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

J. Majhi, Subir Ghosh

Published Jul 20, 2021• 0 citations
OpenAlex

Abstract

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.

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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, Δx2Δy2=Θσ{\sqrt{\Delta x^2\Delta y^2}}=\Theta\sigmaΔx2Δy2​=Θσ, 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.
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 (J ackiw R. and N air V. P. Phys. Rev. D , 43 (1991) 1933) 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 non-commutative space. We have computed the non-trivial uncertainty relation between anyon coordinates, , using the recently constructed anyon wave function (M ajhi J. et al. , Phys. Rev. Lett. , 123 (2019) 164801), in the framework of B ialynicki -B irula I. and B ialynicka -B irula Z., New J. Phys. , 21 (2019) 07306. We also compute the Heisenberg (position-momentum) uncertainty relation for anyons. Lastly we show that the identical formalism when applied to electrons, yield a trivial position uncertainty relation, consistent with their living in a 3-dimensional commutative space.
Anyons and fractional statistics are by now well established in two-dimensional systems. In one dimension, fractional statistics has been established so far only through Haldane's fractional exclusion principle, but not via a fractional phase the wave function acquires as particles are interchanged. At first sight, the topology of the configuration space appears to preclude such phases in one dimension. Here we argue that the crossings of one-dimensional anyons are always unidirectional, which makes it possible to assign phases consistently and hence to introduce a statistical angle $\ensuremath{\theta}$. The fractional statistics then manifests itself in fractional spacings of the single-particle momenta of the anyons when periodic boundary conditions are imposed. These spacings are given by $\ensuremath{\Delta}p=2\ensuremath{\pi}\ensuremath{\hbar}/L(|\ensuremath{\theta}|/\ensuremath{\pi}+\text{non-negative}\text{ }\text{integer})$ for a system of length LLL. This condition is the analog of the quantization of relative angular momenta according to ${l}_{\text{z}}=\ensuremath{\hbar}(\ensuremath{-}\ensuremath{\theta}/\ensuremath{\pi}+2\ifmmode\times\else\texttimes\fi{}\text{integer})$ for two-dimensional anyons.
The structure of the state-vector space of identical bosons in noncommutative spaces is investigated. To maintain Bose–Einstein statistics the commutation relations of phase space variables should simultaneously include coordinate-coordinate non-commutativity and momentum-momentum non-commutativity, which leads to a kind of deformed Heisenberg-Weyl algebra. Although there is no ordinary number representation in this state-vector space, several set of orthogonal and complete state-vectors can be derived which are common eigenvectors of corresponding pairs of commuting Hermitian operators. As a simple application of this state-vector space, an explicit form of two-dimensional canonical coherent state is constructed and its properties are discussed.
This work completes the program started by I. Bialynicki-Birula and Z. Bialynicka-Birula [Uncertainty relation for photons, Phys. Rev. Lett. 108, 140401 (2012); Heisenberg uncertainty relation for photons, Phys. Rev. A 86, 022118 (2012); Heisenberg uncertainty relation for relativistic electrons, New J. Phys. 21, 073036 (2019)] to derive the Heisenberg uncertainty relation for relativistic particles. Sharp uncertainty relations for massive relativistic particles with spin 0 and spin 1 are derived. The main conclusion is that the uncertainty relations for relativistic bosons are markedly different from those for relativistic fermions. The uncertainty relations for bosons are based on the energy density. It is shown that the uncertainty relations based on the time component of the four-current, as we have done previously for electrons, are untenable because they lead to contradictions.