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Acta Physica Polonica B

GENERALIZED CALOGERO MODELS AND THEIR COLLECTIVE FIELD FORMULATION

Joshua Feinberg

Published Dec 1, 2007• 0 citations
OpenAlex

Abstract

The relation between the Calogero model, namely, a system of N identical particles in one dimension with inverse-square interactions, and the three classical types of quantum-mechanical matrix models is well-known. In this talk I explore various generalized Calogero models and identify the quantum mechanical matrix model they correspond to at special values of their couplings. I also present and briefly discuss the collective field formulation of these generalized Calogero models.

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The relation between the Calogero model, namely, a system of N identical particles in one dimension with inverse-square interactions, and the three classical types of quantum-mechanical matrix models is well-known. In this talk I explore various generalized Calogero models and identify the quantum mechanical matrix model they correspond to at special values of their couplings. I also present and briefly discuss the collective field formulation of these generalized Calogero models.
Matrix models and related Spin-Calogero-Sutherland models are of major relevance in a variety of subjects, ranging from condensed matter physics to QCD and low dimensional string theory. They are characterized by integrability and exact solvability. Their continuum, field theoretic representations are likewise of definite interest. In this paper we describe various continuum, field theoretic representations of these models based on bosonization and collective field theory techniques. We compare various known representations and describe some nontrivial applications.
We construct generalizations of the Calogero-Sutherland-Moser system by appropriately reducing a model involving many unitary matrices. The resulting systems consist of particles on the circle with internal degrees of freedom, coupled through modifications of the inverse-square potential. The coupling involves SU(M) non-invariant (anti)ferromagnetic interactions of the internal degrees of freedom. The systems are shown to be integrable and the spectrum and wavefunctions of the quantum version are derived.
This thesis is devoted to the study of various examples of exactly solved quantum many-body systems in one-dimension. It is divided into two parts: the first provides background and complementary results to the second, which consists of three scientific papers. The first paper concerns a particu- lar extension, corresponding to the root system CN, of the delta-interaction model. We prove by construction that its exact solution, even in the gen- eral case of distinguishable particles, can be obtained by the coordinate Bethe ansatz. We also elaborate on the physical interpretation of this model. It is well known that the delta-interaction is included in a four parameter family of local interactions. In the second paper we interpret these parameters as cou- pling constants of certain momentum dependent interactions and determine all cases leading to a many-body system of distinguishable particles which can be solved by the coordinate Bethe ansatz. In the third paper we consider the so-called rational Calogero-Sutherland model, describing an arbitrary number of particles on the real line, confined by a harmonic oscillator potential and interacting via a two-body interaction proportional to the inverse square of the inter-particle distance. We construct a novel solution algorithm for this model which enables us to obtain explicit formulas for its eigenfunctions. We also show that our algorithm applies, with minor changes, to all extensions of the rational Calogero-Sutherland model which correspond to a classical root system.
A multispecies model of Calogero type in D≥1D\geq 1D≥1 dimensions is constructed. The model includes harmonic, two-body and three-body interactions. Using the underlying conformal SU(1,1) algebra, we find the exact eigenenergies corresponding to a class of the exact global collective states. Analysing corresponding Fock space, we detect the universal critical point at which the model exhibits singular behaviour.
A collective field method is extended to obtain all the explicit solutions of the generalized Calogero-Sutherland models that are characterized by the roots of all the classical groups, including the solutions corresponding to spinor representations for BNB_NBN​ and DND_NDN​ cases.