Quadratic Poisson algebras for two dimensional classical superintegrable systems and quadratic associative algebras for quantum superintegrable systems

TL;DR

This paper explores the algebraic structures underlying two-dimensional superintegrable systems, linking classical quadratic Poisson algebras to quantum quadratic associative algebras, and provides a method to determine energy spectra via algebraic equations.

Contribution

It introduces a unified algebraic framework for classical and quantum superintegrable systems with quadratic integrals of motion, including the calculation of energy eigenvalues through algebraic equations.

Findings

Finite dimensional representations are determined by energy eigenvalues.

Quantum quadratic algebras are deformations of classical Poisson algebras.

Energy spectra can be obtained by solving universal algebraic equations.

Abstract

The integrals of motion of the classical two dimensional superintegrable systems with quadratic integrals of motion close in a restrained quadratic Poisson algebra, whose the general form is investigated. Each classical superintegrable problem has a quantum counterpart, a quantum superintegrable system. The quadratic Poisson algebra is deformed to a quantum associative algebra, the finite dimensional representations of this algebra are calculated by using a deformed parafermion oscillator technique. It is shown that, the finite dimensional representations of the quadratic algebra are determined by the energy eigenvalues of the superintegrable system. The calculation of energy eigenvalues is reduced to the solution of algebraic equations, which are universal for all two dimensional superintegrable systems with quadratic integrals of motion.