Mean Field Approaches to Lattice Gauge Theories: A Review

TL;DR

This review explores mean field methods applied to lattice gauge theories, discussing their advantages, limitations, and how different formulations affect results, providing a comprehensive comparison of past and recent findings.

Contribution

It offers a pedagogical comparison of mean field approaches to lattice gauge theories across various formulations, unifying older and newer results in the literature.

Findings

Mean field methods provide approximate insights into lattice gauge theories.

Results depend significantly on the formulation of the gauge theory.

The review highlights the challenges and potential of mean field approaches in this context.

Abstract

Due to their broad applicability, gauge theories (GTs) play a crucial role in various areas of physics, from high-energy physics to condensed matter. Their formulations on lattices, lattice gauge theories (LGTs), can be studied, among many other methods, with tools coming from statistical mechanics lattice models, such as mean field methods, which are often used to provide approximate results. Applying these methods to LGTs requires particular attention due to the intrinsic local nature of gauge symmetry, how it is reflected in the variables used to formulate the theory, and the breaking of gauge invariance when approximations are introduced. This issue has been addressed over the decades in the literature, yielding different conclusions depending on the formulation of the theory under consideration. In this article, we focus on the mean field theoretical approach to the analysis of GTs and LGTs, connecting both older and more recent results that, to the best of our knowledge, have not been compared in a pedagogical manner. After a brief overview of mean field theory in statistical mechanics and many-body systems, we examine its application to pure LGTs with a generic compact gauge group. Finally, we review the existing literature on the subject, discussing the results obtained so far and their dependence on the formulation of the theory.