Discrete Noether Currents

TL;DR

This work extends Noether's first theorem to finite discrete symmetries within four-dimensional relativistic field theories by formulating a discrete current from sequences of infinitesimal shifts and a spacetime-dependent deformation parameter. The resulting conserved discrete current $\mathcal{J}^\mu$ satisfies $\partial_\mu \mathcal{J}^\mu=0$ and provides a charge $Q(t)$ that acts as the symmetry generator; the current can also be expressed as a sum of an emergent-continuous part $\mathcal{J}^\mu_{\text{a}}$ and a nonlocal piece $\mathcal{J}^\mu_{\text{b}}$. The authors illustrate the construction with cyclic $Z_n$ symmetries, including charge-conjugation cases, and discuss the quantum implications via Ward–Takahashi–type identities and Schwinger–Dyson equations for discrete symmetries. The approach opens pathways to applying discrete-Noether currents in Standard Model contexts and finite groups, with potential relevance to flavor structures and emergent symmetries.

Abstract

A simple implementation of Noether's theorem for discrete symmetries in relativistic continuum field theories is presented. The associated conserved current is exemplified by charge conjugation and a cyclic symmetry. In addition, the quantum version of current conservation for discrete symmetries is briefly discussed.