Removing redundancy in relativistic quantum mechanics

The Dirac equation has a hidden geometric structure that is made manifest by reformulating it in terms of a real spacetime algebra. This reveals an essential connection between spin and complex numbers with profound implications for the interpretation of quantum mechanics. Among other things, it suggests that to achieve a complete interpretation of quantum mechanics, spin should be identifled with an intrinsic zitterbewegung.

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2003

The nilpotent Dirac formalism has been shown, in previous publications, to generate new physical explanations for aspects of particle physics, with the additional possibility of calculating some of the parameters involved in the Standard Model. The applications so far obtained are summarised, with an outline of some more recent developments.

This paper presents a new Relativistic Symmetrical Interpretation (RSI) of the Dirac equation in (1+1)D which postulates: quantum mechanics is intrinsically time-symmetric, with no arrow of time; the fundamental objects of quantum mechanics are transitions; a transition is fully described by a complex transition amplitude density with specified initial and final boundary conditions ; and transition amplitude densities never collapse. This RSI is compared to the Copenhagen Interpretation (CI) for the analysis of Einstein's bubble experiment with a spin-1 2 particle. The RSI can both retrodict the past and predict the future, has no zitterbewegung in the particle's rest frame, resolves a few inconsistencies of the CI, and eliminates some of the conceptual problems of the CI.

Physica Scripta, 2012

This paper reexamines the key aspects of Dirac's derivation of his relativistic equation for the electron in order advance our understanding of the nature of quantum field theory. Dirac's derivation, the paper argues, follows the key principles behind Heisenberg's discovery of quantum mechanics, which, the paper also argues, transformed the nature of both theoretical and experimental physics vis-à-vis classical physics and relativity. However, the limit theory (a crucial consideration for both Dirac and Heisenberg) in the case of Dirac's theory was quantum mechanics, specifically, Schrödinger's equation, while in the case of quantum mechanics, in Heisenberg's version, the limit theory was classical mechanics. Dirac had to find a new equation, Dirac's equation, along with a new type of quantum variables, while Heisenberg, to find new theory, was able to use the equations of classical physics, applied to different, quantum-mechanical variables. In this respect, Dirac's task was more similar to that of Schrödinger in his work on his version of quantum mechanics. Dirac's equation reflects a more complex character of quantum electrodynamics or quantum field theory in general and of the corresponding (high-energy) experimental quantum physics vis-à-vis that of quantum mechanics and the (low-energy) experimental quantum physics. The final section examines this greater complexity and its implications for fundamental physics.

International Letters of Chemistry, Physics and Astronomy, 2013

The Dirac equation consistent with the principles of quantum mechanics and the special theory of relativity, introduces a set of matrices combined with the wave function of a particle in motion to give rise to the relativistic energy-momentum relation. In this paper a new hypothesis, the wave function of a particle in motion is associated with a pair of complementary waves is proposed. This hypothesis gives rise to the same relativistic energy-momentum relation and achieves results identical to those of Dirac. Additionally, both the energy-time and momentum-position uncertainty relations are derived from the complementary wave interpretation. How the complementary wave interpretation of the Dirac equation is related to the time-arrow and the four-vectors are also presented.

Advanced Studies in Theoretical Physics

The two-component form of the new Dirac equation is obtained for a zero mass particle using a unitary transformation.

European Journal of Physics, 42, 055404 , 2021

The fundamentals of a quasi-relativistic wave equation, whose solutions match the Schrödinger results for slow-moving particles but are also valid when the particle moves at relativistic speeds, are discussed. This quasi-relativistic wave equation is then used for examining some interesting quantum problems where the introduction of relativistic considerations may produce remarkable consequences. We argue in favor of the academic use of this equation, for introducing students to the implications of the special theory of relativity in introductory quantum mechanics courses.

arXiv: Mathematical Physics, 2014

AbstractIn thispaper the inverse problem of the correspondence between the so-lutions of the Dirac equation and the electromagnetic 4-potentials, is fullysolved. The Dirac solutions are classified into two classes. The first oneconsists of degenerate Dirac solutions corresponding to an infinite num-ber of 4-potentials while the second one consists of non-degenerate Diracsolutions corresponding to one and only one electromagnetic 4-potential.Explicit expressions for the electromagnetic 4-potentials are provided inboth cases. Further, in the case of the degenerate Dirac solutions, it isproven that at least two 4-potentials are gauge inequivalent, and con-sequently correspond to different electromagnetic fields. An example isprovided to illustrate this case.PACS: 03.65.Pm, 03.50.De, 41.20.-q 1 Introduction The Dirac equation has been the first electron equation in quantum mechanicsto satisfy the Lorentz covariance [2], initiating the beginning of one of the mostpowerful theories ever formula...

2017

Dirac’s seminal 1928 paper “The Quantum Theory of the Electron” is the foundation of how we presently understand the behavior of fermions in electromagnetic fields, including their magnetic moments. In sum, it is, as titled, a quantum theory of individual electrons, but in classical electromagnetic fields comprising innumerable photons. Based on the electrodynamic time dilations which the author has previously presented and which arise by geometrizing the Lorentz Force motion, there arises an even-richer “hyper-canonical” variant of the Dirac equation which reduces to the ordinary Dirac equation in the linear limits. This advanced Dirac theory naturally enables the magnetic moment anomaly to be entirely explained without resort to renormalization and other ad hoc add-ons, and it also permits a detailed, granular understanding of how individual fermions interact with individual photons strictly on the quantum level. In sum, it advances Dirac theory to a quantum theory of the electron...

Journal of Physics G: Nuclear Physics, 1983

ABSTRACT The authors discuss two-component reductions of Dirac phenomenology. The presence of strong single-particle potentials results in effective transition operators in the two-component space that are qualitatively different from those that occur in conventional Schrodinger phenomenology.

New Dirac Equation from the View Point of Particle, 2012

Solution of the Dirac equation with pseudospin symmetry for a new harmonic oscillatory ring-shaped noncentral potential J. Math. Phys. 53, 082104 (2012) Effect of tensor interaction in the Dirac-attractive radial problem under pseudospin symmetry limit J. Math. Phys. 53, 082101 (2012) Asymptotic stability of small gap solitons in nonlinear Dirac equations J. Math. Phys. 53, 073705 (2012) On Dirac-Coulomb problem in (2+1) dimensional space-time and path integral quantization J. Math. Phys. 53, 063503 (2012) Quasi-exact treatment of the relativistic generalized isotonic oscillator Abstract. According to the classical approach, especially the Lorentz Invariant Dirac Equation, when particles are bound to each other, the interaction term appears as a quantity belonging to the "field". In this work, as a totally new approach, we propose to alter the rest masses of the particles due to their interaction, as much as their respective contributions to the static binding energy. Thus we rewrite and solve the Dirac Equation for the hydrogen atom, and amazingly, obtain practically the same numerical results for the ground states, as those obtained from the Dirac Equation.

2018

There are several 3 + 1 parameter quantities in physics (like vector + scalar potentials, four-currents, space-time, four-momentum, …). In most cases (but space-time), the three-and the one-parameter characterised elements of these quantities differ in the field-sources (e.g., inertial and gravitational masses, Lorentz-and Coulomb-type electric charges, …) associated with them. The members of the field-source pairs appear in the vector-and the scalar potentials, respectively. Sections 1 and 2 of this paper present an algebra what demonstrates that the members of the fieldsource siblings are subjects of an invariance group that can transform them into each other. (This includes, e.g., the conservation of the isotopic field-charge spin (IFCS), proven in previous publications by the author.) The paper identifies the algebra of that transformation and characterises the group of the invariance; it discusses the properties of this group, shows how they can be classified in the known nomenclature, and why is this pseudo-unitary group isomorphic with the SU(2) group. This algebra is denoted by tau (). The invariance group generated by the tau algebra is called hypersymmetry (HySy). The group of hypersymmetry had not been described. The defined symmetry group is able to make correspondence between scalars and vector components that appear often coupled in the characterisation of physical states. In accordance with conclusions in previous papers, the second part (Sections 3 and 4) shows that the equations describing the individual fundamental physical interacions are invariant under the combined application of the Lorentztransformation and the here explored invariance group at high energy approximation (while they are left intact at lower energies). As illustration, the paper presents a simple form for an extended Dirac equation and a set of matrices to describe the combined transformation in QED. The paper includes a short reference illustration (in Section 2.2) to another applicability of this algebra in the mathematical description of regularities for genetic matrices.

2010

In this paper we present for the first time a complete description of the Bohm model of the Dirac particle. This result demonstrates again that the common perception that it is not possible to construct a fully relativistic version of the Bohm approach is incorrect. We obtain the fully relativistic version by using an approach based on Clifford algebras outlined in two earlier papers by Hiley and by Hiley and Callaghan. The relativistic model is different from the one originally proposed by Bohm and Hiley and by Doran and Lasenby. We obtain exact expressions for the Bohm energy-momentum density, a relativistic quantum Hamilton-Jacobi for the conservation of energy which includes an expression for the quantum potential and a relativistic time development equation for the spin vectors of the particle. We then show that these reduce to the corresponding non-relativistic expressions for the Pauli particle which have already been derived by Bohm, Schiller and Tiomno and in more general form by Hiley and Callaghan. In contrast to the original presentations, there is no need to appeal to classical mechanics at any stage of the development of the formalism. All the results for the Dirac, Pauli and Schroedinger cases are shown to emerge respectively from the hierarchy of Clifford algebras C(13),C(30), C(01) taken over the reals as Hestenes has already argued. Thus quantum mechanics is emerging from one mathematical structure with no need to appeal to an external Hilbert space with wave functions.