Galerkin formulation of path integrals in lattice field theory

Tran, Brian Kha; Southworth, Ben Scott

doi:10.1088/1751-8121/ae1273

Galerkin formulation of path integrals in lattice field theory

Journal Article · Thu Oct 23 00:00:00 EDT 2025 · Journal of Physics. A, Mathematical and Theoretical

DOI:https://doi.org/10.1088/1751-8121/ae1273· OSTI ID:2999954

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Univ. of Colorado, Boulder, CO (United States); Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

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 (DOFs) 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 DOFs. 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, this two-point function satisfies a weak propagator (or Green’s function) identity, in analogy to the continuum case, as well as a convergence estimate obtained from the standard finite element techniques. 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.

View Accepted Manuscript (DOE)

Research Organization:: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Sponsoring Organization:: USDOE National Nuclear Security Administration (NNSA); USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)

Grant/Contract Number:: 89233218CNA000001

OSTI ID:: 2999954

Report Number(s):: LA-UR--24-32282; 10.1088/1751-8121/ae1273

Journal Information:: Journal of Physics. A, Mathematical and Theoretical, Journal Name: Journal of Physics. A, Mathematical and Theoretical Journal Issue: 43 Vol. 58; ISSN 1751-8121; ISSN 1751-8113

Publisher:: IOP PublishingCopyright Statement

Country of Publication:: United States

Language:: English

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