Fields and Particles: Proceedings of the XXIX Int. Universitätswochen für Kernphysik, Schladming, Austria, March 1990

292 Pages·1990·14.345 MB·English

by  H. Grosse (auth.), Professor Dr. Heinrich Mitter, Dr. Wolfgang Schweiger (eds.)

Fields and Particles
H. Mitter W. Schweiger (Eds.)
Fields and Particles
Proceedings of the
XXIX Int. Universitatswochen fOr Kernphysik
Schladming, Austria, March 1990
With 23 Figures
Springer-Verlag
Berlin Heidelberg New York London
Paris Tokyo Hong Kong Barcelona
Professor Dr. Heinrich Mitter
Dr. Wolfgang Schweiger
Institut fOr Theoretische Physik, Karl-Franzens-Universitat, Universitatsplatz 5,
A-8010 Graz, Austria
ISBN-13:978-3-642-76092-1 e-ISBN-13:978-3-642-76090-7
001: 10.1007/978-3-642-76090-7
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© Springer-Verlag Berlin Heidelberg 1990
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Preface
This volume contains the written versions of invited lectures presented at the
29th "Internationale Universitatswochen fiir Kernphysik" in Schladming, Aus
tria, in March 1990.
The generous support of our sponsors, the Austrian Ministry of Science
and Research, the Government of Styria, and others, made it possible to invite
expert lecturers. In choosing the topics of the course we have tried to select
some of the currently most fiercely debated aspects of quantum field theory. It
is a pleasure for us to thank all the speakers for their excellent presentations
and their efforts in preparing the lecture notes.
After the school the lecture notes were revised by the authors and partly
rewritten ~n '!EX. We are also indebted to Mrs. Neuhold for the careful typing
of those notes which we did not receive in '!EX.
Graz, Austria H. Mitter
July 1990 W. Schweiger
Contents
An Introduction to Integrable Models
and Conformal Field Theory
By H. Grosse (With 6 Figures) .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1
1. Introduction ............................................. . 1
1.1 Continuous Integrable Models .......................... . 1
1.2 "Solvable" Models of Statistical Physics ................. . 2
1.3 The Yang-Baxter Relation ............................. . 3
1.4 Braids and I(nots .................................... . 3
1.5 Confonnal Field Theory d = 2 ......................... . 3
2. Integrable Continuum Models -
The Inverse Scattering Method - Solitons .................... . 4
2.1 A General Scheme for Solving (Linear) Problems ......... . 4
2.2 The Direct Step ...................................... . 6
2.3 The Inverse Step ..................................... . 7
2.4 Solutions of the GLM Equation for R == 0 ............... . 8
2.5 Solving the KdV Equation ............................. . 9
2.6 Lax Pairs ........................................... . 9
2.7 Remarks ............................................ . 10
3. Integrable Lattice Systems ................................. . 11
3.1 Introduction ......................................... . 11
3.2 Ising and Potts Models ................................ . 13
3.3 The Vertex Model .................................... . 14
3.4 Connection to Quantum Spin Models ................... . 15
3.5 Integrability of the Lattice Model ...................... . 16
3.6 Bethe States ......................................... . 17
3.7 The Algebraic Bethe Ansatz ........................... . 18
3.8 Knots, Links and Braids .............................. . 19
4. Conformal Field Theory ................................... . 22
4.1 Introduction ......................................... . 22
4.2 Confonnal Invariance ................................. . 23
4.3 Local Conformal 'Transformations d = 2 ................. . 24
4.4 Three Implications ................................... . 24
4.5 The Virasoro Algebra .................................. . 26
4.6 Correlation FUnctions ................................. . 28
References ...................................................... 30
VII
An Introduction to the Renormalization of Theories
with Continuous Symmetries, to the Chiral Models
and to Their Anomalies By C. Becchi ... . . . . . . . . . . . . . . . . . . . . . . .. 31
Introduction ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 31
1. The Renormalization of Field Equations . . . . . . . . . . . . . . . . . . . . . . 32
2. The Renormalization of Models with Continuous Symmetries 41
3. The Chiral Models and Their Current Algebra ................. 44
References ...................................................... 51
Quantum Field Theory in Low Dimensional Space Time
By K. Fredenhagen .............................................. 53
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2. The Algebraic Approach to Quantum Field Theory . . . . . . . . . . . . 54
3. Composition of Sectors ,..................................... 59
4. Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5. Left Inverses, Markov Traces
and the Possible Braid Group Representations .... . . . . . . . . . . . . . 66
6. The 2-Channel Situation .................................... 69
7. Exchange Algebras and R-Matrices ........................... 71
8. Rehren's Derivation of the Verlinde Algebra ................... 74
9. Braid Group Statistics in 3 Dimensions . . . . . . . . . . . . . . . . . . . . . . . 76
10. Conformal Light Cone Theories .............................. 81
11. Soliton Sectors in 2d Minkowski Space ........................ 84
References ...................................................... 85
From Integrable Models to Quantum Groups
By L. Faddeev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 89
1. Introduction to the Quantum Inverse Scattering Method ........ 89
1.1 The Higher Spin Chain ................................ 93
1.2 Complex Spin ......................................... 93
1.3 The Spin 1/2 XXZ Model .............................. 93
1.4 The Higher Spin XXZ Model ........................... 93
1.5 The Complex Spin XXZ Model ... . . . . . . . . . . . . . . . . . . . . . . 94
1.6 The Liouville Limit ................................... . 94
2. Quantum Groups ......................................... . 95
3. The Liouville Model ....................................... . 101
4. The Wess-Zumino-Novikov-Witten Model 108
References .................................................... . 115
Topics in Planar Physics
By R. J ackiw (With 3 Figures) ................... . . . . . . . . . . . . . . .. 117
1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
2. Planar Gauge Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
2.1 Topologically Massive Gauge Theories ................... 118
2.2 Non-Abelian Chern-Simons Gauge Theories ............... 124
VIII
2.3 Abelian Chern-Simons Gauge Theory with Sources ....... . 132
2.4 Quantum Holonomy .................................. . 135
2.5 Anomalous Statistics and the Spin of Charged Particles ... . 140
2.6 Point-Particles with Abelian Chern-Simons Gauge Fields .. . 142
2.7 Quantum Dynamics .................................. . 149
3. Planar Gravity ........................................... . 155
3.1 Introduction ......" . .................................. . 155
3.2 Classical Space-Time ................................. . 156
3.3 Quantum Dynamics .................................. . 160
3.4 Topological Elaborations .............................. . 165
References .................................................... . 167
Boundary Terms, Long Range Effects,
and Chiral Symmetry Breaking
By G. Morchio and F. Strocchi .................................. . 171
1. Introduction ............................................. . 171
2. The Hamiltonian Approach:
Coupling to the Boundary and Variables at Infinity ........... . 176
3. The Lagrangean and Functional Integral Approach.
Boundary Ward Identities .................................. . 187
4. The Schwinger Model and the 8 Angle Problem ............... . 193
5. Fermionic Integration, Boundary Conditions, and Chiral Symmetry 202
References .................................................... . 213
Two-Dimensional Nonlinear Sigma Models:
Orthodoxy and Heresy
By A. Patrascioiu and E. Seiler ................................. . 215
1. Introduction ............................................. . 215
2. Beliefs ................................................... . 216
3. Critique ................................................. . 218
4. Heresy: Strategy for a Proof ............................... . 222
4.1 The FK Representation ............................... . 222
4.2 Interlude on Percolation Theory ........................ . 226
4.3 The H Clusters of the O( N)
Model on the Square Lattice at Large (3 ................. . 227
4.4 From H to FK Clusters ............................... . 228
5. Conclusions 229
References .................................................... . 229
Gauge-Independence of Anomalies
By W. Kummer ............................................... . 231
1. Introduction ............................................. . 231
2. Gauge-Invariance, Gauge-Dependence, External Symmetry 233
3. Quantization ............................................. . 235
4. Extended BRS-Identity, Internal Anomaly ................... . 237
IX
5. External Symmetries, External Anomalies ..................... 242
6. Ghost Number Anomaly for the Bosonic String ................ 243
7. The Symmetry Extended BRS-Technique .................... . 245
8. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
8.1 Chiral Anomaly ....................................... 248
8.2 Horizontal Symmetry .................................. 248
8.3 The Bosonic String .................................... 248
8.4 The Lorentz Anomaly in Noncovariant Gauges ............ 249
8.5 Chiral Breaking in SUSY YM Theory (Trivial CaSe) ....... 249
8.6 Superconformal Invanance .............................. 250
Appendix A: A Toy Model for [8, raj", 0 . . . . . . . . . . . . . . . . . . . . . . . . .. 251
References ..................................................... 253
LEP: The First Hundred Days
By F. Dydak (With 14 Figures) .................................. 255
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
2. Electroweak Physics Results ................................. 256
2.1 Outline of the Programme .............................. 256
2.2 The Z Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
2.3 Hadronic Peak Cross-Section and Total Width ............ 258
2.4 Hadronic and Leptonic Partial Widths ................... 259
2.5 Constraints on the t-Quark Mass ........................ 261
2.6 The Invisible Width and the Number of Neutrino Families .. 262
2.7 Forward-Backward Asymmetry of Leptons ................ 263
2.8 Summary .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
3. QCD Results .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
3.1 Analysis of Global Event Variables ...................... 265
3.2 Analysis of Single-Particle Inclusive Variables ............. 267
3.3 Is a. Running? ........ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
4. Searches and Limits ........................................ 270
4.1 Heavy Sequential Quarks and Charged Leptons ........... 270
4.2 Heavy Neutral Leptons ................................. 271
4.3 The Neutral Higgs Boson ............................... 272
4.4 Charged Higgs Bosons ................................. 273
4.5 Supersymmetric Particles .............................. 274
5. LEP: What Next? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
5.1 A Short-Term Perspective .............................. 275
5.2 Plans for the Future ................................... 276
References ..................................................... 277
x
QeD and Nuclear Structure
By K. Bleuler ..................................... . . . . . . . . . . . .. 279
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
2. The Realm of Light Nuclei .................................. 282
3. A New Interpretation of Conventional Shell Structure . . . . .. .. . . 283
4. Conclusion ................................................. 285
References .......... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 285
XI
An Introduction to Integrable Models
and Conformal Field Theory
H. Grosse
Institut fUr Theoretische Physik, Universitat Wien
Boltzmanngasse 5, A - 1090 Wien, Austria
Abstract: We review first the steps which lead to solutions of continuous in
tegrable models in two dimensions. Next we discuss in more detail solvable
models of statistical physics. The central role of the Yang Baxter equation
is emphasized. The connection to Braids and Knots and certain algebras is
mentioned.
The continuum limit of lattice models may yield a conformal invariant field
theory. The Virasoro algebra is realized in a special manner. The central exten
sion is related to the conformal anomaly. Unitary highest weight representations
restrict both the coefficient in front of the anomaly and the conformal weights.
The latter are directly related to the critical exponents. The operator product
expansion as well as the classification of field operators is mentioned finally.
1. Introduction
Various subjects became more and more interrelated recently (Fig. 1):
1.1 Continuous Integrable Models
More than 150 years ago J. Scott-Russell observed solitons interacting with
vet,
each other. 1898 Korteweg and de Vries wrote down an equation for x):
Vt = 6vv., - v.,.,., (1.1)
which was supposed to describe water waves in a channel. Fermi, Pasta and
Ulam observed already by numerical experiments that certain modes of a dy
namical system may dominate. But it was not before 1967 when Gardner,
Green, Kruskal and Miura invented the inverse scattering method to solve
the KdV equation. Especially the soliton solutions were obtained explicitly.
Lax reformulated their scheme. Soon after these discoveries many hierarchies
were obtained. Among them there are the Nonlinear-Schrodinger equation, the
Sine-Gordon equation, the Toda lattices but more recently the Kadomtsev
Petviashvili type equations in three dimensions were also shown to be inte
grable. Especially during these more recent developments many more insights
have been obtained. Special vertex operators "create" solitons. The Bose-Fermi
Fields and Particles Editors: H. Miner· W. Schweiger
© Springer-Verlag Berlin, Heidelberg 1990