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

Physical Review

Quantization of Relativistic Schrödinger Equations for Arbitrary Spin

P. M. Mathews

Published Mar 25, 1967• 15 citations
Publisher

Abstract

It is shown that a recently derived relativistic Schr\"odinger equation for free particles of arbitrary spin can be consistently quantized in the case of half-integer spins (but not for integer spins) by invoking the microcausality condition and using the role of certain expectation values of the $c$-number theory as generators of transformations of the Poincar\'e group in the $q$-number theory. The correct type of statistics (Fermi-Dirac) for half-integer-spin particles is obtained as a consequence of the theory. The way to handle the integer-spin case is indicated, but details are left for future presentation.

DOI
10.1103/physrev.155.1415

Related papers

The Spinning Electron

John Lekner

The notion introduced by Ohanian that spin is a wave property is implemented, both in Dirac and in Schrödinger quantum mechanics. We find that half-integer spin is the consequence of azimuthal dependence in two of the four spinor components, relativistically and non-relativistically. In both cases the spinor components are free particle wavepackets; the total wavefunction is an eigenstate of the total angular momentum in the direction of net particle motion. In the non-relativistic case we make use of the Lévy-Leblond result that four coupled non-relativistic wave equations, equivalent to the Pauli-Schrödinger equation, represent particles of half-integer spin, with g-factor 2. An example of an exact Gaussian solution of the non-relativistic equations is illustrated.

Substantia•May 24, 2022•0 citations

Stochastic Theory of Quantum Mechanics for Particles with Spin

L. de la Peña-Auerbach

The stochastic theory of quantum mechanics is further developed to include the problem of extended rigid particles, thus allowing the introduction of spin. It is demonstrated that the stochastic equations for the system's center of mass give rise to a generalized Schrödinger equation for integral or half-integral spin; in the particular case of spin ½, upon elimination of the internal variables, Pauli's equation is obtained. A formal simplified relativistic extension of the theory is worked out and shown to lead to Dirac's equation in the case of spin ½ and for a gyromagnetic ratio equal to 2; in the case of arbitrary spin, the theory gives an equation of the Feynman-Gell-Mann type.

Journal of Mathematical Physics•Mar 1, 1971•31 citations

On the Connection Between the Spin and Statistics of Elementary Particles

J. S. de Wet

It is shown that Fermi-Dirac quantization by the procedure of Heisenberg and Pauli cannot be carried out for tensor wave equations. Since the general wave equations for particles with integral spin are tensor equations, it follows that for these integral spin equations Fermi-Dirac quantization cannot be carried out. During the course of the discussion it appears that for equations derived from Lagrangians which are nonlinear in the derivatives of the functions, Fermi-Dirac quantization cannot be carried out. Since the Heisenberg-Pauli theory applies only to nonlinear Lagrangians, a special discussion of linear Lagrangians (linear in the derivatives of the functions) is given. It is shown how equations derived from such Lagrangians can be put into Hamiltonian form. Lagrangians of this type occur for the equations for half-odd spin.

Physical Review•Apr 1, 1940•12 citations

The Connection Between Spin and Statistics

Wolfgang Ernst Friederich Pauli

In the following paper we conclude for the relativistically invariant wave equation for free particles: From postulate (I), according to which the energy must be positive, the necessity of Fermi-Dirac statistics for particles with arbitrary half-integral spin; from postulate (II), according to which observables on different space-time points with a space-like distance are commutable, the necessity of Einstein-Bose statistics for particles with arbitrary integral spin. It has been found useful to divide the quantities which are irreducible against Lorentz transformations into four symmetry classes which have a commutable multiplication like $+1,\ensuremath{-}1,+\ensuremath{\epsilon},\ensuremath{-}\ensuremath{\epsilon}$ with ${\ensuremath{\epsilon}}^{2}=1$.

Physical Review•Oct 15, 1940•813 citations

Case of an Electron

Lukong Cornelius Fai

It is not possible to use the Schrödinger equation to describe the motion of a particle with relativistic effects since it is not invariant relative to the Lorentz transformation. However, the attempt to generalize quantum mechanics in the relativistic domain on the basis of the one particle theory is somehow difficult. So we find the appropriate equations on the basis of the relativistic invariance. To obtain the non-relativistic equations, we take the limit as the speed of light tends to infinity. This obviously recovers the Schrödinger equation. Under this heading, we extend the non-relativistic Schrödinger equation in the description of particles with speeds comparable with that of light. This extension should be consistent with the Lorentz transformation [ 12 ]. The characteristic feature of the relativistic wave equations is the spin which provides a useful gauge in the applicability of particular equations that describe particular kinds of particles. We have two types of relativistic equations, i.e., the equation with spin-zero due to Schrödinger and spin-half due to Dirac that describes the electron.

CRC Press eBooks•Apr 19, 2022•0 citations

Invariant Scalar Product and Observables in a Relativistic Theory of Particles of Arbitrary Spin

P. M. Mathews

In a recent paper a relativistically covariant Schr\"odinger equation was derived for particles of arbitrary spin $s$, locally covariant wave functions without redundant components being used to describe states of a particle. Here we determine the invariant scalar product with respect to which the representation of Poincar\'e transformations on these wave functions is unitary. It is shown that the conventional position and spin operators, not being Hermitian with respect to this scalar product, cannot be observables. New operators which can represent these observables are constructed with the aid of a generalized Foldy-Wouthuysen transformation which is determined explicitly for arbitrary spin.

Physical Review•Mar 25, 1966•27 citations
It is shown that a recently derived relativistic Schr"odinger equation for free particles of arbitrary spin can be consistently quantized in the case of half-integer spins (but not for integer spins) by invoking the microcausality condition and using the role of certain expectation values of the ccc-number theory as generators of transformations of the Poincar'e group in the qqq-number theory. The correct type of statistics (Fermi-Dirac) for half-integer-spin particles is obtained as a consequence of the theory. The way to handle the integer-spin case is indicated, but details are left for future presentation.
The notion introduced by Ohanian that spin is a wave property is implemented, both in Dirac and in Schrödinger quantum mechanics. We find that half-integer spin is the consequence of azimuthal dependence in two of the four spinor components, relativistically and non-relativistically. In both cases the spinor components are free particle wavepackets; the total wavefunction is an eigenstate of the total angular momentum in the direction of net particle motion. In the non-relativistic case we make use of the Lévy-Leblond result that four coupled non-relativistic wave equations, equivalent to the Pauli-Schrödinger equation, represent particles of half-integer spin, with g-factor 2. An example of an exact Gaussian solution of the non-relativistic equations is illustrated.
The stochastic theory of quantum mechanics is further developed to include the problem of extended rigid particles, thus allowing the introduction of spin. It is demonstrated that the stochastic equations for the system's center of mass give rise to a generalized Schrödinger equation for integral or half-integral spin; in the particular case of spin ½, upon elimination of the internal variables, Pauli's equation is obtained. A formal simplified relativistic extension of the theory is worked out and shown to lead to Dirac's equation in the case of spin ½ and for a gyromagnetic ratio equal to 2; in the case of arbitrary spin, the theory gives an equation of the Feynman-Gell-Mann type.
It is shown that Fermi-Dirac quantization by the procedure of Heisenberg and Pauli cannot be carried out for tensor wave equations. Since the general wave equations for particles with integral spin are tensor equations, it follows that for these integral spin equations Fermi-Dirac quantization cannot be carried out. During the course of the discussion it appears that for equations derived from Lagrangians which are nonlinear in the derivatives of the functions, Fermi-Dirac quantization cannot be carried out. Since the Heisenberg-Pauli theory applies only to nonlinear Lagrangians, a special discussion of linear Lagrangians (linear in the derivatives of the functions) is given. It is shown how equations derived from such Lagrangians can be put into Hamiltonian form. Lagrangians of this type occur for the equations for half-odd spin.
In the following paper we conclude for the relativistically invariant wave equation for free particles: From postulate (I), according to which the energy must be positive, the necessity of Fermi-Dirac statistics for particles with arbitrary half-integral spin; from postulate (II), according to which observables on different space-time points with a space-like distance are commutable, the necessity of Einstein-Bose statistics for particles with arbitrary integral spin. It has been found useful to divide the quantities which are irreducible against Lorentz transformations into four symmetry classes which have a commutable multiplication like $+1,\ensuremath{-}1,+\ensuremath{\epsilon},\ensuremath{-}\ensuremath{\epsilon}$ with ${\ensuremath{\epsilon}}^{2}=1$.
It is not possible to use the Schrödinger equation to describe the motion of a particle with relativistic effects since it is not invariant relative to the Lorentz transformation. However, the attempt to generalize quantum mechanics in the relativistic domain on the basis of the one particle theory is somehow difficult. So we find the appropriate equations on the basis of the relativistic invariance. To obtain the non-relativistic equations, we take the limit as the speed of light tends to infinity. This obviously recovers the Schrödinger equation. Under this heading, we extend the non-relativistic Schrödinger equation in the description of particles with speeds comparable with that of light. This extension should be consistent with the Lorentz transformation [ 12 ]. The characteristic feature of the relativistic wave equations is the spin which provides a useful gauge in the applicability of particular equations that describe particular kinds of particles. We have two types of relativistic equations, i.e., the equation with spin-zero due to Schrödinger and spin-half due to Dirac that describes the electron.
In a recent paper a relativistically covariant Schr\"odinger equation was derived for particles of arbitrary spin sss, locally covariant wave functions without redundant components being used to describe states of a particle. Here we determine the invariant scalar product with respect to which the representation of Poincar\'e transformations on these wave functions is unitary. It is shown that the conventional position and spin operators, not being Hermitian with respect to this scalar product, cannot be observables. New operators which can represent these observables are constructed with the aid of a generalized Foldy-Wouthuysen transformation which is determined explicitly for arbitrary spin.