Parity and C-parity are multiplicative quantities.
Parity and C-parity are conserved in strong and electromagnetic interactions.
The parity of a state of angular momentum L is (-1)L.
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:
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:
Another pair of useful operators is
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:
If we combine a quark and an anti-quark to form a meson, what are the possible states of total angular momentum (spin of the meson) that can be formed? Begin by assuming that the q and qbar have orbital angular momentum L=0, so that the problem is simplified to combining to spin 1/2 particles. The combination of two spin 1/2 particles can yield J = 0 or 1. Begin with both particles aligned with spins down. This combination has M = -1, and can only be in the J = 1 state:
Now apply the raising operator to this relation:
Applying the raising operator a second time yields:
Finally, the J=0, M=0 state is obtained by noting that there is a combination of momenta that is orthogonal to the J=1, M=0 state, namely:
Let's determine the angular momentum states possible in the decay of a pseudoscalar meson to a pseudoscalr meson and a vector meson (abbreviated as P to PV). A pseudoscalar meson has JP = 0- and a vector meson has JP = 1-. The initial state is