Parity

Parity follows a product rule. Ptot = P1*P2…

Parity changes x è -x , y è -y , z è -z

The space-time characteristics of functions and vectors are defined.

• Scalars don’t change sign.
• Vectors change sign, position, velocity, momentum, and force.
• Psuedo or axial vectors don’t change sign, Angular momentum and spin (j=1/2,1,3/2….). The spin vector that defines the spin for these states doesn’t change sign and behaves in space-time the same as angular momentum. (For a transformation of an experiment, spinning wheel, the direction of the spinning wheel doesn’t change. For a transition to a new coordinate system the RHRàLHR. Thus the angular momentum vector changes direction along with the coordinate system).

The particle and/or fields can carry with them an intrinsic parity. This has to do with the possibility that the parity transformation can add an additional phase to the states without changing the observable physics. Because the parity transformations return you to the original state the intrinsic parity of a particle may be +1 or –1.

Photon

Spatial part depends on L. Parity=(-1)^L.

Spin=1, vector is even. Parity =+1. {more complete discussion}

Intrinsic is defined to match the classical conventions for E,B,A. Parity =-1.

More discussion.

Fermion

Spatial part depends on L. Parity=(-1)^L.

Spin=1/2, spin (angular momentum) is even. Parity =+1.

Intrinsic

• fermion parity +1
• antifermion parity –1.

More discussion.

Composite systems mesons, hadrons and final states

Spatial part depends on L. Parity=(-1)^L. Relative orbital angular momentum carried by the all of the constituents.

Intrinsic part is the product of each constituent.

Examine the parity rules for a composite system e.g. parity of a meson or the parity for the final state of two photons.

Meson

π

• product of intrinsic parity of each constituent and the spatial parity.
• L is the total orbital angular momentum è Parity=(-1)⁰=1.
• Intrinsic parity convention is given above. è Parity=(-1).

π meson will have an overall parity –1.

π_o à γγ

Two photon γγ

intrinsic = -1* -1 = +1

The drawing show the two allowable photon final states A, B. The photons can have their spin parallel or anti parallel to the photon momentum. Photons must combine so that the spin projection cancels because the π_o is a spin 0 particle. Also note that parity shifts the momentum direction but not the spin. Therefore the two states that have a definite parity are

State 1 è A+B

State 2 è A-B

Each state has a different parity. State 2 has the required negative parity.

This clearly demonstrates the correct parity. However it doesn’t provide insight into the general rule that allows you to build composite states.

State 2 is an anti symmetric combination of spins.

(see also http://www.hep.caltech.edu/~fcp/ph195/problemSets/p1950313ss.pdf)

Clebsh Gordon Couplings for two spin 1

Therefore

Which shows that the final 2 photon state is a spin 1 state. The spin must couple to a state of orbital angular momentum L=1 in order to reach a state of J=0. So the relative orbital angular momentum is L=1 and the parity is (-1)^L= -1.