Invariance Principles and Conservation Laws

Recall from last lecture:

Parity is a multiplicative quantity.

Parity is conserved in strong and electromagnetic interactions.

The parity of a state of angular momentum L is (-1)L.

Charge Conjugation

The operation of charge conjugation in the classical sense is to change every positive charge with a negative charge, and vice versa. The directions of electric fields reverse under charge conjugation. The directions of magnetic fields also reverse since magnetic fields are created by moving charges -- the directions of the charges don't change, but the signs do, resulting in an overall change of sign. These changes leave Maxwell's equations unchanged since they are linear in the electric and magnetic fields.

In quantum field theory the operation of charge conjugation exchanges particle and anti-particle. For charged particles, this is the same thing as changing the sign. The difference occurs only for neutral particles which are not their own anti-particle, and examples abound: K0, D0, B0, neutrinos, Σ0, and Ξ0 are a few.

The charge conjugation or C-pairty eigenstates must be neutral, since C|π+> --> |π-> Beyond that, the C-parity operation is like parity.

C-parity of the photon

Both E and B fields change sign under C-parity. Therefore, photons have negative C-parity, C=-1.

C-parity of particle--anti-particle pairs

The C-parity of a particle--anti-particle pair can be calculated if we know the wave function of the pair. Charge conjugation switches the particles in the wave function, and the resulting change to the state is the C-parity. An example for spin 0 particles, consider a π+π- pair in a state of relative orbital angular momentum L. Under charge conjugation, the state becomes:

C | π+π-; L > = (-1)L | π+π-; L >

The case for fermions is slightly trickier, because there are extra quantum mechanical factors of (-1)S+1, arising from the interchange of particles in a spin state with spin S, and (-1), arising from the interchange of a fermion and an anti-fermion. Under charge conjugation, an ffbar state becomes:

C | ffbar; J,L,S > = (-1)L+S | ffbar; J,L,S >

C-parity of the π0

The π0 decays electromagnetically to two photons π0 --> γγ and therefore has even C-parity.

The decay of a π0 to one photon is ruled out by energy and momentum conservation. The decay of a π0 to three photons violates C-parity. Of course, C-parity is not a truly fundamental symmetry, and even seems to be violated since the world around us is matter dominated, and certainly not symmetric in particles and anti-particles. Therefore we'd like to know how well C-parity works. An indication of this is from limits to forbidden processes such as the 3γ decay of the pion. The limit, expressed as a fraction of the 2γ branching ratio is: < 3×10-8

CPT Invariance

An important result from field theory is that all interactions are invariant under the simultaneous operations of C (charge conjugation), P (parity), and T (time reversal). A consequence is that particles and anti-particles must have the same mass and lifetime, and equal but opposite electric charges and magnetic moments. Up to now, the consequences of the CPT theorem have agreed with experimental tests.

CP violation and T violation

While the combination CPT is thought to be absolutely invariant, no other combination seems to have that property. C and P are violated in the weak interaction.

Discuss C and P violation in the weak interaction using neutrinos as an example.

The combination CP was believed to be conserved by the weak interaction, until:

Addition of Angular Momenta

Review of Rotation Operators

There is a discussion of this in appendix C of Perkins.

Recall that in quantum mechanics, to the measurable angular momentum is associated an operator, just like any other measurable quantity. In fact, angular momentum is more complex because there is a group of operators associated with it. Conceptually, it is easiest to begin with the operators for the angular momentum about each of the x, y, and z axes; I'll call these three operators Jx, Jy, and Jz.

Earlier I wrote out Jz in terms of the coordinates:

Jz = -i hbar(x d/dy - y d/dx)
The three angular momentum operators don't commute:
[Jx, Jy] = i Jz
[Jy, Jz] = i Jx
[Jz, Jx] = i Jy
Notice the cyclic nature of this relation.

The fact that the operators don't commute means that a simultaneous measurement of more than one is impossible. Another operator that does commute with any of the three is the square of the total angular momentum:

J2 = Jx2 + Jy2 + Jz2
The usual choice for simultaneous measurement is J2 with eigenvalue J(J+1), and Jz with eigenvalue M.

Another pair of useful operators is

J± = Jx ± i Jy
These are known as raising (+) and lowering (-) operators of angular momentum because when acting on a state with angular momentum quantum numbers j and m:
J+ | j, m > = C+jm | j, m+1 > = sqrt{j(j+1)-m(m+1)} | j, m+1 >
J- | j, m > = C-jm | j, m-1 > = sqrt{j(j+1)-m(m-1)} | j, m-1 >
The C±jm are coefficients that are given by the square root expression. Notice that if m=j, then the coefficient C+jm = 0, and if m=-j, C-jm = 0. Thus the raising and lowering operators "obey the rules" for angular momentum quantum numbers and do not produce states with m>j or m<-j.

Example of Adding Angular Momenta States

Now suppose we have a system with two (or more) angular momenta given by the states | j1, m1 > and | j2, m2 >, where the j's and m's are the corresponding quantum numbers (m's measured for a common axis), and we want to determine the angular momentum of the whole system, | J, M >. Classically we would simply add the vectors to get J = J1 + J2. But quantum mechanically, we can't simultaneously know all the components of angular momentum, so we can't perform the vector addition. In fact, the rules of quantum mechanics require that the x and y components of angular momenta take on all possible allowed values.

Therefore, the addition of | j1, m1 > and | j2, m2 > can result in a state with | j1 - j2 | < J < j1 + j2. The z components are known though, and their addition yields the z component of the total angular momentum, M = m1 + m2. The result is that the whole system is a superposition of states with the same M and different J's:

| j1, m1; j2, m2 > = ΣJ=|j1-j2|j1+j2 | J, M > < J, M | j1 j2 m1 m2 >
where the quantities < J, M | j1 j2 m1 m2 > are numbers called Clebsch-Gordon coefficients.

Copyright © Robert Harr 2005