Galerkin Formulation of Path Integrals in Lattice Field Theory

TL;DR

This paper introduces a Galerkin-based mathematical framework for lattice field theory, enabling more flexible and higher-order discretizations that improve the approximation of field observables and extend to gauge theories.

Contribution

It develops a novel Galerkin formulation for lattice field theories using finite element spaces, allowing for higher-order discretizations and more general correlation function definitions.

Findings

Higher-order finite element spaces improve observable approximations.

Two-point correlation functions can be defined between any two points.

Framework extends to gauge field theories.

Abstract

We present a mathematical framework for Galerkin formulations of path integrals in lattice field theory. The framework is based on using the degrees of freedom associated to a Galerkin discretization as the fundamental lattice variables. We formulate standard concepts in lattice field theory, such as the partition function and correlation functions, in terms of the degrees of freedom. For example, using continuous finite element spaces, we show that the two-point spatial correlation function can be defined between any two points on the domain (as opposed to at just lattice sites) and furthermore, that this satisfies a weak propagator (or Green's function) identity, in analogy to the continuum case. Furthermore, this framework leads naturally to higher-order formulations of lattice field theories by considering higher-order finite element spaces for the Galerkin discretization. We consider analytical and numerical examples of scalar field theory to investigate how increasing the order of piecewise polynomial finite element spaces affect the approximation of lattice observables. Finally, we sketch an outline of this Galerkin framework in the context of gauge field theories.