Noncompact lattice formulation of gauge theories

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 K--> restricted to the Brillouin zone. A noncompact formulation of lattice QCD (or QED) can be derived by restricting the expansion only to the zeroth-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≠0 bands, without requiring the lattice size l-->0. This makes it possible, at a nonzero l, for the lattice coupling g_l to act as the renormalized continuum coupling. All physical results in the continuum are, of course, independent of l.