Lattice and Continuum Theories

TL;DR

This paper explores representing continuum path integrals as lattice theories, enabling improved lattice calculations by accounting for hard modes and maintaining symmetries like translation and rotary invariance.

Contribution

It introduces a novel lattice representation of continuum path integrals with nonlocal potential terms, facilitating better lattice computations and symmetry considerations.

Findings

Lattice representation of continuum path integrals with nonlocal potentials.

Estimation of contributions from nonlocal terms in the continuum limit.

Potential for improved lattice calculations incorporating hard modes.

Abstract

We investigate path integral formalism for continuum theory. It is shown that the path integral for the soft modes can be represented in the form of a lattice theory. Kinetic term of this lattice theory has a standard form and potential term has additional nonlocal terms which contributions should tend to zero in the limit of continuum theory. Contributions of these terms can be estimated. It is noted that this representation of path integral may be useful to improve lattice calculations taking into account hard modes contribution by standard perturbative expansion. We discuss translation invariance of correlators and the possibility to construct a lattice theory which keeps rotary invariance also.