Advanced Quantum Mechanics Schwabl · Problem 13.14

Quantization of the Radiation Field

Physics — Quantum Mechanics · dirac-field-charge-conjugation-operator

Problem

Show that the operator of the charge conjugation transformation C=C1C2\mathcal{C} = \mathcal{C}_1\mathcal{C}_2 C1=exp⁡[−i∑k,rmVEkλ(bkr†bkr−dkr†dkr)]\mathcal{C}_1 = \exp\left[-\mathrm{i}\sum_{\mathbf{k},r}\frac{m}{VE_{\mathbf{k}}}\lambda(b^\dagger_{\mathbf{k}r}b_{\mathbf{k}r} - d^\dagger_{\mathbf{k}r}d_{\mathbf{k}r})\right] C2=exp⁡[iπ2∑k,rmVEk(bkr†−dkr†)(bkr−dkr)]\mathcal{C}_2 = \exp\left[\frac{\mathrm{i}\pi}{2}\sum_{\mathbf{k},r}\frac{m}{VE_{\mathbf{k}}}(b^\dagger_{\mathbf{k}r} - d^\dagger_{\mathbf{k}r})(b_{\mathbf{k}r} - d_{\mathbf{k}r})\right] transforms the creation and annihilation operators of the Dirac field and the field operator as follows: CbkrC†=ηCdkr ,\mathcal{C}b_{\mathbf{k}r}\mathcal{C}^\dagger = \eta_\mathcal{C} d_{\mathbf{k}r}\,, Cdkr†C†=ηC∗bkr† ,\mathcal{C}d^\dagger_{\mathbf{k}r}\mathcal{C}^\dagger = \eta^*_\mathcal{C} b^\dagger_{\mathbf{k}r}\,, Cψ(x)C†=ηCCψˉT(x),\mathcal{C}\psi(x)\mathcal{C}^\dagger = \eta_\mathcal{C} C\bar{\psi}^T(x), where the transpose only refers to the spinor indices, and C=iγ2γ0C = \mathrm{i}\gamma^2\gamma^0. The factor C1\mathcal{C}_1 yields the phase factor ηC=eiλ\eta_\mathcal{C} = \mathrm{e}^{\mathrm{i}\lambda}. The transformation C\mathcal{C} exchanges particles and antiparticles with the same momentum, energy, and helicity. Show also that the vacuum is invariant with respect to this transformation and that the current density jμ=e:ψˉγμψ:j^\mu = e:\bar{\psi}\gamma^\mu\psi: changes sign.

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