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arXiv:2107.09342 · Europhysics Letters (EPL)

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

Joydeep Majhi, Subir Ghosh

Published Oct 1, 2023• 0 citations
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Abstract

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.

DOI
10.1209/0295-5075/ad0670

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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.
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.
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.
Heisenberg guessed, after he established the matrix quantum mechanics, that the non-commutativity of the matrices of position and momentum implied that the position and momentum of a particle could not be precisely simultaneously determined. He consequently conjectured that time and energy should also have a similar relationship. Soon after, Robertson derived an inequality concerning the space coordinate and momentum, which was thought to be the mathematical expression of the uncertainty relation guessed by Heisenberg. Since then, people have tried various devices to prove the correctness of these two relations. However, no one conducted a careful analysis of Heisenberg’s primary paper. In this work, we point out some serious problems in Heisenberg’s paper and the literature talking about the uncertainty relationships: the physical concepts involved in the uncertainty relations are not clear; one physical concept had more than one explanation, i.e., switching concepts; there has never been measurement experiment to support the relations. The conclusions are that the so-called coordinate–momentum uncertainty relation has never been related to actual measurement and there does not exist a time–energy uncertainty relation.
The focus of the present investigation is uncertainty relations for quantum particles, which quantify the fundamental limitations on some of their properties due to their incompatibility. In the first, longer part, we are concerned with preparational uncertainty \nrelations, while in the second we touch upon measurement inequalities through a generalisation of a model of joint measurement of position and momentum. \nSpecifically, starting from a triple of canonical operators, we prove product and sum inequalities for their variances with bounds larger than those following from combining the pairwise ones. We extend these results to N observables for a quantum particle and prove uncertainty relations for the sums and products of their variances \nin terms of the commutators. \nFurthermore, we present a general theory of preparational uncertainty relations for a quantum particle in one dimension and derive conditions for a smooth function of the second moments to assume a lower bound. The Robertson-Schroedinger inequality \nis found to be of special significance and we geometrically study the space of second moments. We prove new uncertainty relations for various functions of the variances and covariance of position and momentum of a quantum particle. Some of our findings are shown to extend to more than one spatial degree of freedom and we derive \nvarious types of inequalities. \nFinally, we propose a generalisation of the Arthurs-Kelly model of joint measurement of position and momentum to incorporate the case of more than two observables and derive joint-measurement inequalities for the statistics of the probes. For the case of \nthree canonical observables and suitable definitions of error and disturbance we obtain a number of error-error-error and error-error-disturbance inequalities and show that the lower bound is identical to the preparational one.