Mixing of Internal and Space-Time Symmetries

Abstract

In a general Yang-Mills theory with self-interacting boson and/or fermion source fields, it is shown that the angular momentum operator is $\overrightarrow{\mathrm{J}}={\overrightarrow{\mathrm{J}}}_0+{\left(2\mathrm{}\pi \right)}^{-1\mathrm{}}{Q}_a{\overrightarrow{\Phi }}_a$, where ${\overrightarrow{\mathrm{J}}}_0$ is the freefield angular momentum (i.e., that for zero gauge coupling constant), $Q_a$ is the total charge, corresponding to the internal symmetry index $a$, and ${{\Phi }_a}^k$ is the total flux of $\nabla \times {\overrightarrow{\mathrm{A}}}_a$ along the $x^k$ direction, in the gauge $\nabla ·{\overrightarrow{\mathrm{A}}}_a=0\mathrm{}$.