The Dynamical Equation of the Spinning Electron (original) (raw)

Field Theory of the Spinning Electron: II—The New Non-Linear Field Equations

One of the most satisfactory picture of spinning particles is the Barut--Zanghi (BZ) classical theory for the relativistic electron, that relates the electron spin to the so-called zitterbewegung (zbw). The BZ motion equations constituted the starting point for two recent works about spin and electron structure, co-authored by us, which adopted the Clifford algebra language. Here, employing on the contrary the tensorial language, more common in the (first quantization) field theories, we "quantize" the BZ theory and derive for the electron field a non-linear Dirac equation (NDE), of which the ordinary Dirac equation represents a particular case. We then find out the general solution of the NDE. Our NDE does imply a new probability current J^\mu, that is shown to be a conserved quantity, endowed (in the center-of-mass frame) with the zbw frequency \om = 2m, where m is the electron mass. Because of the conservation of J^\mu, we are able to adopt the ordinary probabilistic interpretation for the fields entering the NDE. At last we propose a natural generalization of our approach, for the case in which an external electromagnetic potential A^\mu is present; it happens to be based on a new system of five first--order differential field equations.

FIELD THEORY OF THE SPINNING ELECTRON: ABOUT THE NEW NON-LINEAR FIELD EQUATIONS

One of the most satisfactory picture of spinning particles is the Barut-Zanghi (BZ) classical theory for the relativistic electron, that relates the electron spin to the so-called zitterbewegung (zbw). The BZ motion equations constituted the starting point for two recent works about spin and electron structure, co-authored by us, which adopted the Clifford algebra language. Here, employing on the contrary the tensorial language, more common in the (first quantization) field theories, we "quantize" the BZ theory and derive for the electron field a non-linear Dirac equation (NDE), of which the ordinary Dirac equation represents a particular case.

On the Kinematics of the Centre of Charge of an Elementary Spinning Particle

AIP Conference Proceedings, 2009

In particle physics, most of the classical models consider that the centre of mass and centre of charge of an elementary particle, are the same point. This presumes some particular relationship between the charge and mass distribution, a feature which cannot be checked experimentally. In this paper we give three different kinds of arguments suggesting that, if assumed different points, the centre of charge of an elementary spinning particle moves in a helical motion at the speed of light, and it thus satisfies, in general, a fourth order differential equation. If assumed a kind of rigid body structure, it is sufficient the description of the centre of charge to describe also the evolution of the centre of mass and the rotation of the body. This assumption of a separation betwen the centre of mass and centre of charge gives a contribution to the spin of the system and also justifies the existence of a magnetic moment produced by the relative motion of the centre of charge. This corresponds to an improved model of a charged elementary particle, than the point particle case. This means that a Lagrangian formalism for describing elementary spinning particles has to depend, at least, up to the acceleration of the position of the charge, to properly obtain fourth order dynamical equations. This result is compared with the description of a classical Dirac particle obtained from a general Lagrangian formalism for describing spinning particles.

Field theory of the spinning electron and internal motions

Physics Letters A, 1994

We present here a field theory of the spinning electron, by writing down a new equation for the 4-velocity field v^mu (different from that of Dirac theory), which allows a classically intelligible description of the electron. Moreover, we make explicit the noticeable kinematical properties of such velocity field (which also result different from the ordinary ones). At last, we analyze the internal zitterbewegung (zbw) motions, for both time-like and light-like speeds. We adopt in this paper the ordinary tensorial language. Our starting point is the Barut-Zanghi classical theory for the relativistic electron, which related spin with zbw. This paper is dedicated to the memory of Asim O. Barut, who so much contributed to clarifying very many fundamental issues of physics, and whose work constitutes a starting point of these articles.

On the kinematics of the centre of charge of a spinning particle

Arxiv preprint arXiv:0807.2512, 2008

Abstract: In particle physics, most of the classical models consider that the centre of mass and centre of charge of an elementary particle, are the same point. This presumes some particular relationship between the charge and mass distribution, a feature which cannot ...

Quantization of generalized spinning particles: New derivation of Dirac’s equation

Journal of Mathematical Physics, 1994

Quantization of generalized Lagrangian systems suggests that wave functions for elementary particles must be defined on the kinematical space rather than on configuration space. For spinning particles the center of mass and center of charge are different points. Their separation is of the order of the Compton wavelength. Spin-1/2 particles arise if the classical model rotates but no half integer spins are obtained for systems with spin of orbital nature. Dirac’s equation is obtained when quantizing the classical relativistic spinning particles whose center of charge is circling around its center of mass at the speed c. Internal orientation of the electron completely characterizes its Dirac’s algebra.

A Classical and Spinorial Description of the Relativistic Spinning Particle

arXiv: High Energy Physics - Theory, 2016

In a previous work we showed that spin can be envisioned as living in a phase space that is dual to the standard phase space of position and momentum. In this work we demonstrate that the second class constraints inherent in this "Dual Phase Space" picture can be solved by introducing a spinorial parameterization of the spinning degrees of freedom. This allows for a purely first class formulation that generalizes the usual relativistic description of spinless particles and provides several insights into the nature of spin and its relationship with spacetime and locality. In particular, we find that the spin motion acts as a Lorentz contraction on the four-velocity and that, in addition to proper time, spinning particles posses a second gauge invariant observable which we call proper angle. Heuristically, this proper angle represents the amount of Zitterbewegung necessary for a spin transition to occur. Additionally, we show that the spin velocity satisfies a causality cons...

Mathisson'spinning electron: noncommutative mechanics & exotic Galilean symmetry, 66 years ago

Arxiv preprint hep-th/0303099, 2003

The acceleration-dependent system with noncommuting coordinates, proposed by Lukierski, Stichel and Zakrzewski [Ann. Phys. 260, 224 (1997)] is derived as the non-relativistic limit of Mathisson's classical electron [Acta Physica Polonica 6, 218 (1937)], further discussed by Weyssenhoff and Raabe [Acta Physica Polonica 9, 7 (1947)]. The two-parameter centrally extended Galilean symmetry of the model is recovered using elementary methods. The relation to Schrödinger's Zitternde Elektron is indicated.

On the equations of motion for particles with arbitrary spin in nonrelativistic mechanics

1975

It is well known that the electron motion in the external electromagnetic field is described by the relativistic Dirac equation. In this case, in the Foldy-Wouthuysen representation, the Hamiltonian includes the terms corresponding to the interaction of the particle magnetic moment with a magnetic field (∼ (1/m)(σH)) and the terms which are interpreted as a spin-orbit coupling (∼ (σ/m 2 ){(p − eA) × E)). Apart from these constituents the Hamiltonian includes the Darwin term (∼ (1/m 2 ) div E) .

Dynamics of the relativistic electron spin in an electromagnetic field

Journal of Physics: Condensed Matter, 2020

A relativistic spin operator cannot be uniquely defined within relativistic quantum mechanics. Previously, different proper relativistic spin operators have been proposed, such as spin operators of the Foldy–Wouthuysen and Pryce type, that both commute with the free-particle Dirac Hamiltonian and represent constants of motion. Here we consider the dynamics of a relativistic electron spin in an external electromagnetic field. We use two different Hamiltonians to derive the corresponding spin dynamics. These two are: (a) the Dirac Hamiltonian in the presence of an external field, and (b) the semirelativistic expansion of the same. Considering the Foldy–Wouthuysen and Pryce spin operators we show that these lead to different spin dynamics in an external electromagnetic field, which offers possibilities to distinguish their action. We find that the dynamics of both spin operators involve spin-dependent and spin-independent terms, however, the Foldy–Wouthuysen spin dynamics additionally ...