Noether currents of locally equivalent symmetries

TL;DR

This paper explores how local symmetry transformations induce relations between different Noether currents, providing insights into energy-momentum tensors and their symmetries in particle theories.

Contribution

It generalizes the relationship between conserved currents of different symmetries and offers a natural interpretation for energy-momentum tensor discrepancies.

Findings

Derived linear relations between Noether currents of distinct symmetries.

Provided a natural interpretation for the difference between canonical and metric energy-momentum tensors.

Showed how to obtain a symmetric energy-momentum tensor without additional constraints.

Abstract

Local symmetry transformations play an important role for establishing the existence and form of a conserved (Noether) current in systems with a global continuous symmetry. We explain how this fact leads to the existence of linear relations between Noether currents of distinct global symmetries that coincide on the local level, thus generalizing the well-known relationship $\vec L=\vec r\times\vec p$ between momentum $\vec p$ and angular momentum $\vec L$. As a byproduct, we find a natural interpretation for the discrepancy between the canonical and metric energy-momentum tensors in theories of particles with spin. A symmetric energy-momentum tensor can thus be obtained from the Noether procedure without adding any ad hoc corrections or imposing additional constraints such as gauge invariance in Maxwell's electrodynamics.