A note on Hilbert Space Representation of Quantum Mechanics with Minimal Length

TL;DR

This paper revisits the Hilbert space formulation of quantum mechanics incorporating a minimal length and maximal momentum, highlighting necessary modifications to previous models for consistency at the Planck scale.

Contribution

It provides a revised framework for quantum mechanics with minimal length and maximal momentum, addressing inconsistencies in earlier models.

Findings

Revised the concept of maximally localized states.

Modified the quasi-position representation.

Identified the need for serious adjustments in recent models.

Abstract

We study some fundamental issues related to the Hilbert space representation of quantum mechanics in the presence of a minimal length and maximal momentum. In this framework, the maximally localized states and quasi-position representation introduced by Kempf et al. are reconsidered and modified. We show that all studies in recent years, including [15] and [16] need serious modification in order to be a consistent framework for quantum mechanics in Planck scale.