Noether currents of locally equivalent symmetries

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 ${\boldsymbol{L}}={\boldsymbol{r}}\times {\boldsymbol{p}}$ between momentum ${\boldsymbol{p}}$ and angular momentum ${\boldsymbol{L}}$. As a byproduct, we find a natural interpretation for the discrepancy between the canonical and metric energy–momentum (EM) tensors in theories of particles with spin. A symmetric EM 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.