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International Journal of Physics

A New Interpretation of the Negative Energy Solutions to Dirac Equation and a Configuration of the Electron

Zhonglin BO

Published Dec 1, 2023• 1 citation
Publisher

Abstract

A thorough mathematic analysis was conducted to the negative energy solutions to the Dirac equation in a complex plane due to the fact that the solutions are vectors of complex number. An opposite vector with negative energy state was identified when a vector with positive energy state was defined in a complex plane. We concluded that any opposite vectors in a complex plane must have opposite energy state because their vector heads move always in opposite direction. We configured an electron as a small ball with dimetric vectors instead of a structureless mass point. Based on the new electron configuration, we explained the particle wave duality as the feature of electron’s spinvector in motion, explained the reason for the electrons with half spin quantum number. And we concluded that de Broglie’s standing waves theory would be more rational than the interpretation by current quantum theory.

DOI
10.12691/ijp-11-6-2

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A thorough mathematic analysis was conducted to the negative energy solutions to the Dirac equation in a complex plane due to the fact that the solutions are vectors of complex number. An opposite vector with negative energy state was identified when a vector with positive energy state was defined in a complex plane. We concluded that any opposite vectors in a complex plane must have opposite energy state because their vector heads move always in opposite direction. We configured an electron as a small ball with dimetric vectors instead of a structureless mass point. Based on the new electron configuration, we explained the particle wave duality as the feature of electron’s spinvector in motion, explained the reason for the electrons with half spin quantum number. And we concluded that de Broglie’s standing waves theory would be more rational than the interpretation by current quantum theory.
The usual choice of an orthogonal set of four plane-wave solutions of the free-particle Dirac equation does not lend itself readily to direct and complete physical interpretation except in low-energy approximation. A different choice of solutions can be made which yields a direct physical interpretation at all energies. Besides the separation of positive and negative energy states there is a further separation of states for which the spin is respectively parallel or antiparallel to the direction of the momentum vector.
A detailed kinematic analysis was conducted based on the hypothesis of the photon as an electric dipole in combined motions, a rotation and a linear uniform motion perpendicular to its rotational axis. A particle wave coefficient(j) of a rotational object in combined motions is defined based on the analysis of the motion quantities, translational momentum and rotational angular momentum. The motion analysis explains the intrinsic nature of particle-wave duality, the relation among momentum, angular momentum and its wavelength when the rotational particle in combined motions. For the photon, j equals Planck constant (h), is the sum of its translational kinetic energy and rotational energy. The derivation processes of the wave equation of the photon, both the first order derivative method and the second order derivative method, demonstrate that the Dirac Equation is applicable to describe the motions of the photon as the dipole. The four wave components of the photon interpret why the negative energy solutions exist to the Dirac equation, which reveals the disk structure of dimetric spinvectors of the electron.
We set up the Maxwell's equations and subsequently the classical wave equations for the electromagnetic waves which together with their generating source, an oscillatory charge of zero rest mass in general travelling, make up a particle travelling similarly as the source at velocity ν in the field of an external scalar and vector potentials. The direct solutions in constant external field are Doppler-displaced plane waves propagating at the velocity of light c ; at the de Broglie wavelength scale and expressed in terms of the dynamically equivalent and appropriate geometric mean wave variables, these render as functions identical to the space-time functions of a corresponding Dirac spinor, and in turn identical to de Broglie phase waves previously obtained from explicit superposition. For two spin-half particles of a common set of space-time functions constrained with antisymmetric spin functions as follows the Pauli principle for same charges and as separately indirectly induced based on experiment for opposite charges, the complete wave functions are identical to the Dirac spinor. The back-substitution of the so explicitly determined complete wave functions in the corresponding classical wave equations of the two particles, subjected further to reductions appropriate for the stationary-state particle motion and to rotation invariance when in three dimensions, give a Dirac equation set; the procedure and conclusion are directly extendible to arbitrarily varying potentials by use of the Furious theorem and to particle motions in three dimensions by virtue of the characteristics of de Broglie particle motion. Through the derivation of the Dirac equation, the study hopes to lend insight into the connections between the Dirac wave functions and the electrodynamic components of simple particles under the government by the well established basic laws of electrodynamics.
Dirac's equation with static external fields is considered as an equation of relativistic quantum theory, which describes the quantum dynamics of an electron in four-dimensional space-time. It is shown that the Dirac electron has only one intrinsic degree of freedom -- spin. The quantum ensemble of an electron with a given energy consists of two inseparable from each other subensembles of heavy and light spin-1/2 quasiparticles to have opposite (relative) intrinsic parities. In the case of electrical scalar potential nonuniform in the direction of a particle motion, these subensembles behave as analogs of the electrical and magnetic components of the plane monochromatic electromagnetic wave propagating in a dispersive medium, nonuniform in the direction of a wave motion. The role of states with negative energies and the properties of the velocity operator are explained.
Solutions of Dirac equation are well known in terms of bi-spinors, but the complete determination of these spinors is always derived using the assumption that the wave function is a plane wave. We investigate in this paper the search of general solutions under the form of standing wave modes, using methods issued from electromagnetism. It is shown that such solutions exist as solutions of a system of 64 equations and 64 unknowns, and that they must be associated with a relativistic relation of energy conservation. Complete exact solutions are presented. It is shown that these solutions highlight in a new manner the duality wave-particle, and the uncertainty principle.
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→0G\to 0G→0, where GGG 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-GGG 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.