The spin of an electron is an intrinsic quantum property that cannot be explained using classical mechanics. Nevertheless, it is possible to conceive semiclassical systems with internal degrees of freedom that exhibit spin-like properties. Building on the analysis of phase modulation by elliptical gears and kinematically equivalent antiparallelogram linkages, we present a novel semiclassical model of Zitterbewegung, a rapid oscillatory motion closely connected with spin. The kinematical analog is based on including the internal rotation of the linkage represented in spacetime. The system’s dynamics is related to two-center electron models, which describe the constant momentum of the center of mass and the lightlike oscillation of the center of charge at twice the Compton frequency for a particle at rest. A two-dimensional extension provides an intuitive illustration of topological spin properties and can be used to calculate the spin and magnetic moment of an electron.
The spin of an electron is an intrinsic quantum property that cannot be explained using classical mechanics. Nevertheless, it is possible to conceive semiclassical systems with internal degrees of freedom that exhibit spin-like properties. Building on the analysis of phase modulation by elliptical gears and kinematically equivalent antiparallelogram linkages, we present a novel semiclassical model of Zitterbewegung, a rapid oscillatory motion closely connected with spin. The kinematical analog is based on including the internal rotation of the linkage represented in spacetime. The system’s dynamics is related to two-center electron models, which describe the constant momentum of the center of mass and the lightlike oscillation of the center of charge at twice the Compton frequency for a particle at rest. A two-dimensional extension provides an intuitive illustration of topological spin properties and can be used to calculate the spin and magnetic moment of an electron.
Nonrelativistic formalism is developed, which allows describing systems with internal degrees of freedom in the scalar potential field U, which is a function both on relative coordinates and time, and on relative speed and accelerations. The equation for energy, which is an integral of motion when U satisfies to certain differential condition, is derived for the general case. For a free mass point all solutions of the equations of motion in the center-of-inertia reference frame, moving with constant velocity, are obtained. As a result, the center of mass follows along a complicated trajectory round a direction of motion of the center of inertia. This motion can be interpreted as Zitterbewegung. It is shown that equation of moments is a consequence of the equation of motion. The general equation of motion for a spin which can be interpreted from positions of a classical mechanics is written down. Application of the obtained results to the electron leads to new conception of electrical charge, sign of which corresponds to a sign of spin polarization.
Starting from a probabilistic model of the electron, which explains spin and spin measurements in terms of a probability density distribution resulting from a rapidly changing angular momentum during an extended Zitterbewegung EZBW, a “light-like” model of the electron and other spin-1/2 particles is formulated. This model describes individual particles in terms of paths of a moving quantum. It is shown that this description allows one to reproduce observable properties as path-averages over a period of the fast EZBW in elementary calculations. The general topology of the paths may be described as a helical path, with a helix axis forming a circle around a fixed point in space. The radius of the helix and of the circle is equal and given by half the reduced Compton wave length of a photon of energy equal to the rest energy of the particle described. The paths depend on the relative velocity between the described “entity” and the observer, and represent the De Broglie wave. The merits of the proposed model are summarized and its role in relation to the established description by quantum mechanics is discussed. It is concluded that it supports the existence of the proposed EZBW, and offers a description of quantum behavior without quantum mechanics.
Zitterbewegung and de Broglie matter waves are both phenomena associated with the quantum-mechanical properties of the electron, but they are usually discussed in different contexts. Zitterbewegung is understood as a rapid oscillatory motion appearing in the Dirac equation, whereas the de Broglie matter wave is understood as a wave corresponding to the linear momentum of a particle. Some previous studies have attempted to derive the de Broglie matter wave from an internal clock of the electron or from Zitterbewegung. In the present study, however, no causal relation in which the matter wave is generated by Zitterbewegung is assumed. Instead, the two are regarded as different geometrical representations of a common phase degree of freedom. Specifically, Zitterbewegung is interpreted as a phenomenon corresponding to an internal rotational degree of freedom, while the de Broglie matter wave is interpreted as spatial phase progression associated with linear momentum. and are interpreted, respectively, as the rates of phase progression in the temporal and spatial directions. Furthermore, using the relation introduced in previous work, and reinterpreting it as the ratio of the internal rotational velocity to the external propagation velocity, this study shows the possibility of describing internal rotation, translational motion, matter waves, and the fine-structure constant within a common phase geometry.
This paper focuses on the understanding of the electron and its intrinsic properties. We analyze the Zitterbewegung motion, that gives information about the internal dynamics of the electron (with internal motion at the speed of light with an amplitude at Compton scale). This approach with field explains the origin of the spin and mass, while being compatible with quantum physics. This article is in the continuity of De Broglie and Einstein ideas to make quantum physics realistic (with real trajectories and physical waves) and to define a geometrical structure to the electron.
We show that the Zitterbewegung of the electron arises as a real internal motion when spin is treated as a classical bivector rather than as a point-fermion of the Dirac equation. In the Bivector Standard Model, physically meaningful dynamics reside in the body-fixed frame where two orthogonal internal angular momentum vectors counter-precess about a torque axis. Their rigid rotation generates a time-dependent chord whose magnitude oscillates at twice the Compton frequency, 2ωC, and whose orientation precesses at ωC. When projected into the laboratory-fixed frame, this internal rotor produces the characteristic trembling motion of the Zitterbewegung and traces a horn torus envelope without additional assumptions. The internal clock defined by this cyclic bivector motion unifies the origin of spin properties and the de Broglie modulation. It distinguishes complementary parity sectors that cannot be related by Lorentz transformations. The Zitterbewegung is therefore not an interference between positive- and negative-energy spinors, but the visible shadow of a real, energy-conserving internal rotation inherent to the bivector structure.
The description of the Zitterbewegung at a classical level indicates possible predictions of the electron quantum properties even before quantization: this quivering motion is restricted to a plane, which leads us to only two possible orientations of the corresponding angular momentum, even in the absence of a quantized theory or external fields. Besides, the angular momentum associated with the Zitterbewegung turns out to be proportional to $\ensuremath{\hbar}$. Namely, using a standard constraint analysis we have proven this last result. Thus, assuming that this is an observable phenomenon, we recognize this oscillatory motion as a classical signature of the spin of the electron. We also propose here an interpretation of the Zitterbewegung based on geometrical grounds: it can be seen as the physical degrees of freedom of position variables constrained to a sphere, which enforces the hypothesis of assuming an electron internal structure.