Flux quantization and angular momentum in non-Abelian gauge field theories

Abstract

We study a Yang-Mills field, with gauge group generators $L_a (a=1, \dots , n)$, coupled to self-interacting source fields through a gauge coupling constant $e$. The following results are obtained: (a) The total angular momentum, defined as Poincaré group generators, is the free-field angular momentum plus $\left(\frac{1}{2\pi }\right)Q_a{\overrightarrow{\Phi }}_a$, where $Q_a$ is the total charge and ${\Phi }_a^k$ is the total flux of $\nabla \times {\overrightarrow{\mathrm{A}}}_a$ in the $x^k$ direction. (b) There is a choice of {$L_a$} such that the flux matrix $\overrightarrow{\Phi }\equiv {\overrightarrow{\Phi }}_a{L}_a$ obeys the quantization condition $\left[\right(\frac{e}{2\pi }){\Phi }^1, (\frac{e}{2\pi }\left){\Phi }^2\right]=\left(\frac{\mathrm{ie}}{2\pi }\right){\Phi }^3$. (c) $\left(\frac{e}{2\pi }\right)\overrightarrow{\Phi }$ generates gauge transformations that are equivalent to spatial rotations of asymptotic Higgs fields. (d) For any solution in which $\left(\frac{e}{2\pi }\right)\overrightarrow{\Phi }\ne 0$, the gauge field is a monopole field with pole strength $g=\frac{1}{e}$.