Crypto-Harmonic Oscillator in Higher Dimensions: Classical and Quantum Aspects

TL;DR

This paper explores complexified harmonic oscillators in higher dimensions, revealing richer constraint structures and comparing classical and quantum aspects, extending prior work on Crypto-gauge invariant and PT-symmetric models.

Contribution

It generalizes Crypto-oscillator models to higher dimensions, analyzing their constraint structures and quantizing the two-dimensional case for the first time.

Findings

Higher-dimensional models have more complex constraints.

Rotational symmetry influences the constraint structure.

Quantum analysis of the 2D Crypto-oscillator is provided.

Abstract

We study complexified Harmonic Oscillator models in two and three dimensions. Our work is a generalization of the work of Smilga \cite{sm} who initiated the study of these Crypto-gauge invariant models that can be related to $PT$-symmetric models. We show that rotational symmetry in higher spatial dimensions naturally introduces more constraints, (in contrast to \cite{sm} where one deals with a single constraint), with a much richer constraint structure. Some common as well as distinct features in the study of the same Crypto-oscillator in different dimensions are revealed. We also quantize the two dimensional Crypto-oscillator.