Zitterbewegung

The electron-proton scattering experiment by the PRad (proton radius) team at Jefferson Lab measured the root mean square (rms) charge radius of the proton as r = 0.831 ± 0.007(stat) ± 0.012(syst) fm.  We offer a theoretical explanation of the new measurement based on a ring current model of a proton. This model further builds on older ring current and/or Zitterbewegung models for an electron and, hence, we will also highlight those results when relevant. We obtain a theoretical radius that is equal to four times the range parameter (ħ/mc) in Yukawa’s formula:
r = 4ħ/mc = 0.841 fm.

The 1/4 factor stems from the energy equipartition theorem: using Wheeler’s ‘mass without mass’ idea, we effectively assume half of the energy of a proton is explained by the electromagnetic, while the other half is attributed to the strong force, which we do not model but isolate from the analysis using the energy equipartition theorem.

As for the small difference between the theoretical and measured radius, we attribute this to the mathematical idealizations that underpin ring current models. While useful and necessary as a concept, we think pointlike electric charges with zero rest mass and/or zero dimension that, therefore, move at lights peed, do not exist: they must have some (very) small dimension which explains the anomaly. We think mathematical idealization also explains the anomalous magnetic moment of an electron.

We think the calculations may offer a model of matter-particles in general.

A new preon model is presented as an extension of the semiclassical Helical Solenoid Electron Model that was previously proposed by the author. This helicon model assumes as postulates both the Atomic Principle and the equality between matter and electric charge. These postulates lead us to a radical reinterpretation of the concepts of antimatter and dark matter and form a new framework for future preon theories.

A natural geometric unit system (abbr. SG) based on the pre-2019 SI system, in which four fundamental physical constants undergo joint numerical and dimensional normalization to unity c = G = k = h = 1, with only one base geometric unit u equal to √|h · G/c^3 | m, is defined. A new relation emerges in the SI m^3 /(kg · s^2) = kg/C · √|k/G^3 | , where |x| denotes the dimensionless, numerical part of property x. A simple electron mass to charge ratio is postulated: in the SI m_e = |G| kg/C e/(2^9 πα) and in the SG m_e = e/(2^9 πα) , with the Newtonian gravitational constant G ≈ 6.673 655 205 · 10^-11 m^3 /(kg · s^2) , which fits well in the range allowed by the current state of research on the subject. Similar to the formula for electron rest mass m_e = 1 / (2^9 π √(2πα)) u , proposed are formulas for stable quarks rest masses: quark u m_u = √(⅔) / (2^7 π √(πα)) u , equivalent of 2.360 MeV/c^2 and quark d m_d = √(⅓)^-1 / (2^7 π √(πα)) u , equivalent of 5.007 MeV/c^2. The finding supports the research area of purely geometric modelling of the fundamental physical forces and their unification.

Through investigating history, evolution of the concept, and development in the theories of electrons, I am convinced that what was missing in our understanding of the electron is a structure, into which all attributes of the electron could be incorporated in a self-consistent way. It is hereby postulated that the topological structure of the electron is a closed two-turn Helix (a so-called Hubius Helix) that is generated by circulatory motion of a mass-less particle at the speed of light. A formulation is presented to describe an isolated electron at rest and at high speed. It is shown that the formulation is capable of incorporating most (if not all) attributes of the electron, including spin, magnetic moment, fine structure constant, anomalous magnetic moment, and charge quantization into one concrete description of the Hubius Helix. The equations for the description emerge accordingly. Implications elicited by the postulate are elaborated. Inadequacy of the formulation is discussed.

The proposed electromagnetic model of spacetime developed from the natural geometric unit system (SG) results in values of Newtonian constant of gravitation, stable elementary particles rest masses and electron magnetic moment anomaly in accordance with experimental data. It differentiates between two action and energy types, since in the SG electromagnetic elements are of order √(2πα) and gravitational elements are of order √(2πα)^3. Thus it helps to avoid zero-point energy implication of QED that is not in agreement with observations. It suggests the hypothesis of existence of the 5th curled dimension involved in creating phenomena of electron magnetic moment anomaly and quantum nonlocality and leaves open the possibility that the model can be used in the future to describe strong and weak interactions as exclusively electromagnetic phenomena.

A new model of the electron with Helical Solenoid geometry is presented. This new model is an extension of the Parson's Ring Electron Model and the Hestenes' Zitter Electron Model. In this new electron model, the g-factor appears as a simple consequence of the geometry of the electron. The calculation of the g-factor is performed in a simple manner and we obtain the value of 1.0011607. This value of the g-factor is more accurate that the value provided by the Schwinger's factor.

Magnetic interactions between relativistically-moving charges are similar in strength to the electric interaction between static charges. If the electron always moves luminally, as the Dirac theory implies, and so moving gives rise to the intrinsic magnetic moment, then magnetic spin-spin interactions stronger than currently recognized are to be expected. It is proposed that spin-spin interactions as expected according to this line of reasoning can properly account for the formation of atoms, if quarks and leptons are composed of more-fundamental spin-half particles. Spin-spin interaction of suitable strength and character to form atoms occurs only between equal-mass particles with parallel spins, while vanishing on average if the spins are anti-parallel. Electron and quark compositeness allows Coulomb-like spin-spin interactions between equal-mass constituents with parallel spins, for either parallel or anti-parallel quark and electron spins, and modulation of the force due to spin-spin interaction that provides an electrodynamic basis for the de Broglie matter wave. Related analysis based on an assumption of photon compositeness obtains an energy-frequency relationship that is similar to the Planck-Einstein law apart from a factor of about one-half, providing an alternative resolution to the spin-orbit coupling anomaly.

An imaginative exploration of space and time in which light mediates the relationship between finitude and the Infinite. Light becomes the creative source through which interiority and exteriority are manifested and brought into synchronicity as time, space and mass.  The exploration probes the relational logic of relativity theory using the meta-physical insights of Augustine, Hegel, Levinas, and Peirce.

In the framework of Special Relativity and in agreement with the electromagnetic approach used to connect Quantum and Relativistic effects in Bridge Theory, a first order wave-equation able to emulate the wave-particle duality in terms of local electromagnetic interactions is proposed. The equation is able to describe the propagation of a fermion as a de Broglie's wave and simultaneously to characterise the wave in motion with a mass energy as a particle. The solution of the wave-equation proposed is compatible with the de Broglie's idea of an empty pilot wave, carrying information about the dynamical properties of a particle materialising during each local interaction with the observers.

Through investigating the history and evolution of the concept and the development of the theories of electrons, I am convinced that what was missing in our understanding of the electron is a structure into which all attributes of the electron could be incorporated in a self-consistent way. It is hereby postulated that the topological structure of the electron is a closed two-turn helix (a so-called Hubius helix) that is generated by circulatory motion of a massless particle at the speed of light. A formulation is presented to describe an isolated electron at rest and at high speed. It is shown that the formulation is capable of incorporating most (if not all) attributes of the electron, including spin, magnetic moment, fine-structure constant α, anomalous magnetic moment (α/π)/2, and charge quantization, into one concrete description of the Hubius helix. The equations for the description emerge accordingly. Implications elicited by the postulate are elaborated. Inadequacies of the formulation are discussed.

A hypothetical equation of motion is proposed for Kerr-Newman particles. It's obtained by analytic continuation of the Lorentz-Dirac equation into complex space-time. A new class of "runaway" solutions are found which are similar to zitterbewegung. Electromagnetic fields generated by these motions are studied, and it's found that the retarded (and advanced) times are multi-sheeted functions of the field points. This leads to non-uniqueness for the fields. With fixed weighting factors for these multiple roots, the solutions radiate. However, position dependent weighting factors can suppress radiation and allow non-radiating solutions. Motion with external forces are also considered, and radiation suppression is possible there too. These results are relevant for the idea that Kerr-Newman solutions provide insight into elementary particles and into emergent quantum mechanics. They illustrate a type of nascent wave-particle duality and complementarity in a purely classical field theory. Metric curvature due to gravitation is ignored. Keywords Kerr-Newman • Lorentz-Dirac • zitterbewegung • complex space-time • electron model • emergent quantum mechanics

This paper recaps our electron model – including our explanation of the anomaly – and offers some reflections on Nature’s fundamental constants. We will also present a theoretical explanation of the radius of the Zitterbewegung charge – aka the classical electron radius – using an electromagnetic mass calculation. While, in the previous version of the paper, we limited ourselves to a classical (non-mainstream) explanation of Schwinger’s α/2π factor, we also offer some reflections on a possible explanation of the higher-order factors in the anomaly of the magnetic moment of an electron. Finally, we offer some reflections on the distinction between the spin and orbital angular momentum of an electron.

Starting from the formal expressions of the hydrodynamical (or``local '') quantities employed in the applications of Clifford algebras to quantum mechanics, we introduce± ± in terms of the ordinary tensorial language± ± a new definition for the field of a generic quantity. By translating from Clifford into tensor algebra, we also propose a new (nonrelativistic) velocity operator for a spin-1 2 particle. This operator appears as the sum of the ordinary part p/m describing the mean motion (the motion of the center-of-mass ), and of a second part associated with the socalled Zitterbewegung, which is the spin``internal '' motion observed in the centerof-mass from (CMF). This spin component of the velocity operator is nonzero not only in the Pauli theoretical framework , i.e., in the presence of external electromagnetic fields with a nonconstant spin function , but also in the Schro È dinger case, when the wavefunction is a spin eigenstate. Thus , one gets even in the latter case a decomposition of the velocity field for the Madelung fluid into two distinct parts, which constitutes the nonrelativistic analogue of the Gordon decomposition for the Dirac current. Explicit calculations are presented for the velocity field in the particular cases of the hydrogen atom, of a spherical well potential, and of an electron in a uniform magnetic field. We find, furthermore , that the Zitterbewegung motion involves a velocity field which is solenoidal, and that the local angular velocity is parallel to the spin vector. In the presence of a nonuniform spin vector (Pauli case) we have, besides the component of the local velocity normal to the spin (present even in the Schro È dinger theory), also a component which is parallel to the curl of the spin vector.

The phenomenon of zitterbewegung (ZB, trembling motion) of electrons is described in zigzag carbon nanotubes (CNT) excited by laser pulses. The tight binding approach is used for the band structure of CNT and the effect of light is introduced by the vector potential. Contrary to the common theoretical practice, no a priori assumptions are made concerning electron wave packet; the latter is determined as a result of illumination. In order to overcome the problem of various electron phases in ZB, a method of two-photon echo (2PE) is considered and described using the density function formalism. The medium polarization of CNT is calculated by computing exact solutions of the time-dependent electron Hamiltonian. The signal of 2PE is extracted and it is shown that, using existing parameters of CNT and laser pulses, one should be able to observe the electron trembling motion. Effects of electron decoherence and relaxation are discussed.

This paper recaps our electron model-including our classical explanation of the anomalous magnetic moment-and slightly revises our interpretation of the elementary wavefunction that describes it. We also add some material from previous papers to combine all of the aspects of our electron model in one single paper.

The electron-proton scattering experiment by the PRad (proton radius) team at Jefferson Lab measured the root mean square (rms) charge radius of the proton as rp = 0.831 ± 0.007(stat) ± 0.012(syst) fm.

Assuming all of the electric charge in the proton is packed into a single pointlike (elementary) charge and applying the ring current model to a proton, one gets a radius for the circular current that is equal to a = 2μ/qc = 0.58736 fm. Using CODATA values for all variables and constants in this equation, and applying a SQRT(2) form factor to, somehow, account for the envelope of the magnetic field around the ring current, yields an electric charge radius of 0.8065 fm. The difference between the PRad point estimate and this theoretical value is 0.00035 fm, which represents 5% of the standard error (0.007 fm) of PRad’s point estimate. It is, therefore, hard to argue this is a mere coincidence.

We can also calculate a proton radius based on the idea of a strong charge. This radius corresponds to the range parameter in Yukawa’s equation and is equal to a = ħ/mpc = 0.2103, which is about 1/4 of the PRad point estimate. This 1/4 factor is, obviously, far more mysterious, and the difference between 0.831 and this strong charge radius multiplied by 4 is 0.01 fm, which is about 50% of the combined statistical and systematic error (0.007 + 0.012 = 0.019). We, therefore, think that, while being somewhat less precise, the 1/4 factor cannot be a coincidence.

We, therefore, feel the new measurement of the proton radius by JLAB’s PRad team may lend credibility to attempts to extend the Zitterbewegung hypothesis from electrons to also include protons and other elementary particles. In contrast, the measurement is hard to fit into a model of oscillating quarks that have partial charge only.

Starting from the formal expressions of the hydrodynamical (or "local") quantities employed in the applications of Clifford algebras to quantum mechanics, we introduce-in terms of the ordinary tensorial language-a new definition for the field of a generic quantity. By translating from Clifford into tensor algebra, we also propose a new (nonrelativistic) velocity operator for a spin-1 2 particle. This operator appears as the sum of the ordinary part p/m describing the mean motion (the motion of the center-of-mass), and of a second part associated with the so-called Zitterbewegung, which is the spin "internal" motion observed in the center-of-mass frame (CMF). This spin component of the velocity operator is nonzero not only in the Pauli theoretical framework, i.e., in the presence of external electromagnetic fields with a nonconstant spin function, but also in the Schrödinger case, when the wavefunction is a spin eigenstate. Thus, one gets even in the latter case a decomposition of the velocity field for the Madelung fluid into two distinct parts, which constitutes the nonrelativistic analogue of the Gordon decomposition for the Dirac current. Explicit calculations are presented for the velocity field in the particular cases of the hydrogen atom, of a spherical well potential, and of an electron in a uniform magnetic field. We

Consequences of implementation of the natural geometric unit system (the SG) based on the pre-2019 SI system, in which four fundamental physical constants undergo joint numerical and dimensional normalization to unity c = G= k = h = 1, with only one base geometric unit u equal to √|h · G/c 3 | m, where the Newtonian gravitational constant G ≈ 6.673 655 205 · 10 -11 m3/(kg · s 2 ), are further explored. In addition to the earlier hypothesized simple electron mass to charge ratio formula me = e/(2 9πα), and formulas for stable quarks rest masses: quark u mu = √(⅔) / (2 7π √(πα)) u, equivalent of 2.360 MeV/c 2 and quark d md = √(⅓) -1 / (2 7π √(πα)) u, equivalent of 5.007 MeV/c 2 , a simple formula for electron magnetic moment anomaly is proposed α/2π - (α/2π) 2 - 2 8 (α/2π) 3 - 2 12 (α/2π) 4 - 2 16 (α/2π) 5 - 2 24 (α/2π) 6 ≈ 0.001 159 652 180. The finding supports the research area of purely geometric modelling of the fundamental physical forces and their unification. It seems plausib...

In this article, we show that the wave equation for a free Dirac electron can be represented in a form that is analogous to Maxwell's electrodynamics. The electron bispinor wavefunction is explicitly expressed in terms of its real and imaginary components. This leads us to incorporate into it appropriate scalar and pseudo-scalar fields in advance, so that a full symmetry may be accomplished. The Dirac equation then takes on a form similar to that of a set of inhomogeneous Maxwell's equations involving a particular self-source. We relate plane wave solutions of these equations to waves corresponding to free Dirac electrons, identifying the longitudinal component of the electron motion, together with the corresponding Zitter-bewegung (" trembling motion ").

Starting from the formal expressions of the hydrodynamical (or “local”) quantities employed in the applications of Clifford algebras to quantum mechanics, we introduce—in terms of the ordinary tensorial language—a new definition for the field of a generic quantity. By translating from Clifford into tensor algebra, we also propose a new (nonrelativistic) velocity operator for a spin- \({\frac{1}{2}}\) particle. This operator appears as the sum of the ordinary part p/m describing the mean motion (the motion of the center-of-mass), and of a second part associated with the so-called Zitterbewegung, which is the spin “internal” motion observed in the center-of-mass frame (CMF). This spin component of the velocity operator is nonzero not only in the Pauli theoretical framework, i.e., in the presence of external electromagnetic fields with a nonconstant spin function, but also in the Schrödinger case, when the wavefunction is a spin eigenstate. Thus, one gets even in the latter case a decomposition of the velocity field for the Madelung fluid into two distinct parts, which constitutes the nonrelativistic analogue of the Gordon decomposition for the Dirac current. Explicit calculations are presented for the velocity field in the particular cases of the hydrogen atom, of a spherical well potential, and of an electron in a uniform magnetic field. We find, furthermore, that the Zitterbewegung motion involves a velocity field which is solenoidal, and that the local angular velocity is parallel to the spin vector. In the presence of a nonuniform spin vector (Pauli case) we have, besides the component of the local velocity normal to the spin (present even in the Schrödinger theory), also a component which is parallel to the curl of the spin vector.

Around 1930, both Gregory Breit and Erwin Schroedinger showed that the eigenvalues of the velocity of a particle described by wavepacket solutions to the Dirac equation are simply +-c, the speed of light.  This led Schroedinger to coin the term Zitterbewegung, which is German for ``trembling motion'', where all particles of matter (fermions) zig-zag back-and-forth at only the speed of light.  The result is that any finite speed less than c, including the state of rest, only makes sense as a long-term average that can be thought of as a drift velocity.  In this paper, we seriously consider this idea that the observed velocities of particles are time-averages of motion at the speed of light and demonstrate how the relativistic velocity addition rule in one spatial dimension is readily derived by considering the probabilities that a particle is observed to move either to the left or to the right at the speed of light.