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

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 (Jackiw R. and Nair 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 (Majhi J. et al., Phys. Rev. Lett., 123 (2019) 164801), in the framework of Bialynicki-Birula I. and Bialynicka-Birula 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.