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On the Possibility of a Relativistic Anyon Beam

J. Majhi, Subir Kumar Ghosh, Santanu K. Maiti

Published Feb 4, 2019• 0 citations
OpenAlex

Abstract

In this paper we propose the construction of a relativistic anyon beam. Following Jackiw and Nair~\cite{jn} we derive explicit form of relativistic plane wave solution of a single anyon. Subsequently we construct the planar anyon beam by superposing these solutions. Explicit expressions for the conserved anyon current are evaluated. Finally, we provide expressions for the anyon beam current using the superposed waves and discuss its features pictorially. We also comment on the possibility of experimental observation of anyon beam.

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In this paper we propose the construction of a relativistic anyon beam. Following Jackiw and Nair~\cite{jn} we derive explicit form of relativistic plane wave solution of a single anyon. Subsequently we construct the planar anyon beam by superposing these solutions. Explicit expressions for the conserved anyon current are evaluated. Finally, we provide expressions for the anyon beam current using the superposed waves and discuss its features pictorially. We also comment on the possibility of experimental observation of anyon beam.
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.
It is shown that the uniform ground state of ideal anyon gas can be constructed in the large-N collective field formalism and the excitation spectrum is given. It indicates that the system has a massless mode. In this treatment we employ a new set of gauge invariant variables to bypass the singularity which inevitably exists in the conventional density-phase variables. We also discuss the system of anyons including the delta function potential. The collective solution corresponding to Jackiw and Pi soliton is constructed.
We consider a quantum particle as a wave packet, and find that the group velocities in the coordinate and momentum spaces are in agreement with the Hamilton equations only when the Lagrangian is considered in the time dependent phases, instead of the Hamiltonian in the conventional forms of these waves as solutions of the Schrödinger equation.We define a relativistic quantum principle, and derive a wave equation for a relativistic quantum particle, the relativistic kinematics and dynamics of the particle waves, the Maxwell equations and the Lorentz force of a field interacting with the particle waves, the relativistic transform of such a field, and the spin as a characteristic of the particle waves.We consider a quantum particle as a distribution of conservative matter propagating according to the General Theory of Relativity.We obtain the dynamics of this matter in a gravitational field, the propagation in plane waves perpendicular to geodesic tracks, and equations of conservation.