Uncertainty relations for the relativistic Jackiw-Nair nyon: A first principles derivation

Abstract In this paper we have explicitly computed the $position-position$ and $position-momentum$ (Heisenberg) Uncertainty Relations for the model of relativistic particles with arbitrary spin, proposed by Jackiw and Nair \cite{jn} as a model for Anyon, in a purely quantum mechanical framework. This supports (via Schwarz inequality) the conjecture that anyons live in a 2-dimensional {\it{noncommutative}} space. We have computed the non-trivial uncertainty relation between anyon coordinates, ${\sqrt{\Delta x^2\Delta y^2}}=\hbar\bar{\Theta}_{xy}$, using the recently constructed anyon wave function \cite{jan}, in the framework of \cite{bel}. We also compute the Heisenberg (position-momentum) uncertainty relation for anyons. Lastly we show that the identical {\it{formalism}} when applied to electrons, yield a trivial position uncertainty relation, consistent with their living in a 3-dimensional commutative space.