Noncompact Lattice Formulation of Gauge Theories

Abstract

We expand the gauge field in terms of a suitably constructed complete set of Bloch wave functions, each labeled by a band designation $n$ and a wave number $\vec{K}$ restricted to the Brillouin zone. A noncompact formulation of lattice QCD (or QED) can be derived by restricting the expansion only to the $0^{th}$-band ($n=0$) functions, which are simple continuum interpolations of discrete values associated with sites or links on a lattice. The exact continuum theory can be reached through the inclusion of all $n=0$ and $n\ne 0$ bands, without requiring the lattice size $\ell \to 0$. This makes it possible, at a nonzero $\ell$, for the lattice coupling $g_{\ell }$ to act as the renormalized continuum coupling. All physical results in the continuum are, of course, independent of $\ell$.

Comments: 72 pages, 3 Postscript figures