Noncompact Lattice Formulation of Gauge Theories

TL;DR

This paper introduces a noncompact lattice formulation of gauge theories by expanding gauge fields in Bloch wave functions, allowing continuum limits without shrinking lattice size and enabling the lattice coupling to serve as the renormalized continuum coupling.

Contribution

It presents a novel noncompact lattice gauge theory formulation using Bloch wave expansions, connecting lattice and continuum theories without taking the lattice size to zero.

Findings

Exact continuum theory achieved with all bands included.

Lattice coupling acts as the renormalized continuum coupling.

Physical results are independent of lattice size .

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 $\,\ell\,$.