Advanced Quantum Mechanics Schwabl · Problem 13.2

Quantization of the Radiation Field

Physics — Quantum Mechanics · scalar-field-four-momentum

Problem

(a) For the scalar field, show that the four-momentum operator

P^\mu =:\int d^3x\,\{\pi\phi^{,\mu} - \delta^\mu_0 \mathcal{L}\}:

can be written in the form (13.1.19)

P^\mu = \sum_{\mathbf{k}} k^\mu a^\dagger_{\mathbf{k}} a_{\mathbf{k}}\,.

(b) Show that the four-momentum operator is the generator of the translation operator:

\mathrm{e}^{\mathrm{i}a_\mu P^\mu} F(\phi(x)) \mathrm{e}^{-\mathrm{i}a_\mu P^\mu} = \phi(x+a)\,.