A Unified Grand Tour of Theoretical Physics, 2nd Edition

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. 0750306041......Page 1 Contents......Page 6 Preface to the Second Edition......Page 12 Preface to the First Edition......Page 14 Glossary of Mathematical Symbols......Page 16 1 Introduction: The Ways of Nature......Page 18 2 Geometry......Page 23 2.0.1 The special theory......Page 24 2.0.2 The general theory......Page 29 2.1 Spacetime as a Differentiable Manifold......Page 32 2.1.1 Topology of the real line R and of R[sup(d)]......Page 33 2.1.2 Differentiable spacetime manifold......Page 36 2.1.3 Summary and examples......Page 38 2.2 Tensors......Page 40 2.3 Extra Geometrical Structures......Page 45 2.3.1 The affine connection......Page 46 2.3.2 Geodesics......Page 50 2.3.3 The Riemann curvature tensor......Page 51 2.3.4 The metric......Page 53 2.3.5 The metric connection......Page 55 2.4 What is the Structure of Our Spacetime?......Page 56 3 Classical Physics in Galilean and Minkowski Spacetimes......Page 62 3.1 The Action Principle in Galilean Spacetime......Page 63 3.2 Symmetries and Conservation Laws......Page 67 3.3 The Hamiltonian......Page 69 3.4 Poisson Brackets and Translation Operators......Page 70 3.5 The Action Principle in Minkowski Spacetime......Page 73 3.6 Classical Electrodynamics......Page 78 3.7 Geometry in Classical Physics......Page 81 3.7.1 More on tensors......Page 82 3.7.2 Differential forms, dual tensors and Maxwell’s equations......Page 84 3.7.3 Configuration space and its relatives......Page 90 3.7.4 The symplectic geometry of phase space......Page 92 4.1 The Principle of Equivalence......Page 100 4.2 Gravitational Forces......Page 101 4.3 The Field Equations of General Relativity......Page 104 4.4.1 The Schwarzschild solution......Page 108 4.4.2 Time near a massive body......Page 110 4.4.3 Distances near a massive body......Page 112 4.4.4 Particle trajectories near a massive body......Page 113 4.5 Black and White Holes......Page 114 5 Quantum Theory......Page 124 5.0 Wave Mechanics......Page 125 5.1 The Hilbert Space of State Vectors......Page 128 5.2 Operators and Observable Quantities......Page 131 5.3 Spacetime Translations and the Properties of Operators......Page 133 5.4 Quantization of a Classical System......Page 138 5.5 An Example: The One-Dimensional Harmonic Oscillator......Page 140 6 Second Quantization and Quantum Field Theory......Page 147 6.1 The Occupation-Number Representation......Page 148 6.2 Field Operators and Observables......Page 151 6.3 Equation of Motion and Lagrangian Formalism for Field Operators......Page 152 6.4 Second Quantization for Fermions......Page 154 7 Relativistic Wave Equations and Field Theories......Page 157 7.1 The Klein–Gordon Equation......Page 158 7.2 Scalar Field Theory for Free Particles......Page 161 7.3.1 The Dirac equation......Page 163 7.3.2 Lorentz covariance and spin......Page 165 7.3.3 Some properties of the γ matrices......Page 169 7.3.5 Probability current and bilinear covariants......Page 170 7.3.6 Plane-wave solutions......Page 172 7.3.7 Massless spin-1⁄2 particles......Page 173 7.4 Spinor Field Theory......Page 174 7.5 Weyl and Majorana Spinors......Page 176 7.6.1 Photons and massive spin-1 particles......Page 180 7.6.2 Gravitons......Page 183 7.7 Wave Equations in Curved Spacetime......Page 185 8.1 Electromagnetism......Page 196 8.2 Non-Abelian Gauge Theories......Page 202 8.3 Non-Abelian Theories and Electromagnetism......Page 209 8.4 Relevance of Non-Abelian Theories to Physics......Page 210 8.5 The Theory of Kaluza and Klein......Page 211 9 Interacting Relativistic Field Theories......Page 216 9.1 Asymptotic States and the Scattering Operator......Page 217 9.2 Reduction Formulae......Page 219 9.3.1 Path integrals in non-relativistic quantum mechanics......Page 222 9.3.2 Functional integrals in quantum field theory......Page 225 9.4 Perturbation Theory......Page 228 9.5 Quantization of Gauge Fields......Page 231 9.6 Renormalization......Page 235 9.7.1 The Coulomb potential......Page 241 9.7.2 Vacuum polarization......Page 244 9.7.4 The running coupling constant......Page 246 9.7.5 Anomalous magnetic moments......Page 248 10 Equilibrium Statistical Mechanics......Page 252 10.1 Ergodic Theory and the Microcanonical Ensemble......Page 253 10.2 The Canonical Ensemble......Page 258 10.3 The Grand Canonical Ensemble......Page 260 10.4 Relation Between Statistical Mechanics and Thermodynamics......Page 262 10.5 Quantum Statistical Mechanics......Page 268 10.6 Field Theories at Finite Temperature......Page 271 10.7 Black Body Radiation......Page 274 10.8 The Classical Lattice Gas......Page 276 10.9 Analogies Between Field Theory and Statistical Mechanics......Page 278 11.1 Bose–Einstein Condensation......Page 283 11.2 Critical Points in Fluids and Magnets......Page 286 11.3 The Ising Model and its Approximation by a Field Theory......Page 291 11.4 Order, Disorder and Spontaneous Symmetry Breaking......Page 293 11.5 The Ginzburg–Landau Theory......Page 296 11.6 The Renormalization Group......Page 298 11.7 The Ginzburg–Landau Theory of Superconductors......Page 304 11.7.1 Spontaneous breaking of continuous symmetries......Page 305 11.7.2 Magnetic effects in superconductors......Page 307 11.7.3 The Higgs mechanism......Page 308 12 Unified Gauge Theories of the Fundamental Interactions......Page 312 12.1 The Weak Interaction......Page 313 12.2 The Glashow–Weinberg–Salam Model for Leptons......Page 318 12.3 Physical Implications of the Model for Leptons......Page 323 12.4.1 Quarks......Page 325 12.4.2 Quarks in the electroweak theory......Page 329 12.5 Colour and Quantum Chromodynamics......Page 331 12.6 Grand Unified Theories......Page 336 12.7 Supersymmetry......Page 345 12.7.1 The Wess–Zumino model......Page 346 12.7.2 Superfields......Page 347 12.7.3 Spontaneous supersymmetry breaking......Page 349 12.7.4 The supersymmetry algebra......Page 352 12.7.5 Supersymmetric gauge theories and supergravity......Page 357 12.7.6 Some algebraic details......Page 360 13 Solitons and So On......Page 363 13.1 Domain Walls and Kinks......Page 364 13.2 The Sine–Gordon Solitons......Page 372 13.3 Vortices and Strings......Page 376 13.4 Magnetic Monopoles......Page 386 14 The Early Universe......Page 396 14.1 The Robertson–Walker Metric......Page 397 14.2 The Friedmann–Lemaître Models......Page 402 14.3 Matter, Radiation and the Age of the Universe......Page 407 14.4 The Fairly Early Universe......Page 410 14.5 Nucleosynthesis......Page 418 14.6 Recombination and the Horizon Problem......Page 421 14.7 The Flatness Problem......Page 422 14.8 The Very Early Universe......Page 423 15 An Introduction to String Theory......Page 442 15.1 The Relativistic Point Particle......Page 444 15.2.1 The string action......Page 448 15.2.2 Weyl invariance and gauge fixing......Page 451 15.2.3 The Euclidean worldsheet and conformal invariance......Page 454 15.2.4 Mode expansions......Page 457 15.2.5 A useful transformation......Page 462 15.3 Quantization of the Free Bosonic String......Page 464 15.3.1 The quantum Virasoro algebra......Page 466 15.3.2 Quantum gauge fixing......Page 471 15.3.3 The critical spacetime dimension......Page 475 15.3.4 The ghost Hilbert space......Page 479 15.3.5 The BRST cohomology......Page 481 15.4.1 The mass spectrum......Page 487 15.4.2 Vertex operators......Page 492 15.4.3 Strings and quantum gravity......Page 495 15.5.1 String interactions......Page 498 15.5.2 Superstrings......Page 502 15.5.3 The ramifications of compactification......Page 506 15.6 The Last Word?......Page 512 Some Snapshots of the Tour......Page 518 A.1 Delta Functions and Functional Differentiation......Page 535 A.2 The Levi-Civita Tensor Density......Page 537 A.3 Vector Spaces and Hilbert Spaces......Page 538 A.4 Gauss’ Theorem......Page 540 A.6 Gaussian Integrals......Page 541 A.7 Grassmann Variables......Page 542 Appendix B: Some Elements of Group Theory......Page 545 Appendix C: Natural Units......Page 557 Appendix D: Scattering Cross-Sections and Particle Decay Rates......Page 561 Bibliography......Page 565 References......Page 569 C......Page 572 E......Page 574 G......Page 575 K......Page 576 N......Page 577 Q......Page 578 S......Page 579 W......Page 580 Y......Page 581 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