Hermitian versus non-Hermitian representations for minimal length uncertainty relations

TL;DR

This paper compares Hermitian and non-Hermitian representations of deformed quantum variables in noncommutative spacetime, analyzing their relationships, solvable models, and physical implications, including metric construction and representation independence.

Contribution

It systematically studies four representations of deformed variables, analyzes three solvable models, and introduces an anti-PT-symmetric modification to address unphysical models.

Findings

Expectation values are representation-independent when metrics are properly implemented.

Jordan twist representations can lead to unphysical models.

An anti-PT-symmetric modification can fix issues with certain representations.

Abstract

We investigate four different types of representations of deformed canonical variables leading to generalized versions of Heisenberg's uncertainty relations resulting from noncommutative spacetime structures. We demonstrate explicitly how the representations are related to each other and study three characteristically different solvable models on these spaces, the harmonic oscillator, the manifestly non-Hermitian Swanson model and an intrinsically noncommutative model with Poeschl-Teller type potential. We provide an analytical expression for the metric in terms of quantities specific to the generic solution procedure and show that when it is appropriately implemented expectation values are independent of the particular representation. A recently proposed inequivalent representation resulting from Jordan twists is shown to lead to unphysical models. We suggest an anti-PT-symmetric modification to overcome this shortcoming.