Unconstrained Variables and Equivalence Relations for Lattice Gauge Theories

TL;DR

This paper reformulates lattice gauge theories using unconstrained variables, demonstrating equivalence in mean field approximation for pure gauge theories with compact, rank 1 groups, and explores implications for phase diagrams and symmetry promotion.

Contribution

It introduces an exact unconstrained variable formulation for lattice gauge theories and shows their equivalence in mean field approximation for certain groups, extending understanding of phase diagrams and symmetry.

Findings

Equivalence of SU(2) and U(1) theories in mean field approximation.

Phase diagrams are identical up to coupling redefinition.

Conditions for promoting global symmetry to gauge symmetry.

Abstract

We write the partition function for a lattice gauge theory, with compact gauge group, exactly in terms of unconstrained variables and show that, in the mean field approximation, the dynamics of pure gauge theories, invariant under compact, continuous,groups of rank 1 is the same for all. We explicitly obtain the equivalence for the case of SU(2) and U(1) and show that it obtains, also, if we consider saddle point configurations that are not,necessarily, uniform, but only proportional to the identity for both groups. This implies that the phase diagrams of the (an)isotropic SU(2) theory and the (an)isotropic U(1) theory in any dimension are identical, within this approximation, up to a re-evaluation of the numerical values of the coupling constants at the transitions. Only nonuniform field configurations, that, also, belong to higher dimensional representations for Yang--Mills fields, will be able to p robe the difference between them. We also show under what conditions the global symmetry of an anisotropic term in the lattice action can be promoted to a gauge symmetry of the theory on layers and point out how deconstruction and flux compactification scenaria may thus be studied on the lattice.