Discrete Field Theory: Symmetries and Conservation Laws

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.

Citation
Skopenkov, M. (2023). Discrete Field Theory: Symmetries and Conservation Laws. Mathematical Physics, Analysis and Geometry, 26(3). https://doi.org/10.1007/s11040-023-09459-4

Acknowledgements
The publication was prepared within the framework of the Academic Fund Program at the National Research University Higher School of Economics (HSE) in 2018–2019 (grant N18-01-0023) and by the Russian Academic Excellence Project “5-100”. The author has also received support from the Simons–IUM fellowship. The author is grateful to E. Akhmedov, L. Alania, D. Arnold, A. Bossavit, V. Buchstaber, D. Chelkak, M. Chernodub, M. Desbrun, M. Gualtieri, F. Günther, I. Ivanov, A. Jivkov, M. Kraus, N. Mnev, F. Müller-Hoissen, S. Pirogov, P. Pylyavskyy, A. Rassadin, R. Rogalyov, I. Sabitov, P. Schröder, I. Shenderovich, B. Springborn, A. Stern, S. Tikhomirov, S. Vergeles for useful discussions.

Publisher
Springer Science and Business Media LLC

Journal
MATHEMATICAL PHYSICS ANALYSIS AND GEOMETRY

DOI
10.1007/s11040-023-09459-4

arXiv
1709.04788

Additional Links
https://link.springer.com/10.1007/s11040-023-09459-4https://arxiv.org/pdf/1709.04788

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