Multivalued Fields

In Condensed Matter, Electromagnetism, And Gravitation

By: Hagen Kleinert

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Paperback | 12 February 2008

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524 Pages

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This book lays the foundations of the theory of fluctuating multivalued fields with numerous applications. Most prominent among these are phenomena dominated by the statistical mechanics of line-like objects, such as the phase transitions in superfluids and superconductors as well as the melting process of crystals, and the electromagnetic potential as a multivalued field that can produce a condensate of magnetic monopoles. In addition, multivalued mappings play a crucial role in deriving the physical laws of matter coupled to gauge fields and gravity with torsion from the laws of free matter. Through careful analysis of each of these applications, the book thus provides students and researchers with supplementary reading material for graduate courses on phase transitions, quantum field theory, gravitational physics, and differential geometry.

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Non-Fiction Science Physics Quantum Physics & Quantum Mechanics & Quantum Field Theory Classical Mathematics Gravity Electricity Mathematical Physics Materials & States of Matter Condensed Matter Physics including Liquid State & Solid State Physics Booktopia Publisher Services World Scientific Publishing

Preface	p. vii
Notes and References	p. xi
Basics	p. 1
Galilean Invariance of Newtonian Mechanics	p. 1
Translations	p. 2
Rotations	p. 2
Galilei Boosts	p. 3
Galilei Group	p. 3
Lorentz Invariance of Maxwell Equations	p. 3
Lorentz Boosts	p. 4
Lorentz Group	p. 6
Infinitesimal Lorentz Transformations	p. 6
Generators of Group Transformations	p. 7
Group Multiplication and Lie Algebra	p. 9
Vector-, Tensor-, and Scalar Fields	p. 11
Discret Lorentz Transformations	p. 14
Poincare group	p. 14
Differential Operators for Lorentz Transformations	p. 15
Vector and Tensor Operators	p. 16
Behavior of Vectors and Tensors under Finite Lorentz Transformations	p. 16
Rotations	p. 17
Lorentz Boosts	p. 18
Lorentz Group	p. 18
Relativistic Point Mechanics	p. 19
Quantum Mechanics	p. 22
Relativistic Particles in Electromagnetic Field	p. 24
Dirac Particles and Fields	p. 29
Energy-Momentum Tensor	p. 32
Point Particles	p. 32
Perfect Fluid	p. 34
Electromagnetic Field	p. 34
Angular Momentum and Spin	p. 36
Spacetime-Dependent Lorentz Transformations	p. 41
Angular Velocities	p. 41
Angular Gradients	p. 43
Tensor Identities	p. 43
Product Formulas	p. 44
Determinants	p. 46
Expansion of Determinants	p. 46
Notes and References	p. 46
Action Approach	p. 48
General Particle Dynamics	p. 49
Single Relativistic Particle	p. 50
Scalar Fields	p. 52
Locality	p. 52
Lorenz Invariance	p. 53
Field Equations	p. 54
Plane Waves	p. 55
Schrodinger Quantum Mechanics as Nonrelativistic Limit	p. 55
Natural Units	p. 56
Hamiltonian Formalism	p. 57
Conserved Current	p. 58
Maxwell's Equation from Extremum of Field Action	p. 59
Electromagnetic Field Action	p. 60
Alternative Action for Electromagnetic Field	p. 61
Hamiltonian of Electromagnetic Fields	p. 61
Gauge Invariance of Maxwell's Theory	p. 63
Maxwell-Lorentz Action for Charged Point Particles	p. 65
Scalar Field with Electromagnetic Interaction	p. 66
Dirac Fields	p. 67
Quantization	p. 69
Notes and References	p. 70
Continuous Symmetries and Conservation Laws. Noether's Theorem	p. 71
Continuous Symmetries and Conservation Laws	p. 71
Group Structure of Symmetry Transformations	p. 71
Substantial Variations	p. 72
Conservation Laws	p. 72
Alternative Derivation of Conservation Laws	p. 73
Time Translation Invariance and Energy Conservation	p. 75
Momentum and Angular Momentum	p. 76
Translational Invariance in Space	p. 77
Rotational Invariance	p. 77
Center-of-Mass Theorem	p. 78
Conservation Laws from Lorentz Invariance	p. 80
Generating the Symmetries	p. 82
Field Theory	p. 84
Continuous Symmetry and Conserved Currents	p. 84
Alternative Derivation	p. 85
Local Symmetries	p. 86
Canonical Energy-Momentum Tensor	p. 88
Electromagnetism	p. 90
Dirac Field	p. 91
Angular Momentum	p. 92
Four-Dimensional Angular Momentum	p. 93
Spin Current	p. 95
Electromagnetic Fields	p. 95
Dirac Field	p. 98
Symmetric Energy-Momentum Tensor	p. 100
Internal Symmetries	p. 102
U(1)-Symmetry and Charge Conservation	p. 103
Broken Internal Symmetries	p. 103
Generating the Symmetry Transformations for Quantum Fields	p. 104
Energy-Momentum Tensor of Relativistic Massive Point Particle	p. 105
Energy-Momentum Tensor of Massive Charged Particle in Electromagnetic Field	p. 107
Notes and References	p. 110
Multivalued Gauge Transformations in Magnetostatics	p. 111
Vector Potential of Current Distribution	p. 111
Multivalued Gradient Representation of Magnetic Field	p. 112
Generating Magnetic Fields by Multivalued Gauge Transformations	p. 119
Magnetic Monopoles	p. 120
Minimal Magnetic Coupling of Particles from Multivalued Gauge Transformations	p. 123
Equivalence of Multivalued Scalar and Singlevalued Vector Fields	p. 125
Multivalued Field Theory of Magnetic Monopoles and Electric Currents	p. 127
Notes and References	p. 129
Multivalued Fields in Superfluids and Superconductors	p. 131
Superfluid Transition	p. 131
Configuration Entropy	p. 133
Origin of Massless Excitations	p. 134
Vortex Density	p. 138
Partition Function	p. 138
Continuum Derivation of Interaction Energy	p. 144
Physical Jumping Surfaces	p. 145
Canonical Representation of Superfluid	p. 146
Yukawa Loop Gas	p. 150
Gauge Field of Superflow	p. 151
Disorder Field Theory	p. 153
Phase Transition in Superconductor	p. 156
Ginzburg-Landau Theory	p. 158
Disorder Theory of Superconductor	p. 160
Order versus Disorder Parameter	p. 163
Superfluid [superscript 4]He	p. 163
Superconductor	p. 168
Order of Superconducting Phase Transition and Tricritical Point	p. 174
Fluctuation Regime	p. 174
First- or Second-Order Transition?	p. 175
Partition Function of Superconductor with Vortex Lines	p. 176
First-Order Regime	p. 177
Vortex Line Origin of Second-Order Transition	p. 179
Tricritical Point	p. 180
Disorder Theory	p. 181
Vortex Lattices	p. 182
Single Vortex Line in Superfluid	p. 183
Notes and References	p. 189
Dynamics of Superfluids	p. 196
Hydrodynamic Description of Superfluid	p. 196
Velocity of Second Sound	p. 201
Vortex-Electromagnetic Fields	p. 202
Simple Example	p. 203
Eckart Theory of Ideal Quantum Fluids	p. 206
Rotating Superfluid	p. 207
Notes and References	p. 208
Dynamics of Charged Superfluid and Superconductor	p. 210
Hydrodynamic Description of Charged Superfluid	p. 211
London Theory of Charged Superfluid	p. 212
Including Vortices in London Equations	p. 214
Hydrodynamic Description of Superconductor	p. 215
Excitation Spectrum of Superconductor	p. 219
Gap Equation	p. 219
Action of Quadratic Fluctuations	p. 223
Long-Wavelength Excitations at Zero Temperature	p. 225
Long-Wavelength Excitations at Nonzero Temperature	p. 227
Bending Energies of Order Field	p. 229
Kinetic Terms of Pair Field at Nonzero Temperature	p. 232
Properties of Ginzburg-Landau Theory of Superconductivity	p. 237
Critical Magnetic Field	p. 239
Two Length Scales and Type I or II Superconductivity	p. 240
Single Vortex Line and Critical Field H[subscript c1]	p. 243
Critical Field H[subscript c2] where Superconductivity is Destroyed	p. 248
Notes and References	p. 250
Relativistic Magnetic Monopoles and Electric Charge Confinement	p. 251
Monopole Gauge Invariance	p. 251
Charge Quantization	p. 256
Electric and Magnetic Current-Current Interactions	p. 258
Dual Gauge Field Representation	p. 259
Monopole Gauge Fixing	p. 261
Quantum Field Theory of Spinless Electric Charges	p. 262
Theory of Magnetic Charge Confinement	p. 263
Second Quantization of the Monopole Field	p. 265
Quantum Field Theory of Electric Charge Confinement	p. 267
Notes and References	p. 270
Multivalued Mapping from Ideal Crystals to Crystals with Defects	p. 274
Defects	p. 274
Dislocation Lines and Burgers Vector	p. 278
Disclination Lines and Frank Vector	p. 281
Interdependence of Dislocation and Disclinations	p. 283
Defect Lines with Infinitesimal Discontinuities in Continuous Media	p. 284
Multivaluedness of Displacement Field	p. 286
Smoothness Properties of Displacement Field and Weingarten's Theorem	p. 287
Integrability Properties of Displacement Field	p. 290
Dislocation and Disclination Densities	p. 292
Mnemonic Procedure for Constructing Defect Densities	p. 295
Defect Gauge Invariance	p. 297
Branching Defect Lines	p. 299
Defect Density and Incompatibility	p. 300
Notes and References	p. 305
Defect Melting	p. 307
Specific Heat	p. 307
Elastic Energy of Solid with Defects	p. 308
Notes and References	p. 314
Relativistic Mechanics in Curvilinear Coordinates	p. 316
Equivalence Principle	p. 316
Free Particle in General Coordinates Frame	p. 317
Minkowski Geometry formulated in General Coordinates	p. 320
Local Basis tetrads	p. 321
Vector- and Tensor Fields in Minkowski Coordinates	p. 323
Vector- and Tensor Fields in General Coordinates	p. 324
Affine Connections and Covariant Derivatives	p. 327
Torsion tensor	p. 329
Covariant Time Derivative and Acceleration	p. 331
Curvature Tensor as Covariant Curl of Affine Connection	p. 331
Riemann Curvature Tensor	p. 335
Curvilinear Versions of Levi-Civita Tensor	p. 337
Notes and References	p. 340
Torsion and Curvature from Defects	p. 342
Multivalued Infinitesimal Coordinate Transformations	p. 343
Examples for Nonholonomic Coordinate Transformations	p. 348
Dislocation	p. 348
Disclination	p. 350
Differential-Geometric Properties of Affine Spaces	p. 350
Integrability of Metric and Affine Connection	p. 351
Local Parallelism	p. 352
Circuit Integrals in Affine Spaces with Curvature and Torsion	p. 355
Closed-Contour Integral over Parallel Vector Field	p. 355
Closed-Contour Integral over Coordinates	p. 356
Closure Failure and Burgers Vector	p. 357
Alternative Circuit Integral for Curvature	p. 357
Parallelism in World Crystal	p. 358
Bianchi Identities for Curvature and Torsion Tensors	p. 359
Special Coordinates in Riemann Spacetime	p. 361
Geodesic Coordinates	p. 361
Canonical Geodesic Coordinates	p. 363
Harmonic Coordinates	p. 365
Coordinates with det(g[subscript mu nu]) = 1	p. 367
Orthogonal Coordinates	p. 367
Number of Independent Components of R[subscript mu nu lambda superscript kappa] and S[subscript mu nu superscript lambda]	p. 369
Two Dimensions	p. 369
Three Dimensions	p. 371
Four or More Dimensions	p. 371
Notes and References	p. 373
Curvature and Torsion from Embedding	p. 374
Spacetimes with Constant Curvature	p. 374
Basis Vectors	p. 375
Torsion	p. 379
Notes and References	p. 379
Multivalued Mapping Principle	p. 380
Motion of Point Particle	p. 381
Classical Action Principle for Spaces with Curvature	p. 381
Autoparallel Trajectories in Spaces with Torsion	p. 381
Equations of Motion For Spin	p. 387
Special Properties of Gradient Torsion	p. 387
Autoparallel Trajectories from Embedding	p. 388
Special Role of Autoparallels	p. 388
Gauss Principle of Least Constraint	p. 389
Maxwell-Lorentz Orbits as Autoparallel Trajectories	p. 390
Bargmann-Michel-Telegdi Equation from Torsion	p. 391
Notes and References	p. 391
Field Equations of Gravitation	p. 393
Invariant Action	p. 393
Energy-Momentum Tensor and Spin Density	p. 396
Symmetric Energy-Momentum Tensor and Defect Density	p. 401
Notes and References	p. 403
Minimally Coupled Fields of Integer Spin	p. 404
Scalar Fields in Riemann-Cartan Space	p. 404
Electromagnetism in Riemann-Cartan Space	p. 406
Notes and References	p. 409
Particles with Half-Integer Spin	p. 410
Local Lorentz Invariance and Anholonomic Coordinates	p. 410
Nonholonomic Image of Dirac Action	p. 410
Vierbein Fields	p. 413
Local Inertial Frames	p. 414
Difference between Vierbein and Multivalued Tetrad Fields	p. 416
Covariant Derivatives in Intermediate Basis	p. 417
Dirac Action in Riemann-Cartan Space	p. 420
Ricci Identity	p. 422
Alternative Form of Coupling	p. 422
Invariant Action for Vector Fields	p. 423
Verifying Local Lorentz Invariance	p. 426
Field Equations with Spinning Matter	p. 427
Notes and References	p. 431
Covariant Conservation Law	p. 432
Spin Density	p. 432
Energy-Momentum Density	p. 434
Covariant Derivation of Conservation Laws	p. 437
Matter with Integer Spin	p. 438
Relations between Conservation Laws and Bianchi Identities	p. 440
Particle Trajectories from Energy-Momentum Conservation	p. 442
Notes and References	p. 443
Gravitation of Spinning Matter as a Gauge Theory	p. 444
Local Lorentz Transformations	p. 444
Local Translations	p. 446
Notes and References	p. 447
Evanescent Properties of Torsion in Gravity	p. 448
Local Four-Fermion Interaction due to Torsion	p. 448
No Need for Torsion in Gravity	p. 451
Scalar Fields	p. 453
Only Spin-1/2 Sources	p. 453
Modified Energy-Momentum Conservation Law	p. 455
Solution for Gradient Torsion	p. 456
Gradient Torsion coupled to Scalar Fields	p. 457
New Scalar Product	p. 458
Self-Interacting Higgs Field	p. 459
Summary	p. 459
Notes and References	p. 460
Teleparallel Theory of Gravitation	p. 463
Torsion Form of Einstein Action	p. 463
Schwarzschild Solution	p. 469
Notes and References	p. 473
Emerging Gravity	p. 474
Gravity in the World Crystal	p. 474
Gravity Emerging from Fluctuations of Matter and Radiation	p. 478
Notes and References	p. 480
Index	p. 481
Table of Contents provided by Ingram. All Rights Reserved.

Multivalued Fields

In Condensed Matter, Electromagnetism, And Gravitation

At a Glance

Paperback

Shipping

How to return your order

Defective items

You Can Find This Book In

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