Hidden symmetries of the Higgs oscillator and the conformal algebra

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 Schrödinger equation and the Klein-Gordon equation in Anti-de Sitter spacetime. From such a point-of-view, constructing the hidden symmetry generators reduces to manipulations within the abstract conformal algebra so(d, 2).

Mapping superintegrable quantum mechanics to resonant spacetimes

“…Before proceeding with novel derivations we would like to demonstrate how the case of the Higgs oscillator, which has motivated our general construction, fits into our present framework. We are essentially just reviewing the derivations in [7][8][9].…”

“…Such systems are exemplified by the one-dimensional Pöschl-Teller potential, and in higher dimensions they are typically superintegrable. In fact, our construction has been developed precisely as a generalization of the correspondence [7][8][9] between the Higgs oscillator [10,11], a particularly simple superintegrable system with a quadratic spectrum, and Klein-Gordon equations on the Anti-de Sitter (AdS) spacetime, the maximally symmetric spacetime of constant negative curvature. This correspondence has emerged in the context of studying selection rules [12][13][14] in the nonlinear perturbation theory targeting the AdS stability problem [15,16].…”

“…This correspondence has emerged in the context of studying selection rules [12][13][14] in the nonlinear perturbation theory targeting the AdS stability problem [15,16]. The correspondence has been useful for both elucidating the structure of AdS perturbation theory [8] and for resolving the old problem of constructing explicit hidden symmetry generators for the Higgs oscillator [9].…”

“…In section 2, we formulate our general geometrization procedure and describe how it simplifies for the case of zero mass in the Klein-Gordon equation one is aiming to construct. In section 3, we describe how the previously known correspondence [7][8][9] between the Higgs oscillator and AdS fits in our general framework. In section 4, we analyze the superintegrable Rosochatius system, which generalizes the Higgs oscillator, and generate a large family of spacetimes perfectly resonant with respect to the massless wave equation.…”

General first-order mass ladder operators for Klein–Gordon fields

“…x µ = (λ, x i ) are the local coordinates. In these coordinates, we have ζ µ (∂/∂x µ ) = f (∂/∂λ) 4 In the case δm 2 = 0, the operator D is called a symmetry operator. While a symmetry operator does not shift the mass, it is still interesting since it can map a solution of KGE into another solution of KGE with the same mass.…”