Particles and Fields

by Semenoff, G. W.; Vinet, Luc

• ISBN13: 9780387984025
• ISBN10: 038798402X
• Format: Hardcover
• Copyright: 1998-07-01
• Publisher: Springer Verlag

Summary

The focus of this volume is on quantum field theory: inegrable theories, statistical systems, and applications to condensed-matter physics. It covers some of the most significant recent advances in theoretical physics at a level accessible to advanced graduate students. The contributions, each by a noted researcher, dicuss such topics as: some remarkable features of integrable Toda field theories (E. Corrigan), properties of a gas of interacting Fermions in a lattice of magnetic ions (J. Feldman &. al.), how quantum groups arise in three-dimensional topological quantum field thory (D. Freed), a method for computing correlation functions of solvable lattice models (T. Miwa), matrix models discussed from the point of view of integrable systems (A. Morozov), localization of path integrals in certain equivariant cohomologies (A. Niemi), Calogero-Moser systems (S. Ruijsenaars), planar gauge theories with broken symmetries (M. de Wild Propitius & F.A. Bais), quantum-Hall fluids (A. Capelli & al.), spectral theory of quantum vortex operators (P.I. Ettinghoff).

Table of Contents

Preface	vii	(8)
List of Contributors	xv
1 Recent Developments in Affine Toda Quantum Field Theory	1	(34)
E. Corrigan
1 Introduction	1	(2)
2 Classical Integrability and Classical Data	3	(10)
2.1 Geometry Associated with the Coxeter Element	8	(5)
3 Aspects of the Quantum Field Theory	13	(7)
4 Dual Pairs	20	(4)
5 A Word on Solitons	24	(5)
6 Other Matters	29	(1)
7 References	30	(5)
2 A Class of Fermi Liquids	35	(28)
J. Feldman
H. Knorrer
D. Lehmann
E. Trubowitz
1 Introduction	35	(4)
2 Four-Legged Diagrams	39	(13)
2.1 The Particle-Particle Bubble	42	(6)
2.2 The Particle-Hole Bubble	48	(2)
2.3 Higher-Order Diagrams	50	(2)
3 A Single-Slice Fermionic Cluster Expansion	52	(10)
4 References	62	(1)
3 Quantum Groups from Path Integrals	63	(46)
Daniel S. Freed
1 Classical Field Theory	65	(9)
1.1 Classical Actions	65	(4)
1.2 The Wess-Zumino-Witten Action	69	(5)
2 Categories, Finite Groups, and Covering Spaces	74	(12)
2.1 Going Further	75	(5)
2.2 Finite-Gauge Theory	80	(6)
3 Generalized Path Integrals	86	(8)
3.1 Path Integral Quantization	86	(3)
3.2 Beyond Quantum Hilbert Spaces	89	(2)
3.3 Quantum Finite-Gauge Theory	91	(3)
4 The Quantum Group	94	(11)
4.1 The 2-Hilbert Space	94	(4)
4.2 Locality and Gluing	98	(7)
5 References	105	(4)
4 Half Transfer Matrices in Solvable Lattice Models	109	(18)
Tetsuji Miwa
1 The Six-Vertex Model	109	(2)
2 The Antiferromagnetic Regime	111	(3)
3 Corner Transfer Matrix	114	(2)
4 Half Transfer Matrix	116	(2)
5 Commutation Relations	118	(2)
6 Correlation Functions	120	(2)
7 Two-Point Functions	122	(2)
8 Discussion	124	(1)
9 References	125	(2)
5 Matrix Models as Integrable Systems	127	(84)
A. Morozov
1 Introduction	127	(2)
2 The Basic Example: Discrete 1-Matrix Model	129	(14)
2.1 Ward Identities	130	(3)
2.2 CFT Interpretation of 1-Matrix Model	133	(7)
2.3 1-Matrix Model in Eigenvalue Representation	140	(2)
2.4 Kontsevich-Like Representation of the 1-Matrix Model	142	(1)
3 Generalized Kontsevich Model	143	(19)
3.1 Kontsevich Integral. The First Step	143	(1)
3.2 Itzykson-Zuber Integral and Duistermaat-Heckmann Theorem	144	(1)
3.3 Kontsevich Integral. The Second Step	145	(1)
3.4 "Phases" of Kontsevich Integral. GKM as the "Quantum Piece" of Fv{L} in the Kontsevich Phase	146	(3)
3.5 Relation Between Time- and Potential-Dependencies	149	(1)
3.6 Kac-Schwarz Problem	150	(1)
3.7 Ward Identities for GKM	151	(11)
4 Kp/Toda Tan-Function in Terms of Free Fermions	162	(22)
4.1 Explicit Definition	163	(2)
4.2 Basic Determinant Formula for the Free-Fermion Correlator	165	(4)
4.3 KP Hierarchy and Other Reductions	169	(4)
4.4 Fermion Correlator in Miwa Coordinates	173	(6)
4.5 1-Matrix Model versus Toda-Chain Hierarchy	179	(5)
5 Tan-Function as a Group-Theoretical Quantity	184	(19)
5.1 From Intertwining Operators to Bilinear Equations	185	(2)
5.2 The Case of KP/Toda Tan-Functions	187	(6)
5.3 Example of SL(2)(q)	193	(5)
5.4 Comments on the Quantum Deformation of KP/Toda Tan-Functions	198	(5)
6 Conclusion	203	(1)
7 References	204	(7)
6 Localization, Equivariant Cohomology, and Integration Formulas	211	(40)
Antti J. Niemi
1 Symplectic Geometry	213	(2)
2 Equivariant Cohomology	215	(1)
3 Duistermaat-Heckman Integration Formula	216	(5)
4 Degeneracies	221	(1)
5 Equivariant Characteristic Classes	222	(1)
6 Loop Space	223	(2)
7 Example: Atiyah-Singer Index Theorem	225	(3)
8 Duistermaat-Heckman in Loop Space	228	(3)
9 General Integrable Models	231	(2)
10 Mathai-Quillen Formalism	233	(2)
11 Short Review of Morse Theory	235	(1)
12 Equivariant Mathai-Quillen Formalism	236	(2)
13 Equivariant Morse Theory	238	(2)
14 Loop Space and Morse Theory	240	(2)
15 Loop Space and Equivariant Morse Theory	242	(2)
16 Poincare Supersymmetry and Equivariant Cohomology	244	(4)
17 References	248	(3)
7 Systems of Calogero-Moser Type	251	(102)
S. N. M. Ruijsenaars
1 Introduction	251	(7)
2 Classical Nonrelativistic Calogero-Moser and Toda Systems	258	(22)
2.1 Background: Classical Mechanics/Symplectic Geometry	258	(8)
2.2 Calogero-Moser Systems	266	(10)
2.3 Toda Systems	276	(4)
3 Relativistic Versions at the Classical Level	280	(14)
3.1 The Defining Dynamics and its Commuting Integrals	280	(8)
3.2 Lax Matrices and Their Interrelationships	288	(6)
4 Quantum Calogero-Moser and Toda Systems	294	(20)
4.1 Background: Quantum Mechanics/Hilbert Space Theory	294	(8)
4.2 The Nonrelativistic Case: Commuting PDOs	302	(5)
4.3 The Relativistic Case: Commuting ADeltaOs	307	(7)
5 Action-Angle Transforms	314	(17)
5.1 Introductory Examples	314	(5)
5.2 Wave Maps and Pure Soliton Systems	319	(2)
5.3 Systems of Type I, II, and III	321	(10)
6 Eigenfunction Transforms	331	(17)
6.1 Preliminaries	331	(3)
6.2 Type III Eigenfunctions for Arbitrary N	334	(7)
6.3 Type II and IV Eigenfunctions for N = 2	341	(7)
7 References	348	(5)
8 Discrete Gauge Theories	353	(88)
Mark de Wild Propitius
F. Alexander Bais
1 Broken Symmetry Revisited	353	(7)
2 Basics	360	(35)
2.1 Introduction	360	(1)
2.2 Braid Groups	361	(3)
2.3 Z(N) Gauge Theory	364	(16)
2.4 Non-Abelian Discrete Gauge Theories	380	(15)
3 Algebraic Structure	395	(13)
3.1 Quantum Double	396	(5)
3.2 Truncated Braid Groups	401	(2)
3.3 Fusion, Spin, Braid Statistics, and All That	403	(5)
4 D(2) Gauge Theory	408	(21)
4.1 Alice in Physics	409	(6)
4.2 Scattering Doublet Charges Off Alice Fluxes	415	(3)
4.3 Non-Abelian Braid Statistics	418	(5)
4.4 Aharonov-Bohm Scattering	423	(3)
4.5 B(3,4) and P(3,4)	426	(3)
5 Concluding Remarks and Outlook	429	(2)
6 References	431	(10)
9 Quantum Hall Fluids as W(1+Infinity]) Minimal Models	441	(28)
Andrea Cappelli
Carlo A. Trugenberger
Guillermo R. Zemba
1 Introduction	441	(1)
2 Dynamical Symmetry and Kinematics of Incompressible Fluids	442	(5)
2.1 Classical Fluids	442	(2)
2.2 Quantum Fluids and Their Edge Excitations	444	(3)
2.3 Classification of QHE Universality Classes	447	(1)
3 Existing Theories of Edge Excitations and Experiments	447	(9)
3.1 Hierarchical Trial Wave Functions	448	(1)
3.2 The Chiral Boson Theory of the Edge Excitations	448	(4)
3.3 The Jain Hierarchy	452	(1)
3.4 Experiments	453	(3)
4 W(1+Infinity) Minimal Models	456	(7)
4.1 The Theory of W(1+Infinity) Representations	456	(1)
4.2 The W(1+Infinity) Minimal Models	457	(2)
4.3 Non-Abelian Fusion Rules and Non-Abelian Statistics	459	(1)
4.4 The Degeneracy of Excitations Above the Ground State	460	(3)
4.5 Remarks on the SU(m) and SU(m)(1) Symmetries	463	(1)
5 Further Developments	463	(1)
6 References	464	(2)
10 On the Spectral Theory of Quantum Vertex Operators	469	(18)
Pavel I. Etingof
1 Basic Definitions	469	(4)
1.1 Quantum Groups	469	(1)
1.2 Representations	470	(1)
1.3 Vertex Operators	470	(1)
1.4 The Fock Space	470	(1)
1.5 Bosonization of U(q)(sl(2))	471	(1)
1.6 Bosonization of Vertex Operators	472	(1)
1.7 Boson-Fermion Correspondence	472	(1)
2 Spectral Properties of Vertex Operators	473	(5)
2.1 Vertex Operators as Power Series in q	473	(1)
2.2 Composition of Vertex Operators	474	(1)
2.3 The Operators F(+-)(0) and F(-+)(0)	474	(2)
2.4 The Highest Eigenvalue of F(-+(q)), F(+-(q))	476	(2)
3 The Semi-Infinite Tensor Product Construction	478	(2)
3.1 The Kyoto Conjecture	478	(1)
3.2 The Kyoto Homomorphism	479	(1)
4 Computation of the Leading Eigenvalue and Eigenvector	480	(4)
5 References	484	(3)
Index	487