Charge conjugation in second quantization

[LayMuon]

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^-1ψ^T
ℂψℂ^-1 = - ψ^TC

where the matrix C has the property

Cγ^μC^-1 = - (γ^μ)^T

From this,

ℂ(ψγ^μψ)ℂ^-1 = - (ψ^TC)γ^μ(C^-1ψ^T) = + ψ^T(γ^μ)^Tψ^T = + (ψγ^μψ)^T = - ψγ^μψ.

 

[LayMuon]

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