An algebraic block-diagonalization of the Dirac Hamiltonian in a time-independent external field reveals a charge-index conservation law which forbids the physical phenomena of the Klein paradox type and guarantees a single-particle nature of the Dirac equation in strong external fields. Simultaneously, the method defines simpler quantum-mechanical objects—paulions and antipaulions, whose 2-component wave functions determine the Dirac electron states through exact operator relations. Based on algebraic symmetry, the presented theory leads to a new understanding of the (...) class='Hi'>Dirac equation physics, including new insight into the Dirac measurements and a consistent scheme of relativistic quantum mechanics of electron in the paulion representation. Along with analysis of the mathematical anatomy of the Klein paradox falsity, a complete set of paradox-free eigenfunctions for the Klein problem is obtained and investigated via stationary solutions of the Pauli-like equations with respective paulion Hamiltonians. It is shown that the physically correct Dirac states in the Klein zone are characterized by the total particle reflection from the potential step and satisfy the fundamental charge-index conservation law. (shrink)

Dirac’s relativistic theory of electron generally results in two possible solutions, one with positive energy and other with negative energy. Although positive energy solutions accurately represented particles such as electrons, interpretation of negative energy solution became very much controversial in the last century. By assuming the vacuum to be completely filled with a sea of negative energy electrons, Dirac tried to avoid natural transition of electron from positive to negative energy state using Pauli’s exclusion principle. However, (...) many scientists like Bohr objected to the idea of sea of electrons as it indicates infinite density of charge and electric field and consequently infinite energy. In addition, till date, there is no experimental evidence of a particle whose total energy (kinetic plus rest) is negative. In an alternative approach, Feynman, in quantum field theory, proposed that particles with negative energy are actually positive energy particles running backwards in time. This was mathematically consistent since quantum mechanical energy operator contains time in denominator and the negative sign of energy can be absorbed in it. However, concept of negative time is logically inconsistent since in this case, effect happens before the cause. To avoid above contradictions, in this paper, we try to reformulate the Dirac’s theory of electron so that neither energy needs to be negative nor the time is required to be negative. Still, in this new formulation, two different possible solutions exist for particles and antiparticles (electrons and positrons). (shrink)

Much debate has ensued regarding the challenge to scientific realism provided by consideration of certain problematic episodes of theory change in the history of science. This paper contends that there is an interesting case which has been overlooked in this debate, namely the prediction of the positron by Dirac from his ‘hole’ theory, and its subsequent replacement by a theory which failed to contain a central, and essential, theoretical posit: the ‘Dirac sea’ of negative energy electrons. Accounting for (...) this case with the ‘divide and conquer’ strategy of contemporary scientific realism proves particularly troublesome, even for the structural realist. (shrink)

In this paper, it is presented a historical account of the formulation of the quantum relativistic wave equation of an electron – the Dirac equation, issues regarding its interpretation that arose from the very beginning, and the later formulation of this equation in relation to a quantized electron-positron field, which implies a new interpretation. The way in which solutions obtained under each interpretation of the equation relate to one another is also considered for the simple case of (...) hydrogen-like problems. (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)

One of the key episodes of history of modern physics – Paul Dirac’s startling contrivance of the relativistic theory of the electron – is elicited in the context of lucid epistemological model of mature theory change. The peculiar character of Dirac’s synthesis of special relativity and quantum mechanics is revealed by comparison with Einstein’s sophisticated methodology of the General Relativity contrivance. The subtle structure of Dirac’s scientific research program and first and foremost the odd principles that (...) put up its powerful heuristics is scrutinized with special emphasis on the highly controversial tenet of “mathematical beauty.” It is contended that despite the relentless Dirac’s remarks denigrating the controversial role of philosophy one can trace its indirect influence through Arthur Eddington’s and Hermann Weyl’s whimsical mathematical models. Accordingly, the milestones of Dirac’s research programme realization in the distinctive context of the applied epistemological doctrine are indicated. (shrink)

This paper employs the ideas of geometric algebra to investigate the physical content of Dirac's electron theory. The basis is Hestenes' discovery of the geometric significance of the Dirac spinor, which now represents a Lorentz transformation in spacetime. This transformation specifies a definite velocity, which might be interpreted as that of a real electron. Taken literally, this velocity yields predictions of tunnelling times through potential barriers, and defines streamlines in spacetime that would correspond to electron (...) paths. We also present a general, first-order diffraction theory for electromagnetic and Dirac waves. We conclude with a critical appraisal of the Dirac theory. (shrink)

This article is concerned with the role of fundamental principles in theoretical physics, especially quantum theory. The fundamental principles of relativity will be addressed as well, in view of their role in quantum electrodynamics and quantum field theory, specifically Dirac’s work, which, in particular Dirac’s derivation of his relativistic equation of the electron from the principles of relativity and quantum theory, is the main focus of this article. I shall also consider Heisenberg’s earlier work leading him to (...) the discovery of quantum mechanics, which inspired Dirac’s work. I argue that Heisenberg’s and Dirac’s work was guided by their adherence to and their confidence in the fundamental principles of quantum theory. The final section of the article discusses the recent work by D’Ariano and coworkers on the principles of quantum information theory, which extend quantum theory and its principles in a new direction. This extension enabled them to offer a new derivation of Dirac’s equations from these principles alone, without using the principles of relativity. (shrink)

I propose three new curved spacetime versions of the Dirac Equation. These equations have been developed mainly to try and account in a natural way for the observed anomalous gyromagnetic ratio of Fermions. The derived equations suggest that particles including the Electron which is thought to be a point particle do have a finite spatial size which is the reason for the observed anomalous gyromagnetic ratio. A serendipitous result of the theory, is that, to of the equation exhibits (...) an asymmetry in their positive and negative energy solutions the first suggestion of which is clear that a solution to the problem as to why the Electron and Moun—despite their acute similarities—exhibit an asymmetry in their mass is possible. The Moun is often thought as an Electron in a higher energy state. Another of the consequences of three equations emanating from the asymmetric serendipity of the energy solutions of two of these equations, is that, an explanation as to why Leptons exhibit a three stage mass hierarchy is possible. (shrink)

El electrón introduce en la mecánica subatómica el nuevo concepto de probabilidad cuántica. Este artículo estudia el proceso de esta introducción, de 1916 a 1927. Einstein lo ocasionó, al elaborar un formalismo de emisión y absorción de radiación, que Bohr incluyó en su principio de correspondencia, como «probabilidades a priori». Heisenberg las recogió en su mecánica matricial, mientras Schrodinger introducía en su mecánica ondulatoria una función de onda, que Born convirtió en amplitud de probabilidad. Dirac fundamentó esos conceptos de (...) probabilidad con su teoría de transformaciones, y Heisenberg, inspirado en ella, formuló las relaciones de indeterminación de la nueva física subatómica. (shrink)

It is usually claimed that the Laguerre polynomials were popularized by Schrödinger when creating wave mechanics; however, we show that he did not immediately identify them in studying the hydrogen atom. In the case of relativistic Dirac equations for an electron in a Coulomb field, Dirac gave only approximations, Gordon and Darwin gave exact solutions, and Pidduck first explicitly and elegantly introduced the Laguerre polynomials, an approach neglected by most modern treatises and articles. That Laguerre polynomials were (...) not very popular before their use in quantum mechanics, probably because they had been little used in classical mathematical physics, is confirmed by the fact that, as we show, they had been rediscovered independently several times during the nineteenth century, in published or unpublished studies of Abel, Murphy, Chebyshev, and Laguerre. (shrink)

A null 4-vector ε°σμε, based on Dirac's relativistic electron equation, is shown explicitly for a plane wave and various Coulomb states. This 4-vector constitutes a mechanical “model” for the electron in those states, and expresses the important spinor quantities represented conventionally byn, f, g, m, j, κ,1, ands. The model for a plane wave agrees precisely with the relation between velocity and phase gradient customarily used in quantum theory, but the models for Coulomb states contradict that relation.

The electronic and muonic hydrogen energy levels are calculated very accurately [1] in Quantum Electrodynamics (QED) by coupling the Dirac Equation four vector (c ,mc2) current covariantly with the external electromagnetic (EM) field four vector in QED’s Interactive Representation (IR). The c -Non Exclusion Principle(c -NEP) states that, if one accepts c as the electron/muon velocity operator because of the very accurate hydrogen energy levels calculated, the one must also accept the resulting electron/muon internal spatial and time (...) coordinate operators (ISaTCO) derived directly from c without any assumptions. This paper does not change any of the accurate QED calculations of hydrogen’s energy levels, given the simplistic model of the proton used in these calculations [1]. The Proton Radius Puzzle [2, 3] may indicate that new physics is necessary beyond the Standard Model (SM), and this paper describes the bizarre, and very different, situation when the electron and muon are located “inside” the spatially extended proton with their Centers of Charge (CoCs) orbiting the proton at the speed of light in S energy states. The electron/muon center of charge (CoC) is a structureless point that vibrates rapidly in its inseparable, non-random vacuum whose geometry and time structure are defined by the electron/muon ISaTCO discrete geometry. The electron/muon self mass becomes finite in a natural way due to the ISaTCOs cutting off high virtual photon energies in the photon propagator. The Dirac-Maxwell-Wilson (DMW) Equations are derived from the ISaTCO for the EM fields of an electron/muon, and the electron/muon “look” like point particles in far field scattering experiments in the same way the electric field from a sphere with evenly distributed charge “e” “looks” like a point with the same charge in the far field (Gauss Law). The electron’s/muon’s three fluctuating CoC internal spatial coordinate operators have eight possible eigenvalues [4, 5, 6] that fall on a spherical shell centered on the electron’s CoM with radius in the rest frame. The electron/muon internal time operator is discrete, describes the rapid virtual electron/positron pair production and annihilation. The ISaTCO together create a current that produce spin and magnetic moment operators, and the electron and muon no longer have “intrinsic” properties since the ISaTCO kinematics define spin and magnetic moment properties. (shrink)

Problems of self-interaction arise in both classical and quantum field theories. To understand how such problems are to be addressed in a quantum theory of the Dirac and electromagnetic fields (quantum electrodynamics), we can start by analyzing a classical theory of these fields. In such a classical field theory, the electron has a spread-out distribution of charge that avoids some of the problems of self-interaction facing point charge models. However, there remains the problem that the electron will (...) experience self-repulsion. This self-repulsion cannot be eliminated within classical field theory without also losing Coulomb interactions between distinct particles. But, electron self-repulsion can be eliminated from quantum electrodynamics in the Coulomb gauge by fully normal-ordering the Coulomb term in the Hamiltonian. After normal-ordering, the Coulomb term contains pieces describing attraction and repulsion between distinct particles and also pieces describing particle creation and annihilation, but no pieces describing self-repulsion. (shrink)

The electron is conceived here as a complex structure composed of a subparticle that is bound to a nearly circular motion. Although in quantum mechanics the spin is not representable, in classical stochastic physics this corresponds to the angular momentum of the subparticle. In fact, assuming Schrödinger-type hydrodynamic equations of motion for the subparticle, the spin-1/2 representation in configuration space and the corresponding Pauli matrices for the electron are obtained. The Hamiltonian of Pauli's theory as the nonrelativistic limit (...) of Dirac's equation is also derived. (shrink)

The two center time dependent Dirac equation, for an electron in the external field of two colliding ultrarelativistic heavy ions is considered. In the ultrarelativistic limit, the ions are practically moving at the speed of light and the electromagnetic fields of the ions are confined to the light fronts by the extreme Lorentz contraction and by the choice of gauge, designed to remove the long-range Coulomb effects. An exact solution to the ultrarelativistic limit of the two-center Dirac (...) equation is found by using light-front variables and a light-fronts representation. Previously unexplained experimental results obtained at CERN's SPS are explained in this way and predictions are made as to where one should look, in momentum space, and in space-time, if one wants to study and observe non-perturbative electromagnetic pair-production effects in extremely relativistic heavy-ion collisions. (shrink)

Este artículo se ocupa de parte de la historia del electrón entre 1928 (aparición de la ecuación del electrón de Dirac) y 1947-49 (reformulación de la electrodinámica cuántica). Durante este período la física del electrón experimenta cambios sustanciales, entre los que destaca el descubrimiento de la primera antipartícula, el positrón, y la trabajosa elaboración de la teoría que seguimos utilizando para describir al electrón y sus interacciones. Son dos décadas cruciales no ya para la física del electrón, sino para (...) la física entera, que se industrializa y militariza. (shrink)

This paper charts P.A.M. Dirac’s development of his theory of the electron, and its radical picture of empty space as an almost-full plenum. Dirac’s Quantum Electrodynamics famously accomplished more than the unification of special relativity and quantum mechanics. It also accounted for the ‘duplexity phenomena’ of spectral line splitting that we now attribute to electron spin. But the extra mathematical terms that allowed for spin were not alone, and this paper charts Dirac’s struggle to ignore (...) or account for them as a sea of strange, negative-energy, particles with positive ‘holes’. This work was not done in solitude, but rather in exchanges with Dirac’s correspondence network. This social context for Dirac’s work contests his image as a lone genius, and documents a community wrestling with the ontological consequences of their work. Unification, consistency, causality, and community are common factors in explanations in the history of physics. This paper argues on the basis of materials in Dirac’s archive that — in addition — mathematical beauty was an epistemological factor in the development of the electron and hole theory. In fact, if we believe that Dirac’s beautiful mathematics captures something of the world, then there is both an epistemology and an ontology of mathematical beauty. (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)

We show that the Bohmian approach in terms of persisting particles that move on continuous trajectories following a deterministic law can be literally applied to quantum field theory. By means of the Dirac sea model—exemplified in the electron sector of the standard model neglecting radiation—we explain how starting from persisting particles, one is led to standard QFT employing creation and annihilation operators when tracking the dynamics with respect to a reference state, the so-called vacuum. Since on the level (...) of wave functions, both formalisms are mathematically equivalent, this proposal provides for an ontology of QFT that includes a dynamics of individual processes, solves the measurement problem, and explains the appearance of creation and annihilation events. 1Bohmian Mechanics from Quantum Mechanics to Quantum Field Theory2The Dirac Sea Model3Equilibrium States and the Vacuum4Excitations of the Vacuum and the Fock Space Formalism5The Appearance of Particle Creations and Annihilations6The Merits of the Bohmian Approach. (shrink)

We show that the Bohmian approach in terms of persisting particles that move on continuous trajectories following a deterministic law can be literally applied to QFT. By means of the Dirac sea model – exemplified in the electron sector of the standard model neglecting radiation – we explain how starting from persisting particles, one is led to standard QFT employing creation and annihilation operators when tracking the dynamics with respect to a reference state, the so-called vacuum. Since on (...) the level of wave functions, both formalisms are mathematically equivalent, this proposal provides for an ontology of QFT that includes a dynamics of individual processes, solves the measurement problem and explains the appearance of creation and annihilation events. (shrink)

The presented enhanced version of Eriksen’s theorem defines an universal transform of the Foldy–Wouthuysen type and in any external static electromagnetic field reveals a discrete symmetry of Dirac’s equation, responsible for existence of a highly influential conserved quantum number—the charge index distinguishing two branches of DE spectrum. It launches the charge-index formalism obeying the charge-index conservation law. Via its unique ability to manipulate each spectrum branch independently, the CIF creates a perfect charge-symmetric architecture of Dirac’s quantum mechanics, which (...) resolves all the riddles of the standard DE theory. Besides the abstract CIF algebra, the paper discusses: the novel accurate charge-symmetric definition of the electric-current density; DE in the true-particle representation, where electrons and positrons coexist on equal footing; flawless “natural” scheme of second quantization; and new physical grounds for the Fermi–Dirac statistics. As a fundamental quantum law, the CICL originates from the kinetic-energy sign conservation and leads to a novel single-particle physics in strong-field situations. Prohibiting Klein’s tunneling in Klein’s zone via the CICL, the precise CIF algebra defines a new class of weakly singular DE solutions, strictly confined in the coordinate space and experiencing the total reflection from the potential barrier. (shrink)

Quantum mechanics teaches that before detection, knowledge of particle position is, at best, probabilistic, and classical trajectories are seen as a feature of the macroscopic world. These comments refer to detected particles, but we are still free to consider the motions generated by the wave equation. Within hydrogen, the Schrodinger equation allows calculation of kinetic energy at any location, and if this is identified as the energy of the wave, then radial momentum, allowing for spherical harmonics, becomes available. The distance (...) across the real zone of radial momentum is found to match semi-integer wavelengths of the adjusted matter wave, consistent with what is expected from a standing wave condition. The approach is extended to include orbital motions, where it is established that the underlying wave, which has direction and wavelength at each location, forms a series of connected trajectories, which are shown to be ellipses orientated at various angles to the equatorial plane. This suggests that wave trajectories, rather than particle trajectories, are still a feature of the hydrogen atom. The finding allows the reason for the coincidence between energy results derived by Sommerfeld’s classical trajectories and the Schrodinger wave equation to be appreciated. The result has implications when the relativistic situation is considered, as Sommerfeld’s correct deduction of the relativistic energy levels of hydrogen well before Dirac derived his wave equation has long been somewhat puzzling. (shrink)

The second-order radial differential equations for the relativistic Dirac hydrogen atom are derived from the Dirac equation treated as a system of partial differential equations. The quantum operators which arise in the development are defined and interpreted as they appear. The splitting in the energy levels is computed by applying the theory of singularities for second-order differential equations to the Klein-Gordon and Dirac relativistic equations. In the Dirac radial equation additional terms appear containing a constant, which (...) is shown to be the “radius of the electron.” It is concluded that the minute perturbation of the radial eigenfunction in the vicinity of the proton brought about by the extension of the elementary particles, which appears naturally out of the Dirac equations, results in the prediction of the observed splitting of the hydrogen atom energy levels by the Dirac theory. The extension of the particles arises even though the Dirac hydrogen atom is originally formulated for point charges. (shrink)

This article compares treatments of the Stern–Gerlach experiment across different physical theories, building up to a novel analysis of electron spin measurement in the context of classical Dirac field theory. Modeling the electron as a classical rigid body or point particle, we can explain why the entire electron is always found at just one location on the detector but we cannot explain why there are only two locations where the electron is ever found. Using non-relativistic (...) or relativistic quantum mechanics, we can explain both uniqueness and discreteness. Moving to more fundamental physics, both features can be explained within a quantum theory of the Dirac field. In a classical theory of the Dirac field, the rotating charge of the electron can split into two pieces that each hit the detector at a different location. In this classical context, we can explain a feature of electron spin that is often described as distinctively quantum but we cannot explain another feature that could be explained within any of the other theories. (shrink)

The magnetic dipole moment of the Kerr–Newman metric, defined by mass \, electrical charge \ and angular momentum \, is \, corresponding, for all values of \, to a gyromagnetic ratio \, which is also the value of the intrinsic gyromagnetic ratio of the electron, as first noted by Carter. Here, we argue that this result can be understood in terms of the particle-wave complementarity principle. For \ can only be defined at asymptotic spatial infinity, where the metric appears (...) to describe a spinning point particle, and therefore setting \, \, we necessarily have a model of the electron. From the Dirac equation we can construct a covariantly conserved four-current \ that is the source of the electromagnetic field generated by the charge \. The result \ then follows from the minimal gauge principle \ which is implicit in the formulation of the spinorial wave equation, and which can also be justified from the line action for a spin-1/2 point particle interacting with an external electromagnetic field, due to Berezin and Marinov. By contrast, analysis of the gyrogravito-magnetic effect, investigated classically by Wald and quantum mechanically by Adler et al., yields the result \ in all non-relativistic cases, which can be explained from the principle of equivalence. The results are in accord with the correspondence principle. (shrink)

Several versions exist of pseudo-classical models of the electron using Grassmann variables. Most of these require additional constraints on the variables, and it is these which, when quantized, lead to Dirac's equation. In addition, the Grassmann variables do not have physical interpretations. In this article a model is constructed which does not require constraints and in which the Grassmann variables can be interpreted as observables. Dirac's equation is obtained directly from quantization.

A cornerstone of physics, Maxwell‘s theory of electromagnetism, apparently contains a fatal flaw. The standard expressions for the electromagnetic field energy and the self-mass of an electron of finite extension do not obey Einstein‘s famous equation, \, but instead fulfill this relation with a factor 4/3 on the left-hand side. Furthermore, the energy and momentum of the electromagnetic field associated with the charge fail to transform as a four-vector. Many famous physicists have contributed to the debate of this so-called (...) 4/3-problem without arriving at a complete solution. It has generally been assumed that, as originally suggested by Poincaré, the problems are connected to the question of stability of the charge distribution, and that relativistic equivalence between energy and self-mass can only be restored by inclusion of stabilizing forces. Alternative solutions to the problems have also been proposed. Nearly a century ago Fermi suggested a covariant definition of the electromagnetic energy and momentum, and sixty years later Kalckar et al. argued that the 4/3 problem is caused by omission of a relativistic correction in the standard evaluation of the self-force from Coulomb self-interaction. However, the relation between these suggestions has not been clear. We show that the relativistic correction implies that the mechanical momentum of an accelerated rigid body must be defined as the sum of the momenta of its parts for fixed time in the momentary rest frame of the body. For the total momentum of particles and field to be conserved, the total energy–momentum tensor must be divergence free, and this then requires that the momentum of the associated electromagnetic field be defined in the same way, consistent with the suggestion by Fermi. This comprehensive solution of the 4/3-problem demonstrates that there is no conflict of Maxwell‘s theory with special relativity and the questions of equivalence of electromagnetic energy and self-mass and of stability of a classical charge distribution are independent. In appendices we discuss the relations of our treatment with Fermi‘s seminal paper and with a classic paper by Dirac where he evaluated the damping self-force on a point electron from transport of energy and momentum in the electromagnetic field. (shrink)

It is a commonplace in the philosophy of physics that any local physical theory can be represented using arbitrary coordinates, simply by using tensor calculus. On the other hand, the physics literature often claims that spinors \emph{as such} cannot be represented in coordinates in a curved space-time. These commonplaces are inconsistent. What general covariance means for theories with fermions, such as electrons, is thus unclear. In fact both commonplaces are wrong. Though it is not widely known, Ogievetsky and Polubarinov constructed (...) spinors in coordinates in 1965, enhancing the unity of physics and helping to spawn particle physicists' concept of nonlinear group representations. Roughly and locally, these spinors resemble the orthonormal basis or "tetrad" formalism in the symmetric gauge, but they are conceptually self-sufficient and more economical. The typical tetrad formalism is de-Ockhamized, with six extra field components and six compensating gauge symmetries to cancel them out. The Ogievetsky-Polubarinov formalism, by contrast, is Ockhamized, with most of the fluff removed. As developed nonperturbatively by Bilyalov, it admits any coordinates at a point, but "time" must be listed first. Here "time" is defined in terms of an eigenvalue problem involving the metric components and the matrix $diag$, the product of which must have no negative eigenvalues in order to yield a real symmetric square root that is a function of the metric. Thus even formal general covariance requires reconsideration; the atlas of admissible coordinate charts should be sensitive to the types and \emph{values} of the fields involved. Apart from coordinate order and the usual spinorial two-valuedness, Ogievetsky-Polubarinov spinors form, with the metric, a nonlinear geometric object, for which important results on Lie and covariant differentiation are recalled. Such spinors avoid a spurious absolute object in the Anderson-Friedman analysis of substantive general covariance. They also permit the gauge-invariant localization of the infinite-component gravitational energy in General Relativity. Density-weighted spinors exploit the conformal invariance of the massless Dirac equation to show that the volume element is absent. Thus instead of an arbitrary nonsingular matrix with 16 components, 6 of which are gauged away by a new local $O$ gauge group and one of which is irrelevant due to conformal covariance, one can, and presumably should, use density-weighted Ogievetsky-Polubarinov spinors coupled to the 9-component symmetric square root of the part of the metric that fixes null cones. Thus $\frac{7}{16}$ of the orthonormal basis is eliminated as surplus structure. Greater unity between spinors and tensors and the like is achieved, such as regarding conservation laws. Regarding the conventionality of simultaneity, an unusually wide range of $\epsilon$ values is admissible, but some extreme values are inadmissible. Standard simultaneity uniquely makes the spinor transformation law linear and independent of the metric, because transformations among the standard Cartesian coordinate systems fall within the conformal group, for which the spinor transformation law is linear. The surprising mildness of the restrictions on coordinate order as applied to the Schwarzschild solution is exhibited. (shrink)

Barut's classical model of the spinning particle having external dynamical variables x and p and internal dynamical variables $\bar z$ and z is taken into account. The path integrations over holomorphic spinors $\bar z$ and z are discussed. This quantization gives the kernel of the relativistic particles with higher spin as well as the Dirac electron. The Green's function of the spin-n/2 particle is obtained.

The 24 components of the relativistic spin tensor consist of 3 + 3 basic spin fields and 9 + 9 constitutive fields. Empirically only three basic spin fields and nine constitutive fields are known. This empirem can be expressed by two spin axioms, one of them denying purely relativistic spin fields, and the other one relating the three additional basic fields and the nine additional constitutive fields to the known (and measurable) ones. This identification by the spin axioms is material-independent (...) and does not mix basic spin fields with constitutive properties. The approaches to the Weyssenhoff fluid and the Dirac-electron fluid found in literature are discussed with regard to these spin axioms. The conjecture is formulated, that another reduction from six to three basic spin fields which does not obey the spin axioms introduces special material properties by not allowed mixing of constitutive and basic fields. (shrink)

The symplectic structures (brackets, Hamilton's equations, and Lagrange's equations) for the Dirac electron and its classical model have exactly the same form. We give explicitly the Poisson brackets in the dynamical variables (x μ,p μ,v μ,S μv). The only difference is in the normalization of the Dirac velocities γμγμ=4 which has significant consequences.

A Clifford-algebraic interpretation is proposed of the charge, mass, spin relationship found recently by Cooperstock and Faraoini, which was based on the Kerr–Newman metric solutions of the Einstein–Maxwell equations. The components of the polymomentum associated with a Clifford polyparticle in four dimensions provide for such a charge, mass, spin relationship without the problems encountered in Kaluza–Klein compactifications which furnish an unphysically large value for the electron charge. A physical reasoning behind such charge, mass, spin relationship is provided, followed by (...) a discussion on the geometrical derivation of the fine structure constant by Wyler, Smith, Gonzalez-Martin and Smilga. To finalize, the renormalization of electric charge is discussed and some remarks are made pertaining the modifications of the charge–scale relationship, when the spin of the polyparticle changes with scale, that may cast some light into the alleged Astrophysical variations of the fine structure constant. (shrink)

in Dirac's classical theory of the electron—is causally non-local. I distinguish two distinct causal locality principles and argue, using Dirac's theory as my main case study, that neither can be reduced to a non-causal principle of local determinism.

The Pauli exclusion principle is interpreted using a geometrical theory of electrons. Spin and spatial motion are described together in an eight dimensional spinor coordinate space. The field equation derives from the assumption of conformal waves. The Dirac wave function is a gradient of the scalar wave in spinor space. Electromagnetic and gravitational interactions are mediated by conformal transformations. An electron may be followed through a sequence of creation and annihilation processes. Two electrons are branches of a single (...) particle. Each satisfies a Dirac equation, but together they are a solution of the curvature condition. As two so identified electrons approach each other, the exclusion principle develops from the boundary conditions in spinor space. The gradient motion does not allow the particles to overlap. Since the spinor-gradient of the scalar wave function is odd in the coordinates, the sign of the wave function must change at the electron-electron boundary. The exclusion principle becomes geometry intrinsic and all electrons are combined into one field. Further applications are proposed including the possibility of improved numerical calculations in atomic and molecular systems. There also may be extensions to nuclear or particle physics. Implications are expected for the properties of rotating objects in a gravitational field. (shrink)

Several theoretical publications on the Dirac equation published during the last decades have shown that, an interpretation is possible, which ascribes the origin of electron spin and magnetic moment to an autonomous circular motion of the point-like charged particle around a fixed centre. In more recent publications an extension of the original so called “Zitterbewegung Interpretation” of quantum mechanics was suggested, in which the spin results from an average of instantaneous spin vectors over a Zitterbewegung period. We argue (...) that, the corresponding autonomous motion of the electron should, if it is real, determine non-relativistic spin measurements. Such a direct connection with the established formal quantum mechanical description of spin measurements, into which spin is introduced as a “non-classical” quantity has, to our knowledge, not been reported. In the present work we show that, under certain “model assumptions” concerning the proposed autonomous motion, results of spin measurements, including measurements of angular correlations in singlet systems, can indeed be correctly described using classical probabilities. The success of the model is evidence for the “reality” of the assumed autonomous motion. The resulting model violates the Bell—inequalities to the same extent as quantum mechanics. (shrink)

The possibility that zitterbewegung opens a window to particle substructure in quantum mechanics is explored by constructing a particle model with structural features inherent in the Dirac equation. This paper develops a self-contained dynamical model of the electron as a lightlike particle with helical zitterbewegung and electromagnetic interactions. The model admits periodic solutions with quantized energy, and the correct magnetic moment is generated by charge circulation. It attributes to the electron an electric dipole moment rotating with ultrahigh (...) frequency, and the possibility of observing this directly as a resonance in electron channeling is analyzed in detail. Correspondence with the Dirac equation is discussed. A modification of the Dirac equation is suggested to incorporate the rotating dipole moment. (shrink)

So-called antimatter in the form of elementary particles such as positive electrons (antielectrons alias positrons) and negative protons (antiprotons) has for long been investigated by physicists. However, atoms or molecules of this exotic kind are conspicuously absent from nature. Since antimatter is believed to be symmetric with ordinary matter, the flagrant asymmetry constitutes a problem that still worries physicists and cosmologists. As first suggested by Paul Dirac in 1933, in distant parts of the universe there might be entire stars (...) and galaxies made of antiparticles alone. Why not? This paper examines how the concepts of antiparticles and antimatter slowly migrated from particle physics to astronomy and cosmology. At around 1970 a few physicists speculated about an anti-universe separate from ours while others looked for the charge asymmetry in quantum processes in the early big-bang explosion of the universe. Others again proposed a ‘plasma cosmology’ that kept our world and the hypothetical world of antimatter apart. Soviet physicists and astronomers were no less interested in the problem than their colleagues in the West. The paper details the development up to the late 1970s, paying attention not only to mainstream scientific works but also to more speculative ideas, some of them very speculative. By that time the antimatter mystery remained mysterious – which is still the situation. (shrink)

1. Introduction Dirac's theory of the electron was the first widely accepted relativistic quantum theory, and it later provided the basis for constructing the modern electromagnetic theory of quantum electrodynamics. Whereas Dirac's theory in its simplest form describes relativistic freely-propagating massive non-chiral particles of spin-½, QED describes how such particles interact with one another electromagnetically, via a dynamical quantum field.

It was shown in Dirac A117, 610; A118, 351, 1928) that the ultra-violet divergence in quantum electrodynamics is caused by a violation of the time-energy uncertainly relationship, due to the implicit assumption of infinitesimal time information. In Wheeler et al. it was shown that Einstein’s special theory of relativity and Maxwell’s field theory have mathematically equivalent dual versions. The dual versions arise from an identity relating observer time to proper time as a contact transformation on configuration space, which leaves (...) phase space invariant. The special theory has a dual version in the sense that, for any set of n particles, every observer has two unique sets of global variables \\) and \\) to describe the dynamics, where \ is the canonical center of mass. In the \\) variables, time is relative and the speed of light is unique, while in the \\) variables, time is unique and the speed of light is relative with no upper bound. In the Maxwell case, the two sets of particle wave equations are not equivalent. The dual version contains an additional longitudinal radiation term that appears instantaneously with acceleration, leading to the prediction that radiation from a cyclotron will not produce photoelectrons. A major outcome is the dual unification of Newtonian mechanics and classical electrodynamics with Einstein’s special theory of relativity, without the need for point particles, without a self-energy divergency and, without need for the problematic Lorentz–Dirac equation. The purpose of this paper is to introduce the dual theory of relativistic quantum mechanics. In our approach, we obtain three distinct dual relativistic wave equations that reduce to the Schrödinger equation when minimal coupling is turned off. We show that the dual Dirac equation provides a new formula for the anomalous magnetic moment of a charged particle. We can obtain the exact value for the electron g-factor and phenomenological values for the muon and proton g-factors. (shrink)

This article reviews Hestenes' work on the Dirac theory, where his main achievement is a real formulation of the theory within thereal Clifford algebra Cl 1,3 ≃ M2 (H). Hestenes invented first in 1966 hisideal spinors $\phi \in Cl_{1,3 _2}^1 (1 - \gamma _{03} )$ and later 1967/75 he recognized the importance of hisoperator spinors ψ ∈ Cl 1,3 + ≃ M2 (C).This article starts from the conventional Dirac equation as presented with matrices by Bjorken-Drell. Explicit mappings are (...) given for a passage between Hestenes' operator spinors and Dirac's column spinors. Hestenes' operator spinors are seen to be multiples of even parts of real parts of Dirac spinors (real part in the decompositionC ⊗ Cl 1,3 andnot inC ⊗ M4 (R)=M4 (C)). It will become apparent that the standard matrix formulation contains superfluous parts, which ought to be cut out by Occam's razor.Fierz identities of bilinear covariants are known to be sufficient to study the non-null case but are seen to be insufficient for the null case ψ†γ0ψ=0, ψ†γ0γ0123ψ=0. The null case is thoroughly scrutinized for the first time with a new concept calledboomerang. This permits a new intrinsically geometric classification of spinors. This in turn reveals a new class of spinors which has not been discussed before. This class supplements the spinors of Dirac, Weyl, and Majorana; it describes neither the electron nor the neutron; it is awaiting a physical interpretation and a possible observation.Projection operators P±, Σ± are resettled among their new relatives in End(Cl 1,3 ). Finally, a new mapping, calledtilt, is introduced to enable a transition from Cl 1,3 to the (graded) opposite algebra Cl 3,1 without resorting to complex numbers, that is, not using a replacement γμ →iγμ. (shrink)

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. (shrink)

The spacetime algebra (STA) is the natural, representation-free language for Dirac's theory of the electron. Conventional Pauli, Dirac, Weyl, and Majorana spinors are replaced by spacetime multivectors, and the quantum σ- and γ-matrices are replaced by two-sided multivector operations. The STA is defined over the reals, and the role of the scalar unit imaginary of quantum mechanics is played by a fixed spacetime bivector. The extension to multiparticle systems involves a separate copy of the STA for each (...) particle, and it is shown that the standard unit imaginary induces correlations between these particle spaces. In the STA, spinors and operators can be manipulated without introducing any matrix representation or coordinate system. Furthermore, the formalism provides simple expressions for the spinor bilinear covariants which dispense with the need for the Fierz identities. A reduction to2+1 dimensions is given, and applications beyond the Dirac theory are discussed. (shrink)

Guidelines for constructing point particle models of the electron withzitterbewegung and other features of the Dirac theory are discussed. Such models may at least be useful approximations to the Dirac theory, but the more exciting possibility is that this approach may lead to a more fundamental reality.

Recently a new attempt to go beyond QM was performed in the form of so-called prequantum classical statistical field theory (PCSFT). In this approach quantum systems are described by classical random fields, e.g., the electron field or the neutron field. Averages of quantum observables arise as approximations of averages of classical variables (functionals of “prequantum fields”) with respect to fluctuations of fields. For classical variables given by quadratic functionals of fields, quantum and prequantum averages simply coincide. In this paper (...) we generalize PCSFT to the composite quantum system. The main discovery is that, opposite to a rather common opinion, a composite system can be described by the Cartesian product of state spaces (like in classical physics) and not by the tensor product of them (like in conventional QM). A natural interpretation of the quantum pure state for a composite system is proposed: it is the nondiagonal block in the covariance matrix of the random field describing a composite system. The interpretation of a pure state due to Dirac and von Neumann seems to be an artifact of the conventional mathematical description of micro-systems. PCSFT provides a new possibility for interpretation of entanglement. (shrink)

Thezitterbewegung is a local circulatory motion of the electron presumed to be the basis of the electron spin and magnetic moment. A reformulation of the Dirac theory shows that thezitterbewegung need not be attributed to interference between positive and negative energy states as originally proposed by Schroedinger. Rather, it provides a physical interpretation for the complex phase factor in the Dirac wave function generally. Moreover, it extends to a coherent physical interpretation of the entire Dirac (...) theory, and it implies azitterbewegung interpretation for the Schroedinger theory as well. (shrink)

In this paper we examine the theoretical foundations underlying the testing of quantum electrodynamics. We show that for the photon propagator (together with the contiguous vertices) it is not necessary to introduce ad hoc modifications in sufficiently accurate scattering experiments. Energy, momentum transfer, and accuracy determine the tested length in a model-independent way. The situation is quite different with the electron propagator. If gauge invariance is taken for granted, the electron propagator cannot be tested with processes where diagrams (...) with open electron lines are important in the lowest order of perturbation theory. These processes can only give limits for anomalous moment and multiphoton parts of the vertices. On the other hand, processes with closed electron loops (vacuum polarization), such as photon-photon and Delbrück scattering, as well as photon splitting or corresponding low-energy, high-precision experiments can give limits also for the electron propagator. But in these cases only less accurate limits can be obtained, which depend on the modification model. Hence testing of the electron propagator, i.e., roughly speaking, the Dirac equation, is much more difficult than testing of the photon propagator, i.e, Maxwell's equations. (shrink)

Quantum electrodynamics is the theory of electrons and other elementary charged particles, interacting through the exchange of light quanta. Albert Einstein introduced the light quantum in 1905, but for about three decades physicists applied quantum ideas mainly in theories of the structure and behavior of matter, not to electromagnetic radiation itself, which was always treated semi-classically. This began to change after 1923 with the discovery of the Compton effect and its kinematic description by Arthur Compton and Peter Debye, based on (...) the light quantum. In this paper we review the study of high-energy radiation that led to Compton's discovery. We discuss the analysis of the intensity distribution of Compton-scattered radiation that together with the ''new'' quantum theory beginning in 1925, resulted in the development, especially by Pascual Jordan and Paul Dirac, of a quantum theory of electromagnetic radiation in interaction with matter. (shrink)