The Path Integral for the Loop Representation of Lattice Gauge Theories

TL;DR

This paper develops a path integral formalism for the Hamiltonian lattice loop representation applicable to gauge theories, enabling the use of statistical algorithms with gauge-invariant loop descriptions, demonstrated on compact QED and non-Abelian Yang-Mills.

Contribution

It introduces a general method to formulate the lattice loop representation as a path integral, facilitating numerical simulations of gauge theories.

Findings

Path integral formalism successfully applied to pure compact QED.

Numerical simulations show the effectiveness of the loop classical action.

Framework applicable to non-Abelian Yang-Mills theory.

Abstract

We show how the Hamiltonian lattice loop representation can be cast straightforwardly in the path integral formalism. The procedure is general for any gauge theory. Here we present in detail the simplest case: pure compact QED. We also analyze the non-Abelian Yang-Mills theory. The lattice loop path integral approach allows to knit together the power of statistical algorithms with the transparency of the gauge invariant loop description. The results produced by numerical simulations with the loop classical action for different lattice models are discused.