Hidden symmetries of deformed oscillators

TL;DR

This paper explores the construction of deformed oscillator systems with symmetries derived from simple Lie algebras, including examples with SO(2,3) and G_{2(2)} symmetry, revealing their invariant structures and actions.

Contribution

It introduces a method to associate Lie algebra symmetries with differential equations of deformed oscillators, including explicit examples and invariant action construction techniques.

Findings

Deformed oscillators exhibit symmetries of non-compact Lie groups.

Invariant actions require semi-dynamical degrees of freedom.

Examples include SO(2,3) and G_{2(2)} symmetric systems.

Abstract

We associate with each simple Lie algebra a system of second-order differential equations invariant under a non-compact real form of the corresponding Lie group. In the limit of a contraction to a Schr\"odinger algebra, these equations reduce to a system of ordinary harmonic oscillators. We provide two clarifying examples of such deformed oscillators: one system invariant under SO(2,3) transformations, and another system featuring $G_{2(2)}$ symmetry. The construction of invariant actions requires adding semi-dynamical degrees of freedom; we illustrate the algorithm with the two examples mentioned.