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In Weinberg’s QFT V1, under equation 5.5.58, he says that an anticommutator ($c$-number) can be ignored when we exchange spinors, $\psi$ and $\bar{\psi}$. I cannot fully appreciate why we can ignore the anti commutator, and it seems that neither Schwartz have mentioned it (see 11.49) nor Peskin & Schroeder (see 3.147).

My main confusion is that for example, if we want to evaluate vacuum expectation $\langle{0|\bar{\psi}\psi|0}\rangle$ under charge conjugation transformation, can we ignore the anticommutator arised from the exchanging of the spinors?

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The reason is that when we consider Dirac bilinear under charge conjugation, we are in the situation of questioning us whether our Lagrangian respect such symmetry. As we know that constants appearing in Lagrangian is trivial, so in Schwartz and Peskin & Schroeder, they don't mention the c-number anticommutator in the text, but when we are concerning the vacuum expectation, we should take the anti commutator into account.

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