Medicine show: A Calogero model with principal series states

TL;DR

This paper modifies the Calogero model to include principal series states, preserving unitarity and symmetry but changing integrability, and provides explicit solutions for small particle numbers.

Contribution

It introduces a deformed Calogero model accommodating principal series states, expanding the model's applicability and offering a method for solving it at any particle number.

Findings

Successfully incorporates principal series states into the Calogero model.

Preserves unitarity and $rak{sl}(2,b R)$-invariance despite deformation.

Provides explicit solutions for $N=2,3$ and outlines a general solution procedure.

Abstract

The Calogero model is an interacting, $N$-particle, $\mathfrak{sl}(2,\mathbb R)$-invariant quantum mechanics, whose Hilbert space is furnished by a tower of discrete series modules. The system enjoys both classical and quantum integrability at any $N$ and at any value of the coupling; this is guaranteed by the existence of $N$ mutually-commuting currents, one of them being the Hamiltonian. In this paper, we alter the Calogero model so that it may accommodate states in the unitary principal series irreducible representation of $\mathfrak{sl}(2,\mathbb R)$. Doing so requires changing the domain of the quantum operators--a procedure which succeeds in preserving unitarity and $\mathfrak{sl}(2,\mathbb R)$-invariance, but alters the integrability properties of the theory. We explicitly solve the deformed model for $N=2,3$ and outline a procedure for solving the model at general $N$. We expect this deformed model to provide us with general lessons that carry over to other systems with states in the principal series, for example, interacting massive quantum field theories on de Sitter space.