Quantum Theory, Deformation and Integrability
North-Holland (Verlag)
978-0-444-50621-4 (ISBN)
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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.
1. QUANTIZATION AND INTEGRABILITY1.1 Algebraic and Geometric Methods1.2 Vertex Operators and Coherent States1.3 Remarks on the Olavo Theory1.4 Trajectory Representations1.5 Miscellaneous2. GEOMETRY AND EMBEDDING2.1 Curves and Surfaces2.2 Surfaces in R3 and Conformal Immersion2.3 Quantum Mechanics on Embedded Objects2.4 Willmore Surfaces, Strings and Dirac2.5 Conformal Maps and Curves3. CLASSICAL AND QUANTUM INTEGRABILITY3.1 Background3.2 R Matrices and PL Structures3.3 Quantization and Quantum Groups3.4 Algebraic Bethe Ansatz3.5 Separation of Variables3.6 Hirota Equations3.7 SOV and Hitchin Systems3.8 Deformation Quantization3.9 Miscellaneous3.10 Summary Remarks4. DISCRETE GEOMETRY AND MOYAL4.1 Introduction4.2 Hirota, Strings, and Discrete Surfaces4.3 A Few Summary Remarks4.4 More on Phase Space Discretization5. WHITHAM THEORY5.1 Background5.2 Isomonodromy Problems5.3 Whitham and Seiberg-Witten5.4 Soft Susy Breaking and Whitham5.5 Renormalization5.6 Whitham, WDVV, and Picard-Fuchs6. GEOMETRY AND DEFORMATION QUANTIZATION6.1 Noncommutative Geometry6.2 Gauge Theories6.3 Berezin Toeplitz QuantizationBibliography. Index.
| Erscheint lt. Verlag | 9.11.2000 |
|---|---|
| Reihe/Serie | North-Holland Mathematics Studies |
| Sprache | englisch |
| Gewicht | 900 g |
| Themenwelt | Naturwissenschaften ► Physik / Astronomie ► Mechanik |
| Naturwissenschaften ► Physik / Astronomie ► Quantenphysik | |
| ISBN-10 | 0-444-50621-7 / 0444506217 |
| ISBN-13 | 978-0-444-50621-4 / 9780444506214 |
| Zustand | Neuware |
| Informationen gemäß Produktsicherheitsverordnung (GPSR) | |
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