Noether's 1st theorem with local symmetries

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.