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Classical and quantum theories of spin

Fabián H. Gaioli, Edgardo T. Garcia Alvarez

Published Jan 1, 1998• 7 citations
OpenAlex

Abstract

A great effort has been devoted to formulate a classical relativistic theory of spin compatible with quantum relativistic wave equations. The main difficulty in order to connect classical and quantum theories rests in finding a parameter which plays the role of proper time at a purely quantum level. We present a partial review of several proposals of classical and quantum spin theories from the pioneer works of Thomas and Frenkel, revisited in the classical BMT work, to the semiclassical model of Barut and Zanghi [Phys. Rev. Lett. 52, 2009 (1984)]. We show that the last model can be obtained from a semiclassical limit of the Feynman proper time parametrization of the Dirac equation. At the quantum level we derive spin precession equations in the Heisenberg picture. Analogies and differences with respect to classical theories are discussed in detail. 1

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Abstract Conventional methods for teaching spin at a university level generally follow four main approaches. The first introduces spin into the Schrödinger equation artificially, leaving its physical meaning unclear. The second connects spin to the representation of the rotation group, which clarifies its nature but requires knowledge of group theory. The third approach derives from field theory, separating angular momentum into orbital and intrinsic parts, the latter corresponding to spin; however, this cannot be done within quantum mechanics alone. The fourth derives electron spin from Dirac’s 1928 relativistic equation, which, though correct, has led to the misconception that spin is a purely relativistic effect. A detailed analysis of Dirac’s 1928 papers shows that the concept of spin arises from the need to construct an equation linear in both energy and momentum to preserve Born’s probabilistic interpretation. This linearization of the Klein–Gordon equation reveals that spin is conceptually independent of relativity. Historical and pedagogical perspectives can together clarify conceptual difficulties and promote a deeper understanding of fundamental ideas in physics. Drawing on selected passages from Dirac’s original work, the present discussion offers broader conceptual and cultural considerations that introduce the notion of spin with limited mathematical requirements and can be applied independently of the didactic approach adopted.

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The classical equations of motion of a charged point-particle with intrinsic spin under the influence of an external electromagnetic field are restated and compared with the Heisenberg equations of motion derived from the Dirac theory. The partition of angular momentum between particle and field in the classical theory is contrasted to the Dirac theory of electron spin. The analogy between the Dirac equation and the theory of parametric amplification is pointed out.A free spinning point particle moving according to the laws of classical relativistic point-particle mechanics may move along a helix. The sum of the intrinsic spin $\ensuremath{\sigma}$ and the angular momentum of the helical motion in this classical analog of zitterbewegung is an effective spin vector S which is a constant of the motion. Because of this internal motion, the effective mass $M$ of the particle differs from the mass $m$ which is ascribed to it in the equations of motion. Solutions are found in which S is parallel or antiparallel to the momentum, and the sign of $M$ is determined by the helicity. When placed in a uniform electromagnetic field, the particle behaves as if it had a rest mass $M$ and a magnetic moment $\frac{e\ensuremath{\sigma}}{\mathrm{Mc}}$, in addition to any explicit magnetic moment that may be ascribed to it.

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The functional form of the wave function with respect to variables appropriate for the WKB approximation is chosen by taking into account the transformation properties of the spinor wave-function.This representation makes it possible, using only the wave equation, to pursue directly the time changes in the variables, i.e. the spin direction, velocity and density, etc., characterizing an electron.These changes correspond straightforwardly to a classical model of the non-relativistic Pauli electron with the proper magnetic moment, and the covariant form of this model, which is just "the Frenkel-Kramers model of the Dirac electron" introduced recently by the author.In this derivation, the existence of freedom of • rotation around the .spinaxis and certain definite angular velocity of this rotation are required.These results and the correspondence obtained may clarify the classical-mechanical significance of the spinor wave-function and also of the spin itself.An explanation is appended of the hyperbolic functions of the velocity metric, which help in the complicated calculations for the Dirac electron. § I. Introduction*) The main part of this paper appeared previously in Rep. Seisanken.No. 22 (1957), 55, in Japanese.

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Classical particles with spin. I. The WKBJ approximation

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This is the first of a series of papers developing the classical theory of a spinning particle. The equations of motion will be derived from a Lagrangian, and solutions for the classical trajectory and spin precession in external fields will be given. In this paper an abstract spin vector is introduced to characterize the spin of a classical particle. Lagrangians for the classical trajectories and for the motion of the abstract spin vector are derived from corresponding quantum-mechanical Lagrangians by the WKBJ approximation method for nonrelativistic and relativistic particles. The equations of motion for the trajectory and the abstract spin vector following from the extremalization of these Lagrangians are given. The equations of motion for the precession in an external electromagnetic field of the spin vector (or tensor) in space–time is derived from the equations of motion for the abstract spin vector. In the relativistic case, they are equivalent to the Bargmann–Michel–Telegdi equations [Phys. Rev. Lett. 2, 435 (1959)]. The relationship between the ensemble and single-particle points of view is also elucidated.

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Relativistic Mechanics Theory for Electrons that Exhibits Spin, Zitterbewegung, Superposition and Produces Dirac's Wave Equation

James L. Beck

A neo-classical relativistic mechanics theory is presented where the spin of an electron is a natural part of its space-time path as a point particle. The fourth-order equation of motion corresponds to the same Lagrangian function in proper time as in special relativity except for an additional spin energy term. The total motion can be decomposed into a sum of a local spin motion about a point and a global motion of this point, called the spin center. The global motion is sub-luminal and obeys Newton's second law in proper time, the time for a clock fixed at the spin center, while the total motion occurs at the speed of light c, consistent with the eigenvalues of Dirac's velocity operators having magnitude c. The local spin motion corresponds to Schrödinger's zitterbewegung and is a perpetual motion, which for a free electron has a circular path in the spin-center frame. In an electro-magnetic field, this spin motion generates magnetic and electric dipole energies through the Lorentz force on the electron's point charge. The corresponding electric dipole energy is consistent with the spin-orbit coupling term in the corrected Pauli non-relativistic Hamiltonian but the magnetic dipole energy is one half of that in Dirac's theory. By defining a spin tensor as the angular momentum of the electron's total motion about its spin center, the fundamental equations of motion can be re-written in an identical form to those of the Barut-Zanghi electron theory. These equations of motion can then be expressed using operators applied to a state function of proper time satisfying a Dirac-Schrödinger spinor equation. The operators produce dynamic variables without any probability implications. For the free electron, the state function satisfies Dirac's relativistic wave equation when the Lorentz transformation is applied to express proper time in terms of an observer's space-time coordinates.

arXiv:2108.07267•Aug 14, 2021•2 citations

Proper time derivatives in quantum mechanics

Juan Pablo Aparicio, Fabián H. Gaioli, Edgardo T. Garcia Alvarez

Several quantum proper time derivatives are obtained from the Beck one in the usual framework of relativistic quantum mechanics (spin-1/2 case). The ``scalar Hamiltonians'' of these derivatives should be thought of as the conjugate variables of the proper time. Then, the Hamiltonians would play the role of mass operators, suggesting the formulation of an adequate extended indefinite mass framework. We propose and briefly develop the framework corresponding to the Feynman parametrization of the Dirac equation. In such a case we derive the other parametrizations known in the literature, linking the extension of the different proposals of quantum proper time derivatives again.

arXiv:hep-th/9807212•Physical Review A•Jan 1, 1995•17 citations
A great effort has been devoted to formulate a classical relativistic theory of spin compatible with quantum relativistic wave equations. The main difficulty in order to connect classical and quantum theories rests in finding a parameter which plays the role of proper time at a purely quantum level. We present a partial review of several proposals of classical and quantum spin theories from the pioneer works of Thomas and Frenkel, revisited in the classical BMT work, to the semiclassical model of Barut and Zanghi [Phys. Rev. Lett. 52, 2009 (1984)]. We show that the last model can be obtained from a semiclassical limit of the Feynman proper time parametrization of the Dirac equation. At the quantum level we derive spin precession equations in the Heisenberg picture. Analogies and differences with respect to classical theories are discussed in detail. 1
Abstract Conventional methods for teaching spin at a university level generally follow four main approaches. The first introduces spin into the Schrödinger equation artificially, leaving its physical meaning unclear. The second connects spin to the representation of the rotation group, which clarifies its nature but requires knowledge of group theory. The third approach derives from field theory, separating angular momentum into orbital and intrinsic parts, the latter corresponding to spin; however, this cannot be done within quantum mechanics alone. The fourth derives electron spin from Dirac’s 1928 relativistic equation, which, though correct, has led to the misconception that spin is a purely relativistic effect. A detailed analysis of Dirac’s 1928 papers shows that the concept of spin arises from the need to construct an equation linear in both energy and momentum to preserve Born’s probabilistic interpretation. This linearization of the Klein–Gordon equation reveals that spin is conceptually independent of relativity. Historical and pedagogical perspectives can together clarify conceptual difficulties and promote a deeper understanding of fundamental ideas in physics. Drawing on selected passages from Dirac’s original work, the present discussion offers broader conceptual and cultural considerations that introduce the notion of spin with limited mathematical requirements and can be applied independently of the didactic approach adopted.
The classical equations of motion of a charged point-particle with intrinsic spin under the influence of an external electromagnetic field are restated and compared with the Heisenberg equations of motion derived from the Dirac theory. The partition of angular momentum between particle and field in the classical theory is contrasted to the Dirac theory of electron spin. The analogy between the Dirac equation and the theory of parametric amplification is pointed out.A free spinning point particle moving according to the laws of classical relativistic point-particle mechanics may move along a helix. The sum of the intrinsic spin $\ensuremath{\sigma}$ and the angular momentum of the helical motion in this classical analog of zitterbewegung is an effective spin vector S which is a constant of the motion. Because of this internal motion, the effective mass MMM of the particle differs from the mass mmm which is ascribed to it in the equations of motion. Solutions are found in which S is parallel or antiparallel to the momentum, and the sign of MMM is determined by the helicity. When placed in a uniform electromagnetic field, the particle behaves as if it had a rest mass MMM and a magnetic moment $\frac{e\ensuremath{\sigma}}{\mathrm{Mc}}$, in addition to any explicit magnetic moment that may be ascribed to it.
The functional form of the wave function with respect to variables appropriate for the WKB approximation is chosen by taking into account the transformation properties of the spinor wave-function.This representation makes it possible, using only the wave equation, to pursue directly the time changes in the variables, i.e. the spin direction, velocity and density, etc., characterizing an electron.These changes correspond straightforwardly to a classical model of the non-relativistic Pauli electron with the proper magnetic moment, and the covariant form of this model, which is just "the Frenkel-Kramers model of the Dirac electron" introduced recently by the author.In this derivation, the existence of freedom of • rotation around the .spinaxis and certain definite angular velocity of this rotation are required.These results and the correspondence obtained may clarify the classical-mechanical significance of the spinor wave-function and also of the spin itself.An explanation is appended of the hyperbolic functions of the velocity metric, which help in the complicated calculations for the Dirac electron. § I. Introduction*) The main part of this paper appeared previously in Rep. Seisanken.No. 22 (1957), 55, in Japanese.
This is the first of a series of papers developing the classical theory of a spinning particle. The equations of motion will be derived from a Lagrangian, and solutions for the classical trajectory and spin precession in external fields will be given. In this paper an abstract spin vector is introduced to characterize the spin of a classical particle. Lagrangians for the classical trajectories and for the motion of the abstract spin vector are derived from corresponding quantum-mechanical Lagrangians by the WKBJ approximation method for nonrelativistic and relativistic particles. The equations of motion for the trajectory and the abstract spin vector following from the extremalization of these Lagrangians are given. The equations of motion for the precession in an external electromagnetic field of the spin vector (or tensor) in space–time is derived from the equations of motion for the abstract spin vector. In the relativistic case, they are equivalent to the Bargmann–Michel–Telegdi equations [Phys. Rev. Lett. 2, 435 (1959)]. The relationship between the ensemble and single-particle points of view is also elucidated.
A neo-classical relativistic mechanics theory is presented where the spin of an electron is a natural part of its space-time path as a point particle. The fourth-order equation of motion corresponds to the same Lagrangian function in proper time as in special relativity except for an additional spin energy term. The total motion can be decomposed into a sum of a local spin motion about a point and a global motion of this point, called the spin center. The global motion is sub-luminal and obeys Newton's second law in proper time, the time for a clock fixed at the spin center, while the total motion occurs at the speed of light c, consistent with the eigenvalues of Dirac's velocity operators having magnitude c. The local spin motion corresponds to Schrödinger's zitterbewegung and is a perpetual motion, which for a free electron has a circular path in the spin-center frame. In an electro-magnetic field, this spin motion generates magnetic and electric dipole energies through the Lorentz force on the electron's point charge. The corresponding electric dipole energy is consistent with the spin-orbit coupling term in the corrected Pauli non-relativistic Hamiltonian but the magnetic dipole energy is one half of that in Dirac's theory. By defining a spin tensor as the angular momentum of the electron's total motion about its spin center, the fundamental equations of motion can be re-written in an identical form to those of the Barut-Zanghi electron theory. These equations of motion can then be expressed using operators applied to a state function of proper time satisfying a Dirac-Schrödinger spinor equation. The operators produce dynamic variables without any probability implications. For the free electron, the state function satisfies Dirac's relativistic wave equation when the Lorentz transformation is applied to express proper time in terms of an observer's space-time coordinates.
Several quantum proper time derivatives are obtained from the Beck one in the usual framework of relativistic quantum mechanics (spin-1/2 case). The ``scalar Hamiltonians'' of these derivatives should be thought of as the conjugate variables of the proper time. Then, the Hamiltonians would play the role of mass operators, suggesting the formulation of an adequate extended indefinite mass framework. We propose and briefly develop the framework corresponding to the Feynman parametrization of the Dirac equation. In such a case we derive the other parametrizations known in the literature, linking the extension of the different proposals of quantum proper time derivatives again.