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I seem to have troubles finding definitions of the charge conjugation operator that are independant of the theory considered.

Weinberg defined it as the operator mapping particle types to antiparticles :

$$\operatorname C \Psi^{\pm}_{p_1 \sigma_1 n_1;p_2 \sigma_2 n_2; ...} = \xi_{n_1} \xi_{n_2} ... \Psi^{\pm}_{p_1 \sigma_1 n_1^c;p_2 \sigma_2 n_2^c; ...}$$

He does not really seem to specify what he means by "antiparticles" around there, but I'm guessing this is the one-particle state that is conjugate to this one. This assumes that it is possible to decompose everything into one-particle states.

Wightman seems to go with $C \gamma^\mu C^{-1} = \bar \gamma^\mu$, which isn't terribly satisfying and also only works for spinor fields.

I've seen thrown around that the $C$ conjugation corresponds roughly to the notion of complex conjugation on the wavefunction but never really expanded upon.

Is there a generic definition of charge conjugation that does not depend on how the theory is constructed? The CPT theorem in AQFT indeed seems to not have any of those extraneous constructions, but the action of the different symmetries is a bit hidden as

$$(\Psi_0, \phi(x_1) ... \phi(x_n) \Psi_0) = (\Psi_0, \phi(-x_n) ... \phi(-x_1) \Psi_0)$$

Is the action of $C$ symmetry $\Psi' = C \Psi$ just a state such that for any operator $A$,

$$(\Psi, A \Psi) = (\Psi', A^\dagger \Psi')$$

or something to that effect? From some parts seems like it may just be $C \phi C^{-1} = \phi^*$.

Slereah
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2 Answers2

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There is no natural definition of charge conjugation that works for all QFTs. Rather, you should understand the CPT theorem instead as a combination of reflection-positivity and Wick rotation. See this paper, Appendix A.2.

Ryan Thorngren
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All of your fields naturally lie in some representation of the group of all symmetries (these include gauge symmetries, global gauge transformations and global Lorentz transformations). Charge conjugation is simply passing to the conjugate representation of that group.

E.g. complex scalars are 1d irreps of $U(1)$, and the conjugate object is $\phi^{*}$. The same logic also works for spinors, gauge fields, etc.

Prof. Legolasov
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