I have seen many equations where $P$ and $C$ (eigenvalues of parity and charge conjugation, resp.) are related to $J$, $L$, $S$ and $I$ (eigenvalues of total angular momentum, angular momentum, spin, isospin, resp.). Unfortunately, I do not understand why some of them seem to be different. The wiki page on Parity states: $$P=(-1)^L$$ whereas this review by the PDG states $$P=(-1)^{L+1}$$I guess it is important to know what you want to describe (wiki: wave function $|\ell,m_\ell\rangle$ whereas PDG: mesons), but this does not help my understanding. Similar confusion arises for $C$.

I would be grateful for some clarification regarding the relation of $C$ and $P$ to $L$, $S$ and $I$ as well as some hints on how to derive these relations!


The extra $+1$ for mesons should be due to the fact that they are formed from a particle-antiparticle pair of fermions; which changes with a negative sign under pairty.

This is alluded to breifly in the linked wikipedia article under the quantum field theory heading when it says :

"This is true even for a complex scalar field. (Details of spinors are dealt with in the article on the Dirac equation, where it is shown that fermions and antifermions have opposite intrinsic parity.)"


In the following reference it is explained how eigenvalues of parity and charge conjugation are related with angular momentum. See sections 6.4.3 and 6.7. link!

I will write briefly what I understood from the above reference (for the Parity part only.)
For a single particle, the parity eigenvalue is only the intrinsic parity. But for multi-particle system it depends on the relative orbital angular momentum and total spin too. For a two particle system the interaction between them depends on r, the relative coordinate. If the interaction is isotropic it will only depend on r (magnitude only).In this case the angular part of the wave function for a stationary state will be Ylm(θ, φ). Under parity P,
PYlm(θ, φ)=$(−1)^l$Ylm(θ, φ)
The total effect of parity transformation on the state will then give the eigenvalue, P = $η_1η_2(−1)^l$. Where $η_1 $and $η_2$ are the intrinsic parities of the two particles. $η_1η_2$ can be either 1 or -1 depending on the particles. Accordingly we will get parity as P=$(−1)^l$ or $P=(-1)^{l+1}$.

Source: An Introductory Course of Particle Physics By Palash B. Pal. Sections 6.4.3 and 6.7 (the link above)


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