Oscillators from nonlinear realizations

TL;DR

This paper develops a method to construct harmonic and Pais-Uhlenbeck oscillators invariant under noncompact Lie algebras using nonlinear realizations, enabling linearization of their equations of motion.

Contribution

It introduces a formalism that allows the equations of motion for these oscillators to be linearized via nonlinear realizations, applicable to arbitrary noncompact Lie algebras.

Findings

Equations of motion can be linearized using nonlinear realizations.

Constructed oscillator systems invariant under specific noncompact Lie algebras.

Provided explicit examples with $so(2,3)$ and $G_{2(2)}$ algebras.

Abstract

We construct the systems of the harmonic and Pais-Uhlenbeck oscillators, which are invariant with respect to arbitrary noncompact Lie algebras. The equations of motion of these systems can be obtained with the help of the formalism of nonlinear realizations. We prove that it is always possible to choose time and the fields within this formalism in such a way that the equations of motion become linear and, therefore, reduce to ones of ordinary harmonic and Pais-Uhlenbeck oscillators. The first-order actions, that produce these equations, can also be provided. As particular examples of this construction, we discuss the $so(2,3)$ and $G_{2(2)}$ algebras.