Nonlinear supersymmetry in the quantum Calogero model
Abstract
It is long known that the rational Calogero model describing n identical particles on a line with inverse-square mutual interaction potential is quantum superintegrable. We review the (nonlinear) algebra of the conserved quantum charges and the intertwiners which relate the Liouville charges at couplings g and g+1. For integer values of g, these intertwiners give rise to additional conserved charges commuting with all Liouville charges and known since the 1990s. We give a direct construction of such a charge, the unique one being totally antisymmetric under particle permutations. It is of order n(n-1)(2g-1)/2 in the momenta and squares to a polynomial in the Liouville charges. With a natural Z_2 grading, this charge extends the algebra of conserved charges to a nonlinear supersymmetric one. We provide explicit expressions for intertwiners, charges and their algebra in the cases of two, three and four particles.
Cite
@article{arxiv.1312.5749,
title = {Nonlinear supersymmetry in the quantum Calogero model},
author = {Francisco Correa and Olaf Lechtenfeld and Mikhail Plyushchay},
journal= {arXiv preprint arXiv:1312.5749},
year = {2015}
}
Comments
1+21 pages; v2: minor corrections, typos in eq.(5.18) fixed, 3 refs. added, published version