Minimal length in Quantum Mechanics and non-Hermitian Hamiltonian systems

TL;DR

This paper explores deformations of quantum mechanics with non-Hermitian operators, demonstrating their treatment within PT-symmetric frameworks and calculating minimal length scales relevant to quantum gravity.

Contribution

It introduces a new exponential deformation of quantum mechanics and computes the associated minimal uncertainty and length, linking non-Hermitian systems to quantum gravity concepts.

Findings

Deformed quantum systems can be analyzed using PT-symmetry frameworks.

A new exponential deformation is proposed and studied.

Minimal length scales are computed for the deformation.

Abstract

Deformations of the canonical commutation relations lead to non-Hermitian momentum and position operators and therefore almost inevitably to non-Hermitian Hamiltonians. We demonstrate that such type of deformed quantum mechanical systems may be treated in a similar framework as quasi/pseudo and/or PT-symmetric systems, which have recently attracted much attention. For a newly proposed deformation of exponential type we compute the minimal uncertainty and minimal length, which are essential in almost all approaches to quantum gravity.