On the Time Dependent Oscillator and the Nonlinear Realizations of the Virasoro Group

TL;DR

This paper explores the nonlinear realizations of the Virasoro group to construct actions for conformal quantum mechanics and time-dependent oscillators, revealing their embedding in the reparametrization group and Virasoro symmetry.

Contribution

It introduces a novel approach to constructing actions for conformal quantum mechanics and oscillators using Virasoro group realizations, highlighting their symmetry embeddings.

Findings

Constructed the action for conformal quantum mechanics with harmonic potential.

Generalized to time-dependent oscillators with Virasoro symmetry.

Showed SL(2,R) invariance is embedded in the reparametrization group.

Abstract

Using the nonlinear realizations of the Virasoro group we construct the action of the Conformal Quantum Mechanics (CQM) with additional harmonic potential. We show that $SL(2,R)$ invariance group of this action is nontrivially embedded in the reparametrization group of the time which is isomorphic to the centerless Virasoro group. We generalize the consideration to the Ermakov systems and construct the action for the time dependent oscillator. Its symmetry group is also the $SL(2,R) \sim SU(1,1)$ group embedded in the Virasoro group in a more complicated way.