Uncertainty Relations for the Relativistic Jackiw-Nair Anyon: A First Principles Derivation

TL;DR

This paper derives the uncertainty relations for relativistic anyons, supporting the idea that they exist in a noncommutative 2D space, and compares these results with electrons in 3D space.

Contribution

It provides a first-principles quantum mechanical derivation of uncertainty relations for relativistic anyons, highlighting their noncommutative spatial properties.

Findings

Anyon position-position uncertainty relation indicates noncommutative space.

Heisenberg uncertainty relation for anyons is explicitly computed.

Electrons exhibit trivial position uncertainty in 3D space.

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, ${\sqrt{\Delta x^2\Delta y^2}}=\hbar\bar{\Theta}_{xy}$, 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.