Quantizations of IR

Quantum real numbers are proposed by performing a quantum deformation of the standard real numbers IR. We start with the q-deformed Heisenberg algebra L̂q which is obtained by the Moyal ∗-deformation of the Heisenberg algebra generated by a and a. By representing L̂q 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 1 N , the quantum real line is composed of fuzzy, i.e., fluctuating points and nontrivial infinitesimal structure appears around every standard real number.