Quantum electrodynamics has always included “velocity zitter” through its velocity operator, c α, and now has finite self-mass by including “position zitter”
The electron model developed by the author is unique and is derived directly from the Dirac equation velocity operator, c α , whose four current c ( α, I) in quantum electrodynamics (QED) is coupled with the external electromagnetic field to produce highly accurate results. The issue of whether “zitterbewegung (zbw)” exists, or is an artifact, is widely misunderstood. There are two connected zbw components. First, the velocity operator, c α , contains the electron’s center of charge (CoC) “velocity zitter.” If the electron’s time averaged center of mass (CoM) velocity, v CoM , were used, QED would predict highly inaccurate estimates. Second, integrating the electron velocity operator, c α , produces the unique electron CoC internal position and time coordinate operators that describe the CoC “position zitter.” We call this QED companion theory QED-position (QED-p) and the very successful theory QED-velocity (QED-v) to emphasize their direct connection. The physical description of the free electron requires both QED-p and its companion QED-v theory. QED-v is based on the “strange” Dirac equation velocity operator c α that has eigenvalues ± c . The c α eigenvalues of ± c are certainly not the instantaneous electron CoM velocity but the velocity zitter of the electron CoC “point” vibrating rapidly over time. QED has always incorporated CoC velocity zitter through its velocity operator, c α since its inception in the 1940s.
The electron model developed by the author is unique and is derived directly from the Dirac equation velocity operator, c α , whose four current c ( α, I) in quantum electrodynamics (QED) is coupled with the external electromagnetic field to produce highly accurate results. The issue of whether “zitterbewegung (zbw)” exists, or is an artifact, is widely misunderstood. There are two connected zbw components. First, the velocity operator, c α , contains the electron’s center of charge (CoC) “velocity zitter.” If the electron’s time averaged center of mass (CoM) velocity, v CoM , were used, QED would predict highly inaccurate estimates. Second, integrating the electron velocity operator, c α , produces the unique electron CoC internal position and time coordinate operators that describe the CoC “position zitter.” We call this QED companion theory QED-position (QED-p) and the very successful theory QED-velocity (QED-v) to emphasize their direct connection. The physical description of the free electron requires both QED-p and its companion QED-v theory. QED-v is based on the “strange” Dirac equation velocity operator c α that has eigenvalues ± c . The c α eigenvalues of ± c are certainly not the instantaneous electron CoM velocity but the velocity zitter of the electron CoC “point” vibrating rapidly over time. QED has always incorporated CoC velocity zitter through its velocity operator, c α since its inception in the 1940s.
The QED-Physical (QED-P) theory described in four previous papers [J. H. Wilson, Phys. Essays 28, 1 (2015); 29, 402 (2016); 31, 59 (2018); 34, 17 (2021)] is combined with QED into a single theory in this paper, since both are based on the same Dirac Equation four current c(α,I). QED couples this four-vector with an external electromagnetic (EM) field and uses covariant perturbation theory to produce results that are very accurate computationally [M. L. Eides et al., Phys. Rep. 342, 63 (2001)], except for the electron self-energy, which is infinite. The reason for QED’s accuracy is its Dirac equation velocity operator, cα, with highly unusual properties compared to classical velocity vectors. QED could not produce highly accurate answers without the 4 × 4 complex matrix cα as the electron “velocity operator.” QED-P starts with the same Dirac Equation four current used so successfully in QED, and the discrete internal spatial and time coordinate operators (ISaTCOs) are derived in QED-P to give the electron internal structure a very specific, but highly, nonclassical geometric description with no ad hoc assumptions. The QED/QED-physical (QED-P) unification theorem is stated and proved in Sec. VII. However, one should ask what new physical measurements are predicted by QED-P that are not determined by the very accurate QED? The answer is none yet, due to the very rapid fluctuation of the ISaTCOs with a period of ∼6.4 × 10−22 s. This induced electron field fluctuation is in addition to any vacuum fluctuation that exists without the electron field present. This paper presents a highly speculative, but testable, experiment, in which the ISaTCO rapid fluctuations may be confirmed indirectly as the physical basis of field quantum entanglement (QE). The electron’s ISaTCO produce a digital internal clock that’s locked out of phase with its entangled positron, and they communicate spin states at the phase speed of 2c2/vCoM. The communication of spin state information is instantaneous for vCoM = 0 but “slows down” to 4c at vCoM = 0.5c, still very fast.
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.
Dirac's Relativistic Wave Equation implies a measured electron velocity of ±c in any direction, in contradiction to Special Relativity and observation. It is shown in this article that this anomalous electron velocity reveals an internal structure of the electron whereby the mass and the charge of the electron cannot be described by the same position operator. The measured velocity of electron mass is always less than c in any direction but charge can be displaced at the speed of light. This speed is realizable only when the electron is in a state that is a superposition of positive and negative energy states, also known as a zitterbewegung state. It is shown that in zitterbewegung it is the charge and not the mass that undergoes rapid spatial oscillation, and that there are measurable consequences of this charge zitterbewegung. Zitterbewegung of charge also occurs in an entangled electron-positron pair created by a strong electric field.
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.
With respect to the principal operators including the sixteen 4-4 matrices of the Dirac electron, two covariant forms of the extended Heisenberg picture can be presented with the “two covariant Hamiltonians HI and HII” of the first and second order, respectively. One form with HII involving the proper electromagnetic moment corresponds perfectly to the Frenkel-Kramers model which was recently treated by the author; the other form with HI containing the Zitterbewegung corresponds partly to the so-called Weyssenhoff-like model. That an amalgamation of the former form and the Zitterbewegung brings immediately the latter is explicitly shown by virtue of a decomposition of the operator of the Dirac electron into two corresponding parts. This decomposition is derived from the author's extension of the Feynman-Bunge-Corben formula for the position operator of the electron and, in case without external fields, it corresponds to the covariant form of the so-called ‘gerade-ungerade’ decomposition. In the case of the free state mixed with positive and negative energy, the Zitterbewegung of operators derived from HI can properly be realized by the expectation values of the operators. In this process, the author's complex hyperbolic metric of velocity which represents a new negative energy state plays an essential role. These results give an important clue to solve the puzzling problem of the construction of a consistent classical model for the Dirac electron, and also may be useful for investigation of ‘internal structure’ of elementary particles as well as their spin.
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.