Exact space-time symmetry conservation and automatic mesh refinement for classical lattice field theory

TL;DR

This paper introduces a method to preserve space-time symmetries and Noether charges exactly in lattice field theory by using dynamical coordinate maps, enabling adaptive mesh refinement while maintaining symmetry invariance.

Contribution

It develops a reparameterization invariant classical action for scalar fields with dynamical coordinate maps, ensuring exact symmetry conservation and adaptive mesh refinement in lattice simulations.

Findings

Space-time symmetries remain intact after discretization.

Noether charges are exactly conserved.

Coordinate maps adapt dynamically to the scalar field.

Abstract

The breaking of space-time symmetries and the non-conservation of the associated Noether charges constitutes a central artifact in lattice field theory. In prior work we have shown how to overcome this limitation for classical actions describing point particle motion, using the world-line formalism of general relativity. The key is to treat coordinate maps (from an abstract parameter space into space-time) as dynamical and dependent degrees of freedom, which remain continuous after discretization of the underlying parameter space. Here we present latest results where we construct a reparameterization invariant classical action for scalar fields, which features dynamical coordinate maps. We highlight the following achievements of our approach: 1) global space-time symmetries remain intact after discretization and the associated Noether charges remain exactly preserved 2) coordinate maps adapt to the dynamics of the scalar field leading to adaptive grid resolution guided by the symmetries.