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A short review on Noether's theorems, gauge symmetries and boundary terms

Máximo Bañados, Ignacio A. Reyes

TL;DR

<3-5 sentence high-level summary> This review traces Noether's theorems from their classical roots to modern applications in gauge theories and gravity. It systematically develops global and gauge symmetries, showing how conserved charges arise, how Hamiltonian constraints encode gauge structure, and how boundary conditions and boundary terms define asymptotic symmetry groups. Through detailed examples—particle mechanics, Maxwell theory, General Relativity, Chern-Simons theory, and two-dimensional conformal field theories—the authors illustrate the practical computation of currents, energy-momentum tensors, and Virasoro-type algebras, including central extensions. The work serves as a pedagogical compendium for students, linking Noether's insights to contemporary topics like AdS/CFT and holography while emphasizing the technical role of boundaries in gauge theories.

Abstract

This review is dedicated to some modern applications of the remarkable paper written in 1918 by E. Noether. On a single paper, Noether discovered the crucial relation between symmetries and conserved charges as well as the impact of gauge symmetries on the equations of motion. Almost a century has gone since the publication of this work and its applications have permeated modern physics. Our focus will be on some examples that have appeared recently in the literature. This review is aim at students, not researchers. The main three topics discussed are (i) global symmetries and conserved charges (ii) local symmetries and gauge structure of a theory (iii) boundary conditions and algebra of asymptotic symmetries. All three topics are discussed through examples.

A short review on Noether's theorems, gauge symmetries and boundary terms

TL;DR

<3-5 sentence high-level summary> This review traces Noether's theorems from their classical roots to modern applications in gauge theories and gravity. It systematically develops global and gauge symmetries, showing how conserved charges arise, how Hamiltonian constraints encode gauge structure, and how boundary conditions and boundary terms define asymptotic symmetry groups. Through detailed examples—particle mechanics, Maxwell theory, General Relativity, Chern-Simons theory, and two-dimensional conformal field theories—the authors illustrate the practical computation of currents, energy-momentum tensors, and Virasoro-type algebras, including central extensions. The work serves as a pedagogical compendium for students, linking Noether's insights to contemporary topics like AdS/CFT and holography while emphasizing the technical role of boundaries in gauge theories.

Abstract

This review is dedicated to some modern applications of the remarkable paper written in 1918 by E. Noether. On a single paper, Noether discovered the crucial relation between symmetries and conserved charges as well as the impact of gauge symmetries on the equations of motion. Almost a century has gone since the publication of this work and its applications have permeated modern physics. Our focus will be on some examples that have appeared recently in the literature. This review is aim at students, not researchers. The main three topics discussed are (i) global symmetries and conserved charges (ii) local symmetries and gauge structure of a theory (iii) boundary conditions and algebra of asymptotic symmetries. All three topics are discussed through examples.

Paper Structure

This paper contains 43 sections, 266 equations, 4 figures.

Table of Contents

  1. Preface
  2. Noether's theorem for global symmetries in classical mechanics and field theory
  3. Noether's theorem in particle mechanics
  4. The theorem
  5. Noether symmetries
  6. Some examples and comments
  7. On-shell variations
  8. Noether's theorem
  9. The 'conformal' particle
  10. Background fields and non-conservation equations
  11. Noether's theorem in Hamiltonian mechanics: Symmetry generators and Lie algebras
  12. The conformal particle in Hamiltonian form
  13. Noether's theorem in Field theory. Derivation and examples.
  14. The proof
  15. Symmetries act on fields: Lie derivatives
  16. ...and 28 more sections

Figures (4)

  • Figure 1: Two functions $q(t)$, $q'(t)$ related by a time translation. At any given time $t$, $\delta q(t)$ represents the difference between both functions.
  • Figure 2: Properties of equations of motion of gauge theories
  • Figure 3: The manifold $\mathcal{M}$
  • Figure :