A Unified Grand Tour of Theoretical Physics, 2nd Edition

A conducted grand tour of the fundamental theories which shape our modern understanding of the physical world. This book covers the central themes of spacetime geometry and the general-relativistic account of gravity; quantum mechanics and quantum field theory; gauge theories and the fundamental forces of nature, statistical mechanics and the theory of phase transitions. The basic structure of each theory is explained in explicit mathematical detail with emphasis on conceptual understanding rather than on the technical details of specialized applications. Straightforward accounts are given of the standard models of particle physics and cosmology, and some of the more speculative ideas of modern theoretical physics are examined. This book is unique in bringing together the diverse areas of physics which are usually treated as independent. Designed to be accessible to final year undergraduates in physics and mathematics and to provide first year graduate students with a broad introductory view of theoretical physics, it will also be of interest to scientists and engineers in other disciplines who need an account of the subject at a level intermediate between semi-popular and technical research. Title......Page 1 Contents......Page 3 Glossary of Mathematical Symbols......Page 11 01 - Introduction: The Ways of Nature......Page 13 02 - Geometry......Page 18 2.0 The Special and General Theories of Relativity......Page 19 2.1 Spacetime as a Differentiable Mani......Page 27 2.2 Tensors......Page 35 2.3 Extra Geometrical Structures......Page 40 2.4 What is the Structure of Our Spacetime?......Page 51 Exercises......Page 54 03 - Classical Physics in Galilean & Minkowski Spacetimes......Page 57 3.1 The Action Principle in Galilean Spacetime......Page 58 3.2 Symmetries and Conservation Laws......Page 62 3.3 The Hamiltonian......Page 64 3.4 Poisson Brackets and Translation Operators......Page 65 3.5 The Action Principle in Minkowski Spacetime......Page 68 3.6 Classical Electrodynamics......Page 73 3.7 Geometry in Classical Physics......Page 76 Exercises......Page 91 4.1 The Principle of Equivalence......Page 95 4.2 Gravitational Forces......Page 96 4.3 The Field Equations of General Relativity......Page 99 4.4 The Gravitational Field of a Spherical Body......Page 103 4.5 Black andWhite Holes......Page 109 Exercises......Page 116 05 -Quantum Theory......Page 119 5.0 Wave Mechanics......Page 120 5.1 The Hilbert Space of State Vectors......Page 123 5.2 Operators and Observable Quantities......Page 126 5.3 Spacetime Translations and the Properties of Operators......Page 128 5.4 Quantization of a Classical System......Page 133 5.5 An Example: The One-Dimensional Harmonic Oscillator......Page 135 Exercises......Page 139 06 - Second Quantization and Quantum Field Theory......Page 142 6.1 The Occupation-Number Representation......Page 143 6.2 Field Operators and Observables......Page 146 6.3 Equation of Motion and Lagrangian Formalism......Page 147 6.4 Second Quantization for Fermions......Page 149 Exercises......Page 151 07 - RelativisticWave Equations & Field Theories......Page 152 7.1 The Klein-Gordon Equation......Page 153 7.2 Scalar Field Theory for Free Particles......Page 156 7.3 The Dirac Equation & Spin-1/2 Particles......Page 158 7.4 Spinor Field Theory......Page 169 7.5 Weyl and Majorana Spinors......Page 171 7.6 Particles of Spin 1 and 2......Page 175 7.7 Wave Equations in Curved Spacetime......Page 180 Exercises......Page 189 8.1 Electromagnetism......Page 191 8.2 Non-Abelian Gauge Theories......Page 197 8.3 Non-Abelian Theories and Electromagnetism......Page 204 8.4 Relevance of Non-Abelian Theories to Physics......Page 205 8.5 The Theory of Kaluza and Klein......Page 206 Exercises......Page 208 09 - Interacting Relativistic Field Theories......Page 211 9.1 Asymptotic States & the Scattering Operator......Page 212 9.2 Reduction Formulae......Page 214 9.3 Path Integrals......Page 217 9.4 Perturbation Theory......Page 223 9.5 Quantization of Gauge Fields......Page 226 9.6 Renormalization......Page 230 9.7 Quantum Electrodynamics......Page 236 Exercises......Page 245 10 - Equilibrium Statistical Mechanics......Page 247 10.1 Ergodic Theory and the Microcanonical Ensemble......Page 248 10.2 The Canonical Ensemble......Page 253 10.3 The Grand Canonical Ensemble......Page 255 10.4 Relation Between Statistical Mechanics......Page 257 10.5 Quantum Statistical Mechanics......Page 263 10.6 Field Theories at Finite Temperature......Page 266 10.7 Black Body Radiation......Page 269 10.8 The Classical Lattice Gas......Page 271 10.9 Analogies Between Field Theory and Statistical Mechanics......Page 273 Exercises......Page 275 11.1 Bose-Einstein Condensation......Page 278 11.2 Critical Points in Fluids and Magnets......Page 281 11.3 The Ising Model and its Approximation by a Field Theory......Page 286 11.4 Order, Disorder and Spontaneous Symmetry Breaking......Page 288 11.5 The Ginzburg-Landau Theory......Page 291 11.6 The Renormalization Group......Page 293 11.7 The Ginzburg-Landau Theory of Superconductors......Page 299 Exercises......Page 305 12 - Unified Gauge Theories of the Fundamental Interactions......Page 307 12.1 The Weak Interaction......Page 308 12.2 The Glashow-Weinberg-Salam Model for Leptons......Page 313 12.3 Physical Implications of the Model for Leptons......Page 318 12.4 Hadronic Particles in the Electroweak Theory......Page 320 12.5 Colour and Quantum Chromodynamics......Page 326 12.6 Grand Unified Theories......Page 331 12.7 Supersymmetry......Page 340 Exercises......Page 356 13 - Solitons and So On......Page 358 13.1 DomainWalls and Kinks......Page 359 13.2 The Sine-Gordon Solitons......Page 367 13.3 Vortices and Strings......Page 371 13.4 Magnetic Monopoles......Page 381 Exercises......Page 389 14 - The Early Universe......Page 391 14.1 The Robertson-Walker Metric......Page 392 14.2 The Friedmann-Lemaıtre Models......Page 397 14.3 Matter, Radiation and the Age of the Universe......Page 402 14.4 The Fairly Early Universe......Page 405 14.5 Nucleosynthesis......Page 413 14.6 Recombination and the Horizon Problem......Page 416 14.7 The Flatness Problem......Page 417 14.8 The Very Early Universe......Page 418 Exercises......Page 434 15 - An Intro 2 String Theory......Page 437 15.1 The Relativistic Point Particle......Page 439 15.2 The Free Classical String......Page 443 15.3 Quantization of the Free Bosonic String......Page 459 15.4 Physics of the Free Bosonic String......Page 482 15.5 Further Developments......Page 493 15.6 The LastWord?......Page 507 Exercises......Page 508 Some Snapshots of the Tour......Page 513 Snapshot of Geometry and Gravitation......Page 514 Snapshot of Field Theory......Page 518 Snapshot of Statistical Mechanics......Page 522 Snapshot of Bosonic String Theory......Page 526 A.1 Delta Functions and Functional Differentiation......Page 530 A.2 The Levi-Civita Tensor Density......Page 532 A.3 Vector Spaces and Hilbert Spaces......Page 533 A.4 Gauss’ Theorem......Page 535 A.6 Gaussian Integrals......Page 536 A.7 Grassmann Variables......Page 537 Appendix B Some Elements of Group Theory......Page 540 Appendix C Natural Units......Page 552 Appendix D Scattering Cross-Sections and Particle Decay Rates......Page 556 Bibliography......Page 560 References......Page 564 Index......Page 567 Title 1 Contents 3 Glossary of Mathematical Symbols 11 01 - Introduction: The Ways of Nature 13 02 - Geometry 18 2.0 The Special and General Theories of Relativity 19 2.1 Spacetime as a Differentiable Mani 27 2.2 Tensors 35 2.3 Extra Geometrical Structures 40 2.4 What is the Structure of Our Spacetime? 51 Exercises 54 03 - Classical Physics in Galilean & Minkowski Spacetimes 57 3.1 The Action Principle in Galilean Spacetime 58 3.2 Symmetries and Conservation Laws 62 3.3 The Hamiltonian 64 3.4 Poisson Brackets and Translation Operators 65 3.5 The Action Principle in Minkowski Spacetime 68 3.6 Classical Electrodynamics 73 3.7 Geometry in Classical Physics 76 Exercises 91 04 - General Relativity and Gravitation 95 4.1 The Principle of Equivalence 95 4.2 Gravitational Forces 96 4.3 The Field Equations of General Relativity 99 4.4 The Gravitational Field of a Spherical Body 103 4.5 Black andWhite Holes 109 Exercises 116 05 -Quantum Theory 119 5.0 Wave Mechanics 120 5.1 The Hilbert Space of State Vectors 123 5.2 Operators and Observable Quantities 126 5.3 Spacetime Translations and the Properties of Operators 128 5.4 Quantization of a Classical System 133 5.5 An Example: The One-Dimensional Harmonic Oscillator 135 Exercises 139 06 - Second Quantization and Quantum Field Theory 142 6.1 The Occupation-Number Representation 143 6.2 Field Operators and Observables 146 6.3 Equation of Motion and Lagrangian Formalism 147 6.4 Second Quantization for Fermions 149 Exercises 151 07 - RelativisticWave Equations & Field Theories 152 7.1 The Klein-Gordon Equation 153 7.2 Scalar Field Theory for Free Particles 156 7.3 The Dirac Equation & Spin-1/2 Particles 158 7.4 Spinor Field Theory 169 7.5 Weyl and Majorana Spinors 171 7.6 Particles of Spin 1 and 2 175 7.7 Wave Equations in Curved Spacetime 180 Exercises 189 08 - Forces, Connections & Gauge Fields 191 8.1 Electromagnetism 191 8.2 Non-Abelian Gauge Theories 197 8.3 Non-Abelian Theories and Electromagnetism 204 8.4 Relevance of Non-Abelian Theories to Physics 205 8.5 The Theory of Kaluza and Klein 206 Exercises 208 09 - Interacting Relativistic Field Theories 211 9.1 Asymptotic States & the Scattering Operator 212 9.2 Reduction Formulae 214 9.3 Path Integrals 217 9.4 Perturbation Theory 223 9.5 Quantization of Gauge Fields 226 9.6 Renormalization 230 9.7 Quantum Electrodynamics 236 Exercises 245 10 - Equilibrium Statistical Mechanics 247 10.1 Ergodic Theory and the Microcanonical Ensemble 248 10.2 The Canonical Ensemble 253 10.3 The Grand Canonical Ensemble 255 10.4 Relation Between Statistical Mechanics 257 10.5 Quantum Statistical Mechanics 263 10.6 Field Theories at Finite Temperature 266 10.7 Black Body Radiation 269 10.8 The Classical Lattice Gas 271 10.9 Analogies Between Field Theory and Statistical Mechanics 273 Exercises 275 11 - Phase Transitions 278 11.1 Bose-Einstein Condensation 278 11.2 Critical Points in Fluids and Magnets 281 11.3 The Ising Model and its Approximation by a Field Theory 286 11.4 Order, Disorder and Spontaneous Symmetry Breaking 288 11.5 The Ginzburg-Landau Theory 291 11.6 The Renormalization Group 293 11.7 The Ginzburg-Landau Theory of Superconductors 299 Exercises 305 12 - Unified Gauge Theories of the Fundamental Interactions 307 12.1 The Weak Interaction 308 12.2 The Glashow-Weinberg-Salam Model for Leptons 313 12.3 Physical Implications of the Model for Leptons 318 12.4 Hadronic Particles in the Electroweak Theory 320 12.5 Colour and Quantum Chromodynamics 326 12.6 Grand Unified Theories 331 12.7 Supersymmetry 340 Exercises 356 13 - Solitons and So On 358 13.1 DomainWalls and Kinks 359 13.2 The Sine-Gordon Solitons 367 13.3 Vortices and Strings 371 13.4 Magnetic Monopoles 381 Exercises 389 14 - The Early Universe 391 14.1 The Robertson-Walker Metric 392 14.2 The Friedmann-Lemaˆıtre Models 397 14.3 Matter, Radiation and the Age of the Universe 402 14.4 The Fairly Early Universe 405 14.5 Nucleosynthesis 413 14.6 Recombination and the Horizon Problem 416 14.7 The Flatness Problem 417 14.8 The Very Early Universe 418 Exercises 434 15 - An Intro 2 String Theory 437 15.1 The Relativistic Point Particle 439 15.2 The Free Classical String 443 15.3 Quantization of the Free Bosonic String 459 15.4 Physics of the Free Bosonic String 482 15.5 Further Developments 493 15.6 The LastWord? 507 Exercises 508 Some Snapshots of the Tour 513 Snapshot of Geometry and Gravitation 514 Snapshot of Field Theory 518 Snapshot of Statistical Mechanics 522 Snapshot of Bosonic String Theory 526 Appendix A Some Mathematical Notes 530 A.1 Delta Functions and Functional Differentiation 530 A.2 The Levi-Civita Tensor Density 532 A.3 Vector Spaces and Hilbert Spaces 533 A.4 Gauss’ Theorem 535 A.5 Surface Area and Volume of a d-Dimensional Sphere 536 A.6 Gaussian Integrals 536 A.7 Grassmann Variables 537 Appendix B Some Elements of Group Theory 540 Appendix C Natural Units 552 Appendix D Scattering Cross-Sections and Particle Decay Rates 556 Bibliography 560 References 564 Index 567 A unified account of the principles of theoretical physics, A Unified Grand Tour of Theoretical Physics, Second Edition stresses the inter-relationships between areas that are usually treated as independent. The profound unifying influence of geometrical ideas, the powerful formal similarities between statistical mechanics and quantum field theory, and the ubiquitous role of symmetries in determining the essential structure of physical theories are emphasized throughout. This second edition conducts a grand tour of the fundamental theories that shape our modern understanding of the physical world. The book covers the central themes of space-time geometry and the general relativistic account of gravity, quantum mechanics and quantum field theory, gauge theories and the fundamental forces of nature, statistical mechanics, and the theory of phase transitions. The basic structure of each theory is explained in explicit mathematical detail with emphasis on conceptual understanding rather than on the technical details of specialized applications. The book gives straightforward accounts of the standard models of particle physics and cosmology. In the eighteenth century, it became fashionable for wealthy young Englishmen to undertake the Grand Tour, an excursion which may have lasted several years, their principal destinations being Paris and the great cultural centres of Italy Venice, Florence and Naples. "The book will appeal to advanced undergraduates, to postgraduate students in search of a broad introduction to modern theoretical physics, and to experienced scientists who are not specialists in the topics covered by the Tour."--BOOK JACKET