Ebook: New Symmetry Principles in Quantum Field Theory

Soon after the discovery of quantum mechanics, group theoretical methods were used extensively in order to exploit rotational symmetry and classify atomic spectra. And until recently it was thought that symmetries in quantum mechanics should be groups. But it is not so. There are more general algebras, equipped with suitable structure, which admit a perfectly conventional interpretation as a symmetry of a quantum mechanical system. In any case, a "trivial representation" of the algebra is defined, and a tensor product of representations. But in contrast with groups, this tensor product needs to be neither commutative nor associative. Quantum groups are special cases, in which associativity is preserved. The exploitation of such "Quantum Symmetries" was a central theme at the Ad­ vanced Study Institute. Introductory lectures were presented to familiarize the participants with the al­ gebras which can appear as symmetries and with their properties. Some models of local field theories were discussed in detail which have some such symmetries, in par­ ticular conformal field theories and their perturbations. Lattice models provide many examples of quantum theories with quantum symmetries. They were also covered at the school. Finally, the symmetries which are the cause of the solubility of inte­ grable models are also quantum symmetries of this kind. Some such models and their nonlocal conserved currents were discussed.

---

Lecturers: Quantum Symmetries in 2D Massive Field Theories; D.Bernard. On Knot and Manifold Invariants; R. Bott. The Metric Aspect of Noncommutative Geometry; A. Connes, J. Lott. Observables, Superselection Sectors, and Gauge Groups; C. Fredenhagen. Non-Compact WZW Conformal Field Theories; K. Gawedzki. S-Matrix Theory for Black Holes; G. 't Hooft. Whitman Theory for Integrable Systems and Topological Quantum Field Theories; I. Krichever. Quantum Groups in Lattice Models; V. Pasquier. Lagrangian Conformal Models; R. Stora.Seminar Speakers: Integrability Properties of the Collective String Field Theory; J. Avan. Correlational Functions of Local Operators in 2D Gravity Coupled to Minimal Matter; Vl.S. Dotsenko. W-Algebras and Langlands-Drinfeld Correspondence; E. Frenkel. Aspects of Quantizing Lorentz Symmetry; A. Schirrmacher. 11 additional articles. Index.

---

Lecturers: Quantum Symmetries in 2D Massive Field Theories; D.Bernard. On Knot and Manifold Invariants; R. Bott. The Metric Aspect of Noncommutative Geometry; A. Connes, J. Lott. Observables, Superselection Sectors, and Gauge Groups; C. Fredenhagen. Non-Compact WZW Conformal Field Theories; K. Gawedzki. S-Matrix Theory for Black Holes; G. 't Hooft. Whitman Theory for Integrable Systems and Topological Quantum Field Theories; I. Krichever. Quantum Groups in Lattice Models; V. Pasquier. Lagrangian Conformal Models; R. Stora.Seminar Speakers: Integrability Properties of the Collective String Field Theory; J. Avan. Correlational Functions of Local Operators in 2D Gravity Coupled to Minimal Matter; Vl.S. Dotsenko. W-Algebras and Langlands-Drinfeld Correspondence; E. Frenkel. Aspects of Quantizing Lorentz Symmetry; A. Schirrmacher. 11 additional articles. Index.
Content:
Front Matter....Pages i-viii
Quantum Symmetries in 2D Massive Field Theories....Pages 1-35
On Knot and Manifold Invariants....Pages 37-52
The Metric Aspect of Noncommutative Geometry....Pages 53-93
Intersection Theory, Integrable Hierarchies and Topological Field Theory....Pages 95-158
Quantum Symmetry in Conformal Field Theory by Hamiltonian Methods....Pages 159-175
Observables, Superselection Sectors and Gauge Groups....Pages 177-194
Incompressible Quantum Fluids, Gauge-Invariance, and Current Algebra....Pages 195-246
Non-compact WZW Conformal Field Theories....Pages 247-274
S-Matrix Theory for Black Holes....Pages 275-294
Non-Commutative Geometry and Mathematical Physics....Pages 295-308
Whitham Theory for Integrable Systems and Topological Quantum Field Theories....Pages 309-327
Quantum Symmetry in Quantum Theory....Pages 329-353
Quantum Groups in Lattice Models....Pages 355-380
Lagrangian Conformal Models....Pages 381-381
Semi-classical Liouville Theory, Complex Geometry of Moduli Spaces, and Uniformization of Riemann Surfaces....Pages 383-406
Integrability Properties of the Collective String Field Theory....Pages 407-412
BRST Analysis of Physical States for 2D (Super) Gravity Coupled to (Super) Conformal Matter....Pages 413-422
Correlation Functions of Local Operators in 2D Gravity Coupled to Minimal Matter....Pages 423-431
Non-Tannakian Categories in Quantum Field Theory....Pages 433-447
Extra States in c < 1 String Theory....Pages 449-482
W(sl(n)) : Existence, Cartan Basis and Infinite Abelian Subalgebras....Pages 483-492
Aspects of Quantizing Lorentz Symmetry....Pages 493-503
Back Matter....Pages 505-512
....Pages 513-521