# Charge conjugation in second quantization

We know that under charge conjugation the current operator reverses the sign:

$$\hat{C} \hat{\bar{\Psi}} \gamma^{\mu} \hat{\Psi} \hat{C} = - \hat{\bar{\Psi}} \gamma^\mu \hat{\Psi}$$

Here $\hat{C}$ is the unitary charge conjugation operator. I was wondering should we consider gamma matrix here as also an entity undergoing transformation (like when we prove form-covariance of Dirac equation under any unitary transformation): $\hat{C} \gamma^{\mu} \hat{C} = \gamma^{\prime \mu}$? Or gamma matrix is something of a structure ensuring element and should not be changed?

Bill_K
(Forgive me for writing ψ to mean the adjoint.)

In second quantization, charge conjugation is represented by a unitary operator ℂ. Associated with it is a 4 x 4 matrix C that acts on the spinor indices. According to Bjorken and Drell vol 2, the action is

ℂψℂ-1 = C-1ψT
ψ-1 = - ψTC

where the matrix C has the property

μC-1 = - (γμ)T

From this,

ℂ(ψγμψ)ℂ-1 = - (ψTC)γμ(C-1ψT) = + ψTμ)TψT = + (ψγμψ)T = - ψγμψ.

Thank you, Bill. But I still have some points to think about.