Position and spin in relativistic quantum mechanics

Liping Zou,Pengming Zhang,Alexander J Silenko

doi:10.1103/physreva.101.032117

Abstract

The problem of the position and spin in relativistic quantum mechanics is analyzed in detail. It is definitively shown that the position and spin operators in the Foldy-Wouthuysen representation (but not in the Dirac one) are quantum-mechanical counterparts of the classical position and spin variables. The probabilistic interpretation is valid only for Foldy-Wouthuysen wave functions. The relativistic spin operators are discussed. The spin-orbit interaction does not exist for a free particle if the conventional operators of the orbital angular momentum and the rest-frame spin are used. Alternative definitions of the orbital angular momentum and the spin are based on noncommutative geometry, do not satisfy standard commutation relations, and can allow the spin-orbit interaction.

Highlights

The position operator is very important for relativistic quantum mechanics (QM)
Newton and Wigner [2] have obtained the form of the position operator having commuting components and localized eigenfunctions in the manifold of positive-energy wave functions based on the Dirac representation
The goal of the present study is a change of the paradoxical contemporary situation in the relativistic QM when the forms of the position and spin operators securely established sixty years ago are “forgotten” while incorrect and unsubstantiated definitions of these operators are widely used

Summary

Introduction

The position operator is very important for relativistic quantum mechanics (QM). In nonrelativistic Schrodinger QM, this operator is equal to the radius vector r. A transition to relativistic QM leads to a dependence of this operator on a representation. It has been shown by Pryce [1] that the form of the position operator for a spin-1/2 particle is nontrivial and some possible forms have been obtained. Newton and Wigner [2] have obtained the form of the position operator having commuting components and localized eigenfunctions in the manifold of positive-energy wave functions based on the Dirac representation. Foldy and Wouthuysen have shown [3] that this operator is equal to the radius vector operator in the Foldy-Wouthuysen (FW) representation

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Methods

Conclusion