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Tempered Distribution Theory on a Lattice

This is an online seminar I presented at the Department of Physics & Astronomy, University of Kentucky, on January 15, 2025. The mathematics underlying Noether’s theorem in classical field theory relies on tempered distributions, a framework developed by Laurent Schwartz. This formalism systematically handles Dirac delta functions (which naturally arise from fundamental Poisson brackets) and their derivatives. Importantly, tempered distribution theory extends the definition of derivatives to objects that are not classical functions—such as the Dirac delta—where the usual limit-based definition breaks down. As argued in the paper below, symmetries can, in fact, be preserved on a lattice—if one uses a discrete derivative that satisfies the same calculus rules as in the continuum: arXiv:2307.16520 – https://arxiv.org/abs/2307.16520 This talk reviews the discovery of two such discrete derivatives, collectively referred to as exact derivatives, and the development of the associated lattice version of tempered distribution theory. This framework shows that the resulting discrete calculus is mathematically identical to its continuum counterpart. Using these exact derivatives to construct lattice field theories makes it possible to preserve the full Poincaré symmetry—alongside gauge symmetry—directly on the lattice. Most of this work is ongoing and yet to be published. Chapters: 0:00 Introduction 1:11 Motivation 13:08 Current Understanding 18:41 Exact Approach 30:53 The Full Problem 36:36 Results 1:06:40 Application 1:13:30 Future 1:15:26 Q&A Keywords: seminars, Q&A, physics, classical field theory, Noether's theorem, quantum field theory, UV regulators, UV divergences, tempered distributions, Schwartz distribution theory, symmetries, Poincaré algebra, discrete calculus, discrete derivative, Stirling numbers. #onlineseminars #physicsseminars #quantumfieldtheory #symmetries #noethertheorem #discretecalculus