We extended the Barut’s classical model of zitterbewegung from 3+1 dimensional spacetime into 2+1 and 1+1 dimensional spacetimes and discussed the symmetry and integrability properties of the model in 2+1, 1+1 and 3+1 dimensions. In these cases, the free particle current or the velocity of the particle can be decomposed as a constant convection current and polarization currents.In 2+1 dimensional spacetime, a velocity of the particle and spin tensor are dependent to each other and the chirality can not be introduced. (...) The free particle has 7 constants of motion: The momentum three vector, the charge, the energy in proper time, the scalar constant spin or magnetic polarization and the two components of total angular momentum. Two component electric polarizations oscillate with Zitterbewegung frequency.In 1+1 dimensional spacetime we have an independent velocity vector and a scalar spin tensor. The free particle has 5 integrals of motion: The momentum two vector, the charge, the energy in proper time, and the scalar total angular momentum. The normal component of the velocity or the scalar electric polarization oscillates with Zitterbewegung frequency.In 3+1 dimensional spacetime, the particle has an independent velocity vector, spin tensor and chirality. The free particle has 12 integrals of motion: The momentum four vector, the charge, the energy in proper time or mass, the three vector spin or magnetic polarizations and three components of total angular momentum. The parallel component of velocity into momentum and the normal components of the spin tensor or the spin three vector are constants of motion for the free particle. The chirality and electric polarizations oscillate with the Zitterbewegung frequency. The system is superintegrable in all dimensions. (shrink)

An analysis is presented of the possible existence of the second anomalous dipole moment of Dirac’s particle next to the one associated with the angular momentum. It includes a discussion why, in spite of his own derivation, Dirac has doubted about its relevancy. It is shown why since then it has been overlooked and why it has vanished from leading textbooks. A critical survey is given on the reasons of its reject, including the failure of attempts to measure and the (...) perceived violations of time reversal symmetry and charge–parity symmetry. It is emphasized that the anomalous electric dipole moment of the pointlike electron is fundamentally different from the quantum field type electric dipole moment of an electron as defined in the standard model of particle physics. The analysis has resulted into the identification of a third type Dirac particle, next to the electron type and the Majorana particle. It is shown that, unlike as in the case of the electron type, its second anomalous dipole moment is real valued and is therefore subject to polarization in a scalar potential field. Examples are given that it may have a possible impact in the nuclear domain and in the gravitational domain. (shrink)

A generalized Schrödinger equation containing correction terms to classical kinetic energy, has been derived in the complex vector space by considering an extended particle structure in stochastic electrodynamics with spin. The correction terms are obtained by considering the internal complex structure of the particle which is a consequence of stochastic average of particle oscillations in the zeropoint field. Hence, the generalised Schrödinger equation may be called stochastic Schrödinger equation. It is found that the second order correction terms are similar to (...) corresponding relativistic corrections. When higher order correction terms are neglected, the stochastic Schrödinger equation reduces to normal Schrödinger equation. It is found that the Schrödinger equation contains an internal structure in disguise and that can be revealed in the form of internal kinetic energy. The internal kinetic energy is found to be equal to the quantum potential obtained in the Madelung fluid theory or Bohm statistical theory. In the rest frame of the particle, the stochastic Schrödinger equation reduces to a Dirac type equation and its Lorentz boost gives the Dirac equation. Finally, the relativistic Klein–Gordon equation is derived by squaring the stochastic Schrödinger equation. The theory elucidates a logical understanding of classical approach to quantum mechanical foundations. (shrink)

Elementary particles are considered as local oscillators under the influence of zeropoint fields. Such oscillatory behavior of the particles leads to the deviations in their path of motion. The oscillations of the particle in general may be considered as complex rotations in complex vector space. The local particle harmonic oscillator is analyzed in the complex vector formalism considering the algebra of complex vectors. The particle spin is viewed as zeropoint angular momentum represented by a bivector. It has been shown that (...) the particle spin plays an important role in the kinematical intrinsic or local motion of the particle. From the complex vector formalism of harmonic oscillator, for the first time, a relation between mass $m$ and bivector spin $S$ has been derived in the form $\varvec{\sigma }_3 mc^2{\mathcal {J}}_{\pm } =\lambda \Omega _{\mathbf{s}} \cdot \mathrm{{S}} {\mathcal {J}}_{\pm }$ . Where, $\Omega _{s}$ is the angular velocity bivector of complex rotations, $c$ is the velocity of light. The unit vector $\varvec{\sigma }_3$ acts as an operator on the idempotents ${\mathcal {J}}_{+}$ and ${\mathcal {J}}_{-}$ to give the eigen values $\lambda =\pm 1.$ The constant $\lambda $ represents two fold nature of the equation corresponding to particle and antiparticle states. Further the above relation shows that the mass of the particle may be interpreted as a local spatial complex rotation in the rest frame. This gives an insight into the nature of fundamental particles. When a particle is observed from an arbitrary frame of reference, it has been shown that the spatial complex rotation dictates the relativistic particle motion. The mathematical structure of complex vectors in space and spacetime is developed. (shrink)

A novel photon-based proton model is developed. A proton’s ground state is assumed to be coherent to the degree that all of its mass-energy precipitates into a single uncharged spherical structure. A quantum vortex, initiated by the strong force, but sustained in the proton’s ground state by the circular Unruh effect and a spherical Rindler horizon, is proposed to confine the proton’s mass-energy in its ground state. A direct connection between the circular Unruh effect, the zitterbewegung effect, spin, and general (...) relativity is proposed. Such a structure acts as an uncharged zitterbewegung fermion, and may explain neutrino mass. A ground-state proton’s central zitterbewegung fermion is assumed to be surrounded by a halo of charge shells of both signs. Virtual photon standing waves are assumed to synchronize the inner shell with the central zitterbewegung fermion. The charge shells are proposed to be associated with isospin and proton g-factor. There are only two model inputs—proton mass and quantized electronic charge—and just one adjustable parameter. The adjustable parameter, reduced only by about 0.4% from an initial estimate, provides the proton’s experimentally determined magnetic moment to arbitrary precision. The resulting modeled proton charge radius agrees very well with the 2018 CODATA value. Magnetic moment and charge radius are calculated algebraically in a manner easily understood by undergraduate physics students. This proposed ground-state proton model could be considered a low-energy approximation to a full quantum chromodynamical proton model. (shrink)

In the de Broglie–Bohm formulation of quantum mechanics, the electron is stationary in the ground state of hydrogenic atoms, because the quantum force exactly cancels the Coulomb attraction of the electron to the nucleus. In this paper it is shown that classical electrodynamics similarly predicts the Coulomb force can be effectively canceled by part of the magnetic force that occurs between two similar particles each consisting of a point charge moving with circulatory motion at the speed of light. Supposition of (...) such motion is the basis of the Zitterbewegung interpretation of quantum mechanics. The magnetic force between two luminally-circulating charges for separation large compared to their circulatory motions contains a radial inverse square law part with magnitude equal to the Coulomb force, sinusoidally modulated by the phase difference between the circulatory motions. When the particles have equal mass and their circulatory motions are aligned but out of phase, part of the magnetic force is equal but opposite the Coulomb force. This raises a possibility that the quantum force of Bohmian mechanics may be attributable to the magnetic force of classical electrodynamics. It is further shown that relative motion between the particles leads to modulation of the magnetic force with spatial period equal to the de Broglie wavelength. (shrink)

By analyzing the Dirac equation with static electric and magnetic fields it is shown that Dirac’s theory is nothing but a generalized one-particle quantum theory compatible with the special theory of relativity. This equation describes a quantum dynamics of a single relativistic fermion, and its solution is reduced to solution of the generalized Pauli equation for two quasiparticles which move in the Euclidean space with their effective masses holding information about the Lorentzian symmetry of the four-dimensional space-time. We reveal the (...) correspondence between the Dirac bispinor and Pauli spinor , and show that all four components of the Dirac bispinor correspond to a fermion . Mixing the particle and antiparticle states is prohibited. On this basis we discuss the paradoxical phenomena of Zitterbewegung and the Klein tunneling. (shrink)

By probability theory the probability space to underlie the set of statistical data described by the squared modulus of a coherent superposition of microscopically distinct (sub)states (CSMDS) is non-Kolmogorovian and, thus, such data are mutually incompatible. For us this fact means that the squared modulus of a CSMDS cannot be unambiguously interpreted as the probability density and quantum mechanics itself, with its current approach to CSMDSs, does not allow a correct statistical interpretation. By the example of a 1D completed scattering (...) and double slit diffraction we develop a new quantum-mechanical approach to CSMDSs, which requires the decomposition of the non-Kolmogorovian probability space associated with the squared modulus of a CSMDS into the sum of Kolmogorovian ones. We adapt to CSMDSs the presented by Khrennikov (Found. Phys. 35(10):1655, 2005) concept of real contexts (complexes of physical conditions) to determine uniquely the properties of quantum ensembles. Namely we treat the context to create a time-dependent CSMDS as a complex one consisting of elementary (sub)contexts to create alternative subprocesses. For example, in the two-slit experiment each slit generates its own elementary context and corresponding subprocess. We show that quantum mechanics, with a new approach to CSMDSs, allows a correct statistical interpretation and becomes compatible with classical physics. (shrink)

In this work, a neo-classical relativistic mechanics theory is presented where the spin of an electron is an inherent part of its world space-time path as a point particle. The fourth-order equation of motion corresponds to the same covariant Lagrangian function in proper time as in special relativity except for an additional spin energy term. The theory provides a hidden-variable model of the electron where the dynamic variables give a complete description of its motion, giving a classical mechanics explanation of (...) the electron’s spin, its dipole moments, and Schrödinger’s zitterbewegung, These features are also described mathematically by quantum mechanics theory, of course, but without any physical picture of an underlying reality. The total motion of the electron can be decomposed into a sum of a local spin motion about a point and a global motion of this point, called here the spin center. The global motion is sub-luminal and described by Newton’s Second Law in proper time, the time for a clock fixed at the spin center, while the total motion occurs at the speed of light c, consistent with the eigenvalues of Dirac’s velocity operators having magnitude c. The local spin motion is an inherent perpetual motion, which for a free electron is periodic at the ultra-high zitterbewegung frequency and its path is circular in a spin-center reference frame. In an electro-magnetic field, this spin motion generates magnetic and electric dipole energies through the Lorentz force on the electron’s point charge. The electric dipole energy corresponds to the spin-orbit coupling term involving the electric field that appears in the corrected Pauli non-relativistic Hamiltonian, which has long been used to explain the doublet structure of the spectral lines of the excited hydrogen atom. Pauli’s spin-orbit term is usually derived, however, from his magnetic dipole energy term, including also the effect of Thomas precession, which halves this energy. The magnetic dipole energy from Pauli’s and Dirac’s theory is twice that in the neo-classical theory, a discrepancy that has not been resolved. By defining a spin tensor as the angular momentum of the electron’s total motion about its spin center, the fundamental equations of motion can be re-written in an identical form to those of the Barut–Zanghi electron theory. This allows the equations of motion to be expressed in an equivalent form involving operators applied to a state function of proper time satisfying a neo-classical Dirac–Schrödinger spinor equation. This state function produces the dynamic variables from the same operators as in Dirac’s theory for the electron but without any probability implications. It leads to a neo-classical wave function that satisfies Dirac’s relativistic wave equation for the free electron by applying the Lorentz transformation to express proper time in the state function in terms of an observer’s space-time coordinates, showing that there is a close connection between the neo-classical theory and quantum mechanics theory for the electron’s dynamics. (shrink)