Separable quantizations of St\"{a}ckel systems

TL;DR

This paper demonstrates that many Hamiltonian systems, previously not separably quantizable using classical methods, can be quantized in a separable manner by extending the class of admissible quantizations through Riemann space adaptation.

Contribution

It introduces a new approach to quantize Stäckel systems by extending admissible quantizations via Riemann space, enabling separable quantization for a broad class of systems.

Findings

Existence of infinitely many quantizations for quadratic in momenta Stäckel systems.

Quantizations are parametrized by n arbitrary functions.

Applicable to systems with monomial Stäckel matrices.

Abstract

In this article we prove that many Hamiltonian systems that can not be separably quantized in the classical approach of Robertson and Eisenhardt can be separably quantized if we extend the class of admissible quantizations through a suitable choice of Riemann space adapted to the Poisson geometry of the system. Actually, in this article we prove that for every quadratic in momenta St\"ackel system (defined on a 2n-dimensional Poisson manifold) for which the St\"ackel matrix consists of monomials in position coordinates there exist infinitely many quantizations - parametrized by n arbitrary functions - that turn this system into a quantum separable St\"ackel system.