A New Derivation of Classical Models of the Spinning Electron from the WKB Solutions to the Pauli and Dirac Equations

Yamasaki, Hisaiti

doi:10.1143/ptp.36.72

Prog. Theor. Phys.

1966

DOI: 10.1143/ptp.36.72

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A New Derivation of Classical Models of the Spinning Electron from the WKB Solutions to the Pauli and Dirac Equations

Hisaiti Yamasaki

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1969

2023

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Cited by 2 publications

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“…(6) and using Eq. (1), one can easily obtain the afore-mentioned equation (2) for Os itself, (2) with…”

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confidence: 99%

Reformulation of the Frenkel-Kramers Model Including the Thomas Precession with the Velocity Metric

Yamasaki

1969

Prog. Theor. Phys.

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The relation between the Frenkel-Kramers model and the Thomas precession is investigated with new representation in which the hyperbolic metric V of velocity as well as the proper spin-direction vector Os are used. Our formulation of the model is entirely without the complicated or too formal and abstract expressions which are usually accompanied with the relativistic theory, and reveals a physical picture hitherto hidden. This formulation shows succinctly and definitely that not only the spin precession but also the acceleration itself undergoes the Thomas precession in electromagnetic fields. On that occasion, the use of a new expression, dV/ dr, of the proper acceleration also produces a simple invariant equation of motion as a substitute for the usual covariant equations of motion, § 1. IntroductionThe Frenkel-Kramers model (or the relativistic classical spmnmg particle with the proper magnetic moment) 1) has been derived from the Dirac equation in two ways by consistently applying the WKB method and by using the Heisenberg picture. 2 ),3) Sutton has appropriately explained a motive for construction of classical models bearing a close relationship to the Dirac equation. 4 ) Recently, from a modern viewpoint of the theory of elementary particles, Corben has written the book entitled "Classical and Quantum Theories of Spinning Particles",") There, he deal[), to some extent, with the physical significance and applications of 'the classical Bargmann-Michel-T elegdi equations' giving the precession of the polarization of particles. These BMT equations are essentially equivalent to the FK model named by us, and this is simply proved in our treatment, as seen later. Thus, we may say that the validity of the FK model has been confirmed as the classical counterpart of the Dirac electron.On the other hand, by utilizing the velocity metric V (or the rapidity), *) we have newly shown succinct expressions of the usual continuous Lorentz transformation and the Thomas precession, etc. 6 ),7)On the basis of this formulation, we may suppose that simple physical pictures are still hidden in the usual covariant representations of any relativistic object as well as in the above FK model. In *) In this paper, for convenience, we denote the vector character of the metric by , V' instead. ~ .

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“…(6) and using Eq. (1), one can easily obtain the afore-mentioned equation (2) for Os itself, (2) with…”

mentioning

confidence: 99%

Reformulation of the Frenkel-Kramers Model Including the Thomas Precession with the Velocity Metric

Yamasaki

1969

Prog. Theor. Phys.

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The Euler–Poisswell/Darwin equation and the asymptotic hierarchy of the Euler–Maxwell equation

Möller,

Mauser

2023

ASY

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In this paper we introduce the (unipolar) pressureless Euler–Poisswell equation for electrons as the O ( 1 / c ) semi-relativistic approximation and the (unipolar) pressureless Euler–Darwin equation as the O ( 1 / c 2 ) semi-relativistic approximation of the (unipolar) pressureless Euler–Maxwell equation. In the “infinity-ion-mass” limit, the unipolar Euler–Maxwell equation arises from the bipolar Euler–Maxwell equation, describing a coupled system for a plasma of electrons and ions. The non-relativistic limit c → ∞ of the Euler–Maxwell equation is the repulsive Euler–Poisson equation with electric force. We propose that the Euler–Poisswell equation, where the Euler equation with electric force is coupled to the magnetostatic O ( 1 / c ) approximation of Maxwell’s equations, is the correct semi-relativistic O ( 1 / c ) approximation of the Euler–Maxwell equation. In the Euler–Poisswell equation the fluid dynamics are decoupled from the magnetic field since the Lorentz force reduces to the electric force. The first non-trivial interaction with the magnetic field is at the O ( 1 / c 2 ) level in the Euler–Darwin equation. This is a consistent and self-consistent model of order O ( 1 / c 2 ) and includes the full Lorentz force, which is relativistic at O ( 1 / c 2 ). The Euler–Poisswell equation also arises as the semiclassical limit of the quantum Pauli–Poisswell equation, which is the O ( 1 / c ) approximation of the relativistic Dirac–Maxwell equation. We also present a local wellposedness theory for the Euler–Poisswell equation. The Euler–Maxwell system couples the non-relativistic Euler equation and the relativistic Maxwell equations and thus it is not fully consistent in 1 / c. The consistent fully relativistic model is the relativistic Euler–Maxwell equation where Maxwell’s equations are coupled to the relativistic Euler equation – a model that is neglected in the mathematics community. We also present the relativistic Euler–Darwin equation resulting as a O ( 1 / c 2 ) approximation of this model.

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A New Derivation of Classical Models of the Spinning Electron from the WKB Solutions to the Pauli and Dirac Equations

Cited by 2 publications

References 1 publication

Reformulation of the Frenkel-Kramers Model Including the Thomas Precession with the Velocity Metric

Reformulation of the Frenkel-Kramers Model Including the Thomas Precession with the Velocity Metric

The Euler–Poisswell/Darwin equation and the asymptotic hierarchy of the Euler–Maxwell equation

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