Mathematical Physics with Differential Equations

## Abstract This book presents a broad range of fundamental topics in theoretical and mathematical physics based on the viewpoint of differential equations. The subject areas covered include classical and quantum many-body problems, thermodynamics, electromagnetism, magnetic monopoles, special relativity, gauge field theories, general relativity, superconductivity, vortices and other topological solitons, and canonical quantization of fields, for which, knowledge and use of linear and nonlinear differential equations are essential for comprehension. With insight of differential equations, the book features itself in several aspects. Firstly, the presentation of the content follows a mathematically thorough and transparent style which serves to provide a handy and direct access to approach each of the subjects discussed. Secondly, it renders a fairly wide selection of themes which may further be tailored to suit individual preferences of a reader. Thirdly, it supplies a balanced pool of topics for seminars. Fourthly, it offers guidance and stimulation to the related contemporary research frontiers and literature. Much emphasis is given to the mathematical and physical content, implication, structures, and challenges of the governing differential equations of the theories encountered and developed, thus offering an appreciation of the interplay of mathematics and theoretical physics from the viewpoint of differential equations. On the other hand, advanced methods and techniques of modern nonlinear functional analysis are kept minimal. The book is comprised of fifteen topic chapters and an appendices chapter. Each chapter is supplemented with a collection of exercises of varied depths. Cover 1 Titlepage 4 Copyright 5 Dedication 6 Contents 8 Preface 12 Notation and Convention 20 1 Hamiltonian systems and applications 24 1.1 Motion of massive particle 24 1.2 Many-body problem 29 1.3 Kepler's laws of planetary motion 31 1.4 Helmholtz–Kirchhoff vortex model 37 1.5 Partition function and thermodynamics 42 1.6 Dynamic modeling of DNA denaturation 45 2 Schrödinger equationand quantum mechanics 52 2.1 Path to quantum mechanics 52 2.2 Schrödinger equation 55 2.3 Quantum many-body problem 64 2.4 Hartree–Fock method 66 2.5 Thomas–Fermi approach 69 2.6 Density functional theory 72 3 Maxwell equations, Dirac monopole, and gauge fields 82 3.1 Maxwell equations and electromagnetic duality 82 3.2 Dirac monopole and strings 87 3.3 Charged particle in electromagnetic field 89 3.4 Removal of Dirac strings and charge quantization 92 3.5 Schwinger dyons and extended charge quantization formula 94 3.6 Aharonov–Bohm effect 96 4 Special relativity 102 4.1 Inertial frames, Minkowski spacetime,and Lorentz boosts 102 4.2 Line element, proper time, and consequences 105 4.3 Relativistic mechanics 109 4.4 Doppler effects 115 5 Abelian gauge field equations 119 5.1 Spacetime, covariance, and invariance 119 5.2 Relativistic field equations 125 5.3 Coupled nonlinear hyperbolic and elliptic equations 132 5.4 Symmetry breaking 134 5.5 Higgs mechanism 137 6 Dirac equations 142 6.1 Pauli matrices, spinor fields, and Diracequation 142 6.2 Action, probability, and current densities 145 6.3 Special solutions 145 6.4 Dirac equation coupled with gaugefield 148 6.5 Dirac equation in Weyl representation 152 6.6 Nonlinear Dirac equations 153 7 Ginzburg–Landau equations for superconductivity 156 7.1 Perfect conductors, superconductors, and London equations 156 7.2 Superconductors and Ginzburg–Landau equations 161 7.3 Classification of superconductivityby surface energy 166 7.4 Mixed state and its magnetic characterizations 172 7.5 Some generalized Ginzburg–Landau equations 175 8 Magnetic vortices in Abelian Higgs theory 180 8.1 Energy partition, flux quantization,and topological properties 180 8.2 Vortex-lines, solitons, and particles 185 8.3 Radially symmetric solutions 191 8.4 From monopole confinement to quarkconfinement 192 9 Non-Abelian gauge field equations 199 9.1 Yang–Mills theory 199 9.2 Georgi–Glashow model 204 9.3 't Hooft–Polyakov monopole and Julia–Zeedyon 209 9.4 Monopoles and dyons in BPS limit 213 9.5 Weinberg–Salam electroweak equations 222 10 Einstein equations and related topics 230 10.1 Einstein field equations 230 10.2 Cosmological consequences 238 10.3 Schwarzschild black-hole solution 251 10.4 Reissner–Nordström solution 261 10.5 Kerr solution 268 10.6 Gravitational mass and Penrose bounds 270 10.7 Gravitational waves 276 10.8 Scalar-wave matters as quintessence 278 11 Charged vortices and Chern–Simons equations 290 11.1 Julia–Zee theorem 290 11.2 Chern–Simons term 293 11.3 Dually charged vortices 294 11.4 Rubakov–Tavkhelidze problem 297 12 Skyrme model and related topics 304 12.1 Derrick theorem and Pohozaev identity 304 12.2 Skyrme model 309 12.3 Knots in Faddeev model 315 12.4 Other fractional-exponent growth lawsand knot energies 320 12.5 Q-balls 325 13 Strings and branes 331 13.1 Motivation and relativistic motion of freeparticle as initial setup 331 13.2 Nambu–Goto strings 333 13.3 p-branes 337 13.4 Polyakov strings and branes 340 13.5 Equations of motion with interactions 344 14 Born–Infeld theory of electromagnetism 346 14.1 Resolution of energy divergence problemof point charges 346 14.2 Some illustrative calculations 352 14.3 Dyonic point charge 357 14.4 Formalism based on invariance 359 14.5 Generalized Bernstein problem 365 14.6 Born–Infeld term and virial identities 369 14.7 Integer-squared law for global Born–Infeld vortices 371 14.8 Electrically charged black hole solutions 376 14.9 Dyonic black hole solutions 383 14.10 Generalized Born–Infeld theoriesand applications 392 14.11 Electromagnetic asymmetry by virtueof point charges 394 14.12 Charged black holes 403 14.13 Relegation of curvature singularities of charged black holes 410 14.14 Cosmology driven by scalar-wave matters as k-essence 418 14.15 Finite-energy dyonic point charge 428 14.16 Dyonically charged black holes with relegated singularities 438 15 Canonical quantization of fields 452 15.1 Quantum harmonic oscillator 452 15.2 Canonical quantization 457 15.3 Field equation formalism 462 15.4 Quantization of Klein–Gordon equation 465 15.5 Quantization of Schrödinger equation 471 15.6 Quantization of electromagnetic fields 478 15.7 Thermodynamics of harmonic oscillator 486 Appendices 493 A.1 Index of vector field and topological degree of map 493 A.2 Linking number and Hopf invariant 507 A.3 Noether theorem 512 A.4 Spectra of angular momentum operators 521 A.5 Spins and spin-statistics theorem 526 A.6 Deflection of light in gravitational field 532 Bibliography 547 Index 579 Traditional literature in mathematical physics is clustered around classical mechanics, especially fluids and elasticity. This book reflects the modern development of theoretical physics in the areas of field theories: classical, quantum, and gravitational, in which differential equations play essential roles and offer powerful insight. Yang here presents a broad range of fundamental topics in theoretical and mathematical physics based on the viewpoint of differential equations. The subject areas covered include classical and quantum many-body problems, thermodynamics, electromagnetism, magnetic monopoles, special relativity, gauge field theories, general relativity, superconductivity, vortices and other topological solitons, and canonical quantization of fields, for which knowledge and use of linear and nonlinear differential equations are essential for comprehension. Much emphasis is given to the mathematical and physical content offering an appreciation of the interplay of mathematics and theoretical physics from the viewpoint of differential equations. Advanced methods and techniques of modern nonlinear functional analysis are kept to a minimum and each chapter is supplemented with a collection of exercises of varied depths making it an ideal resource for students and researchers alike. This textbook uses insight from differential equations to analyse fundamental subjects of modern theoretical physics, including classical and quantum mechanics, thermodynamics, electromagnetism, superconductivity, gravitational physics, and quantum field theories.