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.
Highlights
We review the algebra of the conserved quantum charges and the intertwiners which relate the Liouville charges at couplings g and g±1
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
For integer values of g, these intertwiners give rise to additional conserved charges commuting with all Liouville charges and known since the 1990s
Summary
The quantum phase space of the n-particle Calogero model, defined by the Hamiltonian. Since (2.6) contains in particular [H, Ik] = 0, the Ik form n involutive constants of motion, whose leading term for large |xi−xj| is i pki. The Jacobi identity implies that H, [Il, Jk] = 0, so [Il, Jk] must be a linear combination of the Im, i k [Ik. in particular i[I1, Jl] = Il−1 and i[I2, Jl] = 2 Il. the shifted operators Lk = Jk+2 satisfy the Witt algebra. A useful set of 2n−1 constants of motion for the n-particle Calogero model at arbitrary coupling g is {P, H, I3, . Inserting the corresponding expressions into (2.24) and (2.25) changes the quadratic algebra to a polynomial one of order 2n−1
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