Physics Lawrie A Unified Grand Tour of A Unified Grand Tour of Theoretical Physics Theoretical Physics A Unified Grand Tour of Third Edition Ian D. Lawrie Theoretical Physics A Unified Grand Tour of Theoretical Physics invites its readers to a guided Third Edition exploration of the theoretical ideas that shape our contemporary understanding of the physical world at the fundamental level. Its central themes, comprising space-time geometry and the general relativistic account of gravity, quantum field theory and the gauge theories of fundamental forces, statistical mechanics and the theory of phase transitions, are developed in explicit mathematical detail, with an emphasis on conceptual understanding. Straightforward treatments of the standard models of particle physics and cosmology are supplemented with introductory accounts of more speculative theories including supersymmetry and string theory.
This third edition of the Tour includes a new chapter on quantum gravity, focusing on the approach known as Loop Quantum Gravity, while new sections provide extended discussions of topics that have become prominent in recent years, such as the Higgs boson, massive neutrinos, cosmological perturbations, dark energy and matter, and the thermodynamics of black holes. Designed for those in search of a solid grasp of the inner workings of these theories, but who prefer to avoid a full-scale assault on the research literature, the Tour assumes as its point of departure a familiarity with basic undergraduate- level physics and emphasizes the interconnections between aspects of physics that are more often treated in isolation. The companion website at www.org provides further resources, including a comprehensive manual of solutions to the end-of- chapter exercises. Lawrie 0918 K13975 2013 Third ISBN: 978-1-4398-8446-1 Edition 90000 9 781439 884461 www.com A Unified Grand Tour of Theoretical Physics Third Edition www.com A Unified Grand Tour of Theoretical Physics Third Edition Ian D.
Lawrie Formerly Professor of Theoretical Physics University of Leeds www.com Taylor & Francis Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2012 by Taylor & Francis Group, LLC Taylor & Francis is an Informa business No claim to original U. Government works Version Date: 20130524 International Standard Book Number-13: 978-1-4398-8447-8 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained.
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Visit the Taylor & Francis Web site at http://www.com and the CRC Press Web site at http://www.com Contents Preface xi Preface to the First Edition xiii Preface to the Second Edition xv Glossary of Mathematical Symbols xvii 1 Introduction: The Ways of Nature 1 2 Geometry 5 2.0 The Special and General Theories of Relativity .1 The Special Theory .2 The General Theory .1 Spacetime as a Differentiable Manifold .1 Topology of the Real Line R and of Rd .2 Differentiable Spacetime Manifold .3 Summary and Examples .3 Extra Geometrical Structures .1 The Affine Connection .3 The Riemann Curvature Tensor .5 The Metric Connection .4 What Is the Structure of Our Spacetime?. 39 3 Classical Physics in Galilean and Minkowski Spacetimes 45 3.1 The Action Principle in Galilean Spacetime .2 Symmetries and Conservation Laws .4 Poisson Brackets and Translation Operators .5 The Action Principle in Minkowski Spacetime .7 *Geometry in Classical Physics .1 More on Tensors .com vi Contents 3.2 Differential Forms, Dual Tensors and Maxwell’s Equa- tions .3 Configuration Space and Its Relatives .4 The Symplectic Geometry of Phase Space .8 *Hamiltonian Dynamics of Constrained Systems .1 A System with Second-Class Constraints .2 A System with a First-Class Constraint .3 General Constrained Systems. 91 4 General Relativity and Gravitation 97 4.1 The Principle of Equivalence .3 The Field Equations of General Relativity .4 The Gravitational Field of a Spherical Body .1 The Schwarzschild Solution .2 Time Near a Massive Spherical Body .3 Distances Near a Massive Spherical Body .4 Particle Trajectories Near a Massive Spherical Body 111 4.1 Schwarzschild Black Holes .2 *Mass and Surface Gravity of a Schwarzschild Black Hole .3 *Rotating Black Holes and Black-Hole Thermodynam- ics .1 The Hilbert Space of State Vectors .2 Operators and Observable Quantities .3 Spacetime Translations and the Properties of Operators .4 Quantization of a Classical System .5 An Example: The One-Dimensional Harmonic Oscillator. 157 6 Second Quantization and Quantum Field Theory 165 6.1 The Occupation-Number Representation .2 Field Operators and Observables .3 Lagrangian Formalism for Field Operators .4 Second Quantization for Fermions.
173 7 Relativistic Wave Equations and Field Theories 177 7.1 The Klein–Gordon Equation .2 Scalar Field Theory for Free Particles .3 The Dirac Equation and Spin- 12 Particles .1 The Dirac Equation .2 Lorentz Covariance and Spin .3 Some Properties of the γ Matrices .com Contents vii 7.4 Conjugate Wavefunction and the Dirac Action .5 Probability Current and Bilinear Covariants .6 Plane-Wave Solutions .7 Massless Spin- 12 Particles .4 Spinor Field Theory .5 Weyl and Majorana Spinors .6 Particles of Spin 1 and 2 .1 Photons and Massive Spin-1 Particles .7 Wave Equations in Curved Spacetime. 208 8 Forces, Connections and Gauge Fields 221 8.2 Non-Abelian Gauge Theories .3 Non-Abelian Theories and Electromagnetism .4 Relevance of Non-Abelian Theories to Physics .5 The Theory of Kaluza and Klein. 236 9 Interacting Relativistic Field Theories 241 9.1 Asymptotic States and the Scattering Operator .1 Path Integrals in Non-Relativistic Quantum Mechanics 247 9.2 Functional Integrals in Quantum Field Theory .5 Quantization of Gauge Fields .1 The Coulomb Potential .3 The Lamb Shift .4 The Running Coupling Constant .5 Anomalous Magnetic Moments. 273 10 Equilibrium Statistical Mechanics 279 10.1 Ergodic Theory and the Microcanonical Ensemble .2 The Canonical Ensemble .3 The Grand Canonical Ensemble .4 Relation between Statistical Mechanics and Thermodynamics 289 10.5 Quantum Statistical Mechanics .6 Field Theories at Finite Temperature .7 Black-Body Radiation .8 The Classical Lattice Gas .9 Analogies between Field Theory and Statistical Mechanics .com viii Contents 11 Phase Transitions 311 11.1 Bose–Einstein Condensation .2 Critical Points in Fluids and Magnets .3 The Ising Model .4 Spontaneous Symmetry Breaking .5 The Ginzburg–Landau Theory .6 The Renormalization Group .7 The Ginzburg–Landau Theory of Superconductors .1 Spontaneous Breaking of Continuous Symmetries .2 Magnetic Effects in Superconductors .3 The Higgs Mechanism.
337 12 Unified Gauge Theories of the Fundamental Interactions 341 12.1 The Weak Interaction .2 The Glashow–Weinberg–Salam Model for Leptons .3 Physical Implications of the Model for Leptons .4 Hadronic Particles in the Electroweak Theory .2 Quarks in the Electroweak Theory .5 Colour and Quantum Chromodynamics .1 The Higgs Boson .7 Grand Unified Theories .1 The Wess–Zumino Model .3 Spontaneous Supersymmetry Breaking .4 The Supersymmetry Algebra .5 Supersymmetric Gauge Theories and Supergravity .6 Some Algebraic Details. 397 13 Solitons and So On 401 13.1 Domain Walls and Kinks .2 The Sine–Gordon Solitons .3 Vortices and Strings. 424 14 The Early Universe 435 14.1 The Robertson–Walker Metric .2 The Friedmann–Lemaı̂tre Models .3 Matter, Energy and the Age of the Universe .4 The Fairly Early Universe .6 Recombination and the Horizon Problem .com Contents ix 14.7 The Flatness Problem .1 Field Equations for Linear Perturbations .2 Perturbations of Ideal Fluids .4 Qualitative Features of the CMBR Anisotropies .9 The Very Early Universe .1 Cosmological Phase Transitions .3 Density Perturbations Generated during Inflation .10 Dark Energy and Dark Matter. 493 15 *An Introduction to String Theory 499 15.1 The Relativistic Point Particle .2 The Free Classical String .1 The String Action .2 Weyl Invariance and Gauge Fixing .3 The Euclidean Worldsheet and Conformal Invariance 511 15.3 Quantization of the Free Bosonic String .1 The Quantum Virasoro Algebra .2 Quantum Gauge Fixing .3 The Critical Spacetime Dimension .4 The Ghost Hilbert Space .5 The BRST Cohomology .4 Physics of the Free Bosonic String .1 The Mass Spectrum .3 Strings and Quantum Gravity .3 The Ramifications of Compactification .4 Large Extra Dimensions.
568 16 *Gravity and Quantum Mechanics 575 16.1 Canonical Quantization of General Relativity .1 Hamiltonian Formulation of General Relativity .2 New Variables: Triads, Holonomies and Fluxes .3 Towards a Quantum Theory of Gravity .1 The Problem of Time .2 Loop Quantum Cosmology .3 Black-Hole Entropy .com x Contents Some Snapshots of the Tour 623 A Some Mathematical Notes 641 A.1 Delta Functions and Functional Differentiation .2 The Levi-Civita Tensor Density .3 Vector Spaces and Hilbert Spaces .5 Surface Area and Volume of a d-Dimensional Sphere. 649 B Some Elements of Group Theory 651 C Natural Units 665 D Scattering Cross Sections and Particle Decay Rates 669 Bibliography 673 References 677 Index 685 www.com Preface When John Navas of Taylor & Francis first suggested a new edition of my Grand Tour, I was not entirely sure that a worthwhile revision could be achieved. After some further thought, I decided to make the attempt any- way: a boost to my retirement fund would not go amiss, and there seemed to be a chance that some higher purpose might also be served. The Tour’s original itinerary does seem to have served a useful purpose, and I have left it more or less intact.
New excursions this season are, necessarily, a little more technical and certainly more selective. In the light of certain well-publicized de- velopments in experimental physics and observational cosmology, I have given somewhat more detailed treatments of the Higgs boson and neutrino masses, and of anisotropies in the cosmic microwave background, though these dis- cussions are by no means comprehensive, because this is not a textbook on particle physics or cosmology per se. Finally, I have tried to convey something of the remarkable insight into quantum geometry that has been gained over the last twenty years or so through the canonical quantization of general rel- ativity; and in the course of doing that I have provided an introduction to the theory of constrained systems and considerably expanded my treatment of black holes. There was never any hope of dealing in one volume with everything that is important in theoretical physics, but I hope that some readers, at least, will find the subjects I have chosen to address worthy of their attention.
I am grateful to Bob Wald and David Wiltshire for advice on matters of which they know much more than I do, and to Randy Burling and Marcus Fontaine at Taylor & Francis for helping me get this edition into print. Lawrie June 2012 xi www.com Preface to the First Edition A few years ago, I decided to undertake some research having to do with the early history of the universe. It soon became apparent that I should have to improve my understanding of several aspects of theoretical physics, and it was from the ensuing process of self-education that the idea of writing this book emerged. I was particularly struck by two things.