Hidden symmetries of the Higgs oscillator and the conformal algebra

TL;DR

This paper constructs explicit nonlocal generators for the hidden SU(d) symmetry of the quantum Higgs oscillator, revealing a deep algebraic structure linked to the conformal algebra and Anti-de Sitter spacetime.

Contribution

It provides the first explicit construction of the quantum hidden symmetry generators for arbitrary dimension d, using conformal algebra techniques.

Findings

Generators satisfy standard su(d) Lie algebra

Construction relies on conformal algebra realization

Connects Higgs oscillator to Anti-de Sitter spacetime

Abstract

We give a solution to the long-standing problem of constructing the generators of hidden symmetries of the quantum Higgs oscillator, a particle on a d-sphere moving in a central potential varying as the inverse cosine-squared of the polar angle. This superintegrable system is known to possess a rich algebraic structure, including a hidden SU(d) symmetry that can be deduced from classical conserved quantities and degeneracies of the quantum spectrum. The quantum generators of this SU(d) have not been constructed thus far, except at d=2, and naive quantization of classical conserved quantities leads to deformed Lie algebras with quadratic terms in the commutation relations. The nonlocal generators we obtain here satisfy the standard su(d) Lie algebra, and their construction relies on a recently discovered realization of the conformal algebra, which contains a complete set of raising and lowering operators for the Higgs oscillator. This operator structure has emerged from a relation between the Higgs oscillator Schroedinger equation and the Klein-Gordon equation in Anti-de Sitter spacetime. From such a point-of-view, the construction of the hidden symmetry generators reduces to manipulations within the abstract conformal algebra so(d,2).