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!


2 Answers 2


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 $Y_{lm}(\theta, \phi)$. Under parity P,

$$ \mathbf{P} \, Y_{lm}(\theta, \phi) = (−1)^l Y_{lm}(\theta, \phi) $$

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)


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.)"


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