Advanced Quantum Mechanics Schwabl · Problem 13.6

Quantization of the Radiation Field

Physics — Quantum Mechanics · charge-conjugation-klein-gordon

Problem

For the quantized, complex Klein–Gordon field, the charge conjugation operation is defined by

\phi'(x) = \mathcal{C}\phi(x)\mathcal{C}^\dagger = \eta_\mathcal{C}\phi^\dagger(x)\,,

where the charge conjugation operator

\mathcal{C}

is unitary and leaves the vacuum state invariant:

\mathcal{C}|0\rangle = |0\rangle

. (a) Show for the annihilation operators that

\mathcal{C}a_{\mathbf{k}}\mathcal{C}^\dagger = \eta_\mathcal{C} b_{\mathbf{k}},\quad \mathcal{C}b_{\mathbf{k}}\mathcal{C}^\dagger = \eta^*_\mathcal{C} a_{\mathbf{k}}

and derive for the single-particle states

|a,\mathbf{k}\rangle \equiv a^\dagger_{\mathbf{k}}|0\rangle\,,\quad |b,\mathbf{k}\rangle \equiv b^\dagger_{\mathbf{k}}|0\rangle\,,

the transformation property

\mathcal{C}|a,\mathbf{k}\rangle = \eta^*_\mathcal{C}|b,\mathbf{k}\rangle\,,\quad \mathcal{C}|b,\mathbf{k}\rangle = \eta_\mathcal{C}|a,\mathbf{k}\rangle\,.

(b) Also show that the Lagrangian density (13.2.1) is invariant under charge conjugation and that the current density (13.2.11) changes sign:

\mathcal{C}j(x)\mathcal{C}^\dagger = -j(x).

Hence, particles and antiparticles are interchanged, with the four-momentum remaining unchanged. (c) Find a representation for the operator

\mathcal{C}