M A T H E M A T I C A L P E R S P E C T I V E S THEORETICAL
ON
PHYSICS
A Journey from Black Holes to Superstrings
M A T H E M A T I C A L P E R S P E C T I V E S THEORETICAL
ON
PHYSICS
A Journey from Black Holes to Superstrings
NIRMALA PRAKASH Visting Scientist Massachusetts Institute of Technology, USA
/flh
Imperial College Press
Published by Imperial College Press 57 Shelton Street Covent Garden London WC2H 9HE Distributed by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: Suite 202,1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
First published in 2000 by Tata McGraw-Hill Publishing Company Limited Copyright © 2000 by Tata McGraw-Hill Publishing Company Limited
MATHEMATICAL PERSPECTIVES ON THEORETICAL PHYSICS A Journey from Black Hole to Superstrings Copyright © 2003 by Imperial College Press All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.
ISBN 1-86094-364-0 ISBN 1-86094-365-0 (pbk)
Printed by Fulsland Offset Printing (S) Pte Ltd, Singapore
"To the readers of the new millennium
PREFACE
This text, unlike others, did not grow out of seminar or classroom lectures; instead it grew out of the author's conviction that present day physicists and mathematicians should know the basics of string and superstring theories, just as they know calculus, linear algebra, geometry and analysis. To reach this goal, however, the theory that is pursued at research level in selected schools has to be made available to a wider audience. This is possible only if the teaching (and a text) focuses not only on the string and superstring theories but also provides: (i) the elements of all the prerequisites; (ii) an overview of other great theories that have preceded it (since it uses their phenomenology); and (iii) a motivational thread to reach the end goal. The present text is organized to fulfill these objectives. Since the target here is a much larger population of physicists and mathematicians, we avoid the mathematical rigor. No theorems (with few exceptions) are proved in the main text, in fact they are stated as 'Results' and 'Facts,' and their proofs (in important cases) are given as solutions to the exercises at the end of a section. This offers the reader (the teacher) the option of choosing the preferred level of in-depth/non-depth coverage (in a class). Besides this, the material is often presented using the two points of view (mathematics and physics) which makes the subject easily comprehensible. The first six chapters provide the mathematical background needed for the theory and thus fulfill the criterion (i). Chapter 0 gives the definitions in topology, differentiable manifolds, analysis and algebraic topology, and Chapter 1 explains the basics of the theory of complex functions, Riemann surfaces, and twodimensional conformal field theory. In Chapter 2 a quick review of group theory which includes algebraic, topological and Lie groups is given. A brief description of bundle theory from two points of view (mathematics and physics) is also part of this chapter. Chapter 3 is devoted to elementary operator theory with emphasis on spectral decomposition of Hermitian and unitary operators, on generalized Schrodinger and Dirac operators, and on the operators formed by the generators of the groups SU(2) and SU(3). Chapter 4 deals with the basics offinite-dimensionalalgebras. Solvable, semi-simple and simple Lie algebras along with their representations are studied, objects such as weights and root systems, Weyl groups, Cartan matrices and Dynkin diagrams are defined. Chapter 5 explains the intricacies of infinitedimensional algebras, in particular those of Kac-Moody algebras, and Heisenberg algebras. Using the Dynkin indices the generalized Casimir operator is defined, and vertex operators (needed especially in string theory) are introduced.
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Preface
Chapter 6, devoted to several aspects of symmetry (e.g., global and gauge) in nature and to symmetry breaking phenomena, is the first chapter that exposes the reader to particle physics. Examples based on different types of Lagrangians are used to explain various gauge theories, namely Maxwell's, YangMills' and GSW's. Chapter 7 is a brief review of all those objects that have emerged, ever since the notion of supersymmetry gained credence amongst physicists and mathematicians. The chapter begins with the definitions of Z2-graded algebras (superalgebras), Lie superalgebras, Clifford algebras and spinors (Dirac, Majorana and Weyl). The concepts of supersymmetry transformations, superspace, supermanifold, superscalar field and supervector field, etc., are introduced in order to write a Lorentzinvariant super Lagrangian. The question of renormalizability is addressed, though rather briefly (by using the Wess-Zumino gauge). The form calculus on supermanifolds and Berezin integrals are also included in this chapter. Chapter 8 gives an overview of the theories of gravitation, relativity and black holes. Beginning with Newton's laws and Einstein's free float frame, the principles of general relativity are explained. Wellknown exact solutions of Einstein's equation (e.g., Schwarzschild), the singularity theorems of Penrose and Hawking's black holes are then studied. Chapter 9 is devoted to quantum theories. Due to the vastness of the subject, it includes four appendices. It introduces the reader to principles of quantum mechanics, the so-called Schrodinger and Heisenberg pictures, the Dirac equation in a non-relativistic as well as relativistic field. The chapter then develops into Feynman's path integral formalism. The Feynman propagator, Green's function, the action principles in quantum mechanics and some examples based on path integrals are given. In Appendix D a brief review of quantum groups is also given. Chapter 10 is an introduction to Yang-Mills (YM) and Yang-Mills-Higgs (YMH) theories, in particular to those aspects that have resulted from applications of index theory, algebraic geometry, and algebraic topology to these two theories. A qualitative study of solutions of YM and YMH equations (known as instantons vortices and monopoles) is done in this chapter. A section on anomalies is also included. Chapter 11, devoted to strings and superstrings, introduces the reader to Regge trajectories, the NambuGoto action of a string, bosonic strings and their quantization, DDF operators, the No-ghost theorem, the Fadeev-Popov ghosts and ghosts in bosonic theory. Some global aspects of string world sheet, the world sheet supersymmetry and super Virasoro operators in string theory are described. The superVirasoro algebra, the anomaly, and superstrings as a theory of unification are briefly overviewed. From the above descriptions it should be apparent that Chapters 6, 8, 9 and 10 are in accordance with (ii). Chapter 7, on the other hand, provides the tools for describing superstring theory. Finally, the solved exercises and examples (approximately 250 in number), more than 60 illustrations, explanatory footnotes and appendices, and a large number of references provide the motivational thread towards our main goal: learning a synthesized theory of 'Mathematical Physics of the 21st Century.' The book fails to be 'consistent' with 'symbols.' The diversity of covered subjects made the 'consistency in symbols' rather impractical. Sometimes alphabets and notations have been used to represent different objects, whereas at other times the very same object (e.g., hermitian conjugate s h.c.) is denoted differently. A list of notations chapterwise and adequate footnotes for describing the symbols (wherever required) should help to alleviate this problem. Since the book comprises of chapters devoted to different subjects, one may have the impression that the chapters are separate entities. This, however, is not the case as one would find a large number of cross-references spread through these chapters. The repeated references which are indicated by decimal
Preface
ix
notation (for instance, Ref. 10.[5] in Chapter 3 stands for Ref [5] of Chapter 10) are another proof of a thread that binds the chapters. Every attempt has been made to make the book self-contained. A few concepts that remained undefined in former chapters as well as concepts e.g. p-branes, D-branes, and dualities that led to 'Second Superstring Revolution' and black holes in string theory, are explained at the end of the book in Appendix ll.B. Similarly some recent titles of interest and the references not covered earlier are added in the form of Reference Addendum at the end of 11 .B. The original inspiration for writing the book came from Raoul Bott's remark to the author. He thought the relation that existed between string theory and Schwartzian derivative used by the author in her work on projective structures was worth examining. A greater motivational inspiration from Isadore Singer that led to the planning of the book followed soon after; the author is indebted to him for sponsoring a visiting position at the MIT mathematics department. The author acknowledges the encouragement received from Sigurdur Helgason, Victor G. Kac, R. A. Gangolli, J. N. Kapur, M. Sharma, Jim Eggleston, and David Ferriero, and the help from Dennis Porche in administrative matters, from Sylvie Besett, Suli Rocha, and Nini Wang in partial typing, from Hong Shu in proof-reading and preparing the references, and from Hayden library staff in searching the material during the course of this long project. Heartfelt thanks are due to Ulrich Gerlach and Scott Axelrod for their precious time to give comments on portions of the book, as well as to those whose excellent texts and original works helped in writing the book. A conversation with Gerlach, Witten and Canizares, that led to the section on black holes is thankfully acknowledged. The author also wishes to acknowledge Giuseppe Castellacci's help in editing, and Jean Morris' outstanding typing of the final version of the book, and Angela Chang's help in typing the subject index. In spite of their good work, and author's commitment to accuracy of information there will still be some errors, for which the latter is fully responsible; she would appreciate, if they are brought to her attention. Finally, a word of gratitude to the staff of Tata McGraw-Hill, New Delhi for their patience and continued support during the writing of the book. It is hoped that the book in spite of its shortcomings will be a useful addition to the scientific literature.
NlRMALA PRAKASH
FOREWORD
I would like to record a few thoughts for the ICP print. I never anticipated that the path of this book would ultimately end where it began. Geetha Nair, Scientific Editor ICP asked for a copy of the book when it came out in 2000. After browsing through — she remarked that the book fulfilled the goals that it set for itself in the preface. Commissioning Editor Anthony Doyle was equally encouraging — and more. It was his patience and superb coordination with TMH and WS that brought about this unusual printing event. The past two years have seen new discoveries in Cosmology, Quantum theory and Superstrings. I was inclined to integrate these new developments, instead I chose to add just a few references (96105) on p. 798 as an aid to the reader, and postpone the integration for a revised edition. The book in its present form remains more than adequate precursor to everchanging landscape of scientific ideas. I hope that readers from all walks of life will find this simple-minded version of mathematical and physical theories along with their historical notes rather appealing. Finally, I wish to thank TMH management for graciously reliquishing their publication rights and for sharing the book with the global community, and to World Scientific for bringing it under their banner. I also take this opportunity to acknowledge James H. Wiborg's generosity for a grant — so that I continue my work in these areas.
Nirmala Prakash 11.19.2002
ACKNOWLEDGEMENTS
The author gratefully acknowledges the grant of permission (with no charges) for the use of material in this book from the following publishers: Cambridge University Press, New York, Princeton University Press, Princeton; W.W. Norton, New York; John-Wiley and Sons, New York; Springer-Verlag GmbH & Co., Heidelberg; Birkhaiiser-Verlag, Basel; Gordon and Breach Publishers, Switzerland; Elsevier Science, Oxford; Kluwer Academic Press, Netherland; Imperial College Press, London; D. Riedel Publishing Co., Holland; W.H. Freeman & Co., New York and World Scientific Publishing Co., Singapore. Ted Gerney of CUP, Loan Osborne of PUP and Sarah Feider of WWN, much against the conventional practice of their respective companies, waived the copyright fee upon the author's request. The author owes her sincere thanks to professors Raoul Bott, Loring Tu, Arthur Jaffe, Cifford Taubes, Christian Kassel, Brian Greene, M.B. Green, John H. Schwarz, Julius Wess, Ashok Das, Thomas Schiiker, Frank Miller Jr., Jan Louis, and Dr Stefan Forste, who graciously permitted her to use their intellectual work, and wished this book every success. NlRMALA PRAKASH
CONTENTS Preface Foreword Acknowledgements Chapter 0. Preliminaries
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1. Basic Definitions 1 2. Topology 1 Exercise (0.2) 7 Hints to Exercise (0.2) 8 3. Differentiable Manifolds 8 3.1 Differentiable Manifolds 8 3.2 Tangent Space 10 3.3 Vector Fields, Tensors and Tensor Fields 10 3.4 Riemannian Metric and Covariant Derivation 11 3.5 Geodesies, Jacobi Fields, Curvature and Torsion 13 4. Measure, Exp H, Dirac 5-function 14 4.1 Measurable Spaces and Measurable Functions 14 4.2 Haar Measure 15 4.3 The Space Exp H 15 4.4 Dirac 5-function 16 5. Examples Based on Differential Geometry 17 5.1 Critical Points 19 6. Basic Definitions in Algebraic Topology 21 6.1 de Rham Complex and de Rham Cohomology 21 6.2 Category and Functors 23 6.3 Mayer- Vietoris Sequence 24 6.4 Homotopy 24 References 26 Chapter 1. Complex Functions, Riemann Surfaces and Two-Dimensional Conformal Field Theory (an Introduction) 1. Complex Functions 27 1.1 Complex Plane 27 1.2 Analytic Function 27 1.3 Harmonic Functions 28 1.4 Laurent Series 29
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1.5 Simply Connected and Multiply Connected Domain 30 1.6 Residues and Poles JO 1.7 Elliptic Curves 32 1. Complex Structure on a Manifold, Kahler Metric 32 2.1 Complex Manifold M 32 2.2 Complex Structure on M 32 2.3 The Tangent and Cotangent Spaces to M 33 2.4 Holomorphic Vector Fields and Holomorphic Forms on M 33 2.5 Some Calculus on M 34 2.6 Kahler Manifold 36 2.7 Harmonic Forms on a Kahler Manifold 36 Exercise (1.2) 37 Hints to Exercise (1.2) 37 3. Riemann Surfaces 37 3.1 Riemann Surface M 37 3.2 Holomorphic Mappings on M 38 3.3 Differential Forms on M, their Algebra and Calculus 39 3.4 The Star (*) Operator on M 42 3.5 Harmonic and Holomorphic Forms on M 42 3.6 Square-integrable 1-forms on M 43 3.7 Abelian Differentials on M 45 3.8 A Few Results Based on Transformation Groups of M 46 Exercise (1.3) 49 Hints to Exercise (1.3) 50 4. The Two-Dimensional Conformal Field Theory 60 4.1 Conformal Group 60 4.2 Light-cone Formalism and the Lorentz Group 61 4.3 Euclidean Space Formalism 62 AA Two-dimensional Conformal Group 63 4.5 Mobius Transformation 64 4.6 Conformal Tensor Calculus 65 4.7 Conserved Currents 66 Exercise (1.4) 67 Hints to Exercise (1.4) 68 References 69 Chapter 2. Elements of Group Theory and Group Representations 1. Introduction 70 1.1 Definition of a Group, Examples, and Conjugate Classes 1.2 Invariant Subgroups, Factor Groups, Simple and Semi-simple Groups 72 1.3 Products of Groups and Homomorphism 72 2. Lie Groups and Topological Groups 73 2.1 Topological Groups 73 2.2 Algebraic Groups 77
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Exercise (2.2) 77 Hints to Exercise (2.2) 79 3. Basics of Group Representation 80 3.1 Relation Between Two Representations 80 3.2 Tensor Product of Representations 81 Exercise (2.3) 82 4. Specific Examples of Group Representation 82 Exercise (2.4) 85 Hints to Exercise (2.4) 86 5. The Theory of Bundles and Related Objects 86 Part A 86 5.1 Fiber Bundle, Bundle Morphism 86 5.2 Tangent Bundle 89 5.3 Lie Group of Transformations, One-parameter Subgroups of a Lie Group, Killing Vector Fields 90 5.4 Parallel Transport and Connection 92 5.5 The Linear and the Metric Connection, and the Torsion Form 94 PartB 95 5.6 Connection and Curvature on a Bundle from a Different Point of View 95 5.7 Associated Bundles 97 5.8 Affine Bundle and Affine Connection 97 5.9 Tensorial and Bundle-valued Forms 98 Exercise (2.5) 100 Hints to Exercise (2.5) 101 References 106 Chapter 3. A Primer on Operators 1. Definitions and Examples 109 1.1 Properties of a Linear Operator 110 1.2 Matrix Representation of a Linear Operator 111 1.4 List of Operators (Commonly in Use) 114 Exercise (3.1) 116 Hints to Exercise (3.1) 117 2. Eigenvalues and Eigenfunctions 119 2.1 The Resolvent and the Spectrum of an Operator 119 2.2 Examples and Results on Eigenvalues and Eigenfunctions of an Operator 119 2.3 Hermitian Operators 720 2.4 Properties of Commuting Operators 121 Exercise (3.2) 122 Hints to Exercise (3.2) 123 3. Some Properties of Operators 124 3.1 Projection Operators and their Properties 124 3.2 More on Hermitian and Unitary Operators 125
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Exercise (3.3) 127 Hints to Exercise (3.3) 128 4. The Spectral Decomposition 129 4.1 Results Based on Spectral Families of Operators 130 Exercise (3.4) 132 Hints to Exercise (3.4) 133 5. Group Theoretic Aspects of Operators 134 6. A Few Important Operators 135 6.1 Laplace Operator 135 6.2 The Riemarinian Measure 137 6.3 Operators other than A 139 6.4 Dirac Operator 139 Exercise (3.6) 140 Hints to Exercise (3.6) 141 7. Representations of SI/(2) and SU(3) Using the Theory of Operators 143 7.1 The Group SI/(2) 143 7.2 The Group SU (3) and its Irreducible Representations 145 Exercise (3.7) 148 Hints to Exercise (3.7) 148 References 152 Chapter 4. Basics of Algebras and Related Concepts 1. Some Definitions and Examples 153 1.1 Associative, Jordan and Lie Algebras 153 1.2 Lattices 154 1.3 Examples of Algebras, and the *- and C*-Algebra 155 1.4 Examples on Lie Algebra 156 Exercise (4.1) 157 Hints to Exercise (4.1) 158 2. Solvable and Semi-simple Lie Algebras 161 2.1 Lie Subalgebras, Ideals and Lattices 161 2.2 Semi-simple and Simple Lie Algebras and their Levi Decomposition 162 2.3 Lie Algebra of Derivations, Adjoint Mapping and Centralizer 163 2.4 Modules, Lower and Upper Central Series 164 Exercise (4.2) 165 Hints to Exercise (4.2) 165 3. Representations of Lie Algebras, Modules over Lie Algebras 767 3.1 Representations of a Lie Algebra 167 3.2 Representations via Modules over a Lie Algebra 168 3.3 Nilpotent Lie Algebras 769 3.4 Weight System and Roots of a Lie Algebra 770 3.5 Lexicographic Ordering, Simple and Highest Root, and Highest Weight 7 72
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Exercise (4.3) 174 Hints to Exercise (4.3) 174 4. Universal Enveloping Algebra, Weyl Group and Cartan Matrix 176 4.1 Universal Enveloping Algebra, Representations on Modules 4.2 Root Systems and the Weyl Group 178 4.3 Cartan Matrices 779 4.4 Dynkin Diagram 180 4.5 Casimir Element and Casimir Operator of L 182 Exercise (4.4) 183 Hints to Exercise (4.4) 184 References 759
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Chapter 5. Infinite-Dimensional Algebras 1. Lie Algebras Associated to Cartan Matrices 790 1.1 Generalized Cartan Matrix and its Realization 797 1.2 Construction of Kac-Moody Algebra g (A) and its Universal Enveloping Algebra 792 Exercise (5.1) 194 Hints to Exercise (5.1) 795 2. Affine Algebras: An Introduction 198 2.1 Construction of Affine Algebra 198 2.2 Derivations and the Affine Algebra 799 2.3 The Root Decomposition of g 200 2.4 Formulation of the Virasoro Algebra 207 2.5 The Chevalley Basis and the Casimir Element in Terms of the Chevalley Basis 203 2.6 Casimir Element of QJ 204 2.7 Canonical Generators of the Affine Algebra g 205 2.8 The Weyl Group of g 206 Exercise (5.2) 207 Hints to Exercise (5.2) 207 3. Modules and Representations 270 3.1 Highest Weight Modules 270 3.2 The Basic Representation of the Affine Algebra 277 4. Heisenberg Systems and Differential Operators 213 4.1 Heisenberg Systems 275 4.2 Fock Spaces Constructed from a Heisenberg System 214 4.3 The Canonical Representation 275 Exercise (5.4) 275 Hints to Exercise (5.4) 218 5. Creation and Annihilation Operators 279 5.1 Creation and Annihilation Operators on Fock Spaces 279 5.2 Hamiltonian in Terms of Creation and Annihilation Operators 220 5.3 Operators on the Fock Space Associated to Heisenberg System 227
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Exercise (5.5) 224 Hints to Exercise (5.5) 225 6. The Vertex and Virasoro Operators 226 6.1 The Vertex Operators 227 6.2 The Virasoro Operators 229 Exercise (5.6) 231 Hints to Exercise (5.6) 231 References 233 Chapter 6. The Role of Symmetry in Physics and Mathematics
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1. What is Symmetry? 234 2. Definitions and Descriptions 235 3. Exact Symmetries, Conservation Laws and Currents 238 3.1 Euler-Lagrange Equations and Currents 238 3.2 Conservation Law and the Conserved Charges as Generators of Symmetry Group 239 3.3 Examples 240 4. Gauge Symmetries—Their Origin 242 4.1 A Historical Perspective 242 4.2 Examples (Physicists' Point of View) 243 Exercise (6.4) 247 Hints to Exercise (6.4) 248 5. Examples of Theories with Gauge Symmetry 250 5.1 Maxwell and Yang-Mills Equations in Classical Form 251 5.2 Other Important Gauge Theories; Spontaneously Broken Symmetry 252 Exercise (6.5) 256 Hints to Exercise (6.5) 257 6. Bundle Theory Formalism in Gauge Theory 261 6.1 Principal Bundles as Tools in Gauge Theory 261 6.2 The Group Aut(P) of Generalized Gauge Transformations 262 6.3 The Gauge Algebra of P(M,G) and the Space of Gauge Potentials on it 264 6.4 The Moduli Space of Gauge Potentials on P(M, G) and Gribov-Ambiguity 266 7. More on Characteristics of Gauge Theories, and Examples Based on Them 267 7.1 A Generalized Maxwell's Field 268 7.2 A Generalized Yang-Mills Field 270 Exercise (6.7) 276 Hints to Exercise (6.7) 276 Table 6.1 Some Symmetries and Conservation Laws 277 References 277 Chapter 7. All That's Super-An Introduction 1. Graded-Algebras 279
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4.
5.
6.
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1.1 Superalgebras and Lie Superalgebras 279 1.2 Other Important Superalgebras and Bose and Fermi Sectors 281 Exercise (7.1) 284 Hints to Exercise (7.1) 284 TheSpinors 285 2.1 The Definitions and Properties of Spinors 285 2.2 Clifford Algebras and Spinors 286 2.3 Dirac, Majorana and Weyl Spinors 289 Exercise (7.2) 290 Hints to Exercise (7.2) 297 More on Spinors 292 3.1 The Poincare Superalgebra 292 3.2 Lorentz Invariance 293 3.3 Dirac Matrices and Dirac and Majorana Spinors 294 Exercise (7.3) 297 Hints to Exercise (7.3) 298 Supersymmetry Algebras and Introduction to Superspaces 301 4.1 Supernumbers 301 4.2 Superanalytic Functions 302 4.3 Real and Imaginary Supernumbers 304 4.4 Supervector Spaces 304 4.5 Supermanifolds; Charts and Atlases 306 4.6 Supersymmetry Generators and Construction of Superalgebras from First Principles 307 4.7 Supersymmetry Transformations on a Superspace 310 Exercise (7.4) 311 Hints to Exercise (7.4) 312 The Calculus on Superspace, the Component Fields and Superfields 313 5.1 Infinitesimal Generators and Covariant Vector Fields 313 5.2 Component Multiplets and Superfields 315 Exercise (7.5) 320 Hints to Exercise (7.5) 321 Differential Forms and Gauge Transformations on Superspaces 328 6.1 Differential Forms 328 6.2 The Gauge Invariant Lagrangian in Superspace 330 6.3 Supergauge Transformations 332 Exercise (7.6) 334 Hints to Exercise (7.6) 335 The Basics of Integration and Conformality in Superspaces 338 1.1 Integration on Superspace 339 7.2 Variation of a Superfield 340 13 Superconformal Transformations 341 Exercise (7.7) 345 Hints to Exercise (7.7) 345
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Appendix 7A 350 A.O Notations and Pauli Matrices 350 A. 1 Standard Bases and Components of a Supervector 351 A.2 Contravariant Vector-fields on Supermanifold M 352 A.3 Super Lie Groups 352 A.4 Conventional Super Lie Groups 353 A.5 Exponential Mapping 353 A.6 Conventions on Structure Constants 354 References 354 Chapter 8. Gravitation, Relativity and Black Holes 1. Gravitation (from Newton to Einstein) and an Overview of Special Relativity 356 1.1 Newton's Theory of Gravitation and his Famous Laws 357 1.2 Einstein's Proposal—the Free-float Frame and the Observer 358 1.3 Acceleration and Spacetime Curvature 359 1.4 The Coordinate Transformations: Distinction Between the Galilean and Special Relativity Theory 360 1.5 Equations of Motion in Newtonian Mechanics 361 1.6 Special Relativity 362 Exercise (8.1) 367 Hints to Exercise (8.1) 367 2. The Einstein Universe 371 2.1 The Mathematical Model 372 2.2 The Matter Fields 373 2.3 Postulate (a): Local Causality 373 2.4 Postulate (b): Local Conservation of Energy and Momentum 374 2.5 Construction of the Energy-momentum Tensor Tab 374 2.6 The Field Equations 376 2.7 Postulate (c): Field Equations 380 Exercise (8.2) 380 Hints to Exercise (8.2) 381 3. Curvature and Energy Conditions 387 3.1 The Separation Vector, Vorticity, Shear and Expansion 387 3.2 Energy Conditions 393 3.2.1 The Weak Energy Condition 393 3.2.2 The Dominant Energy Condition 395 3.3 Results Based on Energy Conditions 395 3.4 Conjugate Points 397 3.5 Results Based on Curvature, Conjugate Points and the Expansions 9,9 398 3.6 Variational Techniques 400 4 Exact Solutions, and the Causal Structure 403 4.1 An Exact Solution 403 4.1.1 Minkowski Spacetime 404
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4.1.2 de-Sitter and Anti-de Sitter Spacetimes 407 4.1.3 Robertson-Walker Space 412 4.1.4 The Schwarzschild and the Reissner-Nordstrom Solution 413 4.1.5 The Kerr Solution 415 4.1.6 Gbdel's Universe 416 AA.l Taub-NUT and Misner Spaces 416 4.2 Causal Structure 418 4.2.1 Orientability 418 4.2.2 Chronological and Causal Future 418 4.2.3 Horismos and Achronal Sets 419 4.2.4 The Concept of Imprisonment 420 4.2.5 Cauchy Developments 421 Exercise (8.4) 422 Hints to Exercise (8.4) 422 5. The Basics of Spacetime Singularities and Black Holes 429 5.1 Singularities and Completeness in Spacetime 429 5.2 Black Holes 433 Exercise (8.5) 441 Hints to Exercise (8.5) 442 Appendix 8A 443 A. 1 Spatially Homogeneous 443 A.2 Geodesically Complete 443 A.3 Normal Coordinates 443 A.4 Open or Closed Universe 443 A.5 Cavendish Constant Gc 444 A.6 Closed Trapped Surface 444 A.7 Particle Horizon 445 A.8 Event Horizon 446 References 446 Chapter 9. Basics of Quantum Theory 1. Introduction 449 2. Passage from Classical to Quantum 450 2.1 The Concept of Amplitude, Observable, and Hamiltonian 451 2.2 Symmetry Group of the Motion of a Particle in 1-Dimension 452 2.3 Two-body Problem with Spherically Symmetric Potential 454 2.4 The Radial Hamiltonian of the Two-body Problem 455 2.5 The Relation between Schrddinger and Heisenberg Equations 457 Table 9.2.1 Classical and Quantum Mechanics 458 Exercise (9.2) 458 Hints to Exercise (9.2) 459 3. Quantum Mechanical Equations and Related Concepts 459 3.1 Hamiltonian in a Relativistic Field, and Klein-Gordon Equation 460 3.2 The Dirac Equation 461
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3.3 Commuting Observables for a Free Relativistic Dirac Particle 464 3.4 The Relationship Between Free Klein-Gordon and Dirac Particles 465 3.5 The Dirac Equation in Rest Frame 466 3.6 The Feynman-Gell-Mann Reduction 467 Exercise (9.3) 468 Hints to Exercise (9.3) 468 4. Gauge Field Quantizations 471 4.1 Feynman's Functional Integral 472 4.2 Functional Integral of a Scalar Field 473 4.3 Green's Function and Generating Functional 474 A A Diagram Technique for Scalar Field Theory 477 4.5 Functional Integral Approach to Bose and Fermi Fields 479 Exercise (9.4) 481 Hints to Exercise (9.4) 481 5. Path Integrals 483 5.1 Path Integral via Operator Formalism 483 5.2 Time Ordered Product of Operators 486 5.3 Correlation Functions Using an External Source J 488 5.4 Vacuum Functional Z[J] and Green's Functions in the Vacuum 489 5.5 Effective Action W[J] 491 5.6 Path Integral Approach to Field Theory 494 5.7 Pi-formalism and Field Theories (with Infinite Degrees of Freedom) 495 5.8 The Faddeev-Popov Ansatz 499 Exercise (9.5) 501 Hints to Exercise (9.5) 502 6. Feynman Graphs 507 6.1 Connected Diagrams 508 6.2 Effective Functional and Feynman Graphs with Vertex-functions 513 Exercise (9.6) 518 Hints to Exercise (9.6) 518 Appendix 9A: Language of Quantum Mechanics 521 A.I State Space, Kets and Bras, Hermitian Operators and Observables 521 A.2 Position and Momentum Operators of a Particle 523 A. 3 Coordinate and Momentum Space Representations 525 A.4 The Complete Set of Commuting Operators 526 Appendix 9B: A Few Definitions and Derivations 527 B.I The Wave Function ynn Quantum Mechanics 527 B.2 The Hamiltonian Operator H{i), and the Time Evolution Operator. U(t) 529 B.3 Dynamical Laws 531 Exercise (9B) 532 Hints to Exercise (9B) 532 Appendix 9C: Tools of Physical Theories 533 C. 1 Test Functions and Distributions 533
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C.2 Properties of Distributions with Respect to Operations on Them 534 C.3 Green's Functions 536 C.4 Fourier Transforms and Related Objects 539 C.5 Functionals and their Calculus 545 Exercise (9C) 548 Hints to Exercise (9C) 548 Appendix 9D: Quantum Groups 554 D.I Algebra, Coalgebra, Bialgebra and Hopf Algebra 554 D.2 The Quantum Plane, the Algebra Mq (2), and Hopf Algebras GLq{2), SLq(2\ Uq{sl{2)) 559 Exercise (9D) 565 Hints to Exercise (9D) 566 References 569 Chapter 10. Theory of Yang-Mills and The Yang-Mills-Higgs Mechanism 1. Introduction 571 2. Yang-Mills and Yang-Mills-Higgs Functional 572 2.1 Yang-Mills-Higgs Action in R" and R"'1 572 2.2 The Variational Equations and Solutions 573 2.3 Instantons, Vortices and Monopoles 573 2.4 An Example on Instantons 574 2.5 An Example on Vortices 576 Exercise (10.2) 579 Hints to Exercise (10.2) 579 3. Self-duality in Yang-Mills Theory and Instantons 584 3.1 Self-duality in 4-dimensions 584 3.2 Examples of Self-(Anti-Self) Dual Manifolds 585 3.3 Self-dual Connection on a 4-manifold 585 3.4 Self-duality in Spinor-bundles 586 3.5 Quaternions and Yang-Mills' Instanton 586 3.6 The BasiG Instanton and its Asymptotic Form 588 3.7 Anti-instanton in Asymptotic Gauge 589 3.8 Application of Conformal Transformations to Basic Anti-instanton 3.9 Construction of Multi-instantons 590 3.10 Projective Spaces and Instantons 592 Exercise (10.3) 594 Hints to Exercise (10.3) 595 4. More on Monopoles 597 4.1 Yang-Mills-Higgs'Configuration Space 597 4.2 Bogomolny Equations 598 4.3 The Solution Space of Monopoles 601 4.4 The Scattering and Spectral Curve 603 4.5 The Metric on Mk 605 4.6 Monopoles in Coordinate Form 607
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Contents
Exercise (10.4) 613 Hints to Exercise (10.4) 674 5. More on Vortices 618 5.1 Characterization of Superconductivity 678 5.2 Superconductivity and Multivortex Configuration 678 5.3 Vortices when A= 1 627 5.4 Some Existence Theorems in the Complex Framework 623 Exercise (10.5) 628 Hint to Exercise (10.5) 628 6. Anomalies 628 6.1 Renormalization (The Physicist's Approach) 628 6.2 Anomaly as an "Obstruction" (A Mathematician's Point of View) 634 6.3 Anomaly as a Welcome Phenomenon 636 6.4 Chiral Gauge Theories 637 6.5 Construction and Computation of an Anomaly 637 Exercise (10.6) 645 Hints for Exercise (10.6) 645 Appendix 10A: Glossary 648 A.I The Projective Space Pn (C) 648 A.2 Vector Bundles over Pn ( 0 if x * y, p(x, y) = 0 if x = y; (ii) p(x, y) = p(y, x): and (iii) p(x, y) < p(x, z) + p(z, y) for any z G X. If A is a subset of metric space (X, p), we call and denote the diameter of A and the distance from x to A respectively by:
diam A = sup [p(x, y): x, ye A] dist (JC, A) = inf {p(x, y) : y e A}. The notation B(x; r) is used to denote a closed ball centred at x with radius r > 0, thus, B{x;r)=[ye
X: p(x, y) < r ) .
The subset of this ball given by {y e X: p{x, y) < r) is called an open ball and is denoted Bo (JC; r). A set 11 in space X is said to be open if for each point x e 11 there exists an open ball centred at x and contained in 11. It is easy to verify that this defines a topology on X. Given a topological space X, we call it metrizable if there exists a metric p such that the open balls form a basis for the topology; such a metric is said to be compatible with the topology on X. It can be checked that the metric p is translation invariant in this case, i.e., p(x, y) = p(x + z,y + z) for x, y, z e X. Recall that the symbol A denotes the closure of A in X. If X has a topology T other than the one induced by the metric, we use A T to denote the closure of A in {X, T). Definition 0.2.9: A subset C c X of a t.v.s. is convex if for any JC, y e C, A x + (1 - A) y e C, where A e [0,1]. Given A c X, the convex: /iw// of A denoted conv A is the smallest convex subset of X that contains A, thus: conv A = n {£ c X : K 3 A, K is convex}. n
n
Thus x e conv A if and only if x = £ A; *,- where JC; e A and X ^, = 1- The closure of conv A is 1=1
i=i
conv A. A well known theorem states: If A is compact so is conv A. Definition 0.2.10: A real valued mapping on a vector space X defined over IR or C (K stands for real or complex number field) is called a norm (denoted || ||) if it satisfies: (a) || Ax|| = |A| ||JC|| for all A e K, x e X; (b) jjjc + v|| < ||x|| + ||y||; and (c) ||x|| = 0 implies x = 0. A normed space is thus a pair (X, || ||). It is easy to check that a distance function between a pair of elements can be defined on this space by the rule: {x, y) —> [|JC - y||. Consequently the vector space X can be given a topology. Hence a normed vector space is a topological vector space. Definition 0.2.11: In a normed linear space X, a sequence {xn} is said to be convergent if there exists an element x in X such that | | x n - x\\ —> 0. The sequence {xn] is said to converge to the element x.
4 Mathematical Perspectives on Theoretical Physics
Definition 0.2.12: A sequence {xn} in X is said to be a Cauchy sequence if given £ > 0 we can find an integer N(e) such that \\xn — xm\\ < e for all n, m > N(e). Evidently every convergent sequence is a Cauchy sequence, but the converse is not always true. Definition 0.2.13: A normed linear space in which every Cauchy sequence is a convergent sequence is said to be complete, and it is called a Banach space after the Polish mathematician Stefan Banach. One of the important spaces that is used in quantum theories is the Hilbert space. In order to define it next we describe the space from which it follows. Pre-Hilbert space: If H is a (complex or real) vector space and < .,. > is a non-degenerate scalar product on H, then we call the pair (H, < . , . > ) a vector space with scalar product or a pre-Hilbert space. Every pre-Hilbert space carries a norm in a natural way, the norm being | | / j | = ( / , / ) l y ' 2 where / € H. If in addition every Cauchy sequence is convergent, then this space in view of Def. (0.2.13) is complete. A complete pre-Hilbert space is called a Hilbert space (see Def. (0.2.14)). Examples of Norm: In Cm (or Rm) define the norms as m
(0 ll/lli = XlJ3 (ii)
| | / | | . . = max { | / - | : i = l ,
-,m}
(iiD 11/11= jlU-lj 2 The last of these gives the Euclidean length1 of the vector/= (fx,---,fm) e C m (Rm), and | | / - g\\ here is the Euclidean distance of points/and g. Let (H,( ., .)) be a pre-Hilbert space. A family M - {ea: a e an index set A} of elements from H is called an orthonormal system (ONS) if ( ea,-ep)
>= 8afifor
a, j8 e A.
An orthonormal system M is called an orthonormal basis (ONB) of a subspace T of H, if M is total in T (i.e., M c Tand L(M)
:D T). If M is an ONB of H, then M is called an orthonormal basis of H.
Definition 0.2.14: A complete normed space whose norm || || is given by a scalar product ( ) is called a Hilbert space. More explicitly a space X is Hilbert if to each-pair of elements x, y in X there is associated a scalar {x, y ) that satisfies: (i) ( ax xx + «2 x2, y ) = ax ( xx, y) + a2 ( x 2 , y )
(ii) ( x , y ) = (y, x) (the complex conjugate) (iii) (x,x) 2
= (|| JC||) is positive definite when x ^ 0.
L(M) = closure of the linear hull L (M) (the set of finite linear combinations of elements of M, or in other words, the smallest subspace of H that contains M).
Preliminaries 5
The scalar product defined above is called a positive definite Hermitian form. It is usual to denote a Hilbert space by 9i. We shall use this notation throughout the text. It is important to note here that elements of Hare arbitrary (e.g., real or complex numbers or real valued or complex valued functions) and scalar product (as obvious from (ii)) is not necessarily real. It can also be checked that the norm and the scalar product on J/'are related by the Schwartz inequality: (iv) | U y ) | £ | H | | | y | | and by the polarization identity: (v) 4( x, y) = \\x+y\\2 - \\x- y\\2 + i\\x + iy\\2 - i\\x -
iyf.
The notion of completeness in 9i means that if a sequence of elements {(j)n} in !H satisfies the condition 110/1 ~ 0mll —> 0 for m, « —> °°, then there exists an element
0 for n —> °°. this context we would like to mention that there are two types of convergence in ?/(in fact in a normed linear space), the strong and the weak. The sequence of elements {0,,} converges to the element (f> strongly if as n —> °°
ik-0ii->o and weakly to an element <j)' if for each element y/ e 9{
The "strong convergence" is also called "convergence in the mean" and it implies:
IKII-HMIEvidently strong convergence implies weak convergence. Let u,veJ{, we say that these elements are orthogonal if ( u, v ) = 0. Suppose that H is a subset of 9(, the set of all elements we 9{fox which (u,v) = 0, v e H forms a subspace of"K,we denote this by HL. If the subset H is the whole space 9{, then the space H' consisting of all elements u in #"such that u e !HL is called the null space of the Hermitian product. We shall in general deal with a separable Hilbert space. Its dimension is either finite or is denumerably infinite. In the former case, we shall sometimes use a familiar nomenclature a unitary space of dimension n. Even when the space is infinite dimensional, an orthonormal basis can be obtained. Since it can be shown that there always exists an infinite complete orthonormal sequence, that can be obtained by applying Gram-Schmidt orthogonalization process to a complete denumerable subset of 9{. We give below a few examples to illustrate the objects that have been defined above. The numbering of the example has an additional letter a, b, etc., added to the definition of the object it represents. Example 0.2.1a: Let X = R be the set of real numbers. Let a subset U of R be called open if for each point x in U there exists an open interval / containing x and contained in U . Obviously R with the set 11 formed by open sets U becomes a topological space. Example 0.2.1b: Let X = {a, b, c, d, e,f) and let 11 be the set formed by subsets (assumed as open) [{a, b, c, d}, {a, b, e,f), {c, d, e,f), 0 , X]. It can be easily checked that X together with 11 forms a topological space. Example 0.2.1c: Among the simplest examples of a compact and non-compact set are respectively an open or a closed disc and an extended plane as shown in Fig. (0.1). Note that the topological space of (0.2.1a) is non-compact and that of (0.2.1b) is compact.
6
Mathematical Perspectives on Theoretical Physics
• Q mg
|x] < r, open disc v-tv
|x| < r, closed »ICD2
Plane extended to infinity on either side
x = ^X-|, X2) fc n
l ^ ^ ^ f l Open and closed discs, and extended plane Example 0.2.1d: Connected and disconnected topological spaces are visually represented as follows.
I^SCTj Connected and disconnected topological spaces Also note that an open, half open, or closed interval / cz R is always connected. Example 0.2.2a: The set Q of rationals is dense in the set R of real numbers. Example 0.2.3a: An open interval (a, b) is a neighbourhood of each of its points. A closed interval [a, b] is a neighbourhood of each point of (a, b). As can be easily seen, [a, b] is not the neighbourhood of a or b. Example 0.2.3b: The set R of real numbers is the neighbourhood of each of its points. The set Q of rational numbers is not the neighbourhood of any of its points. Example 0.2.5a: If the set U of subsets in Exp. (0.2.1b) is replaced by [{a} {b} ••• { / } ; {a, b] •••; {a, b, c} •••; {a, b, c, d] •••; {a, b, c, d, e} • • ; 0 , X], the topology defined will be the discrete topology. Example 0.2.6a: Every metric space is Hausdorff, for if d(x, y) = S> 0 defines a metric on X, then the sets defined as: Va : = {x\d(a, x) < 5/2}, Vh : = {x\d(b, x) < 6/2} a * b are disjoint neighbourhoods of points a, b in X. Example 0.2.6b: A topological space endowed with the discrete topology is a Hausdorff space. Example 0.2.7a: The cartesian product Rn = R x • • • x R over the field of reals is the simplest example of a topological vector space; the vector in this case is x = (xlt ••• , xn), the mappings Pl and P2 are pointwise addition and scalar multiplication of vectors e R". Example 0.2.7b: A similar example is offered by the collection of nxn of real or complex numbers.
matrices defined over the field
Example 0.2.8a: Let X = R2 be a topological space whose metric is the Euclidean distance: p(x, y) : = •/(*! - y\ ) 2 + (x2 ~ y2 ) 2 •
Preliminaries 7
Denote it by (X, T). Define another metric p' on X = R2 given by: p'{x,
y): = max {|*, - y,|, \x2-
y2\}.
It can be easily verified that both p and p ' given above satisfy the postulates of a metric on R2 and the two topologies defined by them are the same, i.e., 1(p) = T(p'), in the sense that every 5-ball constructed for T(p) can be shown to be contained in a ball in T(p') and vice-versa. Example 0.2.10a: Let X be the space of continuous functions that are defined on the interval (a, b) and let m denote a general measure on X. The mapping / s u c h that
f^{jjf{x)\» dXy {l\f(x)\"dm) defines a norm for 1 < p < °°. The space X is called a Lp-space. Example 0.2.11a: Let X be the real line R, the sequence defined by fn (x) = —j=— (x e R, n = 1, 2, 3, • • •) is a convergent sequence, since f(x) = lim fn (x) = 0. But the sequence given by f'n (x) = -Jri H-»°o
cos rix is not convergent to any / ' , since f'n (0) = 4n
—> °° whereas / ' (0) = 0.
1
Example 0.2.13a: The vector space R and the space C(K) of continuous functions defined on a compact set K of R1 are complete spaces (obviously it is true when 1 is replaced by n). Example 0.2.13b: The space of square (summable) functions defined on a measurable space (Exp. 2.10a for p = 2) is a complete space. Example 0.2.14a: The space formed by arbitrary sequences (
which is linear in the first variable and is semilinear in the second variable ({ x, ay) a (x,y)). 2. Let the function u i-> ||M|| be a seminorm on X i.e., \\u\\ > 0 for any u € !tf. Show that \\u\\ = 0 if and only if u e 9{Q the null space corresponding to the Hermitian product. 3. Prove property (iv) of Def. (0.2.14) by choosing R as the Hilbert space. 4. Let (X, {Xa}, T) be a topological space and let Y be a subset of X. A topology formed by open sets Oa = Y n X a of Y is called a subspace (or relative) topology on Y and is denoted as TY.
8 Mathematical Perspectives on Theoretical Physics
Show that the real line R c R2 has the relative topology which is induced by the usual topology on R2 given by the Euclidean metric. 5. Show that the sequence {sin nx}\ 0 < x < it converges weakly to 0 but does not converge in the mean. 6. Given an orthonormal sequence {n will converge in the mean l
l
to an element g of the Z2-space where g satisfies: (g,
<j)n) = an{n = 1, 2 , 3 , •••)
(Riesz-Fischer theorem).
Hints to Exercise 0.2 1. Since {x, y) = (y, x), we have ( x, a1yl + a2y2 ) = {^I^I entry implies ax{yx,
x) + a2{y2,x)
= a , {yl,x)
+ a2
+ a
2y2^x)
(y2,x)
= al
anc
* linearity in the first
{ x, yx ) + a j (x,
y2).
2
3. Consider the function <j): R - » R defined by <j>(t) = ||JC + fy|| = (x + ty, x + ty) = (x, x) + 2t( x, y ) + t2( y, y ) . Since
