How to show that the charge conjugation operator reverses the charge(s) of a (fermionic or bosonic) state?
Let us consider a spin-12 fermionic state of momentum k and spin projection s given by a†s(k)|0⟩. The normal ordered charge operator associated with the symmetry ψ→eiqψ is given by Q=q∫d3p∑s=1,2[a†s(p)as(p)−b†s(p)bs(p)]
Now, the charge conjugation operator is given by C=iγ2γ0.
Am I on the right track? If yes, any hint how to act C on the state a†s(k)|0⟩?
Answer
Yes, you are on the right track.
The rest of this answer is about the distinction between the charge conjugation operator Cop and the charge conjugation matrix Cmat. If this misses the point of the question, let me know and I'll revise (or delete) the answer.
The field operator ψ is an array of operators — a Dirac spinor whose four components each act as an individual operator on the Hilbert space. A Dirac matrix γμ does not act on the Hilbert space; it only mixes the components of the field operator ψ. When we write an expression like γ0ψ|0⟩, we are mixing matrix notation (it is a column matrix in which each component is a whole state-vector) and operator notation (because each component of γ0ψ is a different operator acting on |0⟩).
The charge conjugation operator Cop exchanges states of the form ∫d3x ∑nfn(x)ψn(x)|0⟩
The key point is that the charge conjugation matrix Cmat is used to define the charge conjugation operator like this: CopψC−1op∼Cmatψ∗
The matrix Cmat can be expressed as some combination of Dirac matrices (as shown in the question), but an explicit expression for the operator Cop requires more. We could presumably express it as some combination of field operators (or creation and annihilation operators), but usually we just define it through its effect on the field operators, as in equation (2). After writing ψ in terms of creation/annihilation operators, equation (2) tells us how Cop affects them, too.
The matrix Cmat should be chosen so that the effect of Cop on the current q¯ψγμψ is equivalent to reversing the sign of the charge q, because this is the current that couples to Aμ in the Lagrangian. According to this requirement, different representations for the γ-matrices lead to different representations for the matrix Cmat. One common representation leads to Cmat∝γ0γ2, as stated in the question.
When the matrix Cmat is chosen to satisfy that requirement, the operator Cop will do something like Copa†C−1op∼b†CopaC−1op∼b
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