# What is the definition of the charge conjugation?

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 $C \phi C^{-1} = \phi^*$.

• Wightman (1-47) defines the action of $C$ on a two-component spinor. A field in an arbitrary representation of Loretnz can always be understood as a tensor with several (dotted and undotted) spinor indices, or direct sums thereof. Therefore, Wightman's definition works for a field of arbitrary spin. Just act on its spinor indices as (1-47) indicates. – AccidentalFourierTransform Mar 5 '18 at 21:40
• What about the case of a scalar field? – Slereah Mar 5 '18 at 21:45
• Well, no indices, no transformation (up to a phase) :-P – AccidentalFourierTransform Mar 5 '18 at 21:47
• Except according to him later on it's $\phi \to \phi^*$! – Slereah Mar 5 '18 at 21:47

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.