Zeta-regularized Lattice Field Theory with Lorentzian background metrics

TL;DR

This paper introduces a zeta-regularization scheme for lattice field theories with Lorentzian metrics, enabling non-perturbative analysis and demonstrating consistency with classical limits through explicit harmonic oscillator examples.

Contribution

It develops a formal zeta-regularization method for lattice theories in Lorentzian backgrounds and verifies its validity with explicit harmonic oscillator cases.

Findings

Regularization scheme reproduces correct ground state energy.

Classical limit is independent of the regularization method.

Numerical solutions confirm theoretical predictions.

Abstract

Lattice field theory is a very powerful tool to study Feynman's path integral non-perturbatively. However, it usually requires Euclidean background metrics to be well-defined. On the other hand, a recently developed regularization scheme based on Fourier integral operator $\zeta$-functions can treat Feynman's path integral non-pertubatively in Lorentzian background metrics. In this article, we formally $\zeta$-regularize lattice theories with Lorentzian backgrounds and identify conditions for the Fourier integral operator $\zeta$-function regularization to be applicable. Furthermore, we show that the classical limit of the $\zeta$-regularized theory is independent of the regularization. Finally, we consider the harmonic oscillator as an explicit example. We discuss multiple options for the regularization and analytically show that they all reproduce the correct ground state energy on the lattice and in the continuum limit. Additionally, we solve the harmonic oscillator on the lattice in Minkowski background numerically.