Liouville Integrability of Classical Calogero-Moser Models

Liouville integrability of classical Calogero-Moser models is proved for models based on any root systems, including the non-crystallographic ones. It applies to all types of elliptic potentials, i.e. untwisted and twisted together with their degenerations (hyperbolic, trigonometric and rational), except for the rational potential models confined by a harmonic force. In this note we demonstrate the Liouville integrability of classical Calogero-Moser models [1] based on any root systems [2] including the non-crystallographic ones. This applies to models with all possible forms of the potentials, in particular, various types (untwisted and twisted) of elliptic potentials, except for the one with rational potential in the confining harmonic force. The quantum version of this result, restricted to non-elliptic potential cases, is reported in [3]. The proof of Liouville integrability is obtained by combining two known facts: The first is the universal Lax pair of Bordner-Corrigan-Sasaki [4]. This is obtained by unifying the known Lax pairs of various types [5], [6] in terms of representations of Coxeter group, which is the symmetry group of Calogero-Moser models. The universal Lax pair provides a complete set of integrals of motion for each Calogero-Moser model with any potential and based on any root system. The second ingredient is a Theorem by Olshanetsky and Perelomov [7, 2] on the structure of the conserved quantities of Calogero-Moser models in general. The latter simply asserts that for the conserved quantities of Calogero-Moser models {Qn} satisfying certain conditions to be listed below, any Poisson brackets among them {Qn, Qm} must vanish. First let us recapitulate the basic ingredients of the Calogero-Moser models, the Hamiltonian, the Lax pair and conserved quantities in order to set the stage and to introduce notation. A Calogero-Moser model is a Hamiltonian system associated with a root system ∆ of rank r. Quantum versions of these models are also integrable, at least for degenerate potential functions [3] for any choice of ∆. The dynamical variables are the coordinates {qj} and their canonically conjugate momenta {pj}, with the Poisson brackets {qj , pk} = δjk, {qj , qk} = {pj, pk} = 0, j, k = 1, . . . , r. (1) These will be denoted by vectors in R q = (q1, . . . , qr), p = (p1, . . . , pr). (2) The Hamiltonian for the Calogero-Moser model is H(p, q) = 1 2 p + 1 2 ∑ α∈∆+ g |α||α| 2 V|α|(α · q), ∆+ : set of positive roots, (3) in which the real coupling constants g|α| and potential functions V|α| are defined on orbits of the corresponding finite reflection (Coxeter, Weyl) group, i.e. they are identical for roots in the same orbit. The generic potential is elliptic. It is given by Weierstrass’ ℘ function: V|α|(α · q) = ℘(α · q|{2ω1, 2ω3}), for all roots, (4) which is called untwisted model. Here the two standard periods of the ℘ function are explicitly displayed. In the twisted model [8, 6, 5] the form of the potentials depends on the length of the roots. For long roots, it is the same as above. For short roots