Some authors have claimed that there exists a minimum size (on the order of the Compton radius) for electron states composed entirely of positive-frequency solutions to the free Dirac equation. Other authors have put forward counterexamples to such claims. This article asks how the counterexamples of A. J. Bracken and G. F. Melloy [J. Phys. A. 32, 6127 (1999)] bypass two arguments against their possibility. The first is an old argument that, because of the prohibition on faster-than-light motion, the electron must be larger than a certain minimum size if it is to have the correct angular momentum and magnetic moment. This challenge can be addressed by analyzing the flow of energy and charge for the counterexample states. The second argument is an explicit proof (presented in C.-P. Chuu et al. [Solid State Commun. 150, 533 (2010)]) that there is a minimum size for purely positive-frequency electron states. This proof hinges on the assumption of a small spread in momentum space, which is violated by the counterexamples that have been put forward.
Some authors have claimed that there exists a minimum size (on the order of the Compton radius) for electron states composed entirely of positive-frequency solutions to the free Dirac equation. Other authors have put forward counterexamples to such claims. This article asks how the counterexamples of A. J. Bracken and G. F. Melloy [J. Phys. A. 32, 6127 (1999)] bypass two arguments against their possibility. The first is an old argument that, because of the prohibition on faster-than-light motion, the electron must be larger than a certain minimum size if it is to have the correct angular momentum and magnetic moment. This challenge can be addressed by analyzing the flow of energy and charge for the counterexample states. The second argument is an explicit proof (presented in C.-P. Chuu et al. [Solid State Commun. 150, 533 (2010)]) that there is a minimum size for purely positive-frequency electron states. This proof hinges on the assumption of a small spread in momentum space, which is violated by the counterexamples that have been put forward.
We consider wave functions in the Hilbert space H=L2(R3,C4) of a single Dirac particle, specifically from the positive-energy subspace H+ of the free Dirac Hamiltonian. Over the decades, various authors hypothesized that for wave functions from H+, there is a positive lower bound to the position uncertainty σx; in other words, that such states cannot be arbitrarily narrow in x. Using a sequence of wave functions introduced by Bracken and Melloy, we show that this hypothesis is false. (In fact, they already stated that it is false, but their proof that their sequence is a counter-example had a gap.)
Highly localized positive-energy states of the free Dirac electron are constructed and shown to evolve in a simple way under the action of Dirac's equation. When the initial uncertainty in position is small on the scale of the Compton wavelength, there is an associated uncertainty in the mean energy that is large compared with the rest mass of the electron. However, this does not lead to any breakdown of the one-particle description, associated with the possibility of pair-production, but rather leads to a rapid expansion of the probability density outwards from the point of localization, at speeds close to the speed of light.
A causally well-behaved solution of the localization problem for the free electron is given, with natural space-time transformation properties, in terms of Dirac’s position operator x. It is shown that, although x is not an observable in the usual sense, and has no positive-energy (generalized) eigenstates, the 4-vector density (ρ(x,t),j(x,t)/c) is observable, and can be localized arbitrarily precisely about any point in space, at any instant of time, using only positive-energy states. A suitable spin operator can be diagonalized at the same time. 1
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.
This paper further explores the structural similarities between the elementary quantum-mechanical wavefunction and circularly polarized electromagnetic waves to further tune a possible physical interpretation of the wavefunction. The interpretation that is offered analyzes the real and the imaginary part of the wavefunction as oscillations which each carry half of the total energy of the particle. These oscillations are perpendicular to each other, and the interplay between both may describe how energy propagates through space over time. The model is based on three fundamental premises: 1.The dimension of the matter-wave field vector is force per unit mass (N/kg), as opposed to the force per unit charge (N/C) dimension of the electric field vector. This dimension is an acceleration (m/s2), which is the dimension of the gravitational field. 2.This gravitational disturbance may cause a charged mass to move about some center, combining linear and circular motion. This interpretation may reconcile the wave-particle duality to some extent: fields interfere but if, at the same time, they do drive a pointlike particle, this may explain why, as Feynman puts it, “when you do find the electron some place, the entire charge is there.” This hybrid hypothesis is supported by an elegant yet simple derivation of the Compton radius of an electron. 3.In light of the direction of the magnetic moment of an electron in an inhomogeneous magnetic field, the plane which circumscribes the circulatory motion of the electron should also comprise the direction of its linear motion. Hence, unlike an electromagnetic wave, the plane of the two-dimensional oscillation cannot be perpendicular to the direction of motion of our electron. Finally, this paper addresses an issue which has hampered other physical interpretations of the wavefunction: amplitude transformations when going from one representation (or reference frame) to another. Indeed, the author re-visits the original mathematical arguments here and shows that the proposed model, and the related physical interpretation of the wavefunction, are not incompatible with the key results and mainstream logic in this regard.