Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Acknowledgements
- Introduction
- Part I General Properties of Fields; Scalars and Gauge Fields
- 1 Short Review of Classical Mechanics
- 2 Symmetries, Groups, and Lie algebras; Representations
- 3 Examples: The Rotation Group and SU(2)
- 4 Review of Special Relativity: Lorentz Tensors
- 5 Lagrangeans and the Notion of Field; Electromagnetism as a Field Theory
- 6 Scalar Field Theory, Origins, and Applications
- 7 Nonrelativistic Examples:WaterWaves and Surface Growth
- 8 Classical Integrability: Continuum Limit of Discrete, Lattice, and Spin Systems
- 9 Poisson Brackets for Field Theory and Equations of Motion: Applications
- 10 Classical Perturbation Theory and Formal Solutions to the Equations of Motion
- 11 Representations of the Lorentz Group
- 12 Statistics, Symmetry, and the Spin-Statistics Theorem
- 13 Electromagnetism and the Maxwell Equation; Abelian Vector Fields; Proca Field
- 14 The Energy-Momentum Tensor
- 15 Motion of Charged Particles and ElectromagneticWaves; Maxwell Duality
- 16 The Hopfion Solution and the Hopf Map
- 17 Complex Scalar Field and Electric Current: Gauging a Global Symmetry
- 18 The Noether Theoremand Applications
- 19 Nonrelativistic and Relativistic Fluid Dynamics: Fluid Vortices and Knots
- Part II Solitons and Topology; Non-Abelian Theory
- Part III Other Spins or Statistics; General Relativity
- References
- Index
18 - The Noether Theoremand Applications
from Part I - General Properties of Fields; Scalars and Gauge Fields
Published online by Cambridge University Press: 04 March 2019
Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Acknowledgements
- Introduction
- Part I General Properties of Fields; Scalars and Gauge Fields
- 1 Short Review of Classical Mechanics
- 2 Symmetries, Groups, and Lie algebras; Representations
- 3 Examples: The Rotation Group and SU(2)
- 4 Review of Special Relativity: Lorentz Tensors
- 5 Lagrangeans and the Notion of Field; Electromagnetism as a Field Theory
- 6 Scalar Field Theory, Origins, and Applications
- 7 Nonrelativistic Examples:WaterWaves and Surface Growth
- 8 Classical Integrability: Continuum Limit of Discrete, Lattice, and Spin Systems
- 9 Poisson Brackets for Field Theory and Equations of Motion: Applications
- 10 Classical Perturbation Theory and Formal Solutions to the Equations of Motion
- 11 Representations of the Lorentz Group
- 12 Statistics, Symmetry, and the Spin-Statistics Theorem
- 13 Electromagnetism and the Maxwell Equation; Abelian Vector Fields; Proca Field
- 14 The Energy-Momentum Tensor
- 15 Motion of Charged Particles and ElectromagneticWaves; Maxwell Duality
- 16 The Hopfion Solution and the Hopf Map
- 17 Complex Scalar Field and Electric Current: Gauging a Global Symmetry
- 18 The Noether Theoremand Applications
- 19 Nonrelativistic and Relativistic Fluid Dynamics: Fluid Vortices and Knots
- Part II Solitons and Topology; Non-Abelian Theory
- Part III Other Spins or Statistics; General Relativity
- References
- Index
Summary
We study the Noether theorem, which relates a conserved charge to any global symmetry, and its applications. We define the Noether current and charge, and the general form of the Noether procedure for “gauging” the global symmetry, as well as the ambiguity of the Noether current. We apply it to translations, the Abelian U(1) symmetry, the O(N) model, and vectors in the fundamental of O(N). We define the Noether charge as an integral in terms of fields, and the transformation of the fields as the Poisson bracket of the charge with the fields.
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- Classical Field Theory , pp. 165 - 171Publisher: Cambridge University PressPrint publication year: 2019