The parametrized relativistic quantum mechanics of Stueckelberg (1942 Helv. Phys. Acta 15 23) represents time as an operator, and has been shown elsewhere to yield the recently observed phenomena of quantum interference in time, quantum diffraction in time and quantum entanglement in time. The Stueckelberg wave equation as extended to a spin-1/2 particle by Horwitz and Arshansky (1982 J. Phys. A: Math. Gen. 15 L659) is shown here to yield the electron g -factor g = 2 (1 + α/2π), to leading order in the renormalized fine structure constant α, in agreement with the quantum electrodynamics of Schwinger (1948 Phys. Rev. 73 416).
The parametrized relativistic quantum mechanics of Stueckelberg (1942 Helv. Phys. Acta 15 23) represents time as an operator, and has been shown elsewhere to yield the recently observed phenomena of quantum interference in time, quantum diffraction in time and quantum entanglement in time. The Stueckelberg wave equation as extended to a spin-1/2 particle by Horwitz and Arshansky (1982 J. Phys. A: Math. Gen. 15 L659) is shown here to yield the electron g -factor g = 2 (1 + α/2π), to leading order in the renormalized fine structure constant α, in agreement with the quantum electrodynamics of Schwinger (1948 Phys. Rev. 73 416).
In the search for a relativistic wave equation for a free particle of rest mass m one must demand that the energy(E)-momentum (p) [dispersion] relation E2 = m2c4 + p2c2 (25.1) is satisfied. By the substitutions E → i~ ∂ ∂t , p→ −i~∇ (25.2) in Eq. (25.1) one obtains an operator, which when acting on a scalar wave function, gives us the Klein-Gordon wave equation for a spinless particle. The Klein-Gordon equation is of second order in the space (r) and time (t) coordinates. If one seeks a covariant first-order differential equation, r and t must play symmetric roles. A natural starting point therefore would be a linear dispersion relation of the form E = βmc2 + cα · p, (25.3) where β and α = (α1, α2, α3) are real and dimensionless. Since Eq. (25.3) must be compatible with Eq. (25.1), squaring of Eq. (25.3) leads to the following relations [already given in Eqs. (10.77)-(10.79)]: β2 = 1, (25.4) αiβ + βαi = 0, (25.5) αiαj + αjαi = 2δij , (25.6) where i, j = 1, 2, 3 (x, y, z). It is clear from Eqs. (25.4)-(25.6) that β and α cannot be numbers, but one can find matrices which satisfy these equations. Since the Hamiltonian appearing upon a substitution of the connections in (25.2) into Eq. (25.3) must be Hermitian, the αi and β matrices also have to be Hermitian, i.e., α†i = αi, i = 1− 3, (25.7) β† = β. (25.8) One can show that the matrices must have even rank and that the rank must be at least 4, see, e.g., [88]. The Dirac equation, which describes the quantum mechanics of a spin1/2 elementary particle, e.g., an electron, relates to a four-dimensional realization of Eqs. (25.4)-(25.8). Representations of a given rank are not unique.
A consistent theory of quantum entanglement requires that constituent single-particle states belong to the same Hilbert space, the coherent eigenstates of a complete set of operators in a given representation, defined with respect to a shared continuous parameterization. Formulating such eigenstates for a single relativistic particle with spin, and applying them to the description of many-body states, presents well-known challenges. In this paper, we review the covariant theory of relativistic spin and entanglement in a framework first proposed by Stueckelberg and developed by Horwitz, Piron, et al. This approach modifies Wigner’s method by introducing an arbitrary timelike unit vector nμ and then inducing a representation of SL(2,C), based on pμ rather than on the spacetime momentum. Generalizing this approach, we construct relativistic spin states on an extended phase space {(xμ,pμ),(ζμ,πμ)}, inducing a representation on the momentum πμ, thus providing a novel dynamical interpretation of the timelike unit vector nμ=πμ/M. Studying the unitary representations of the Poincaré group on the extended phase space allows us to define basis quantities for quantum states and develop the gauge invariant electromagnetic Hamiltonian in classical and quantum mechanics. We write plane wave solutions for free particles and construct stable singlet states, and relate these to experiments involving temporal interference, analogous to the spatial interference known from double slit experiments.
In 1965, the Nobel Prize was awarded for the modern formulation of Quantum Electrodynamics (QED) to R.P. Feynman for his diagrammatic method, J. Schwinger for the operator method, and S.-I. Tomonaga for his relativistic derivation. Subsequently, P. Kusch performed a precise measurement of the electron's anomalous magnetic moment, providing a critical test validating QED's computational methods, which proved to be extremely accurate. Nevertheless, QED is based on a physically unrealistic assumption—mathematically circumvented—that the electron is a dimensionless point particle endowed with physical properties "ex abrupto." This work, developed within a deterministic and non-local dBBZ (modified de Broglie–Bohm) theory, models the electron as an entangled and distributed structure, as proposed in previous studies. It enables the calculation of the electron's anomalous magnetic moment as a consequence of its intrinsic structure, without employing QED techniques. The method involves calculating, for each orbital, the anomalous moment modified to account for the influence of existing magnetic fields. Subsequently, entanglement is imposed on the sum of these moments, and from the total orbital anomalous moment thus obtained, the theoretical magnetic anomaly is derived and compared with the corresponding experimental value, yielding relative errors on the order of 10^-10. This procedure, which allows for greater theoretical precision , need a similar , albeit more complex, calculation of the muon's and tau anomalous magnetic moment. This is because a specific parameter, termed the "source parameter," selected within an allowable range requires , at least , dual verification to be adopted with high precision. The theoretical determination of the muon's anomalous magnetic moment is also presented in a subsequent document, employing analogous computational procedures.
Quantum Electrodynamics may be formulated as a Quantum Field Theory , and also as relativistic quantum mechanics by introduction of the Feynman-Stueckelberg parameter. As stated by M. Srednicki ({\it Quantum Field Theory}, Cambridge University Press, Cambridge and New York, 2007), "\dots any relativistic quantum physics that can be treated in one formalism can also be treated in the other". Entanglement of electrons at macroscopic spacelike separation has been observed (B. Hensen {\it et al.}, Nature, 526, 2015, doi:10.1038.nature/15759), yet according to Quantum Field Theory is restricted to separations of the order of the Compton wavelength ℏ/mec (see e.g., S. J. Summers and R. Werner, Commun. Math. Phys. 110,247,1987). There is no restriction on entanglement in parametrized relativistic quantum mechanics (A. F. Bennett, Ann. Phys. 345, 1, 2014). The parametrized formalism is extended here to pair annihilation and pair creation. The {\it ansatz} used in the formalism to develop the correct higher order corrections is justified here with plane wave summations. The conflict over entanglement is further discussed in terms of the Klein-Gordon propagator, and an invariant Bell's inequality is developed for spin -1/2.
A novel interpretation is given of Dirac's "wave equation for the relativistic electron" as a quantum-mechanical one-particle equation. In this interpretation the electron and the positron are merely the two different "topological spin" states of a single more fundamental particle, not distinct particles in their own right. The new interpretation is backed up by the existence of such bi-particle structures in general relativity, in particular the ring singularity present in any spacelike section of the maximal analytically extended, topologically non-trivial, electromagnetic Kerr-Newman spacetime in the zero-gravity limit (here, "zero-gravity" means the limit G→0, where G is Newton's constant of universal gravitation). This novel interpretation resolves the dilemma that Dirac's wave equation seems to be capable of describing both the electron and the positron in "external" fields in many relevant situations, while the bi-spinorial wave function has only a single position variable in its argument, not two - as it should if it were a two-particle equation! A Dirac equation is formulated for a ring-like bi-particle which interacts with a static point charges located elsewhere in the topologically non-trivial physical space associated with the moving ring bi-particle, the motion being governed by a de-Broglie-Bohm type law extracted from the Dirac equation. As an application, the pertinent general-relativistic zero-gravity Hydrogen problem is studied in the usual Born-Oppenheimer approximation. Its spectral results suggest that the zero-G Kerr-Newman magnetic moment be identified with the so-called "anomalous magnetic moment of the physical electron," not with the Bohr magneton, so that the ring radius is only a tiny fraction of the electron's Compton wave length.
This paper is a progress report on the foundations for the canonical proper-time approach to relativistic quantum theory. We first review the the standard square-root equation of relativistic quantum theory, followed by a review of the Dirac equation, providing new insights into the physical properties of both. We then introduce the canonical proper-time theory. For completeness, we give a brief outline of the canonical proper-time approach to electrodynamics and mechanics, and then introduce the canonical proper-time approach to relativistic quantum theory. This theory leads to three new relativistic wave equations. In each case, the canonical generator of proper-time translations is strictly positive definite, so that it represents a particle. We show that the canonical proper-time extension of the Dirac equation for Hydrogen gives results that are consistently closer to the experimental data, when compared to the Dirac equation. However, these results are not sufficient to account for either the Lamb shift or the anomalous magnetic moment.