Deformation quantization of the Pais-Uhlenbeck fourth order oscillator

TL;DR

This paper applies deformation quantization to the Pais-Uhlenbeck oscillator, using symmetries and quantum transformations to derive the Wigner function and wave function, ensuring unitary evolution even at equal frequencies.

Contribution

It introduces a novel deformation quantization approach for the Pais-Uhlenbeck oscillator utilizing Noether symmetries and quantum canonical transformations.

Findings

Successfully derived the Wigner function and wave function for the system.

Demonstrated unitary evolution in the equal frequency limit.

Confirmed consistency with recent theoretical results.

Abstract

We analyze the quantization of the Pais-Uhlenbeck fourth order oscillator within the framework of deformation quantization. Our approach exploit the Noether symmetries of the system by proposing integrals of motion as the variables to obtain a solution to the -genvalue equation, namely the Wigner function. We also obtain, by means of a quantum canonical transformation the wave function associated to the Schr\"odinger equation of the system. We show that unitary evolution of the system is guaranteed by means of the quantum canonical transformation and via the properties of the constructed Wigner function, even in the so called equal frequency limit of the model, in agreement with recent results.