Quantum Theory, Deformation and Integrability

Description

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. Front Cover Quantum Theory, Deformation and Integrability Copyright Page Contents Preface CHAPTER 1. QUANTIZATION AND INTEGRABILITY 1.1 ALGEBRAIC AND GEOMETRIC METHODS 1.2 VERTEX OPERATORS AND COHERENT STATES 1.3 REMARKS ON THE OLAVO THEORY 1.4 TRAJECTORY REPRESENTATIONS 1.5 MISCELLANEOUS CHAPTER 2. GEOMETRY AND EMBEDDING 2.1 CURVES AND SURFACES 2.2 SURFACES IN R3 AND CONFORMAL IMMERSION 2.3 QUANTUM MECHANICS ON EMBEDDED OBJECTS 2.4 WILLMORE SURFACES, STRINGS, AND DIRAC 2.5 CONFORMAL MAPS AND CURVES CHAPTER 3. CLASSICAL AND QUANTUM INTEGRABILITY 3.1 BACKGROUND 3.2 R MATRICES AND PL STRUCTURES 3.3 QUANTIZATION AND QUANTUM GROUPS 3.4 ALGEBRAIC BETHE ANSATZ 3.5 SEPARATION OF VARIABLES 3.6 HIROTA EQUATIONS 3.7 SOV AND HITCHIN SYSTEMS 3.8 DEFORMATION QUANTIZATION 3.9 MISCELLANEOUS 3.10 SUMMARY REMARKS CHAPTER 4. DISCRETE GEOMETRY AND MOYAL 4.1 INTRODUCTION 4.2 HIROTA, STRINGS, AND DISCRETE SURFACES 4.3 A FEW SUMMARY REMARKS 4.4 MORE ON PHASE SPACE DISCRETIZATION CHAPTER 5. WHITHAM THEORY 5.1 BACKGROUND 5.2 ISOMONODROMY PROBLEMS 5.3 WHITHAM AND SEIBERG-WITTEN 5.4 SOFT SUSY BREAKING AND WHITHAM 5.5 RENORMALIZATION 5.6 WHITHAM, WDVV, AND PICARD-FUCHS CHAPTER 6. GEOMETRY AND DEFORMATION QUANTIZATION 6.1 NONCOMMUTATIVE GEOMETRY 6.2 GAUGE THEORIES 6.3 BEREZIN TOEPLITZ QUANTIZATION Bibliography Index