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

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.

75 pages, 4 figures

References in corpus (4)

• Generalized Black Holes in Three-dimensional Spacetime
• On singular Lagrangian underlying the Schrödinger equation
• Variational Methods in AdS/CFT
• A Note on Conserved Charges of Asymptotically Flat and Anti-de Sitter Spaces in Arbitrary Dimensions

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• Weyl Charges in Asymptotically Locally AdS${}_3$ Spacetimes
• Asymptotic Symmetries in $p$-Form Theories
• Observable Supertranslations
• Asymptotic dynamics on the worldline for spinning particles
• Double-soft graviton amplitudes and the extended BMS charge algebra
• Conserved Charges in Extended Theories of Gravity
• The constraints of post-quantum classical gravity
• Surface Charges for Gravity and Electromagnetism in the First Order Formalism
• Noether currents for higher-order variational problems of Herglotz type with time delay
• Integrable Lambda Models And Chern-Simons Theories
• Surface Charges Toolkit for Gravity
• From physical symmetries to emergent gauge symmetries
• Symmetries and Charges in Weyl-Fefferman-Graham Gauge
• Noether currents of locally equivalent symmetries
• Closing the gaps in quantum space and time: Conformally augmented gauge structure of gravitation
• Lambda Models From Chern-Simons Theories
• Emergent gauge symmetries: Yang-Mills theory
• Spacetime quantum and classical mechanics with dynamical foliation
• Gauge Invariant Derivation of Zeroth and First Laws of Black Hole Thermodynamics
• Noether's theorems and conserved currents in gauge theories in the presence of fixed fields
• Symmetric space $\lambda$-model exchange algebra from 4d holomorphic Chern-Simons theory
• Consistency Problems of Conformal Killing Gravity
• Improvement of a conserved current density versus adding a total derivative to a Lagrangian density
• Scale invariance and constants of motion
• First order flow equations for nonextremal black holes in AdS (super)gravity
• Energy-momentum tensor and duality symmetry of linearized gravity in the Fierz formalism
• Optimally scrambling chiral spin-chain with effective black hole geometry
• The stepwise path integral of the relativistic point particle
• Non-relativistic transport from frame-indifferent kinetic theory
• Canonical aspects of pregeometric vector-based first order gauge theory
• Classical double copy in the black hole mini-superspace
• Conserved charges for rational electromagnetic knots
• Trace and diffeomorphism anomalies of the classical Liouville theory, Virasoro algebras, Weyl-gauging and all that
• 1/c deformations of AdS${}_3$ boundary conditions and the Dym hierarchy
• AdS gravity, SO(2,d) gauge theory and Holography
• Large Gauge Symmetries of an Asymptotically de Sitter Horizon: An Extended First Law of Thermodynamics
• The chiral SYK model in three-dimensional holography
• Entanglement, anomalies and Mathisson's helices
• On the Meaning of Localization in Non-Local Quantum Field Theory and On the Limits of a Space-Time Description and the Physical Meaning of Phase Space in a Nonlocal Continuum
• Quantised helicity in optical media
• Radial canonical $\Lambda <0$ gravity
• Non-Abelian gauge theories invariant under diffeomorphisms