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Noether's Second Theorem and Ward Identities for Gauge Symmetries

Steven G. Avery, Burkhard U. W. Schwab

TL;DR

The paper develops a path-integral framework based on Noether's second theorem to derive Ward identities for residual (large) gauge symmetries across Maxwell, Yang–Mills, p-form fields, and gravity. It connects these Ward identities to a two-form current k^{μν} that encodes charges on codimension-2 surfaces and ties them to soft theorems and asymptotic symmetry structures such as BMS. The authors discuss gauge fixing, spontaneous breaking of residual gauge symmetries, and interpret soft states as Goldstone modes, with links to memory effects and Wald entropy. Through explicit Abelian, non-Abelian, p-form, and gravity examples, the work provides a general, practical framework for deriving Ward identities in gauge theories and gravity, with notes on anomalies and potential subregion BV/BRST extensions.

Abstract

Recently, a number of new Ward identities for large gauge transformations and large diffeomorphisms have been discovered. Some of the identities are reinterpretations of previously known statements, while some appear to be genuinely new. We use Noether's second theorem with the path integral as a powerful way of generating these kinds of Ward identities. We reintroduce Noether's second theorem and discuss how to work with the physical remnant of gauge symmetry in gauge fixed systems. We illustrate our mechanism in Maxwell theory, Yang-Mills theory, p-form field theory, and Einstein-Hilbert gravity. We comment on multiple connections between Noether's second theorem and known results in the recent literature. Our approach suggests a novel point of view with important physical consequences.

Noether's Second Theorem and Ward Identities for Gauge Symmetries

TL;DR

The paper develops a path-integral framework based on Noether's second theorem to derive Ward identities for residual (large) gauge symmetries across Maxwell, Yang–Mills, p-form fields, and gravity. It connects these Ward identities to a two-form current k^{μν} that encodes charges on codimension-2 surfaces and ties them to soft theorems and asymptotic symmetry structures such as BMS. The authors discuss gauge fixing, spontaneous breaking of residual gauge symmetries, and interpret soft states as Goldstone modes, with links to memory effects and Wald entropy. Through explicit Abelian, non-Abelian, p-form, and gravity examples, the work provides a general, practical framework for deriving Ward identities in gauge theories and gravity, with notes on anomalies and potential subregion BV/BRST extensions.

Abstract

Recently, a number of new Ward identities for large gauge transformations and large diffeomorphisms have been discovered. Some of the identities are reinterpretations of previously known statements, while some appear to be genuinely new. We use Noether's second theorem with the path integral as a powerful way of generating these kinds of Ward identities. We reintroduce Noether's second theorem and discuss how to work with the physical remnant of gauge symmetry in gauge fixed systems. We illustrate our mechanism in Maxwell theory, Yang-Mills theory, p-form field theory, and Einstein-Hilbert gravity. We comment on multiple connections between Noether's second theorem and known results in the recent literature. Our approach suggests a novel point of view with important physical consequences.

Paper Structure

This paper contains 21 sections, 110 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Noether's theorems
  3. Notation and terminology
  4. Noether's first
  5. Noether's second
  6. Two-form currents
  7. Implications for QFT
  8. Gauge fixing
  9. Ward identities
  10. Global identities
  11. Commutators
  12. Central terms
  13. Examples
  14. Abelian gauge field
  15. The soft theorem
  16. ...and 6 more sections

Figures (3)

  • Figure 1: A depiction of the indicator $\mathbf{1}_R(x)$ for $R$ having support on the boundary of the spacetime manifold $M$. The support of $R$ on $\partial M$ is denoted $B = R \cap \partial M = \partial R \cap \partial M$. The interior portion of $\partial R$ is denoted $\Sigma$.
  • Figure 2: A depiction of the region $R$ and its complement $\tilde{R}$ as used to find the global Ward identity \ref{['eq:global-ward-id']}. $R$ encloses all interior insertions and the entire past boundary $\Sigma_0$.
  • Figure 3: A depiction of the boundaries of $R_1 \subset R_2 \subset R_3$ as used to compute commutators using the Ward identity. It is useful to consider three surfaces to show \ref{['eq:second-comm-id']}, which relates the commutator to one symmetry transformation of the other charge.