Constants of motion of the four-particle Calogero model
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ELEMENTARY PARTICLES AND FIELDS Theory
Constants of Motion of the Four-Particle Calogero Model* A. Saghatelian** Yerevan State University, Armenia Received March 5, 2012
Abstract—We present the explicit expressions of the complete set of constants of motion of four-particle Calogero model with excluded center of mass, i.e. of the A3 rational Calogero model. Then we find the constants of motion of its spherical part, defining two-dimensional 12-center spherical oscillator, with the force centers located at the vertexes of cuboctahedron. DOI: 10.1134/S1063778812100171
1. INTRODUCTION The Calogero model [1, 2] plays a distinguished role among other multi-particle integrable systems. Since its invention it attracts much attention due to their rich internal structure and numerous applications in many areas of physics (see, e.g., the recent review [3] and references therein). The initial N particle Calogero model is given by the following simple Hamiltonian p2 g2 i + . (1) H= 2 (xi − xj )2 i = j
Soon after its invention, the Calogero model has been generalized for all Lie algebras [4], for the trigonometric [5] and elliptic potentials, as well as for particles with spin degrees of freedom [6]. The Calogero model and its modifications appear in matrix models [7], W∞ algebras [8], Yangian quantum groups [9], random matrices [10] and many other areas of physics and mathematics. In the continuum limit, i.e. for large number of particles, it gives rise to a Yang–Mills theory on a cylinder [11], while its superconformal extension describes a black hole in the near-horizon limit [12]. An important feature of the Calogero model is the dynamical conformal symmetry so(1, 2), which defines many of its important properties. In particular, in terms of its generators, one can formulate [13] the decoupling transformation from the Calogero model to the free particle systems [14]. Unfortunately, the standard (and most convenient) approach to the study of the Calogero model is the Lax pair technique, which distinguishes it from the textbook systems of ∗ **
The text was submitted by the author in English. E-mail: [email protected]
classical and quantum mechanics [15]. This yields many complications in the establishing the parallels with known mechanical systems. Resent years an interesting approach in the study of the Calogero system was developed, based on common differential-geometrical methods [16–20]. The idea of that approach is to use the dynamical symmetry algebra of the Calogero model for the separation of the “spherical” and radial part of the Calogero model. The separation of the spherical part of the Calogero model from the radial one gave transparent explanation of the superintegrability property of the Calogero model, as well as allowed to present the explicit construction of N = 4 superconformal extension. In this way some interesting observation has been made. It was found that the spherical part of N particle Calogero model with excluded center of mass (i.e. of the AN −1 rational Calogero model) defines the (N −
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