (on a particular system) have the following properties to define a group, where Ra is an element belonging in it. For Ra is notation through a. Dayalbagh Educational Institute

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Symmetry and Conservation Laws

Particle Physics

a) Ra Rb = Rc

b) Identity: The set has an element I such that I Ra= RaI= Ra for all the elements Ra.

c) Inverse: For every element Ra, there is an inverse Ra-1, such that RaRa-1= Ra-1Ra= I

d) Associate: Ra (RbRc ) = (RaRb) Rc Generally, the groups useful in physics, can be combined as groups of matrices. In practice physics, the common groups are U(n): U(n): collection of all unitary nxn matrices. [unitary matrix : U-1 = Ĺ¨*] SU(n): unitary matrices with determinant = 1 [special unitary matrix] O(n): real unitary matrices, O-1 = Ă• SO(n): real, orthogonal, nxn matrices of det = 1. e.g. an ordinary scalar belongs to one-dimensional representation of the rotation group, SO(3). A sector belongs to the three-dimensional representation; four-sectors belong to the fourdimensional representation of Lorentz group. Dayalbagh Educational Institute

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Symmetry and Conservation Laws

Particle Physics

Space-Time Symmetries Many conservation laws have their origin in the symmetries and invariance properties of the underlying interactions

Translational invariance When a closed system of particles is moved from from one position in space to another, its physical properties do not change Considering an infinitesimal translation: the Hamiltonian of the system transforms as In the simplest case of a free particle, (1) From Equation (1) it is clear that (2) which is true for any general closed system: the Hamiltonian is invariant under the translation operator DË†, which is defined as an action onto an arbitrary wave function such as (3) For a single-particle state (3) one obtains:

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, by definition

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Symmetry and Conservation Laws

Particle Physics

further, since the Hamiltonian is invariant under translation, and using definitions once again, (4) This means that DË† commutes with Hamiltonian (a standard notation for this is [DË† , H] = 0) Since is an infinitely small quantity, translation (3) can be expanded as (5) Form (5) includes explicitly the momentum operator hence the translation operator Ë†D can be rewritten as (6) Substituting (6) to (4), one obtains (7) which is nothing but the momentum conservation law for a single-particle state whose Hamiltonian in invariant under translation. Generalization of (6) and (7) for the case of multiparticle state leads to the general momentum conservation law for the total momentum

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Symmetry and Conservation Laws

Particle Physics

Rotational Invariance When a closed system of particles is rotated about its centre-of-mass, its physical properties remain unchanged Under the rotation about, for example, z-axis through an angle θ, coordinates xi, yi, zi transform to new coordinates x'i, y'i, z'i as following:

(8) Correspondingly, the new Hamiltonian of the rotated system will be the same as the initial one, Considering rotation through an infinitesimal angle δθ, equations (8) transforms to A rotational operator is introduced by analogy with the translation operator : (9) Expansion to first order in δθ gives where is the z-component of the orbital angular momentum operator : Dayalbagh Educational Institute

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Symmetry and Conservation Laws

Particle Physics

For the general case of the rotation about an arbitrary → direction specified by a unit vector n, LˆZ, has to be ˆ Lˆ · n→, replaced by the corresponding projection of L: hence

(10) ˆ acting on a single-particle state Considering R n and repeating same steps as for the translation case, one gets: (11) (12) This applies for a spin-0 particle moving in a central potential, i.e., in a field which does not depend on a direction, but only on the absolute distance. If a particle possesses a non-zero spin, the total angular momentum is the sum of the orbital and spin angular momenta: (13) and the wavefunction is the product of [independent] space wavefuncion and spin wavefunction χ: For the case of spin-1/2 particles, the spin operator is represented in terms of Pauli matrices σ: (14) where σ has components: (15) Dayalbagh Educational Institute

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Symmetry and Conservation Laws

Particle Physics

Let us denote now spin wavefunction for spin “up” state as χ = α (Sz = 1/2) and for spin “down” state as χ = β (Sz = –1/ 2 ), so that (16) Both α and β satisfy the eigenvalue equations for operator (14): Analogously to (10), the rotation operator for the spin-1/2 particle generalizes to (17) When the rotation operator acts onto the wave function , components and of act independently on the corresponding wave functions:

That means that although the total angular momentum has to be conserved, but the rotational invariance does not in general lead to the conservation of and separately: However, presuming that the forces can change only orientation of the spin, but not its absolute value ⇒ Good quantum numbers are those which are associated with conserved observables (operators commute with the Hamiltonian) Dayalbagh Educational Institute

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Symmetry and Conservation Laws

Particle Physics

Spin is one of the quantum numbers which characterize any particle - elementary or composite. Spin of the particle is the total angular momentum of its constituents in their centre-of-mass frame Quarks are spin-1/2 particles ⇒ the spin quantum number SP=J can be either integer or half-integer Its projections on the z-axis – Jz – can take any of

2J+1 values, from -J to J with the “step” of 1, depending on the particle’s spin orientation

Usually, it is assumed that L and S are “good” quantum numbers together with J=SP , while Jz depends on the spin orientation.

Figure 2: A naive illustration of possible Jz values for spin-1/2 and spin-1 particles

Using “good” quantum numbers, one can refer to a particle via spectroscopic notation, like (18) Following chemistry traditions, instead of numerical values of L=0,1,2,3..., letters S,P,D,F... are used correspondingly Dayalbagh Educational Institute

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Symmetry and Conservation Laws

Particle Physics

In this notation, the lowest-lying (L=0) bound state of two particles of spin-1/2 will be 1S0 or 3S1

Fig 3: Quark-antiquark states for L=0

For mesons with L ≥ 1, possible states are: Baryons are bound states of 3 quarks⇒ there are two orbital angular momenta connected to the relative motion of quarks. total orbital angular momentum is L=L12+L3 .

Fig 4: Internal orbital angular momenta of a three-quark state

spin of a baryon S=S1+S2+S3 ⇒S=1/2 or S=3/2 Possible baryon states:

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