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

TL;DR

This review explains the modern significance of Noether's 1918 work on symmetries, gauge theories, and boundary conditions, illustrating their applications in current physics through accessible examples for students.

Contribution

It provides a simplified overview of Noether's theorems and their modern applications in gauge symmetries and boundary conditions, targeting students rather than experts.

Findings

Clarifies the relation between symmetries and conserved charges.

Explains the role of gauge symmetries in equations of motion.

Discusses boundary conditions and asymptotic symmetry algebra.

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.