Calogero—Moser— Sutherland Models

571 Pages·2000·43.649 MB·English

by  J. Avan (auth.), Jan Felipe van Diejen, Luc Vinet (eds.)

CRM Series in Mathematical Physics
Springer Science+Business Media, LLC
CRM Series in Mathematical Physics
Conte, The Painleve Property: One Century Later
MacKenzie, Paranjape, and Zakrzewski, Solitons: Properties,
Dynamics, Interactions, Applications
Semenoff and Vinet, Particles and Fields
van Diejen and Vinet, Calogero-Moser-Sutherland Models
Ja n Felipe van Diejen
Luc Vinet
Editors
Calogero-Moser
Sutherland Models
With 27 Illustrations
, Springer
Jan Felipe van Diejen Luc Vinet
Departamento de Matematicas McGill University
Universidad de Chile James Administration Building, Room 504
Las Palmeras, 3425, Nunoa Montreal, Quebec H3A 2T5
Santiago Canada
Chile [email protected]
[email protected]
Editorial Board
Joel S. Feldman Duong H. Phong
Department of Mathematics Department of Mathematics
University of British Columbia Columbia University
Vancouver, BC V6T lZ2 New York, NY 10027-0029
Canada USA
[email protected] [email protected]
Yvan Saint-Aubin Luc Vinet
Departement de Mathematiques McGill University
et Statistique James Administration Building, Room 504
Universite de Montreal Montreal, Quebec H3A 2T5
c.P. 6128, Succursale Centre-ville Canada
Montreal, Quebec H3C 317 [email protected]
Canada
[email protected]
Library of Congress Cataloging-in-PubJieation Data
van Diejen, Jan Felipe, 1965-
Calogero-Moser-Sutherland models I Jan Felipe van Diejan, Luc Vinet.
p. em. - (CRM series in mathematieal physics)
lncludes bibliographieal referenees.
ISBN 978-1-4612-7043-0 ISBN 978-1-4612-1206-5 (eBook)
DOI 10.1007/978-1-4612-1206-5
1. Nuclear reaetions. 2. Many-body problem. I. Vinet, Lue. II. Title. III. CRM series
on mathematical physies
QC793.9.V36 2000
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Series Preface
The Centre de recherches matMmatiques (CRM) was created in 1968 by the
Universite de Montreal to promote research in the mathematical sciences.
It is now a national institute that hosts several groups; holds special theme
years, summer schools, workshops; and offers a postdoctoral program. The
focus of its scientific activities ranges from pure to applied mathematics,
and includes statistics, theoretical computer science, mathematical meth
ods in biology and life sciences, and mathematical and theoretical physics.
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dian Institute for Advanced Research, and has private endowments. Cur
rent activities, fellowships, and annual reports can be found on the CRM
web page at http://www . CRM. UMontreal. CAl.
The CRM Series in Mathematical Physics includes monographs, lecture
notes, and proceedings based on research pursued and events held at the
Centre de recherches matMmatiques.
Yvan Saint-Aubin
Montreal
Preface
In the early seventies Francesco Calogero made the following remarkable
observation: for a quantum system that consists of N identical particles on
the line interacting pairwise via an inverse-square potential and confined
by a harmonic well, the Schrodinger eigenvalue problem is exactly solvable!
(Calogero, J. Math. Phys., 1971). Soon thereafter, it was demonstrated by
Bill Sutherland that the corresponding quantum problem on a circle is also
exactly solvable (Sutherland, Phys. Rev. A, 1971, 1972). The revolutionary
nature of both discoveries is well illustrated by the fact that, until these
works of Calogero and Sutherland, the only known nontrivial example of
an exactly solvable quantum N-particle problem consisted of (apart from
the textbook case of coupled harmonic oscillators) the quantized nonlinear
Schrodinger equation, i.e., the system of N identical particles on the line
with a pairwise interaction via a "delta function potential."
The next big advance was made by Jiirgen Moser, who proved among
other things that the classical mechanical counterpart of the Calogero-Suth
erland N-particle model constitutes a completely integrable Hamiltonian
system in the sense of Liouville-Arnold, thereby showing that the study of
the classical dynamics of the model is also amenable to an exact analytical
approach (Moser, Adv. Math., 1975).
Since the appearance of the seminal papers of Calogero, Moser, and
Sutherland, the field opened by their works has developed considerably.
Important contributions were made in the late seventies by Olshanetsky
and Perelomov, who pointed out an intimate relation between the Calo
gero-Moser-Sutherland (CMS) systems and the study of geodesic motion
(at the classical level) and harmonic analysis (at the quantum level) on
symmetric spaces of simple Lie groups. This observation led them to a
generalization of the CMS systems associated with integral root systems
(Olshanetsky and Perelomov, Invent. Math. 1976; Phys. Rep., 1981; Phys.
Rep., 1983). Other highly influential developments include the introduction
of a relativistic deformation of the CMS system by Ruijsenaars and Schnei
der (Ruijsenaars and Schneider, Ann. Phys., 1986; Ruijsenaars, Commun.
Math. Phys., 1987) and the discovery of a one-dimensional CMS-type long-
lThat is: the spectrum of the Hamiltonian can be exhibited in closed form and the
problem of constructing the eigenfunctions has been reduced to elementary operations
involving linear algebra and the use of properties of classical special functions.
viii Preface
range spin model by Haldane and Shastry (Haldane, Phys. Rev. Lett., 1988;
Shastry, Phys. Rev. Lett., 1988).
The last decade of the 20th century has witnessed a true explosion of
activities involving CMS models. These investigations have revealed the
extraordinary ubiquitous character of these systems. By now it is well es
tablished that CMS systems play a role in investigations in research areas
ranging from theoretical physics (such as, e.g., soliton theory, quantum
field theory, string theory, solvable models of statistical mechanics, con
densed matter physics, quantum chaos, etc.) to pure mathematics (such
as representation theory, harmonic analysis, theory of special functions,
combinatorics of symmetric functions, dynamical systems, random matrix
theory, complex geometry, etc.).
During the period March 10-15, 1997, the Centre de recherches mathe
matiques in Montreal organized and hosted a workshop on Calogero-Moser
Sutherland models. One of the principal aims of this workshop was to bring
together leading researchers from various disciplines in whose work CMS
systems appear, either as the topic of investigation itself or as a tool for
further applications. This is an attempt to get some kind of an overview of
the many branches into which research on CMS systems has diversified in
recent years. Apart from the lectures delivered by two of the three found
ing fathers of the field, F. Calogero (Italy) and B. Sutherland (USA), the
workshop program contained 40 more presentations by J. A van (France),
H. Awata (USA), T. Baker (Australia), Y. Berest (Canada), R. Bhaduri
(Canada), O. Bogoyavlenskij (Canada), H. Braden (UK), Ph. Choquard
(Switzerland), J. van Diejen (Canada), M. Dijkhuizen (Japan), B. En
riquez (France), R. Floreanini (Italy), Ph. Di Francesco (USA), E. Gutkin
(USA), F. D. Haldane (USA), K. Hasegawa (Japan), V. Inozemtsev (Rus
sia), A. Kasman (Canada), A. N. Kirillov (Canada), I. Krichever (USA),
F. Lesage (USA), P. Mathieu (Canada), N. Nekrasov (USA), M. Olshanet
sky (Russia), A. Polychronakos (Greece), C. Quesne (Belgium), S. Ruij
senaars (Netherlands), D. Sen (India), E. Sklyanin (Japan), T. Shiota
(Japan), K. Taniguchi (Japan), C. Tracy (USA), A. Turbiner (Mexico),
D. Uglov (Japan), K. Vaninsky (USA), A. Varchenko (USA), A. Veselov
(UK), M. Wadati (Japan), G. Wilson (UK), and A. Zhedanov (Ukraine).
We thank all attendants for their active participation and the speakers
for their interesting state-of-the-art lectures. Special thanks and apprecia
tion goes out to those speakers who took the time and effort to prepare a
contribution for these proceedings.
J. F. van Diejen
Santiago
Contents
Series Preface v
Preface vii
Contributors xxi
1 Classical Dynamical r-Matrices for Calogero-Moser
Systems and Their Generalizations 1
J. Avan
1 Introduction...... 1
2 Preliminaries . . . . . 2
2.1 Liouville Theorem 2
2.2 Lax Pair Formulation 3
2.3 The r-Matrix Structure 3
2.4 The Classical Yang-Baxter Equation. 3
3 Hamiltonian Reduction and r-Matrices . 4
3.1 General Hamiltonian Reduction. 4
3.2 The Case N = T*G ..... 6
3.3 The Calogero-Moser Models .. 7
3.4 Two Examples . . . . . . . . . . 10
4 The Dynamical r-Matrices of Calogero and Ruijsenaars
Models. . . 12
5 References.......................... 17
2 Hidden Algebraic Structure of the Calogero-Sutherland
Model, Integral Formula for Jack Polynomial and Their
Relativistic Analog 23
Hidetoshi Awata
1 Introduction..................... 23
2 Calogero-Sutherland Model and Jack Polynomial 24
3 Integral Formula for Jack Polynomial. . . . . . 26
4 Relation with Virasoro Singular Vectors . . . . . 28
5 Macdonald Polynomial and q-Virasoro Algebra . 30
6 Boson Realization for q-Virasoro Algebra and Level One
Elliptic Affine Lie Algebra . 32
7 References.......................... 34
x Contents
3 Polynomial Eigenfunctions of the Calogero-Sutherland-
Moser Models with Exchange Terms 37
T. H. Baker, C. F. Dunkl, and P. J. Forrester
1 Introduction....... 37
1.1 The Periodic Model . . . . . . . . . 37
1.2 The Linear Model .. . . . . . . . . 40
2 Eigenfunctions of Prescribed Symmetry 41
2.1 Expansions in Terms of ETf 41
2.2 Normalization 45
3 q-Analogs 46
4 References..... 50
4 The Theory of Lacunas and Quantum Integrable Systems 53
Yuri Yu. Berest
1 Introduction........................... 53
2 Hyperbolic Operators on Root Systems ........... 54
2.1 Hyperbolic Polynomials and Associated Riesz Integrals 54
2.2 Invariant Hyperbolic Operators. . . . . . 56
2.3 The Representation Theorem . . . . . . . 57
3 Lacunas and Topology of Algebraic Surfaces . 58
3.1 Vector Fields and Cycles ......... 58
3.2 The Herglotz-Petrovsky-Leray Formulas 60
3.3 Products of Wave Operators 62
4 References.................... 63
5 Canonical Forms for the C-Invariant Tensors 65
Oleg I. Bogoyavlenskij
1 Introduction............. 65
2 C-Invariant Differential I-Forms 68
3 General C-Invariant (0,2) Tensors 70
4 Characteristic Polynomial of Any C-Invariant (1,1) Tensor Is
a Perfect Square .................. 71
5 Nijenhuis Tensor and C-Invariant (1,2) Tensors. 73
6 References...................... 75
6 R-Matrices, Generalized Inverses, and Calogero-Moser-
Sutherland Models 77
H. W. Bmden
1 Introduction. 77
2 R-Matrices 80
3 Four Results. 82
4 Application 85
5 Discussion. 89
6 References. 90
Contents xi
7 Tricks of the Trade: Relating and Deriving Solvable and
Integrable Dynamical Systems 93
Francesco Calogero
1 Introduction........................ 93
2 Survey of Solvable and Integrable Many-Body Models 94
3 '!'ricks . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.1 Time-Dependent Rescalings of Dependent Variables and
of Time . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.2 From One to More Kinds of Particles by Shifting the
Dependent Variables. . . . . . . . . . . . . . . . . 100
3.3 "Duplications": Configurations that are Preserved
Throughout the Motion . . . . . . . . . . . . . 101
3.4 How to Get Rid of Velocity-Dependent Forces. . . 107
3.5 From Two-Body to Nearest-Neighbor Forces ... 108
3.6 Infinite Rescalings, and the Use of Special Solutions, to
Identify Solvable Nonautonomous Models . . . 109
3.7 Two-Dimensional Models via Complexification 111
4 Envoi ... 114
5 References....................... 115
8 Classical and Quantum Partition Functions of the
Calogero-Moser-Sutherland Model 117
Ph. Choquard
1 Preamble and Summary . . . . . 117
2 The CSM Model . . . . . . . . . 118
3 Thermodynamics: Classical Case 119
4 Thermodynamics: Quantum Case . 121
5 References.............. 124
9 The Meander Determinant and Its Generalizations 127
P. Di Francesco
1 Introduction................. 127
2 Meanders: Definitions and Reformulations 128
2.1 Definitions ........ 128
2.2 Temperley-Lieb Algebra. 129
2.3 Meander Determinant . 131
3 Road/River Generalizations . 133
3.1 Semimeanders ...... 133
3.2 Crossing Meanders and Brauer Algebra 134
4 SU (N) Generalizations . . . . . . . . 136
4.1 Hecke Algebra and Its Quotients 136
4.2 SU(N) Meanders. . . . . . . . 137
4.3 SU(N) Meander Determinants 140
5 References.............. 143