Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Mathematical Physics

Position and Polarization Operators in Relativistic and Nonrelativistic Mechanics

H. Bacry

Published Jan 1, 1964• 32 citations
Publisher

Abstract

The author shows how the position operator and the polarization operator are obtained in nonrelativistic quantum mechanics from their equivalents in relativistic quantum mechanics. This problem is investigated with the aid of group methods; it is furthermore shown that the two operators cannot be studied separately.
DOI
10.1063/1.1704054

Related papers

Galilei Group and Nonrelativistic Quantum Mechanics

Jean-Marc Lévy-Leblond

This paper is devoted to the study of the Galilei group and its representations. The Galilei group presents a certain number of essential differences with respect to the Poincaré group. As Bargmann showed, its physical representations, here explicitly constructed, are not true representations but only up-to-a-factor ones. Consequently, in nonrelativistic quantum mechanics, the mass has a very special role, and in fact, gives rise to a superselection rule which prevents the existence of unstable particles. The internal energy of a nonrelativistic system is known to be an arbitrary parameter; this is shown to come also from Galilean invariance, because of a nontrivial concept of equivalence between physical representations. On the contrary, the behavior of an elementary system with respect to rotations, is very similar to the relativistic case. We show here, in particular, how the number of polarization states reduces to two for the zero-mass case (though in fact there are no physical zero-mass systems in nonrelativistic mechanics). Finally, we study the two-particle system, where the orbital angular momenta quite naturally introduce themselves through the decomposition of the tensor product of two physical representations.

Journal of Mathematical Physics•Jun 1, 1963•241 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

Группа Пуанкаре и операторы местоположения и спина: Аннотации

Alexander J. Silenko, Pengming Zhang, Liping Zou +1 more

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.

ФИЗИКА ЭЛЕМЕНТАРНЫХ ЧАСТИЦ И АТОМНОГО ЯДРА•May 31, 2024•0 citations

ON NON-SELF-ADJOINT OPERATORS FOR OBSERVABLES IN QUANTUM MECHANICS AND QUANTUM FIELD THEORY

Erasmo Recami, V. S. Olkhovsky, Sergei P. Maydanyuk

The aim of this paper is to show the possible significance, and usefulness, of various non-self-adjoint operators for suitable Observables in nonrelativistic and relativistic quantum mechanics, and in quantum electrodynamics. More specifically, this work deals with: (i) the maximal Hermitian (but not self-adjoint) time operator in nonrelativistic quantum mechanics and in quantum electrodynamics; (ii) the problem of the four-position and four-momentum operators, each one with its Hermitian and anti-Hermitian parts, for relativistic spin-zero particles. Afterwards, other physically important applications of non-self-adjoint (and even non-Hermitian) operators are discussed: in particular, (iii) we reanalyze in detail the interesting possibility of associating quasi-Hermitian Hamiltonians with (decaying) unstable states in nuclear physics. Finally, we briefly mention the cases of quantum dissipation, as well as of the nuclear optical potential.

arXiv:0903.3187•International Journal of Modern Physics A•Apr 10, 2010•14 citations

Relativistic Schrödinger equations

Ll. Bel, E. Ruiz

A representation of the Poincaré group by means of operators acting on the function space of the nonrelativistic quantum mechanics is introduced. The conditions that the Hamiltonian of an interacting particle system must verify in order to define a relativistic invariant Schrödinger equation are found out. The relation with usual quantum constraint theories is also studied.

Journal of Mathematical Physics•Aug 1, 1988•4 citations

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
This paper is devoted to the study of the Galilei group and its representations. The Galilei group presents a certain number of essential differences with respect to the Poincaré group. As Bargmann showed, its physical representations, here explicitly constructed, are not true representations but only up-to-a-factor ones. Consequently, in nonrelativistic quantum mechanics, the mass has a very special role, and in fact, gives rise to a superselection rule which prevents the existence of unstable particles. The internal energy of a nonrelativistic system is known to be an arbitrary parameter; this is shown to come also from Galilean invariance, because of a nontrivial concept of equivalence between physical representations. On the contrary, the behavior of an elementary system with respect to rotations, is very similar to the relativistic case. We show here, in particular, how the number of polarization states reduces to two for the zero-mass case (though in fact there are no physical zero-mass systems in nonrelativistic mechanics). Finally, we study the two-particle system, where the orbital angular momenta quite naturally introduce themselves through the decomposition of the tensor product of two physical representations.
The aim of this paper is to show the possible significance, and usefulness, of various non-self-adjoint operators for suitable Observables in nonrelativistic and relativistic quantum mechanics, and in quantum electrodynamics. More specifically, this work deals with: (i) the maximal Hermitian (but not self-adjoint) time operator in nonrelativistic quantum mechanics and in quantum electrodynamics; (ii) the problem of the four-position and four-momentum operators, each one with its Hermitian and anti-Hermitian parts, for relativistic spin-zero particles. Afterwards, other physically important applications of non-self-adjoint (and even non-Hermitian) operators are discussed: in particular, (iii) we reanalyze in detail the interesting possibility of associating quasi-Hermitian Hamiltonians with (decaying) unstable states in nuclear physics. Finally, we briefly mention the cases of quantum dissipation, as well as of the nuclear optical potential.
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.