Noether's 1st theorem with local symmetries

TL;DR

This paper introduces a method to derive conserved matter currents linked to hidden global symmetries in theories with local gauge invariances, demonstrating on-shell conservation in various fundamental theories.

Contribution

It presents a novel approach to identify conserved currents from hidden global symmetries within local gauge theories, clarifying their physical significance.

Findings

Conserved currents are derived for hidden matter symmetries in gauge theories.

In general relativity, the conserved charge matches recent proposals.

The method applies to U(1), non-abelian gauge theories, and gravity.

Abstract

Noether's 2nd theorem applied to a total system states that a global symmetry which is a part of local symmetries does not provide a physically meaningful conserved charge but it instead leads to off-shell constraints as a form of conserved currents. In this paper, we propose a general method to derive a matter conserved current associated with a special global symmetry in the presence of local symmetries. While currents derived from local symmetries of a matter sector with a covariant background gauge field are not conserved in general, we show that the current associated with a special type of a global symmetry, called a hidden matter symmetry, is on-shell conserved. We apply this derivation to a $U(1)$ gauge theory, general relativity and a non-abelian gauge theory. In general relativity, the associated conserved charge agrees with the one recently proposed from a different point of view.