Complete integrability of relativistic Calogero-Moser systems and elliptic function identities
type: Item
instance of: scholarly article
title: Complete integrability of relativistic Calogero-Moser systems and elliptic function identities
zbMATH Open document ID: 0673.58024
DOI: 10.1007/BF01207363
published in: Communications in Mathematical Physics
publication date: 1987-01-01T00:00:00Z
review text: 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)].
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 81S99
Mathematics Subject Classification ID: 33E05
Mathematics Subject Classification ID: 15A15
zbMATH DE Number: 4102130
zbMATH Keywords: quantization of classical systems
zbMATH Keywords: Weierstrass p-function
zbMATH Keywords: \(\sigma\)- function
zbMATH Keywords: Cauchy's determinantal identity
zbMATH Keywords: Calogero-Moser system
zbMATH Keywords: classical integrability
zbMATH Keywords: hyperbolic Lax matrix
author: Simon Ruijsenaars
reviewed by: Hugo H. Torriani
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4102130