Quantum $SU(2,2)$-Harmonic Oscillator

TL;DR

This paper quantizes the $SU(2,2)$-harmonic oscillator using coherent states, computes the quantum Hamiltonian as a Toeplitz operator, and determines its spectrum based on different $SU(2,2)$ representations.

Contribution

It introduces a novel quantization approach for the $SU(2,2)$-harmonic oscillator on a specific phase space using coherent states and computes the spectrum for various representations.

Findings

Spectrum depends on the chosen $SU(2,2)$ representation.

Quantum Hamiltonian is a Toeplitz operator related to the invariant Kähler metric.

Provides explicit spectral calculations for the quantized system.

Abstract

The $SU(2,2)$-harmonic oscillator on the phase space ${\cal A}(2,2)= {SU(2,2)}/{S(U(2)\times U(2))}$ is quantized using the coherent states. The quantum Hamiltonian is the Toeplitz operator corresponding to the square of the distance with respect to the $SU(2,2)$-invariant K\"ahler metric on the phase space. Its spectrum, depending on the choice of representation of $SU(2,2)$, is computed.