Generalized symmetries and Noether's theorem in QFT

TL;DR

This paper explores the limitations of generalized symmetries in quantum field theory, revealing topological obstructions to Noether's theorem and extending foundational results like Weinberg-Witten's theorem to broader contexts.

Contribution

It introduces a refined classification of twist operators, demonstrating that only non-compact generalized symmetries can be charged under continuous global symmetries, and generalizes key theorems in QFT.

Findings

Generalized symmetries cannot be charged under continuous global symmetries with Noether currents.

Only non-compact generalized symmetries can be charged under continuous global symmetries.

Reinterpretation and extension of Weinberg-Witten's theorem to arbitrary dimensions and representations.

Abstract

We show that generalized symmetries cannot be charged under a continuous global symmetry having a Noether current. Further, only non-compact generalized symmetries can be charged under a continuous global symmetry. These results follow from a finer classification of twist operators, which naturally extends to finite group global symmetries. They unravel topological obstructions to the strong version of Noether's theorem in QFT, even if under general conditions a global symmetry can be implemented locally by twist operators (weak version). We use these results to rederive Weinberg-Witten's theorem within local QFT, generalizing it to massless particles in arbitrary dimensions and representations of the Lorentz group. Several examples with local twists but without Noether currents are described. We end up discussing the conditions for the strong version to hold, dynamical aspects of QFT's with non-compact generalized symmetries, scale vs conformal invariance in QFT, connections with the Coleman-Mandula theorem and aspects of global symmetries in quantum gravity.