Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Acknowledgements
- Introduction
- Part I General Properties of Fields; Scalars and Gauge Fields
- Part II Solitons and Topology; Non-Abelian Theory
- 20 Kink Solutions in ϕ4 and Sine-Gordon, Domain Walls and Topology
- 21 The Skyrmion Scalar Field Solution and Topology
- 22 Field Theory Solitons for Condensed Matter: The XY and Rotor Model, Spins, Superconductivity, and the KT Transition
- 23 Radiation of a Classical Scalar Field: The Heisenberg Model
- 24 Derrick’s Theorem, Bogomolnyi Bound, theAbelian-Higgs System, andSymmetryBreaking
- 25 The Nielsen-Olesen Vortex, Topology and Applications
- 26 Non-Abelian Gauge Theory and the Yang–Mills Equation
- 27 The Dirac Monopole and Dirac Quantization
- 28 The ’t Hooft–Polyakov Monopole Solution and Topology
- 29 The BPST-’t Hooft Instanton Solution and Topology
- 30 General Topology and Reduction on an Ansatz
- 31 Other Soliton Types. Nontopological Solitons: Q-Balls; Unstable Solitons: Sphalerons
- 32 Moduli Space; Soliton Scattering in Moduli Space Approximation; Collective Coordinates
- Part III Other Spins or Statistics; General Relativity
- References
- Index
30 - General Topology and Reduction on an Ansatz
from Part II - Solitons and Topology; Non-Abelian Theory
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
- Part II Solitons and Topology; Non-Abelian Theory
- 20 Kink Solutions in ϕ4 and Sine-Gordon, Domain Walls and Topology
- 21 The Skyrmion Scalar Field Solution and Topology
- 22 Field Theory Solitons for Condensed Matter: The XY and Rotor Model, Spins, Superconductivity, and the KT Transition
- 23 Radiation of a Classical Scalar Field: The Heisenberg Model
- 24 Derrick’s Theorem, Bogomolnyi Bound, theAbelian-Higgs System, andSymmetryBreaking
- 25 The Nielsen-Olesen Vortex, Topology and Applications
- 26 Non-Abelian Gauge Theory and the Yang–Mills Equation
- 27 The Dirac Monopole and Dirac Quantization
- 28 The ’t Hooft–Polyakov Monopole Solution and Topology
- 29 The BPST-’t Hooft Instanton Solution and Topology
- 30 General Topology and Reduction on an Ansatz
- 31 Other Soliton Types. Nontopological Solitons: Q-Balls; Unstable Solitons: Sphalerons
- 32 Moduli Space; Soliton Scattering in Moduli Space Approximation; Collective Coordinates
- Part III Other Spins or Statistics; General Relativity
- References
- Index
Summary
A general topological classification accross theories and dimensions is described. First for scalars, then for gauge fields, in terms of various homotopy groups, we classify all the solitons described.A generalization of Derrick's argument in the presence of gauge fields is given. We then show, with examples, how to reduce the theory in the presence of a specific ansatz for a solution.
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- Classical Field Theory , pp. 278 - 284Publisher: Cambridge University PressPrint publication year: 2019