Quantum and Non-Commutative Analysis: Past, Present and Future Perspectives / Edition 1by Huzihiro Araki
Pub. Date: 11/30/1993
Publisher: Springer Netherlands
In the past decade, there has been a sudden and vigorous development in a number of research areas in mathematics and mathematical physics, such as theory of operator algebras, knot theory, theory of manifolds, infinite dimensional Lie algebras and quantum groups (as a new topics), etc. on the side of mathematics, quantum field theory and statistical mechanics on the side of mathematical physics. The new development is characterized by very strong relations and interactions between different research areas which were hitherto considered as remotely related. Focussing on these new developments in mathematical physics and theory of operator algebras, the International Oji Seminar on Quantum Analysis was held at the Kansai Seminar House, Kyoto, JAPAN during June 25-29, 1992 by a generous sponsorship of the Japan Society for the Promotion of Science and the Fujihara Foundation of Science, as a workshop of relatively small number of (about 50) invited participants. This was followed by an open Symposium at RIMS, described below by its organizer, A. Kishimoto. The Oji Seminar began with two key-note addresses, one by V.F.R. Jones on Spin Models in Knot Theory and von Neumann Algebras and by A. Jaffe on Where Quantum Field Theory Has Led. Subsequently topics such as Subfactors and Sector Theory, Solvable Models of Statistical Mechanics, Quantum Field Theory, Quantum Groups, and Renormalization Group Ap proach, are discussed. Towards the end, a panel discussion on Where Should Quantum Analysis Go? was held.
Table of ContentsForeword and Preface; H. Araki, K.R. Ito, A. Kishimoto, I. Ojima. Part 1: Quantum Field Theory. 1. Local Quantum Physics and Beyond; R. Haag. 2. A Non-Commuting Realization of Minkowski Space; H.J. Borchers. 3. Goldstone's Theorem Revisited; J.E. Roberts. 4. Global Observables in Local Quantum Physics; K. Fredenhagen. 5. State of the Art of Alain Connes' Version of the Standard Model; D. Kastler. 6. Supersymmetric Extension of Quantum Scalar Field Theories; A. Arai. Part 2: Statistical and Solid State Physics. 7. Optimal Two-Uniform Convexity and Fermion Hypercontractivity; E. Lieb, E. Carlen. 8. Renormalization Group and Nonlinear Media; A. Kupiainen, J. Bricmont. 9. Random Wall Representations and Mayer Expansion; K.R. Ito. 10. The Free Energy Theorem; R.F. Streater. 11. Dynamical Entropy of Quasi-Local Algebras; Y.M. Park. 12. The Weak Coupling Limit for a Fermi Gas in a Random Potential; L.J. Landau. 13. The Gap Labelling Theorem: the Case of Automatic Sequences; J. Bellisard. 14. Operator Algebra Approach to Soluble Models of Quantum Spin Lattice Systems; H. Araki. 15. On Conservation Laws of XY Model; T. Matsui. Part 3: Quantum Groups. 16. Reflection Equation Algebras and Quantum Groups; P.P. Kulish. 17. Quantum Groups, Star Products and Cyclic Cohomology; M. Flato, D. Sternheimer. 18. Quantum Group Symmetry in Conformal Field Theory; K. Gawedzki. 19. Spectrum of an Operator appears in the Quantum SU(1,1) Group; T. Kakehi, K. Ueno, T. Masuda. 20. Takesaki Duality for the Crossed Product by Quantum Groups; Y. Nakagami. Part 4: Subfactors and Index Theory. 21. A New Role of Graph Projections in Index Theory; T. Natsume. 22. Endomorphisms and Automorphisms for Factor Inclusions; M. Choda. 23. Automorphisms in the Irreducible Decomposition of Sectors; H. Kosaki. 24. Fusion Rules and Classification of Subfactors; M. Izumi. 25. Vector Bundles and Bimodules; S. Yamagami. 26. Lattice Structure of Intermediate Subfactors; Y. Watatani. 27. Minimal Index Unimodular Sectors; R. Longo. 28. Subfactors and Conformal Field Theory; J. Evans, Y. Kawahigashi. Part 5: Operator Algebras and Related Topics. 29. A Classification of Certain Simple C*-Algebra; G. Elliott. 30. K-Theoretic Classifications for Certain Real Rank Zero C* Algebras; H. Su. 31. Inductive Limits of Interval Algebras; the Simple Case; K. Thomsen. 32. Operator Algebras and Abstract Duals: Progress and Problems; S. Doplicher. 33. Regular Actions of Compact Groups on Cuntz Algebras; C. Pinzari. 34. Almost Shift Invariant Projections in Infinite Tensor Products; O. Bratteli, D. Evans, A. Kishimoto. 35. Strongly Elliptic and Subelliptic Operators on Lie Groups; D.W. Robinson.
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