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.
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 \ensuremathℏ. 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.
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.
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 found. As a result, the center of mass follows along helical and more complicated trajectories round a direction of motion of the center of inertia. This motion can be interpreted as trembling movement (Zitterbewegung). On this basis a conclusion is done that Zitterbewegung has purely classical origin, arising even in a nonrelativistic case if internal degrees of freedom are taken into account. 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 electric charge, sign of which corresponds to a sign of spin polarization.
The fact that the Dirac equation predicts Zitterbewegung (ZBW) for a free electron is well known from Schrodinger [E. Schrodinger, Sitzungsber Preuss. Akad. Wiss. Berlin (Math. Phys.) 24, 418 (1930)] from the very early days of Dirac's relativistic quantum theory. Schrodingerdescribed ZBW qualitatively as a persistent interference between positive and negative energy states, and the objective of this and previous papers by this author [Phys. Essays 28, 1 (2015); Phys. Essays 29, 402 (2016)] is to describe ways to measure the presence or absence ofZBW for a free (or nearly free) electron. If ZBW measurements show that it is not present for a free electron, then the Dirac Equation and Schrodinger's analysis [E. Schrodinger, Sitzungsber Preuss. Akad. Wiss. Berlin (Math. Phys.) 24, 418 (1930)] are inaccurate. If ZBW ispresent as Schrodinger's analysis [E. Schrodinger, Sitzungsber Preuss. Akad. Wiss. Berlin (Math. Phys.) 24, 418 (1930)] of the Dirac Equation asserts, then a description of a free electron emerges that is significantly different than the present very small, static point electronwith “intrinsic” properties of spin and magnetic moment accepted by most physicists [Eides et al., Phys. Reports 342, 63 (2001).] today. This paper develops a quantitative theory of the free electron directly from the Dirac Equation and Schrodinger's analyses[E. Schrodinger, Sitzungsber Preuss. Akad. Wiss. Berlin (Math. Phys.) 24, 418 (1930)] without any other assumptions. The fact that ZBW exists for the electron in the hydrogen atom is confirmed by the Darwin term [J. J. Sakurai, Advanced Quantum Mechanics (Addison-Wesley,Reading, MA, 1967)]. It seems unnatural to the author that ZBW would vanish when the electron is removed from the hydrogen atom, and becomes free. The unanswered question is whether ZBW can be confirmed or refuted by measurement for a free electron, and possible ways to measure its presenceor absence are described in this paper. The remainder of the paper describes the free electron, assuming ZBW exists. The Dirac Equation [J. H. Wilson, Phys. Essays 28, 1 (2015); Phys. Essays 29, 402 (2016)] produces a physical description of a free electron's ZBW as rapidly oscillating,quantized Center of Charge (CoC) point rotating about a fictitious Center of Mass (CoM) at a radius of 3/ 2 times the electron's Compton wavelength. The results in the remainder of this paper, as well as Schrodinger's analyses [E. Schrodinger, Sitzungsber Preuss. Akad. Wiss. Berlin (Math. Phys.)24, 418 (1930).], are only true if ZBW can be confirmed by measurement. In case measurements show no ZBW exists for a free electron, the Dirac Equation is not a correct description of a single free electron, and more serious problems arise. The static point like electron deduced fromscattering experiments [Eides et al., Phys. Rep. 342, 63 (2001)] that give estimates of the free electron's internal structure based on far field measurements can be misleading in estimating the free electron's internal structure. In Classical Electrodynamics, when one observesthe electric field from a point electron and from a sphere with the same charge spread uniformly over its surface from the far field, Gauss' Law shows that the electric fields are identical. When one tries to “observe” or “measure” the free electron's internal structurewith the high-energy photons required, the single particle free electron is completely destroyed by the measurement process. In fact, any attempt to observe or measure a free electron's structure ensures that the electron is no longer a totally free single particle, and one must be carefulto use only elastic observation methods [J. H. Wilson, Phys. Essays 28, 1 (2015); Phys. Essays 29, 402 (2016)] for this very difficult measurement. Observing a free Quark, much less its internal structure, has proven very difficult, if not impossible.
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.
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.
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.