Monday, 30 December 2019

homework and exercises - How to show that the charge conjugation reverses the charge of a state?


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 as(k)|0. The normal ordered charge operator associated with the symmetry ψeiqψ is given by Q=qd3ps=1,2[as(p)as(p)bs(p)bs(p)]

where a,b are respectively the particle and antiparticle creation operators. It is a trivial matter to show that Qas(k)|0=qas(k)|0  and  Qbs(k)|0=qbs(k)|0.


Now, the charge conjugation operator is given by C=iγ2γ0.

I want to show that Q(Cas(k)|0)=q(Cas(k)|0)
or something like Q(Cas(k)|0)=bs(k)|0.


Am I on the right track? If yes, any hint how to act C on the state as(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

with states of the form d3x ngn(x)ψn(x)|0,
where f and g are ordinary functions with the same number of components as ψ, and where ψn denotes the operator adjoint of ψn. (If the components of ψ are ψn, then the components of ψ are ψn.) We want Cop to exchange these because ψ has components that create antiparticles (and components that annihilate particles), and ψ has components that create particles (and components that annihilate antiparticles).


The key point is that the charge conjugation matrix Cmat is used to define the charge conjugation operator like this: CopψC1opCmatψ

where ψ denotes the component-wise operator adjoint of ψ, without any matrix transpose. Again, this equation mixes matrix notation and operator notation. On the left-hand side, ψ is a column-matrix, and the operator transformation CopC1op acts on each individual element of this matrix. On the right-hand side, ψ is again a column matrix, the transpose of the row matrix ψ, and Cmat is a square matrix of ordinary numbers; this matrix product mixes the components of ψ with each other.


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 Cmatγ0γ2, as stated in the question.


When the matrix Cmat is chosen to satisfy that requirement, the operator Cop will do something like CopaC1opbCopaC1opb

as anticipated by the question. Then, since the vacuum is invariant under Cop, we have CopaC1op|0=Copa|0b|0.
The fact that Cop reverses the charge of the state is guaranteed by the fact that it reverses the sign of the charge density operator: CopQC1op=Qwith Qq¯ψγ0ψ=qψψq(aabb).
I glossed over a lot of details in this answer. The main point is that Cop and Cmat are two different things — related, but different.


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