© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Physical Review

Lorentz-Covariant Position Operators for Spinning Particles

Thomas F. Jordan, Narasimhaiengar Mukunda

Published Nov 15, 1963• 77 citations

Abstract

An examination is made of the consequences for the quantum mechanics of spinning particles of equations characteristic of Lorentz-covariant position variables. These equations are commutator analogs of the Poisson bracket equations that express the familiar transformation properties of space-time events in classical mechanics. For a particle of zero spin it is found that the usual canonical coordinate is the unique solution of these equations. For a particle with positive spin there is no position operator which satisfies these equations and has commuting components. For a particle and antiparticle there is a unique solution with commuting components which is valid for all values of the spin and reduces for zero spin to the canonical coordinate. For spin \textonehalf{} this is the Foldy-Wouthuysen transform of the position operator of the Dirac equation. A generalization of the inverse Foldy-Wouthuysen transformation, valid for any value of the spin, appears as a unique unitary transformation which takes this generalized Dirac position to the canonical coordinate. The application of this transformation to the canonical form of the Hamiltonian gives a generalization of the Dirac equation Hamiltonian. This is developed and compared with the literature for spin 1. It gives a nonlocal equation as the spin 1 analog of the Dirac equation.

DOI: 10.1103/physrev.132.1842

Related papers

Position and spin in relativistic quantum mechanics

Liping Zou, Pengming Zhang, Alexander J. Silenko

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.

arXiv:2003.06572•Physical Review A•Mar 26, 2020•45 citations

Position and Intrinsic Spin Operators in Quantum Theory

R. A. Berg

The problem of defining position and intrinsic spin operators in terms of the generators of the inhomogeneous Lorentz group is considered. These operators are taken to have the properties usually attributed to them in nonrelativistic quantum theory: their commutation relations must have the commonly accepted form. These commutation relations are then shown to define the intrinsic spin uniquely. The position operator is shown to be also essentially uniquely determined. Explicit forms of the spin and position operators for some special representations are exhibited. The relation of these operators to the spin and position operators of the spin-½ Dirac theory in the Foldy-Wouthuysen representation is considered. An Appendix gives a heuristic derivation of the spin operator.

Journal of Mathematical Physics•Jan 1, 1965•26 citations

Relativistic corrections to the algebra of position variables and spin-orbital interaction

Alexei A. Deriglazov, Andrey M. Pupasov-Maksimov

In the framework of vector model of spin, we discuss the problem of a covariant formalism [35] concerning the discrepancy between relativistic and Pauli Hamiltonians. We show how the spin-induced non-commutativity of a position accounts the discrepancy on the classical level, without appeal to the Dirac equation and Foldy–Wouthuysen transformation.

arXiv:1609.00043•Physics Letters B•Aug 22, 2016•36 citations

Fundamental operators in Dirac quantum mechanics

Alexander J. Silenko, Pengming Zhang, Liping Zou

Abstract Old achievements and more recent results in a solution of problem of the position and spin in relativistic quantum mechanics are considered. It is definitively shown that quantum-mechanical counterparts of the classical position and spin variables are the position and spin operators in the Foldy-Wouthuysen representation (but not in the Dirac one). The probabilistic interpretation is valid only for Foldy-Wouthuysen wave functions.

Journal of Physics Conference Series•Jan 1, 2020•1 citation

Definition of Position and Spin Operators for Massive Elementary Relativistic Systems of Arbitrary Spin

J. A. de Azcárraga, Luis Oliver, J. Pascual

Since the important work of Newton and Wigner on the position operator and the localized states, several methods have been developed in the literature to deal with the problem of finding a position operator for relativistic systems. One of the most relevant has been the method based on the use of "canonical" transformations such as the Foldy-Wouthuysen transformation. In this paper, we strictly consider the Foldy-Wouthuysen transformation as a procedure which allows one to write the Bargmann-Wigner equations in a form which in the nonrelativistic limit leads to the Galilei-invariant Schr\"odinger equations for arbitrary spin. As a result of this interpretation, we derive for the position operator and the localized states of elementary systems the same expressions as Newton and Wigner; the spin operator is also obtained. Finally, the Chakrabarti transformation is considered in the same spirit.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Dec 15, 1973•6 citations

Poincaré Group and Operators of Position and Spin

Alexander J. Silenko, Pengfei Zhang, Liping Zou

Abstract Fundamental properties of the position and spin operators in relativistic quantum mechanics are defined with the Poincaré group. Quantum-mechanical counterparts of the classical position and spin variables are the corresponding operators in the Foldy–Wouthuysen representation but not in the Dirac one. The probabilistic interpretation is valid only for Foldy–Wouthuysen wave functions. The relativistic operators of the position and spin are discussed.

Physics of Particles and Nuclei•Nov 29, 2023•0 citations

An examination is made of the consequences for the quantum mechanics of spinning particles of equations characteristic of Lorentz-covariant position variables. These equations are commutator analogs of the Poisson bracket equations that express the familiar transformation properties of space-time events in classical mechanics. For a particle of zero spin it is found that the usual canonical coordinate is the unique solution of these equations. For a particle with positive spin there is no position operator which satisfies these equations and has commuting components. For a particle and antiparticle there is a unique solution with commuting components which is valid for all values of the spin and reduces for zero spin to the canonical coordinate. For spin \textonehalf{} this is the Foldy-Wouthuysen transform of the position operator of the Dirac equation. A generalization of the inverse Foldy-Wouthuysen transformation, valid for any value of the spin, appears as a unique unitary transformation which takes this generalized Dirac position to the canonical coordinate. The application of this transformation to the canonical form of the Hamiltonian gives a generalization of the Dirac equation Hamiltonian. This is developed and compared with the literature for spin 1. It gives a nonlocal equation as the spin 1 analog of the Dirac equation.

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.

The problem of defining position and intrinsic spin operators in terms of the generators of the inhomogeneous Lorentz group is considered. These operators are taken to have the properties usually attributed to them in nonrelativistic quantum theory: their commutation relations must have the commonly accepted form. These commutation relations are then shown to define the intrinsic spin uniquely. The position operator is shown to be also essentially uniquely determined. Explicit forms of the spin and position operators for some special representations are exhibited. The relation of these operators to the spin and position operators of the spin-½ Dirac theory in the Foldy-Wouthuysen representation is considered. An Appendix gives a heuristic derivation of the spin operator.

Since the important work of Newton and Wigner on the position operator and the localized states, several methods have been developed in the literature to deal with the problem of finding a position operator for relativistic systems. One of the most relevant has been the method based on the use of "canonical" transformations such as the Foldy-Wouthuysen transformation. In this paper, we strictly consider the Foldy-Wouthuysen transformation as a procedure which allows one to write the Bargmann-Wigner equations in a form which in the nonrelativistic limit leads to the Galilei-invariant Schr\"odinger equations for arbitrary spin. As a result of this interpretation, we derive for the position operator and the localized states of elementary systems the same expressions as Newton and Wigner; the spin operator is also obtained. Finally, the Chakrabarti transformation is considered in the same spirit.