Multicenter Higgs oscillators and Calogero model
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ELEMENTARY PARTICLES AND FIELDS Theory
Multicenter Higgs Oscillators and Calogero Model* V. Yeghikyan** Yerevan State University, Armenia Received April 17, 2009
Abstract—We show that the spherical part of N -particle Calogero model describes, after exclusion of the center of mass, the motion of the particle on (N − 2)-dimensional sphere interacting with N (N − 1)/2 force centers with Higgs oscillator potential. In the case of four-particle system these force centers are located at the vertexes of cuboctahedron. The geometry of the five-particle case is also investigated. DOI: 10.1134/S1063778810030208
1. INTRODUCTION The Calogero model plays a distinguished role among other multiparticle integrable systems. It is a one-dimensional multiparticle integrable system with inverse-square interaction [1] g 1 2 pi + , H= 2 (xi − xj )2 N
i=1
(1)
i 3: the “angular” part of the Calogero model has not been properly studied for these cases. Such a study would be an interesting problem from few viewpoints. First of all, it is the transport of the discrete symmetries of the one-dimensional multiparticle system to the higherdimensional one-particle one. This would provide us with the a´ priori integrable higher-dimensional oneparticle system with some discrete symmetry. The purpose of the present paper is the investigation of the “angular” part of the N -particle Calogero model with ∗ **
The text was submitted by the author in English. E-mail: [email protected]
the excluded center of mass. We will show that it describes a particle on the (N − 2)-dimensional sphere, which interacts with the N (N − 1)/2 force centers by Higgs oscillator law. In other words, it corresponds to N (N − 1)/2-center (N − 2)-dimensional Higgs oscillator. For the N = 4 case, corresponding to the particle on the two-dimensional sphere, the force centers are located at the vertexes of the Archimedean solid called cuboctahedron. This observation opens new horizons in the further study of the Calogero model, particularly, the possibility of its applications in the solid state physics. 2. GENERAL CONSIDERATION For our purpose we need to introduce N -dimensional vectors bij as follows: 1 (2) bij k = √ (δik − δjk ). 2 They satisfy the relations ij (bk )2 = 1,
(e, ba ) = 0,
(3)
k
where e = (1, 1, . . . , 1), and round brackets define scalar product. After this denotation, we can rewrite the Hamiltonian (1) as follows: 1 2 = pi + 2 N
HN
i=1
N (N −1)/2
a=1
2
g
N a k=1 bk xk
2 , (4)
{pi , xj } = δij , where a ≡ (i, j) is N (N − 1)/2-valued index, which enumerates pairs of interacting particles, pi are the 555
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corresponding momenta. The second relation in (3) means that all vectors ba lie in a hypersurface, which is orthogonal to the vector e. This mean, that the set of vectors is not N -dimensional one and can be put in N − 1-dimensional space by an orthogonal space rotation using an appropriate matrix Aik . Dynamically, this rotation is equivalent to the transition to the center-of-mass system. Let us write down the
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