Title of lectures
Calogero-like systems: physics and mathematics
Lecturer:
Alexios POLYCHRONAKOS, University of Ioannina & Uppsala University
Outline:
In these lectures the physics and mathematics of the Calogero
model and its various generalizations will be exposed. The
mathematical properties of these models will be examined in
some detail, but the emphasis will be on issues of interest
to physics rather than the more formal aspects. Some open
questions and possible directions for future research will
be indicated.
Lecture 1: Introduction to the Calogero-Moser-Sutherland model.
Definition and overview; interesting physical properties;
Quantization, duality, connection to generalized statistics.
Lecture 2: Algebraic approach.
The exchange (Dunkle) operator formalism; proof of integrability
and quantum states; multicomponent generalizations; the `freezing trick'
and spin chain models; spectrum and properies of these models;
open questions.
Lecture 3: Matrix model approach.
Definition and properties of the hermitian and unitary matrix models;
classical reduction to particle systems; quantization and spectrum;
reduction to particle-spin systems; comparison to the operator
approach and open questions.
Lecture 4: Many-matrix models and anisotropic systems.
Multi-matrix models and SU(n)-non-invariant generalizations;
multidimensional models and frustrated hopes.
Lecture 5: Further connections and generalizations.
Classical reductions and generalized `twisted' models; theta-angle
generalizations; connection to two-dimensional (Yang-Mills) field
theories; continuous limit and solitons; open questions and epilogue.
References:
[1] The canonical, and still one of the best, reference for the
`old art' of these models is:
M.A. Olshanetskii and A.M. Perelomov, Phys.Rep. 71 (1981) 314
(classical systems) and Phys.Rep. 94 (1983) 6 (quantum systems).
[2] A review with partial overlap with the present lectures and a
much stronger physics slant is:
A.P. Polychronakos, `Generalized Statistics in One Dimension,'
Les Houches 1998 Lectures, hep-th/9902157
[3] A.P. Polychronakos, `Exchange Operator Formalism for Integrable
Systems,' Phys.Rev.Lett. 69 (1992) 703.
[4] J. Minahan and A.P. Polychronakos, `Integrable Systems for Particles
with Internal Degrees of Freedom,' Phys.Lett. B302 (1993) 265.
[5] A.P. Polychronakos, `Lattice Integrable Systems of the
Haldane-Shastry Type,' Phys.Rev.Lett. 70 (1993) 2329.
[6] A.P. Polychronakos, `Exact Spectrum of $SU(n)$ Spin Chain with
Inverse-Square Exchange,' Nucl.Phys. B419 [FS] (1994) 553.
[7] A.P. Polychronakos, `Generalized Calogero-Sutherland Systems from
Many-Matrix Models,' Nucl.Phys. B (in print), hep-th/9806189.
[8] A.P. Polychronakos, `Generalized Calogero Models through Reductions
by Discrete Symmetries,' Nucl.Phys. B543 (1999) 485.
[9] A.P. Polychronakos, `Waves and Solitons in the Continuum Limit
of the Calogero-Sutherland Model,' Phys.Rev.Lett. 74 (1995) 5153.