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

arXiv:1109.4899 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Spin operator and spin states in Galilean covariant Fermi field theories

Fuad M. Saradzhev

Published Dec 20, 2011• 2 citations
arXivHTMLPDF

Abstract

Spin degrees of freedom of the Galilean covariant Dirac field in $(4+1)$ dimensions and its nonrelativistic counterpart in $(3+1)$ dimensions are examined. Two standard choices of spin operator, the Galilean covariant and Dirac spin operators, are considered. It is shown that the Dirac spin of the Galilean covariant Dirac field in $(4+1)$ dimensions is not conserved, and the role of non-Galilean boosts in its nonconservation is stressed out. After reduction to $(3+1)$ dimensions the Dirac field turns into a nonrelativistic Fermi field with a conserved Dirac spin. A generalized form of the L\'evy-Leblond equations for the Fermi field is given. One-particle spin states are constructed. A particle-antiparticle system is discussed.

DOI
10.1103/physrevd.84.125025

Related papers

A spin Hamiltonian for non-relativistic electrons and their interaction with an external field

E. S. Santos, Roberto Rivelino, M. de Montigny +1 more

We investigate the spin of the electron in a non-relativistic context by using the Galilean covariant Pauli-Dirac equation. From a non-relativistic Lagrangian density, we find an appropriate Dirac-like Hamiltonian in the momentum representation, which includes the spin operator in the Galilean covariant framework. Within this formalism, we show that the total angular momentum appears as a constant of motion. Additionally, we propose a non-minimal coupling that describes the Galilean interaction between an electron and the electromagnetic field. Thereby, we obtain, in a natural way, the Hamiltonian including all the essential interaction terms for the electron in a general vector field.

Journal of Physics A Mathematical and Theoretical•Jul 16, 2010•4 citations

Path-integral quantization of Galilean Fermi fields 1

M. de Montigny, F. C. Khanna, Fuad M. Saradzhev

The Galilei-covariant fermionic field theories are quantized by using the path-integral method and five-dimensional Lorentz-like covariant expressions of non-relativistic field equations. Firstly we review the five-dimensional approach to the Galilean Dirac equation, which leads to the Lévy-Leblond equations, and define the Galilean generating functional and Green’s functions for positive- and negative-energy/mass solutions. Then, as an example of interactions, we consider the quartic self-interacting potential λ(ΨΨ) 2, and we derive expressions for the 2- and 4-point Green’s functions. Our results are compatible with those found in the literature on non-relativistic many-body systems. The extended manifold allows for compact expressions of the contributions in (3 + 1) space-time. This is particularly apparent when we represent the results with diagrams in the extended (4 + 1) manifold, since they usually encompass more diagrams in Galilean (3 + 1) space-time. 1

•Dec 4, 2012•5 citations

Lorentz-Covariant Position Operators for Spinning Particles

Thomas F. Jordan, Narasimhaiengar Mukunda

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.

Physical Review•Nov 15, 1963•77 citations

On the Galilean Duffin–Kemmer–Petiau equation in arbitrary dimensions

M. de Montigny, E. S. Santos

We obtain the representations of the Galilean covariant Duffin–Kemmer–Petiau equation in an arbitrary number of dimensions. Their purpose is to facilitate the study of nonrelativistic many-body systems with spinless and spin-one fields. A Galilean covariant formalism exploits the tensor structure of relativistic Lorentz-invariant theories by adding one extra spacelike coordinate and working with light-cone coordinates.

International Journal of Modern Physics A•Jun 17, 2020•2 citations

Covariant Abstract Description for a Dirac Particle

Paweł Caban, Jakub Rembieliński, Marta Włodarczyk

We give a direct link between covariant description of Dirac particles in the abstract framework of unitary representation of the Poincaré group and description with the help of the Dirac equation. In this context we discuss the spin operator for a relativistic Dirac particle. We show also that the spin operator used in quantum field theory for spin s = 1/2 corresponds to the Foldy-Wouthuysen mean-spin operator. We hope that this formalism will be useful in the theory of relativistic quantum information.

Open Systems & Information Dynamics•Dec 1, 2012•4 citations

Charge Conjugation in the Galilean Limit

M. Socolovsky

Strictly working in the framework of the nonrelativistic quantum mechanics of a spin 1/2 particle coupled to an external electromagnetic field, we show, by explicit construction, the existence of a charge conjugation operator matrix which defines the corresponding antiparticle wave function and leads to the galilean and gauge invariant Schroedinger-Pauli equation satisfied by it.

arXiv:hep-th/0603137•Mar 17, 2006•1 citation
Spin degrees of freedom of the Galilean covariant Dirac field in (4+1)(4+1)(4+1) dimensions and its nonrelativistic counterpart in (3+1)(3+1)(3+1) dimensions are examined. Two standard choices of spin operator, the Galilean covariant and Dirac spin operators, are considered. It is shown that the Dirac spin of the Galilean covariant Dirac field in (4+1)(4+1)(4+1) dimensions is not conserved, and the role of non-Galilean boosts in its nonconservation is stressed out. After reduction to (3+1)(3+1)(3+1) dimensions the Dirac field turns into a nonrelativistic Fermi field with a conserved Dirac spin. A generalized form of the L'evy-Leblond equations for the Fermi field is given. One-particle spin states are constructed. A particle-antiparticle system is discussed.
We investigate the spin of the electron in a non-relativistic context by using the Galilean covariant Pauli-Dirac equation. From a non-relativistic Lagrangian density, we find an appropriate Dirac-like Hamiltonian in the momentum representation, which includes the spin operator in the Galilean covariant framework. Within this formalism, we show that the total angular momentum appears as a constant of motion. Additionally, we propose a non-minimal coupling that describes the Galilean interaction between an electron and the electromagnetic field. Thereby, we obtain, in a natural way, the Hamiltonian including all the essential interaction terms for the electron in a general vector field.
The Galilei-covariant fermionic field theories are quantized by using the path-integral method and five-dimensional Lorentz-like covariant expressions of non-relativistic field equations. Firstly we review the five-dimensional approach to the Galilean Dirac equation, which leads to the Lévy-Leblond equations, and define the Galilean generating functional and Green’s functions for positive- and negative-energy/mass solutions. Then, as an example of interactions, we consider the quartic self-interacting potential λ(ΨΨ) 2, and we derive expressions for the 2- and 4-point Green’s functions. Our results are compatible with those found in the literature on non-relativistic many-body systems. The extended manifold allows for compact expressions of the contributions in (3 + 1) space-time. This is particularly apparent when we represent the results with diagrams in the extended (4 + 1) manifold, since they usually encompass more diagrams in Galilean (3 + 1) space-time. 1
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.
We give a direct link between covariant description of Dirac particles in the abstract framework of unitary representation of the Poincaré group and description with the help of the Dirac equation. In this context we discuss the spin operator for a relativistic Dirac particle. We show also that the spin operator used in quantum field theory for spin s = 1/2 corresponds to the Foldy-Wouthuysen mean-spin operator. We hope that this formalism will be useful in the theory of relativistic quantum information.