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 $\vec{K}$ 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,withoutrequiringthelatticesize$,\ell \to 0,$\mathrm{.}Thismakesitpossible,atanonzero$,\ell,$,forthelatticecoupling$,g_\ell,$ to act as the renormalized continuum coupling. All physical results in the continuum are, of course, independent of $\mathrm{\ell }$.