Lie Groupoids in Classical Field Theory I: Noether's Theorem

TL;DR

This paper introduces a novel framework for classical field theory using Lie groupoids to better handle local symmetries and gauge transformations, extending Noether's theorem within this context.

Contribution

It reformulates Noether's theorem using Lie groupoids and algebroids, advancing the mathematical tools for local symmetry analysis in classical field theory.

Findings

Reformulation of Noether's theorem with Lie groupoids

Framework for local symmetries in gauge theories

Foundation for further development of symmetry concepts

Abstract

In the two papers of this series, we initiate the development of a new approach to implementing the concept of symmetry in classical field theory, based on replacing Lie groups/algebras by Lie groupoids/algebroids, which are the appropriate mathematical tools to describe local symmetries when gauge transformations are combined with space-time transformations. Here, we outline the basis of the program and, as a first step, show how to (re)formulate Noether's theorem about the connection between symmetries and conservation laws in this approach.