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Discrete Field Theory: Symmetries and Conservation Laws

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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.

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Acknowledgements

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.

Funding

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.

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  1. HSE University and King Abdullah University of Science and Technology, Thuwal, Saudi Arabia

    M. Skopenkov

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Communicated by Alexander I. Bobenko.

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Skopenkov, M. Discrete Field Theory: Symmetries and Conservation Laws. Math Phys Anal Geom 26, 19 (2023). https://doi.org/10.1007/s11040-023-09459-4

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