Non-Hermitian noncommutative quantum mechanics

TL;DR

This paper develops a phase-space formalism to analyze non-Hermitian and noncommutative quantum systems, providing a unified approach to compute expectation values and applying it to a harmonic oscillator with amplification.

Contribution

It introduces a general formalism connecting non-Hermitian and noncommutative Hamiltonians with their phase-space representations, enabling systematic analysis of such systems.

Findings

Established robust maps between Hamiltonians and Wigner functions for various quantum structures.

Provided a method to compute expectation values in complex non-Hermitian, noncommutative systems.

Applied the formalism to a harmonic oscillator with linear amplification, illustrating the effects of non-Hermitian and noncommutative features.

Abstract

In this work we present a general formalism to treat non-Hermitian and noncommutative Hamiltonians. This is done employing the phase-space formalism of quantum mechanics, which allows to write a set of robust maps connecting the Hamitonians and the associated Wigner functions to the different Hilbert space structures, namely, those describing the non-Hermitian and noncommutative, Hermitian and noncommutative, and Hermitian and commutative systems. A general recipe is provided to obtain the expected values of the more general Hamiltonian. Finally, we apply our method to the harmonic oscillator under linear amplification and discuss the implications of both non-Hermitian and noncommutative effects.