Multivalued fields in condensed matter, electromagnetism, and gravitation

Preface
	Notes and References
Contents
List of Figures
	4.1 Infinitesimally thin closed current loop L and magnetic field
	4.2 Single- and multi-valued definitions of arctanϕ
	5.1 Specific heat of superfluid 4He
	5.2 Energies of the elementary excitations in superfluid 4He
	5.3 Rotons join side by side to form surfaces whose boundary appears as a large vortex loop
	5.4 Vortex loops in XY-model for different β = 1/kBT
	5.5 Lattice Yukawa potential at the origin and the associated Tracelog
	5.6 Specific heat of Villain model in three dimensions
	5.7 Critical temperature of a loop gas with Yukawa interactions
	5.8 Specific heat of superconducting aluminum
	5.9 Potential for the order parameter ρ with cubic term
	5.10 Phase diagram of a two-dimensional layer of superfluid 4He
	5.11 Order parameter ¯ρ = |φ|/|φ0| around a vortex line of strength n = 1, 2, 3, . . . as a function of the reduced distance ¯r = r/ξ
	7.1 Energy gap of superconductor as a function of temperature
	7.2 Energies of the low-energy excitations in superconductor
	7.3 Contour plot of zeros for energy eigenvalues in superconductor
	7.4 Temperature behavior of superfluid density ρs/ρ = φ(Δ/T) (Yoshida function) and the gap function ¯ρs/ρ = ¯π(Δ/T)
	7.5 Temperature behavior of the functions governing the kinetic term of the pair field in the BCS superconductor
	7.6 Spatial variation of order parameter ρ and magnetic field H in the neighborhood of a planar domain wall between normal and superconducting phases N and S
	7.7 Order parameter ρ and magnetic field H for a vortex line
	7.8 Critical field Hc1 as a function of the parameter κ
	7.9 Lines of equal size of order parameter ρ(x) in a typical mixed state in which the vortex lines form a hexagonal lattice
	7.10 Temperature behavior of the critical magnetic fields of a type-II superconductor
	7.11 Magnetization curve as a function of the external magnetic field
	9.1 Intrinsic point defects in a crystal
	9.2 Formation of a dislocation line (of the edge type) from a disc of missing atoms
	9.3 Naive estimate of maximal stress supported by a crystal under shear stress
	9.4 Dislocation line permitting two crystal pieces to move across each other
	9.5 Formation of a disclination from a stack of layers of missing atoms
	9.6 Grain boundary where two crystal pieces meet with different orientations
	9.7 Two typical stacking faults
	9.8 Definition of Burgers vector
	9.9 Screw dislocation which arises from tearing a crystal
	9.10 Volterra cutting and welding process
	9.11 Lattice structure at a wedge disclination
	9.12 Three different possibilities of constructing disclinations
	9.13 In the presence of a dislocation line, the displacement field is defined only modulo lattice vectors
	9.14 Geometry used in the derivation of Weingarten’s theorem
	9.15 Illustration of Volterra process
	9.16 Defect line L branching into L′ and L′′
	9.17 Generation of dislocation line from a pair of disclination lines
	10.1 Specific heat of various solids
	10.2 Specific heat of melting model
	10.3 Phase diagram in the T-ℓ-plane in two-dimensional melting
	10.4 Separation of first-order melting transition into two successive Kosterlitz-Thouless transitions in two dimensions
	11.1 Illustration of crystal planes before and after elastic distortion
	12.1 Edge dislocation in a crystal associated with a missing semi-infinite plane of atoms as a source of torsion
	12.2 Edge disclination in a crystal associated with a missing semi-infinite section of atoms as a source of curvature
	12.3 Illustration of parallel transport of a vector around a closed circuit
	12.4 Illustration of non-closure of a parallelogram after inserting an edge dislocation
	14.1 Images under holonomic and nonholonomic mapping of δ-function variation
1 Basics
	1.1 Galilean Invariance of Newtonian Mechanics
		1.1.1 Translations
		1.1.2 Rotations
		1.1.3 Galilei Boosts
		1.1.4 Galilei Group
	1.2 Lorentz Invariance of Maxwell Equations
		1.2.1 Lorentz Boosts
		1.2.2 Lorentz Group
	1.3 Infinitesimal Lorentz Transformations
		1.3.1 Generators of Group Transformations
		1.3.2 Group Multiplication and Lie Algebra
	1.4 Vector-, Tensor-, and Scalar Fields
		1.4.1 Discrete Lorentz Transformations
		1.4.2 Poincar´e group
	1.5 Differential Operators for Lorentz Transformations
	1.6 Vector and Tensor Operators
	1.7 Behavior of Vectors and Tensors under Finite Lorentz Transformations
		1.7.1 Rotations
		1.7.2 Lorentz Boosts
		1.7.3 Lorentz Group
	1.8 Relativistic Point Mechanics
	1.9 Quantum Mechanics
	1.10 Relativistic Particles in Electromagnetic Field
	1.11 Dirac Particles and Fields
	1.12 Energy-Momentum Tensor
		1.12.1 Point Particles
		1.12.2 Perfect Fluid
		1.12.3 Electromagnetic Field
	1.13 Angular Momentum and Spin
	1.14 Spacetime-Dependent Lorentz Transformations
		1.14.1 Angular Velocities
		1.14.2 Angular Gradients
	Appendix 1 A Tensor Identities
		1 A.1 Product Formulas
		1 A.2 Determinants
		1 A.3 Expansion of Determinants
	Notes and References
2 Action Approach
	2.1 General Particle Dynamics
	2.2 Single Relativistic Particle
	2.3 Scalar Fields
		2.3.1 Locality
		2.3.2 Lorenz Invariance
		2.3.3 Field Equations
		2.3.4 Plane Waves
		2.3.5 Schr¨odinger Quantum Mechanics as Nonrelativistic Limit
		2.3.6 Natural Units
		2.3.7 Hamiltonian Formalism
		2.3.8 Conserved Current
	2.4 Maxwell’s Equation from Extremum of Field Action
		2.4.1 Electromagnetic Field Action
		2.4.2 Alternative Action for Electromagnetic Field
		2.4.3 Hamiltonian of Electromagnetic Fields
		2.4.4 Gauge Invariance of Maxwell’s Theory
	2.5 Maxwell-Lorentz Action for Charged Point Particles
	2.6 Scalar Field with Electromagnetic Interaction
	2.7 Dirac Fields
	2.8 Quantization
	Notes and References
3 Continuous Symmetries and Conservation Laws Noether’s Theorem
	3.1 Continuous Symmetries and Conservation Laws
		3.1.1 Group Structure of Symmetry Transformations
		3.1.2 Substantial Variations
		3.1.3 Conservation Laws
		3.1.4 Alternative Derivation of Conservation Laws
	3.2 Time Translation Invariance and Energy Conservation
	3.3 Momentum and Angular Momentum
		3.3.1 Translational Invariance in Space
		3.3.2 Rotational Invariance
		3.3.3 Center-of-Mass Theorem
		3.3.4 Conservation Laws from Lorentz Invariance
	3.4 Generating the Symmetries
	3.5 Field Theory
		3.5.1 Continuous Symmetry and Conserved Currents
		3.5.2 Alternative Derivation
		3.5.3 Local Symmetries
	3.6 Canonical Energy-Momentum Tensor
		3.6.1 Electromagnetism
		3.6.2 Dirac Field
	3.7 Angular Momentum
	3.8 Four-Dimensional Angular Momentum
	3.9 Spin Current
		3.9.1 Electromagnetic Fields
		3.9.2 Dirac Field
	3.10 Symmetric Energy-Momentum Tensor
	3.11 Internal Symmetries
		3.11.1 U(1)-Symmetry and Charge Conservation
		3.11.2 Broken Internal Symmetries
	3.12 Generating the Symmetry Transformations for Quantum Fields
	3.13 Energy-Momentum Tensor of Relativistic Massive Point Particle
	3.14 Energy-Momentum Tensor of Massive Charged Particle in Electromagnetic Field
	Notes and References
4 Multivalued Gauge Transformations in Magnetostatics
	4.1 Vector Potential of Current Distribution
	4.2 Multivalued Gradient Representation of Magnetic Field
	4.3 Generating Magnetic Fields by Multivalued Gauge Transformations
	4.4 Magnetic Monopoles
	4.5 Minimal Magnetic Coupling of Particles from Multivalued Gauge Transformations
	4.6 Equivalence of Multivalued Scalar and Singlevalued Vector Fields
	4.7 Multivalued Field Theory of Magnetic Monopoles and Electric Currents
	Notes and References
5 Multivalued Fields in Superfluids and Superconductors
	5.1 Superfluid Transition
		5.1.1 Configuration Entropy
		5.1.2 Origin of Massless Excitations
		5.1.3 Vortex Density
		5.1.4 Partition Function
		5.1.5 Continuum Derivation of Interaction Energy
		5.1.6 Physical Jumping Surfaces
		5.1.7 Canonical Representation of Superfluid
		5.1.8 Yukawa Loop Gas
		5.1.9 Gauge Field of Superflow
		5.1.10 Disorder Field Theory
	5.2 Phase Transition in Superconductor
		5.2.1 Ginzburg-Landau Theory
		5.2.2 Disorder Theory of Superconductor
	5.3 Order versus Disorder Parameter
		5.3.1 Superfluid 4He
		5.3.2 Superconductor
	5.4 Order of Superconducting Phase Transition and Tricritical Point
		5.4.1 Fluctuation Regime
		5.4.2 First- or Second-Order Transition?
		5.4.3 Partition Function of Superconductor with Vortex Lines
		5.4.4 First-Order Regime
		5.4.5 Vortex Line Origin of Second-Order Transition
		5.4.6 Tricritical Point
		5.4.7 Disorder Theory
	5.5 Vortex Lattices
	Appendix 5 A Single Vortex Line in Superfluid
	Notes and References
6 Dynamics of Superfluids
	6.1 Hydrodynamic Description of Superfluid
	6.2 Velocity of Second Sound
	6.3 Vortex-Electromagnetic Fields
	6.4 Simple Example
	6.5 Eckart Theory of Ideal Quantum Fluids
	6.6 Rotating Superfluid
	Notes and References
7 Dynamics of Charged Superfluid and Superconductor
	7.1 Hydrodynamic Description of Charged Superfluid
	7.2 London Theory of Charged Superfluid
	7.3 Including Vortices in London Equations
	7.4 Hydrodynamic Description of Superconductor
	Appendix 7 A Excitation Spectrum of Superconductor
		7 A.1 Gap Equation
		7 A.2 Action of Quadratic Fluctuations
		7 A.3 Long-Wavelength Excitations at Zero Temperature
		7 A.4 Long-Wavelength Excitations at Nonzero Temperature
		7 A.5 Bending Energies of Order Field
		7 A.6 Kinetic Terms of Pair Field at Nonzero Temperature
	Appendix 7 B Properties of Ginzburg-Landau Theory of Superconductivity
		7 B.1 Critical Magnetic Field
		7 B.2 Two Length Scales and Type I or II Superconductivity
		7 B.3 Single Vortex Line and Critical Field Hc1
		7 B.4 Critical Field Hc2 where Superconductivity is Destroyed
	Notes and References
8 Relativistic Magnetic Monopoles and Electric Charge Confinement
	8.1 Monopole Gauge Invariance
	8.2 Charge Quantization
	8.3 Electric and Magnetic Current-Current Interactions
	8.4 Dual Gauge Field Representation
	8.5 Monopole Gauge Fixing
	8.6 Quantum Field Theory of Spinless Electric Charges
	8.7 Theory of Magnetic Charge Confinement
	8.8 Second Quantization of the Monopole Field
	8.9 Quantum Field Theory of Electric Charge Confinement
	Notes and References
9 Multivalued Mapping from Ideal Crystals to Crystals with Defects
	9.1 Defects
	9.2 Dislocation Lines and Burgers Vector
	9.3 Disclination Lines and Frank Vector
	9.4 Defect Lines with Infinitesimal Discontinuities in Continuous Media
	9.5 Multivaluedness of Displacement Field
	9.6 Smoothness Properties of Displacement Field andWeingarten’s Theorem
	9.7 Integrability Properties of Displacement Field
	9.8 Dislocation and Disclination Densities
	9.9 Mnemonic Procedure for Constructing Defect Densities
	9.10 Defect Gauge Invariance
	9.11 Branching Defect Lines
	9.12 Defect Density and Incompatibility
	9.13 Interdependence of Dislocation and Disclinations
	Notes and References
10 Defect Melting
	10.1 Specific Heat
	10.2 Elastic Energy of Solid with Defects
	Notes and References
11 Relativistic Mechanics in Curvilinear Coordinates
	11.1 Equivalence Principle
	11.2 Free Particle in General Coordinates Frame
	11.3 Minkowski Geometry formulated in General Coordinates
		11.3.1 Local Basis tetrads
		11.3.2 Vector- and Tensor Fields in Minkowski Coordinates
		11.3.3 Vector- and Tensor Fields in General Coordinates
		11.3.4 Affine Connections and Covariant Derivatives
	11.4 Torsion Tensor
	11.5 Covariant Time Derivative and Acceleration
	11.6 Curvature Tensor as Covariant Curl of Affine Connection
	11.7 Riemann Curvature Tensor
	Appendix 11 A Curvilinear Versions of Levi-Civita Tensor
	Notes and References
12 Torsion and Curvature from Defects
	12.1 Multivalued Infinitesimal Coordinate Transformations
	12.2 Examples for Nonholonomic Coordinate Transformations
		12.2.1 Dislocation
		12.2.2 Disclination
	12.3 Differential-Geometric Properties of Affine Spaces
		12.3.1 Integrability of Metric and Affine Connection
		12.3.2 Local Parallelism
	12.4 Circuit Integrals in Affine Spaces with Curvature and Torsion
		12.4.1 Closed-Contour Integral over Parallel Vector Field
		12.4.2 Closed-Contour Integral over Coordinates
		12.4.3 Closure Failure and Burgers Vector
		12.4.4 Alternative Circuit Integral for Curvature
		12.4.5 Parallelism in World Crystal
	12.5 Bianchi Identities for Curvature and Torsion Tensors
	12.6 Special Coordinates in Riemann Spacetime
		12.6.1 Geodesic Coordinates
		12.6.2 Canonical Geodesic Coordinates
		12.6.3 Harmonic Coordinates
		12.6.4 Coordinates with det(gμν) = 1
		12.6.5 Orthogonal Coordinates
	12.7 Number of Independent Components of Rμνλκ and Sμνλ
		12.7.1 Two Dimensions
		12.7.2 Thee Dimensions
		12.7.3 Four or More Dimensions
	Notes and References
13 Curvature and Torsion from Embedding
	13.1 Spacetimes with Constant Curvature
	13.2 Basis Vectors
	13.3 Torsion
	Notes and References
14 Multivalued Mapping Principle
	14.1 Motion of Point Particle
		14.1.1 Classical Action Principle for Spaces with Curvature
		14.1.2 Autoparallel Trajectories in Spaces with Torsion
		14.1.3 Equations of Motion For Spin
		14.1.4 Special Properties of Gradient Torsion
	14.2 Autoparallel Trajectories from Embedding
		14.2.1 Special Role of Autoparallels
		14.2.2 Gauss Principle of Least Constraint
	14.3 Maxwell-Lorentz Orbits as Autoparallel Trajectories
	14.4 Bargmann-Michel-Telegdi Equation from Torsion
	Notes and References
15 Field Equations of Gravitation
	15.1 Invariant Action
	15.2 Energy-Momentum Tensor and Spin Density
	15.3 Symmetric Energy-Momentum Tensor and Defect Density
	Notes and References
16 Minimally Coupled Fields of Integer Spin
	16.1 Scalar Fields in Riemann-Cartan Space
	16.2 Electromagnetism in Riemann-Cartan Space
	Notes and References
17 Particles with Half-Integer Spin
	17.1 Local Lorentz Invariance and Anholonomic Coordinates
		17.1.1 Nonholonomic Image of Dirac Action
		17.1.2 Vierbein Fields
		17.1.3 Local Inertial Frames
		17.1.4 Difference between Vierbein and Multivalued Tetrad Fields
		17.1.5 Covariant Derivatives in Intermediate Basis
	17.2 Dirac Action in Riemann-Cartan Space
	17.3 Ricci Identity
	17.4 Alternative Form of Coupling
	17.5 Invariant Action for Vector Fields
	17.6 Verifying Local Lorentz Invariance
	17.7 Field Equations with Spinning Matter
	Notes and References
18 Covariant Conservation Law
	18.1 Spin Density
	18.2 Energy-Momentum Density
	18.3 Covariant Derivation of Conservation Laws
	18.4 Matter with Integer Spin
	18.5 Relations between Conservation Laws and Bianchi Identities
	18.6 Particle Trajectories from Energy-Momentum Conservation
	Notes and References
19 Gravitation of Spinning Matter as a Gauge Theory
	19.1 Local Lorentz Transformations
	19.2 Local Translations
	Notes and References
20 Evanescent Properties of Torsion in Gravity
	20.1 Local Four-Fermion Interaction due to Torsion
	20.2 No Need for Torsion in Gravity
	20.3 Scalar Fields
		20.3.1 Only Spin-1/2 Sources
	20.4 Modified Energy-Momentum Conservation Law
		20.4.1 Solution for Gradient Torsion
		20.4.2 Gradient Torsion coupled to Scalar Fields
		20.4.3 New Scalar Product
		20.4.4 Self-Interacting Higgs Field
	20.5 Summary
	Notes and References
21 Teleparallel Theory of Gravitation
	21.1 Torsion Form of Einstein Action
	21.2 Schwarzschild Solution
	Notes and References
22 Emerging Gravity
	22.1 Gravity in the World Crystal
	22.2 New Symmetry of Einstein Gravity
	22.3 Gravity Emerging from Fluctuations of Matter and Radiation
	Notes and References
Index