A lattice Maxwell system with discrete space-time symmetry and local energy-momentum conservation

TL;DR

This paper develops a lattice Maxwell system that preserves gauge symmetry, symplectic structure, and discrete space-time symmetry, leading to local energy-momentum conservation and improved numerical algorithms.

Contribution

It generalizes Noether's theorem to discrete symmetries and constructs a lattice Maxwell system with all conservation laws and geometric structures.

Findings

The lattice Maxwell system admits a discrete local energy-momentum conservation law.

The system is effective for numerically solving Maxwell's equations.

The lattice model is as good as or better than the continuous Maxwell model.

Abstract

A lattice Maxwell system is developed with gauge-symmetry, symplectic structure and discrete space-time symmetry. Noether's theorem for Lie group symmetries is generalized to discrete symmetries for the lattice Maxwell system. As a result, the lattice Maxwell system is shown to admit a discrete local energy-momentum conservation law corresponding to the discrete space-time symmetry. These conservative properties make the discrete system an effective algorithm for numerically solving the governing differential equations on continuous space-time. Moreover, the lattice model, respecting all conservation laws and geometric structures, is as good as and probably more preferable than the continuous Maxwell model. Under the simulation hypothesis by Bostrom and in consistent with the discussion on lattice QCD by Beane et al., the two interpretations of physics laws on space-time lattice could be essentially the same.