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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Causal action at a distance in a relativistic system of two bound charged spinless particles: Hydrogenlike models

Christopher Dewdney, Peter Holland, Anastasios Kyprianidis, Jean Pierre Vigier

Published May 15, 1985• 10 citations
Publisher

Abstract

We analyze the causal quantum behavior of two charged spinless particles having different masses interacting via an action-at-a-distance relativistic scalar Coulomb potential. Developing de Broglie's idea of splitting the potential energy between the particles, we show that the predictivity constraints of relativistic predictive mechanics established by Droz-Vincent and Sudarshan are satisfied. In this way we develop a relativistic theory of hydrogenlike atoms in the framework of the causal interpretation of quantum mechanics.

DOI
10.1103/physrevd.31.2533

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We analyze the causal quantum behavior of two charged spinless particles having different masses interacting via an action-at-a-distance relativistic scalar Coulomb potential. Developing de Broglie's idea of splitting the potential energy between the particles, we show that the predictivity constraints of relativistic predictive mechanics established by Droz-Vincent and Sudarshan are satisfied. In this way we develop a relativistic theory of hydrogenlike atoms in the framework of the causal interpretation of quantum mechanics.
New relativistic mechanics of two-particle system is set forth, where the two constituent particles are interacting by an arbitrary (central) action-at-a-distance. The fundamental equations are presented in a form covariant under general transformation of parameters parametrizing the world lines of constituent particles. The theory represents the proper relativistic generalization of the usual Newtonian mechanics in the sense that it tends in the non-relativistic (and weak interaction) limit to the usual mechanics of two particles moving under a corresponding non-relativistic potential. For the analysis of theory it is convenient to choose a certain particular gauge (i.e., parametrization) fixed by two gauge relations. This brings the theory to a canonical formalism accompanied by two weak equations, and in this gauge quantization can be performed. The result verifies that the relativistic quantum mechanics for two particles interacting by an action-at-a-distance is just represented by a bilocal wave equation and a subsidiary condition, with the clarification of its correspondence-theoretical fundation and internal dynamics. As an example the case of Hooke-type force is illustrated, where the internal motions are elliptic oscillations in the center-of-mass frame. Its quantum theory just reproduces the original form of bilocal theory giving bound states lying on a straightly rising trajectory and on its daughter trajectories.
A nonrelativistic equation for the system of two interacting particles within the framework of a model with noncommuting operators of coordinates and momenta of different particles is proposed, and a self-consistent system of equations for the wave function of every quantum state is deduced. Solutions for the lowest states of a hydrogenlike atom are found, and the comparison with analogous solutions of the Klein-Gordon equation for the relativistic spinless problem is performed. In the case where the size of a two-particle system and the Compton wavelengths of particles forming it are of the same order, the essential differences with solutions of the Schrödinger nonrelativistic equation are revealed.
An approximate quantum-mechanical two-body equation for spinless particles incorporating relativistic kinematics is derived. The derivation is based on the relativistic energy-momentum relation mc2+ε=m2c4+p2c2+Vmc^{2}+ε= \sqrt{m^{2}c^{4}+p^{2}c^{2}}+Vmc2+ε=m2c4+p2c2​+V for each single particle, where mc2mc^2mc2 is the particle rest mass energy, ppp its linear momentum, εεε its dynamical energy, and VVV being the time-like vector interaction potential. The resulting two-body equation assumes rapid wave oscillations in a single, slowly varying potential well. A Bohr-Sommerfeld-type quantization condition is obtained. The approximation is compared to exact results for the harmonic potential.
We introduce the de Broglie-Bohm causal interpreation of quantum mechanics and compare it to the standard interpretation of quantum mechanics, the Copenhagen interpretation. We examine the possibility of experimentally distinguishing between the two theories, as well as the potential for the causal interpretation to more easily bridge the gap between the physics of the quantum and classical worlds. We then use the causal interpretation to construct a deterministic model of the helium atom in which the two electrons move along trajectories through space and time about a stationary nucleus. The dynamics are governed by the non-relativistic Schrödinger equation and the spin vectors of both electrons are assumed to be constant along their respective trajectories. We examine the Bohmian trajectories associated with (approximations to) eigenstates of the helium Hamiltonian as well as the trajectories associated with some non-eigenstates. We also compute an approximation to the ground state energy of the helium atom using a representation of the helium wavefunction in terms of hydrogenic eigenfunctions which is motivated by a perturbation approach.
The Hamiltonian formulation of the Wheeler-Feynman theory of action at a distance in classical electrodynamics as carried out by Kerner, who has derived the nonrelativistic classical Hamiltonian for the interaction of two spinless charged particles interacting by means of half-retarded plus half-advanced potentials, has been generalized to obtain the corresponding relativistic classical Hamiltonian to order e4{e}^{4}e4 and ${c}^{\ensuremath{-}4}$. Using the procedure of Daughaday and Nigam, the classical Hamiltonian is transformed into the quantum-mechanical Hamiltonian operator.