Discrete field theory: symmetries and conservation laws

TL;DR

This paper introduces a general algorithm for discretizing classical field theories, establishing a discrete Noether theorem linking symmetries to conservation laws, and demonstrating exact conservation laws in lattice gauge theories.

Contribution

It presents a novel discretization method based on topological concepts and proves a new discrete Noether theorem, enabling exact conservation laws in lattice field theories.

Findings

Exact conservation laws for lattice electrodynamics and gauge theories

Construction of a discrete energy-momentum tensor approximating the continuum version

A general algorithm for discretizing classical field theories from a Lagrangian

Abstract

We present a general algorithm constructing a discretization of a classical field theory from a Lagrangian. We prove a new discrete Noether theorem relating symmetries to conservation laws and an energy conservation theorem not based on any symmetry. This gives exact conservation laws for several theories, e.g., lattice electrodynamics and gauge theory. In particular, we construct a conserved discrete energy-momentum tensor, approximating the continuum one at least for free fields. The theory is stated in topological terms, such as coboundary and products of cochains.