Hilbert space representation of maximal length and minimal momentum uncertainties

TL;DR

This paper introduces a new deformed algebra incorporating a maximal length and minimal momentum uncertainties, deriving their implications, constructing Fourier transforms, and extending the framework to n-dimensional spaces.

Contribution

It proposes a novel deformed algebra with maximal length and minimal momentum, deriving associated uncertainties and mathematical representations, extending previous models.

Findings

Derived maximal length and minimal momentum uncertainties from the new algebra.

Constructed Fourier transform and inverse for the deformed algebra.

Extended the algebra's representation to n-dimensional spaces.

Abstract

Perivolaropoulos has recently proposed a position-deformed Heisenberg algebra which includes a maximal length [Phys.Rev.95, 103523 (2017)]. He has shown that this length scale naturally emerges in the context of cosmological particle's horizon or cosmic topology. Following this work, we propose a new deformed algebra and derive the maximal length uncertainty and its corresponding minimal momentum uncertainty from the generalized uncertainty principle. We also construct the corresponding Fourier transform and its inverse representations. Finally, we propose n-dimensional representation of this algebra