A spatial model of a free electron (or a positron) is formed by a proposed superluminally circulating point-like charged superluminal quantum. The model includes the Dirac equation’s electron spin 12 and magnetic moment / 2e m as well as three Dirac equation measures of the electron’s Zitterbewegung (“jittery motion”): a speed of light velocity c, a frequency of 2 202 / 2.5 10mc h hz, and a size of 131 2 / 1.9 10mc m. The electron’s superluminal quantum has a closed double-looped helical trajectory whose circular axis ’ double-looped length is one Compton wavelength h/mc. In the electron’s rest frame, the equations for the superluminal quantum’s position are:
A spatial model of a free electron (or a positron) is formed by a proposed superluminally circulating point-like charged superluminal quantum. The model includes the Dirac equation’s electron spin 12 and magnetic moment / 2e m as well as three Dirac equation measures of the electron’s Zitterbewegung (“jittery motion”): a speed of light velocity c, a frequency of 2 202 / 2.5 10mc h hz, and a size of 131 2 / 1.9 10mc m. The electron’s superluminal quantum has a closed double-looped helical trajectory whose circular axis ’ double-looped length is one Compton wavelength h/mc. In the electron’s rest frame, the equations for the superluminal quantum’s position are:
A semiclassical model of the electron is presented based on the Principle of Helical Motion (“A free electron always moves at the speed of light following a helical motion, with a constant radius, and with the direction of movement perpendicular to the rotation plane”). This model interprets the Zitterbewegung as a real motion that causes rotation of the electron spin and its magnetic moment. Based on this model, the quantum magnetic flux and quantum Hall resistance are obtained as parameters of the electron and special relativity theory is derived from the helical motion of the electron. Finally, a fix is proposed for the De Broglie’s wavelength that questions the very validity of the Dirac equation.
A neo-classical relativistic mechanics theory is presented where the spin of an electron is a natural part of its space-time path as a point particle. The fourth-order equation of motion corresponds to the same Lagrangian function in proper time as in special relativity except for an additional spin energy term. The total motion can be decomposed into a sum of a local spin motion about a point and a global motion of this point, called the spin center. The global motion is sub-luminal and obeys 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 corresponds to Schrödinger's zitterbewegung and is a perpetual motion, which for a free electron has a circular path in the spin-center 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 corresponding electric dipole energy is consistent with the spin-orbit coupling term in the corrected Pauli non-relativistic Hamiltonian but the magnetic dipole energy is one half of that in Dirac's theory. 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. These equations of motion can then be expressed using operators applied to a state function of proper time satisfying a Dirac-Schrödinger spinor equation. The operators produce dynamic variables without any probability implications. For the free electron, the state function satisfies Dirac's relativistic wave equation when the Lorentz transformation is applied to express proper time in terms of an observer's space-time coordinates.
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.
The spherical model of electron cannot explain Bohr magneton, de Broglie wave, etc. To explain these properties physically, we propose that spin, one core concept of quantum theory, is a reality, and introduce this alternative solely based on experimental findings, Planck’s, Einstein’s, and Dirac’s insights, and follows classical laws, that an electron is an electrical wave with a literal 21 spin (Dirac), i.e., spins two circles in one cycle, both angular momentum are ℏ. Its wavelength is determined by E=hνm0=λm0hc=m0c2 (Plank and Einstein), λ m 0 is the Compton wavelength. As E⃗ ⊥ c⃗ for a transverse wave, it has a net electricity due to the first spin, f(Eq)=21E0(1−cos(2ω0t)), half of its cycle, i.e., the minimal meaningful unit as F = E q , is related with its charge, q = E q ϵ 0 . Moreover, an electron has a ring shape due to the second spin, 2 π R 0 = λ m 0 , λ m 0 is the electron-impact cross section. As ℏ=λm0h2πλm0=m0cR0, it has a quantized classical angular momentum, with a speed of c . The Bohr magneton, μB=2m0qℏ, g -factor is 2. Critically, the full Dirac’s equation of an electron with a constant velocity is mc=ℏ⊥+p⊥+ℏ//+p//, and its electromagnetic properties due to charge are f(Eq)=f(Eq)cc2−v′2+f(Eq)cv′=Enet//+Eq⊥, v′//ℏ⊥ or v′//ℏ//, and v′//Enet. Assuming heavy charged particles have equivalent though shall not identical mechanisms, mathematically this model is equivalent to Special Relativity, and supports ALL its formulas, i.e., v < c , E = m c 2 , direction-insensitive m = γ m 0 and t=γt0; Moreover, the net electricity Enet// and Bohr magneton is orientation-quantized, direction-sensitive and follows Lorentz transformation, leading to Lorentz transformation of length contraction; while following Relativity, magnetism can be defined by f(B)=f(Eq)c2v′, which is not circularly symmetric and self-evidently does not follow Lorentz transformation. The temporospatial averaged magnetic intensity of a current is relativistic, Bˉ=∑f(Eq)c2Δv=2πϵ0c2I=2πμ0I, and μ 0 ϵ 0 c 2 = 1, supporting Maxwell’s equations. Further implications include chirality therefore magnetic spatial quantization, de Broglie wave, etc. The spherical model should be just an observation due to orientation quantization of Enet. We conclude this model of electron may provide one step further to the ultimate Quantum Theory in a nonrelativistic spacetime.
Hestenes has presented an integration of Schrödinger's zitterbewegung with the spin matrices of the Dirac equation, suggesting the electron can be modeled by a rapidly rotating dipole moment and a frequency related to the de Broglie frequency. He presents an elegant spacetime algebra that provides a reformulation of the Dirac equation that incorporates these real spin characteristics. A similar heuristic model for quantum particles has been derived by this author from a different, quasi-classical premise: That the most fundamental subcomponents of quantum particles all travel at a constant speed of light. Time is equated with the spatial displacement of these subcomponents – the speed of light is the speed of time. This approach suggests a means of integrating special relativity and quantum mechanics with the same concept of time. The relativistic transformation of spinning quantum particles create the appearance of additional, compactified spatial dimensions that can be correlated with the complex phase of the spin matrices as in the Dirac formalism. This paper further examines the convergence on such new models for quantum particles built on this rapid motion of particle subcomponents. The modeling leverages a string-like heuristic for particle subcomponents and a revised description for the wave-like properties of particles. This examination provides useful insights to the real spatial geometries and interactions of electrons and photons.
The Dirac equation is presented as a complete theory of the electron as a gyromagnetic particle clock with precise physical interpretation for all degrees of freedom. The electron is modeled as a point charge with toroidal zitter and variable zilch. That is to say, the charge oscillates at the speed of light on a torus centered on a circular orbit around its center of massz=z(t), with an axis at a variable angleβ=β(z(t))with respect to its spin vectors=s(t). The Dirac wave functionΨ=Ψ(ct+x)has a unique factorizationΨ=(ρeiβ)21U, whereU=U(ct+x)is a spatial rotor with magnetic degrees of freedom, and the quantityΨΨ=ρeiβspecifies an embedding of electron paths in the vacuum, where the zilch functionβ=β(ct+x−z(t))is a measure of electron energy density. It culminates in a new synthesis of Dirac electron theory with Maxwell’s electrodynamics by identifying zilch as a common factor that binds them together.