Quantum Real Lines,Infinitesimal Structure of $\R$

TL;DR

This paper introduces quantum real lines as noncommutative, discrete spaces derived from q-deformations of the real numbers, revealing structures like minimal length and fuzziness.

Contribution

It constructs two types of quantum real lines from q-deformed algebras, exploring their noncommutative, discrete, and infinitesimal properties.

Findings

Quantum lines are discrete noncommutative spaces.

Identified minimal length and fuzzy structures.

Revealed infinitesimal structures in quantum lines.

Abstract

We present in this paper quantum real lines as quantum defomations of the real numbers $\R$.Upon deforming the Heisenberg algebra $\cL$ generated by $(a, a^\dagger)$ in terms of the Moyal $\ast$-product,we first construct q-deformed algebras of q-differentiable functions in two cases where q is generic (not a root of unity) and q is the N-th root of unity. We then investigate these algebras and finally propose two quantum real lines as the base spaces of these algebras. It is turned out that both quantum lines are discrete spaces and have noncommutative structures.We further find, minimal length, fuzzy structure and infinitesimal structure.