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arXiv · 1312.5749

Nonlinear supersymmetry in the quantum Calogero model

Francisco Correa·Olaf Lechtenfeld·Mikhail Plyushchay
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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.

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Francisco Correa, Olaf Lechtenfeld, Mikhail Plyushchay. 2014-04-25. Nonlinear supersymmetry in the quantum Calogero model. https://doi.org/10.1007/jhep04(2014)151

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