Noether's Theorems and Gauge Symmetries

TL;DR

This paper reviews Noether's theorems in gauge theories, highlighting the well-known and lesser-known results, and clarifies their relationships for both Abelian and non-Abelian cases.

Contribution

It provides a comprehensive, general presentation of all main Noether-related theorems in gauge theories, including lesser-known results and their interconnections.

Findings

Identifies three key theorems related to gauge invariance

Clarifies relationships between Noether's theorems in gauge theories

Applies results to both Abelian and non-Abelian gauge theories

Abstract

Consideration of the Noether variational problem for any theory whose action is invariant under global and/or local gauge transformations leads to three distinct theorems. These include the familiar Noether theorem, but also two equally important but much less well-known results. We present, in a general form, all the main results relating to the Noether variational problem for gauge theories, and we show the relationships between them. These results hold for both Abelian and non-Abelian gauge theories.