Complete integrability of relativistic Calogero-Moser systems and elliptic function identities
We recall that the potential energy of the (classical, nonrelativistic) Calogero-Moser system on the line is expressed in terms of the Weierstrass function \({\mathfrak p}(x)\), degenerate cases of which are \(x^{-2}\), \(\sin^{-2}x\) and \(sh^{-2}x\). In a previous paper with \textit{H. Schneider} [Ann. Phys. 170, 370-405 (1986; Zbl 0608.35071)] the author exhibited sufficiently many independent constants of the motion in involution for Poincaré-invariant generalizations of those systems. In the present paper a formal quantization of those classical integrals is considered. The definition of the associated operators involves exponentials of the quantized rapidity variables and a meromorphic function that must satisfy a set of functional equations. In an appendix it is proved that a suitable quotient of \(\sigma\)-functions satisfies that set, and the vanishing of the quantum commutators is obtained as a consequence. A corollary shows that a translate of the \({\mathfrak p}\)- function satisfies a set of functional equations obtaind as a limit of the previous set, and the vanishing of the Poisson brackets of the classical integrals follows. Thus, for these models quantum integrability implies classical integrability. This paper includes identities for the \(\sigma\)-function and generalized Cauchy determinantal identities that would seem to be of independent interest. The latter are used in the author's extension to the elliptic case of the results obtained in op.cit. for the hyperbolic Lax matrix. The author has kindly informed the reviewer that item [10] in the bibliography has already appeared; cf. the author in Commun. Math. Phys. 115, No.1, 127-165 (1988; Zbl 0667.58016)].
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