Classical Field Theory 1 Hardcover 1st by Horatiu Nastase 1108753213 9781108477017

ISBN 10: 1108753213

ISBN 13:9781108477017

Author: Horatiu Nastase

Classical field theory predicts how physical fields interact with matter, and is a logical precursor to quantum field theory. This introduction focuses purely on modern classical field theory, helping graduates and researchers build an understanding of classical field theory methods before embarking on future studies in quantum field theory. It describes various classical methods for fields with negligible quantum effects, for instance electromagnetism and gravitational fields. It focuses on solutions that take advantage of classical field theory methods as opposed to applications or geometric properties. Other fields covered includes fermionic fields, scalar fields and Chern–Simons fields. Methods such as symmetries, global and local methods, Noether theorem and energy momentum tensor are also discussed, as well as important solutions of the classical equations, in particular soliton solutions.

1 Short Review of Classical Mechanics

1.1 A Note on Conventions

1.2 Lagrangean and Equations of Motion

1.3 Systems with Constraints

1.4 Canonically Conjugate Momentum and Conservation Laws

1.5 The Hamiltonian Formalism

1.6 Canonical Transformations

1.7 Poisson Brackets

1.8 Hamilton–Jacobi Theory

Further Reading

Exercises

2 Symmetries, Groups, and Lie algebras; Representations

2.1 Groups and Invariance in the Simplest Case

2.2 Generalizations: Cyclic Groups and Their Representations

2.3 Lie Groups and Abelian Lie Group Invariance

2.4 Lie Algebra

2.5 Representations for Lie Groups

Further Reading

Exercises

3 Examples: The Rotation Group and SU(2)

3.1 Rotational Invariance

3.2 Example: SO(2)

3.3 SO(3) and Its General Parametrization

3.4 Isomorphism of SO(3) with SU(2) (Modulo Z[sub(2)])

3.5 Construction of Representations of SU(2) and Invariant Theories

Further Reading

Exercises

4 Review of Special Relativity: Lorentz Tensors

4.1 Special Relativity and the Lorentz Group

4.2 Characterizing the Lorentz Group

4.3 Kinematics of Special Relativity

4.4 Dynamics of Special Relativity

4.5 Relativistically Covariant Lagrangeans

Further Reading

Exercises

5 Lagrangeans and the Notion of Field; Electromagnetism as a Field Theory

5.1 Fields and Lagrangean Densities

5.2 Maxwell’s Equations in Covariant Formalism

5.3 Euler–Lagrange Equations in Field Theory

5.4 Lagrangean for the Gauge Field A[sub(μ)]

5.5 Adding Sources to Maxwell’s Equations and Their Lagrangean

Further Reading

Exercises

6 Scalar Field Theory, Origins, and Applications

6.1 Scalar Fields and Their Equations of Motion

6.2 Constructing Lagrangeans

6.3 Specific Models (Applications)

Further Reading

Exercises

7 Nonrelativistic Examples: Water Waves and Surface Growth

7.1 Nonrelativistic Scalar Action

7.2 Hydrodynamics

7.3 Water Waves

7.4 Korteweg–de Vries (KdV) Equation and Solitonic Water Wave

7.5 The Kuramoto–Sivashinsky (KS) Equation

7.6 Growth of Surfaces

Further Reading

Exercises

8 Classical Integrability: Continuum Limit of Discrete, Lattice, and Spin Systems

8.1 Classical Integrability

8.2 Examples of Integrable Systems

8.3 Classical Integrable Fields

8.4 Spin Systems and Discretization

Further Reading

Exercises

9 Poisson Brackets for Field Theory and Equations of Motion: Applications

9.1 Symplectic Formulation

9.2 Generalization to Fields

9.3 Examples

Further Reading

Exercises

10 Classical Perturbation Theory and Formal Solutions to the Equations of Motion

10.1 General Formalism

10.2 Polynomial Potential

10.3 Diagrammatic Procedure

10.4 Noncanonical Case: Nonlinear Sigma Model

Further Reading

Exercises

11 Representations of the Lorentz Group

11.1 Characterization of the Lie Algebra of the Lorentz Group

11.2 The Poincaré Group

11.3 The Universal Cover of the Lorentz Group

11.4 The Wigner Method and the Little Group

11.5 Representations in Terms of Fields

11.6 The Double Cover of SO(3,1) as Sl(2,C) and the Definition of Spinors

Further Reading

Exercises

12 Statistics, Symmetry, and the Spin-Statistics Theorem

12.1 Statistics

12.2 Rotation Matrices in Different Lorentz Representations

12.3 The Spin-Statistics Theorem

12.4 Symmetries

Further Reading

Exercises

13 Electromagnetism and the Maxwell Equation; Abelian Vector Fields; Proca Field

13.1 Electromagnetism as a U(1) Gauge Field

13.2 Electromagnetism in p-Form Language

13.3 General p-Form Fields

13.4 The Proca Field

Further Reading

Exercises

14 The Energy-Momentum Tensor

14.1 Defining the Energy-Momentum Tensor

14.2 Conservation Equations

14.3 An Ambiguity in T[sub(μν)] and Ways to Fix It

14.4 Interpretation of T[sup(μν)]

14.5 The Belinfante Tensor

14.6 Example: The Electromagnetic Field

Further Reading

Exercises

15 Motion of Charged Particles and Electromagnetic Waves; Maxwell Duality

15.1 Set of Static Charges

15.2 Uniformly Moving Charges

15.3 Electrostatics Methods

15.4 The Electric and Magnetic Fields Away from a Collection of Particles: Multipole Expansions

15.5 Electromagnetic Waves

15.6 Arbitrary Moving Charges

15.7 Generation of Electromagnetic Waves by Dipoles

15.8 Maxwell Duality

Further Reading

Exercises

16 The Hopfion Solution and the Hopf Map

16.1 Sourceless Maxwell’s Equations and Conserved “Helicities”

16.2 Hopf Map and Hopf Index

16.3 Bateman’s Construction

16.4 The Hopfion Solution

16.5 More General Solutions and Properties

Further Reading

Exercises

17 Complex Scalar Field and Electric Current: Gauging a Global Symmetry

17.1 Complex Scalar Field

17.2 Electric Current and Charge

17.3 Gauging a Global Symmetry

Further Reading

Exercises

18 The Noether Theorem and Applications

18.1 Setup

18.2 Noether’s Theorem

18.3 The Noether Procedure

18.4 A Subtlety and a General Form: Extra Terms in the Current

18.5 Applications

18.6 The Charge as a Function of Fields

Further Reading

Exercises

19 Nonrelativistic and Relativistic Fluid Dynamics: Fluid Vortices and Knots

19.1 Ideal Fluid Equations

19.2 Viscous Fluid and Navier–Stokes Equation

19.3 Relativistic Generalization of Fluid Dynamics

19.4 Vorticity and Helicity of Ideal Fluid

19.5 Small Fluctuations and Fluid Waves

19.6 Fluid Vortices and Knots

Further Reading

Exercises

Part II Solitons and Topology; Non-Abelian Theory

20 Kink Solutions in [phi[sup(4)]] and Sine-Gordon, Domain Walls and Topology

20.1 Setup

20.2 Analysis of Classical Solutions

20.3 Topology

20.4 Domain Walls

20.5 Sine-Gordon System

Further Reading

Exercises

21 The Skyrmion Scalar Field Solution and Topology

21.1 Defining the Skyrme Model

21.2 Analysis of the Model

21.3 Topological Numbers

21.4 Hedgehog Configuration and Skyrmion Solution

21.5 Generalizations of the Skyrme Model

Further Reading

Exercises

22 Field Theory Solitons for Condensed Matter: The XY and Rotor Model, Spins, Superconductivity, and

22.1 The XY and Rotor Model

22.2 Field Theory and Vortices

22.3 Kosterlitz–Thouless Phase Transition

22.4 Landau–Ginzburg Model

Further Reading

Exercises

23 Radiation of a Classical Scalar Field: The Heisenberg Model

23.1 Scalar Models

23.2 Shock Wave Solutions for Relativistic Sources: The Heisenberg Model

23.3 Radiation from Scalar Waves

Further Reading

Exercises

24 Derrick’s Theorem, Bogomolnyi Bound, the Abelian-Higgs System, and Symmetry Breaking

24.1 Derrick’s Theorem

24.2 Bogomolnyi Bound

24.3 Abelian-Higgs System and Symmetry Breaking

24.4 Unitary Gauge

24.5 Non-Abelian Higgs System

Further Reading

Exercises

25 The Nielsen-Olesen Vortex, Topology and Applications

25.1 Setup

25.2 Bogomolnyi Bound and BPS Limit

25.3 BPS Equations and the Vortex Solution

25.4 Applications

Further Reading

Exercises

26 Non-Abelian Gauge Theory and the Yang–Mills Equation

26.1 Non-Abelian Gauge Groups

26.2 Coupling the Gauge Field to Other Fields

26.3 Pure Yang–Mills Theory

26.4 The Yang–Mills Equation

Further Reading

Exercises

27 The Dirac Monopole and Dirac Quantization

27.1 Dirac Monopole from Maxwell Duality

27.2 Gauge Fields on Patches

27.3 Topology and Dirac Quantization

27.4 Dirac String Singularity and Dirac Quantization from It

27.5 Dirac Quantization from Semiclassical Nonrelativistic Considerations

Further Reading

Exercises

28 The ’t Hooft–Polyakov Monopole Solution and Topology

28.1 Setup for Georgi–Glashow Model

28.2 Vacuum Manifold and ’t Hooft–Polyakov Solution

28.3 Topology of the Monopole

28.4 Bogomolnyi Bound and BPS Monopole

28.5 Topology for the BPS and the Dirac Monopole

Further Reading

Exercises

29 The BPST-’t Hooft Instanton Solution and Topology

29.1 Setup for Euclidean Yang–Mills and Self-Duality Condition

29.2 Chern Forms

29.3 The Instanton Solution

29.4 Quantum Interpretation of the Instanton

Further Reading

Exercises

30 General Topology and Reduction on an Ansatz

30.1 Topological Classification of Scalars and Homotopy Groups

30.2 Gauge Field Classifications

30.3 Derrick’s Argument with Gauge Fields

30.4 Reduction on an Ansatz

Further Reading

Exercises

31 Other Soliton Types. Nontopological Solitons: Q-Balls; Unstable Solitons: Sphalerons

31.1 Q-Balls

31.2 Sphalerons

31.3 Sphaleron on a Circle

31.4 Other Sphalerons

Further Reading

Exercises

32 Moduli Space; Soliton Scattering in Moduli Space Approximation; Collective Coordinates

32.1 Moduli Space and Soliton Scattering in Moduli Space Approximation

32.2 Example: ANO Vortices in the Abelian-Higgs Model

32.3 Collective Coordinates and Their Quantization

32.4 Change of Basis in a Hamiltonian and Quantization

32.5 Application to Collective Coordinate Quantization

Further Reading

Exercises

Part III Other Spins or Statistics; General Relativity

33 Chern–Simons Terms: Emergent Gauge Fields, the Quantum Hall Effect (Integer and Fractional), An

33.1 Chern–Simons Gauge Field

33.2 Topological Response Theory and the Quantum Hall Effect

33.3 Emergent (Statistical) Chern–Simons Gauge Field in Solids

33.4 Anyons and Fractional Statistics

33.5 Anyons and the Fractional Quantum Hall Effect (FQHE)

Further Reading

Exercises

34 Chern–Simons and Self-Duality in Odd Dimensions, Its Duality to Topologically Massive Theory an

34.1 Vector Theories in 2+1 Dimensions

34.2 Self-Duality in Odd Dimensions

34.3 Duality of Self-Dual Action and Topologically Massive Action

34.4 Duality in 1+1 Dimensions: “T-Duality”

34.5 Comments on Chern–Simons Theories in Higher Dimensions

Further Reading

Exercises

35 Particle–Vortex Duality in Three Dimensions, Particle–String Duality in Four Dimensions, and

35.1 Maxwell Duality in 3+1 Dimensions

35.2 Particle–Vortex Duality in 2+1 Dimensions

35.3 Particle–String Duality in 3+1 Dimensions

35.4 Poincaré Duality and Applications to 3+1 Dimensions

Further Reading

Exercises

36 Fermions and Dirac Spinors

36.1 Spinors, Dirac and Weyl

36.2 Gamma Matrices

36.3 Majorana Spinors

Further Reading

Exercises

37 The Dirac Equation and Its Solutions

37.1 Lorentz Invariant Lagrangeans

37.2 The Dirac Equation

37.3 Solutions of the Dirac Equation

37.4 Normalizations

Further Reading

Exercises

38 General Relativity: Metric and General Coordinate Invariance

38.1 Intrinsically Curved Spaces and General Relativity

38.2 Einstein’s Theory of General Relativity

38.3 Kinematics

38.4 Motion of Free Particles

Further Reading

Exercises

39 The Einstein Action and the Einstein Equation

39.1 Riemann Tensor and Curvature

39.2 Turning Special Relativity into General Relativity and Einstein–Hilbert Action

39.3 Einstein’s Equations

39.4 Examples of Energy-Momentum Tensor

39.5 Interpretation of the Einstein Equation

Further Reading

Exercises

40 Perturbative Gravity: Fierz-Pauli Action, de Donder Gauge and Other Gauges, Gravitational Waves

40.1 Perturbative Gravity and Fierz-Pauli Action

40.2 Gauge Invariance, de Donder Gauge and Other Gauges

40.3 Gravitational Waves

40.4 Exact Cylindrical Gravitational Wave (Einstein–Rosen)

Further Reading

Exercises

41 Nonperturbative Gravity: The Vacuum Schwarzschild Solution

41.1 Newtonian Limit

41.2 Ansatz and Equations of Motion

41.3 Final Solution of Equations of Motion

Further Reading

Exercises

42 Deflection of Light by the Sun and Comparison with General Relativity

42.1 Motion of Light As Motion in a Medium with Small, Position-Dependent Index of Refraction

42.2 Formal Derivation Using the Hamilton–Jacobi Equation

42.3 Comparison with Special Relativity

Further Reading

Exercises

43 Fully Linear Gravity: Parallel Plane (pp) Waves and Gravitational Shock Wave Solutions

43.1 PP Waves

43.2 The Penrose Limit

43.3 Gravitational Shock Waves in Flat Space

43.4 Gravitational Shock Waves in Other Backgrounds

43.5 The Khan–Penrose Interacting Solution

Further Reading

Exercises

44 Dimensional Reduction: The Domain Wall, Cosmic String, and BTZ Black Hole Solutions

44.1 The Domain Wall: Ansatz

44.2 The Domain Wall: Einstein’s Equation

44.3 The Domain Wall Solutions

44.4 Cosmic String: Ansatz

44.5 Cosmic String: Einstein’s Equations

44.6 Cosmic String Solution via Dimensional Reduction

44.7 Cosmic String: Alternative Weak Field Derivation

44.8 The BTZ Black Hole Solution in 2+1 Dimensions and Anti–de Sitter Space

Further Reading

Exercises

45 Time-Dependent Gravity: The Friedmann-Lemaître-Robertson-Walker (FLRW) Cosmological Solution

45.1 Metric Ansatz

45.2 Christoffel Symbols and Ricci Tensors

45.3 Solution to the Einstein’s Equations

Further Reading

Exercises

46 Vielbein-Spin Connection Formulation of General Relativity and Gravitational Instantons

46.1 Vielbein-Spin Connection Formulation of General Relativity

46.2 Taub-NUT Solutions

46.3 Hawking’s Taub-NUT Gravitational Instanton

46.4 The Eguchi–Hanson Metric and Yang–Mills-Like Ansatz

46.5 Instanton Ansätze and Solutions

46.6 Gibbons–Hawking Multi-Instanton Solution