Deformed Heisenberg algebra and minimal length

TL;DR

This paper investigates how deforming the Heisenberg algebra affects the existence of a minimal length scale, providing explicit formulas for minimal uncertainty in position for various deformation functions.

Contribution

It characterizes the conditions on the deformation function for minimal length to exist and derives explicit expressions for this minimal length.

Findings

Identifies conditions on $f(P)$ for nonzero minimal length

Provides explicit formulas for minimal length in deformed algebra

Analyzes the relationship between deformation functions and minimal uncertainty

Abstract

A one-dimensional deformed Heisenberg algebra $[X,P]=if(P)$ is studied. We answer the question: For what function of deformation $f(P)$ there exists a nonzero minimal uncertainty in position (minimal length). We also find an explicit expression for the minimal length in the case of arbitrary function of deformation.