Uncertainty Relations for the Relativistic Jackiw-Nair Anyon: 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 ref.[1] 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 \textit{noncommutative} space. We have computed the non-trivial uncertainty relation between anyon coordinates, $\sqrt{\Delta x^2\Delta y^2}=\hslash {\bar{\Theta }}_{xy}$, using the recently constructed anyon wave function ref.[6], in the framework of ref.[7]. We also compute the Heisenberg (position-momentum) uncertainty relation for anyons. Lastly we show that the identical \textit{formalism} when applied to electrons, yield a trivial position uncertainty relation, consistent with their living in a 3-dimensional commutative space.

arXiv comment7 pages, 5 figures; title changed, change in presentation, no channge in final results and conclusions, accepted in EPL

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