Quantizations of R(eal numbers)

Details Quantizations of R(eal numbers) Quantizations of R(eal numbers)

2003-11-17

arxiv.org/abs/hep-th/0311140v1

Physics

High Energy Physics

High Energy Physics - Theory

23 pages, no figures

Scientific paper

Quantum real numbers are proposed by performing a quantum deformation of the standard real numbers $\R$. We start with the q-deformed Heisenberg algebra $\cLLq$ which is obtained by the Moyal $\ast$-deformation of the Heisenberg algebra generated by $a$ and $\ad$. By representing $\cLLq$ as the algebras of $q$-differentiable functions, we derive quantum real lines from the base spaces of these functional algebras. We find that these quantum lines are discrete spaces. In particular, for the case with $q = e^{2\pi i \frac{1}{N}} $, the quantum real line is composed of fuzzy, i.e., fluctuating points and nontrivial infinitesimal structure appears around every standard real number.

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