Quantum Theory, Deformation and Integrability (eBook)

R. Carroll (Autor)

eBook Download: PDF

2000 | 1. Auflage
420 Seiten
Elsevier Science (Verlag)
978-0-08-054008-5 (ISBN)

Infos

About four years ago a prominent string theorist was quoted as saying that it might be possible to understand quantum mechanics by the year 2000. Sometimes new mathematical developments make such understanding appear possible and even close, but on the other hand, increasing lack of experimental verification make it seem to be further distant. In any event one seems to arrive at new revolutions in physics and mathematics every year. This book hopes to convey some of the excitment of this period, but will adopt a relatively pedestrian approach designed to illuminate the relations between quantum and classical. There will be some discussion of philosophical matters such as measurement, uncertainty, decoherence, etc. but philosophy will not be emphasized, generally we want to enjoy the fruits of computation based on the operator formulation of QM and quantum field theory. In Chapter 1 connections of QM to deterministic behavior are exhibited in the trajectory representations of Faraggi-Matone. Chapter 1 also includes a review of KP theory and some preliminary remarks on coherent states, density matrices, etc. and more on deterministic theory. We develop in Chapter 4 relations between quantization and integrability based on Moyal brackets, discretizations, KP, strings and Hirota formulas, and in Chapter 2 we study the QM of embedded curves and surfaces illustrating some QM effects of geometry. Chapter 3 is on quantum integrable systems, quantum groups, and modern deformation quantization. Chapter 5 involves the Whitham equations in various roles mediating between QM and classical behavior. In particular, connections to Seiberg-Witten theory (arising in N = 2 supersymmetric (susy) Yang-Mills (YM) theory) are discussed and we would still like to understand more deeply what is going on. Thus in Chapter 5 we will try to give some conceptual background for susy, gauge theories, renormalization, etc. from both a physical and mathematical point of view. In Chapter 6 we continue the deformation quantization then by exhibiting material based on and related to noncommutative geometry and gauge theory.

Inhalt

Front Cover 1
Quantum Theory, Deformation and Integrability 4
Copyright Page 5
Contents 6
Preface 10
CHAPTER 1. QUANTIZATION AND INTEGRABILITY 14
1.1 ALGEBRAIC AND GEOMETRIC METHODS 14
1.2 VERTEX OPERATORS AND COHERENT STATES 36
1.3 REMARKS ON THE OLAVO THEORY 58
1.4 TRAJECTORY REPRESENTATIONS 62
1.5 MISCELLANEOUS 69
CHAPTER 2. GEOMETRY AND EMBEDDING 76
2.1 CURVES AND SURFACES 76
2.2 SURFACES IN R3 AND CONFORMAL IMMERSION 86
2.3 QUANTUM MECHANICS ON EMBEDDED OBJECTS 99
2.4 WILLMORE SURFACES, STRINGS, AND DIRAC 111
2.5 CONFORMAL MAPS AND CURVES 123
CHAPTER 3. CLASSICAL AND QUANTUM INTEGRABILITY 126
3.1 BACKGROUND 126
3.2 R MATRICES AND PL STRUCTURES 132
3.3 QUANTIZATION AND QUANTUM GROUPS 137
3.4 ALGEBRAIC BETHE ANSATZ 143
3.5 SEPARATION OF VARIABLES 149
3.6 HIROTA EQUATIONS 156
3.7 SOV AND HITCHIN SYSTEMS 160
3.8 DEFORMATION QUANTIZATION 162
3.9 MISCELLANEOUS 173
3.10 SUMMARY REMARKS 177
CHAPTER 4. DISCRETE GEOMETRY AND MOYAL 180
4.1 INTRODUCTION 180
4.2 HIROTA, STRINGS, AND DISCRETE SURFACES 207
4.3 A FEW SUMMARY REMARKS 240
4.4 MORE ON PHASE SPACE DISCRETIZATION 247
CHAPTER 5. WHITHAM THEORY 268
5.1 BACKGROUND 268
5.2 ISOMONODROMY PROBLEMS 283
5.3 WHITHAM AND SEIBERG-WITTEN 306
5.4 SOFT SUSY BREAKING AND WHITHAM 315
5.5 RENORMALIZATION 322
5.6 WHITHAM, WDVV, AND PICARD-FUCHS 327
CHAPTER 6. GEOMETRY AND DEFORMATION QUANTIZATION 338
6.1 NONCOMMUTATIVE GEOMETRY 338
6.2 GAUGE THEORIES 354
6.3 BEREZIN TOEPLITZ QUANTIZATION 372
Bibliography 376
Index 414

Produkt-Details

• Erscheint lt. Verlag: 9.11.2000
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Mechanik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Quantenphysik
• Themenwelt: Technik
• ISBN-10: 0-08-054008-2 / 0080540082
• ISBN-13: 978-0-08-054008-5 / 9780080540085