Angular momentum of the quantized electromagnetic field with periodic boundary conditions

Abstract

When an electromagnetic field is quantized within a box of side $L$ with periodic boundary conditions, the total angular momentum $\widehat{\overrightarrow{\mathrm{J}}}$ is not strictly a constant of the motion even when $L\to \infty$. As a result, the conditions for isotropy of such a field involve subtle differences from the usual conditions. The same constraints apply to the orbital part of $\widehat{\overrightarrow{\mathrm{J}}}$, but not to the spin part. The expectation value of $\widehat{\overrightarrow{\mathrm{J}}}$ for any monochromatic plane wave is shown to vanish. Commutation relations are derived between $\widehat{\overrightarrow{\mathrm{J}}}$ and an arbitrary field vector, which show explicitly how $\widehat{\overrightarrow{\mathrm{J}}}$ generates rotation.