Clarifying the relation between covariantly conserved currents and Noether's second theorem

TL;DR

This paper clarifies how Noether's second theorem can be used to derive covariant conservation laws associated with local symmetries, using accessible calculus methods to enhance understanding for physicists.

Contribution

It provides a clear, accessible explanation of the conditions under which Noether's second theorem generates covariant conservation laws, avoiding complex mathematical tools.

Findings

Conditions for using Noether's second theorem to generate covariant conservation laws

Simplified calculus-based derivation of covariant conservation laws

Enhanced understanding of the relation between local symmetries and conservation laws

Abstract

Proper conservation laws, that is, the existence of local functions of the fields whose divergence vanishes on-shell, are associated with global symmetries and are a consequence of Noether's first theorem. Covariant conservation laws, i.e., the same type of laws but featuring a covariant divergence, are associated with local symmetries and are a consequence (under certain conditions) of Noether's second theorem. In this paper, we clarify under which conditions Noether's second theorem, which is mostly known for providing relations between the equations of motion of a theory that possesses some local symmetries, can be used to generate covariantly conservation laws. We do it using mostly the tools of Calculus, in the fashion of Noether's original article, avoiding the somewhat complicated mathematical tools that plague much of the literature on this topic, thus making the article readable for the general physicist.