Symmetries and conservation laws in discrete spacetime

TL;DR

This paper extends Noether's theorem to discrete spacetime, deriving exact conservation laws and energy-momentum tensors, which enable the development of explicit, energy-conserving algorithms with implications for physics and numerical methods.

Contribution

It introduces a method to obtain discrete conservation laws analogous to continuous Noether's theorem, applicable to electromagnetism and particle algorithms.

Findings

Derived exact energy-momentum conservation laws in discrete spacetime

Developed explicit energy-conserving particle-in-cell algorithms

Connected discrete symmetries to nonlocal conservation channels

Abstract

Noether's theorem connects symmetries to invariants in continuous systems, however its extension to discrete systems has remained elusive. Recognizing the lowest-order finite difference as the foundation of local continuity, a viable method for obtaining discrete conservation laws is developed by working in exact analogy to the continuous Noether's theorem. A detailed application is given to electromagnetism, where energy-momentum conservation laws are rapidly obtained in highly generalized forms that disrupt conventional notions regarding conservative algorithms. Field-matter couplings and energy-momentum tensors with optional deviations at the discreteness scale properly reduce in the continuous limit. Nonlocal symmetries give rise to an additional conservation channel for each spacetime displacement, permitting generalized nonlocal couplings. Prescriptions for conservative particle integrators emerge directly from field-matter coupling terms, enabling the development of fully explicit, energy-conserving particle-in-cell algorithms. The demonstration of exact conservation laws in discrete spacetime that preserve canonical structure has deep implications for numerical algorithms and fundamental physics.