Non-Abelian gauge theory on a finite-element lattice.

Abstract

We formulate the equations of motion of a non-Abelian gauge field coupled to fermions on a finite-element lattice. This is done by a straightforward iterative approach, in which successive interaction terms are added to the Dirac and Yang-Mills equations of motion, and to the field strength, in order to preserve gauge invariance, yielding a series in powers of $\mathrm{gh} A$. Here $g$ is the coupling constant, $h$ is the lattice spacing, and $A$ is the gauge potential. A simple, nonlocal, iterative formula is obtained for the interaction terms in the equations of motion. Difference equations which are satisfied by the full interaction terms are derived. On the other hand, the field strength is expressed locally in terms of the potential, in terms of nested commutators. The transformations of the gauge potentials are similarly determined to be series of nested commutators.