Anyons Live in Non-Commutative Space: A First Principles Demonstration

In this paper we have successfully established (from first principles) that anyons do live in a 2-dimensional {\it{noncommutative}} space. We have directly computed the non-trivial uncertainty relation between anyon coordinates, ${\sqrt{\Delta x^2\Delta y^2}}=\Theta\sigma$, using the recently constructed anyon wave function [J. Majhi, S. Ghosh and S.K. Maiti, Phys. Rev. Lett. \textbf{123}, 164801 (2019)] \cite{jan}, in the framework of I. Bialynicki-Birula and Z. Bialynicka-Birula, New J. Phys. \textbf{21}, 07306 (2019) \cite{bel}. Furthermore we also compute the Heisenberg uncertainty relation for anyon and as a consistency check, show that the results of \cite{bel} prove that, as expected, electrons live in 3-dimensional commutative space.