Ebook: Quantum and Non-Commutative Analysis: Past, Present and Future Perspectives

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.

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This volume contains the proceedings of two international colloquia held in Japan in 1992. The various contributions by pre-eminent scientists cover the fields of quantum field theory, statistical and solid state physics, quantum groups and subfactors and index theory, and operator algebras and related topics. Together they present an authoritative overview of the latest developments by pioneers in these fields. Most of the contributions are self-contained.
For graduate students and researchers in mathematics and mathematical physics.

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This volume contains the proceedings of two international colloquia held in Japan in 1992. The various contributions by pre-eminent scientists cover the fields of quantum field theory, statistical and solid state physics, quantum groups and subfactors and index theory, and operator algebras and related topics. Together they present an authoritative overview of the latest developments by pioneers in these fields. Most of the contributions are self-contained.
For graduate students and researchers in mathematics and mathematical physics.

Content:
Front Matter....Pages i-xii
Front Matter....Pages 1-1
Local Quantum Physics and Beyond....Pages 3-10
A Non-commuting Realization of Minkowski Space....Pages 11-30
Goldstone’s Theorem Revisited....Pages 31-39
Global Observables in Local Quantum Physics....Pages 41-51
State of the art of Alain Connes’ version of the standard model....Pages 53-71
Supersymmetric Extension of Quantum Scalar Field Theories....Pages 73-90
Front Matter....Pages 91-91
Optimal Two—Uniform Convexity and Fermion Hypercontractivity....Pages 93-111
Renormalization Group and Nonlinear PDE’s....Pages 113-118
Random Walk Rperesentations and Mayer Expansion....Pages 119-135
The Free-energy Theorem....Pages 137-147
Dynamical Entropy of Quasi-local Algebras....Pages 149-165
The Weak Coupling Limit for a Fermi Gas in a Random Potential....Pages 167-178
The Gap Labelling Theorem: the case of Automatic Sequences....Pages 179-181
Operator Algebra Approach to Soluble Models of Quantum Spin Lattice Systems....Pages 183-196
On Conservation Laws of the XY Model....Pages 197-204
Front Matter....Pages 205-205
Reflection Equation Algebras and Quantum Groups....Pages 207-220
Quantum Groups, Star Products and Cyclic Cohomology....Pages 221-237
Quantum Group Symmetries in Conformal Field Theory....Pages 239-251
Spectrum of an operator appears in the quantum SU(1,1) group....Pages 253-261
Takesaki Duality for the Crossed Product by Quantum Groups....Pages 263-281
Front Matter....Pages 283-283
A New Role of Graph Projections in Index Theory....Pages 285-290
Endomorphisms and Automorphisms for Factor Inclusions....Pages 291-304
Automorphisms in the Irreducible Decomposion of Sectors....Pages 305-316
Fusion Rules and Classification of Subfactors....Pages 317-320
Vector Bundles and Bimodules....Pages 321-329
Lattice Structure of Intermediate Subfactors....Pages 331-333
Minimal Index and Unimodular Sectors....Pages 335-340
Subfactors and Conformal Field Theory....Pages 341-369
Front Matter....Pages 371-371
A Classification of Certain Simple C*-Algebras....Pages 373-385
K-theoretic Classification for Certain Real Rank Zero C*-algebras....Pages 387-397
Inductive Limits of Interval Algebras: The Simple Case....Pages 399-404
Operator Algebras and Abstract Duals: Progress and Problems....Pages 405-418
Regular Actions of Compact Groups on Cuntz Algebras....Pages 419-426
Almost Shift Invariant Projections in Infinite Tensor Products....Pages 427-434
Strongly Elliptic and Subelliptic Operators on Lie Groups....Pages 435-453
Back Matter....Pages 455-464

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This volume contains the proceedings of two international colloquia held in Japan in 1992. The various contributions by pre-eminent scientists cover the fields of quantum field theory, statistical and solid state physics, quantum groups and subfactors and index theory, and operator algebras and related topics. Together they present an authoritative overview of the latest developments by pioneers in these fields. Most of the contributions are self-contained.
For graduate students and researchers in mathematics and mathematical physics.

Content:
Front Matter....Pages i-xii
Front Matter....Pages 1-1
Local Quantum Physics and Beyond....Pages 3-10
A Non-commuting Realization of Minkowski Space....Pages 11-30
Goldstone’s Theorem Revisited....Pages 31-39
Global Observables in Local Quantum Physics....Pages 41-51
State of the art of Alain Connes’ version of the standard model....Pages 53-71
Supersymmetric Extension of Quantum Scalar Field Theories....Pages 73-90
Front Matter....Pages 91-91
Optimal Two—Uniform Convexity and Fermion Hypercontractivity....Pages 93-111
Renormalization Group and Nonlinear PDE’s....Pages 113-118
Random Walk Rperesentations and Mayer Expansion....Pages 119-135
The Free-energy Theorem....Pages 137-147
Dynamical Entropy of Quasi-local Algebras....Pages 149-165
The Weak Coupling Limit for a Fermi Gas in a Random Potential....Pages 167-178
The Gap Labelling Theorem: the case of Automatic Sequences....Pages 179-181
Operator Algebra Approach to Soluble Models of Quantum Spin Lattice Systems....Pages 183-196
On Conservation Laws of the XY Model....Pages 197-204
Front Matter....Pages 205-205
Reflection Equation Algebras and Quantum Groups....Pages 207-220
Quantum Groups, Star Products and Cyclic Cohomology....Pages 221-237
Quantum Group Symmetries in Conformal Field Theory....Pages 239-251
Spectrum of an operator appears in the quantum SU(1,1) group....Pages 253-261
Takesaki Duality for the Crossed Product by Quantum Groups....Pages 263-281
Front Matter....Pages 283-283
A New Role of Graph Projections in Index Theory....Pages 285-290
Endomorphisms and Automorphisms for Factor Inclusions....Pages 291-304
Automorphisms in the Irreducible Decomposion of Sectors....Pages 305-316
Fusion Rules and Classification of Subfactors....Pages 317-320
Vector Bundles and Bimodules....Pages 321-329
Lattice Structure of Intermediate Subfactors....Pages 331-333
Minimal Index and Unimodular Sectors....Pages 335-340
Subfactors and Conformal Field Theory....Pages 341-369
Front Matter....Pages 371-371
A Classification of Certain Simple C*-Algebras....Pages 373-385
K-theoretic Classification for Certain Real Rank Zero C*-algebras....Pages 387-397
Inductive Limits of Interval Algebras: The Simple Case....Pages 399-404
Operator Algebras and Abstract Duals: Progress and Problems....Pages 405-418
Regular Actions of Compact Groups on Cuntz Algebras....Pages 419-426
Almost Shift Invariant Projections in Infinite Tensor Products....Pages 427-434
Strongly Elliptic and Subelliptic Operators on Lie Groups....Pages 435-453
Back Matter....Pages 455-464
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