Ein Existenzbeweis für Lösungen des klassischen Zweielektronenproblems

A method due to Hale and Stokes [3] proving the existence of solutions of the classical relativistic equation of motion of an electron in a given electromagnetic field is extended to the two-electron problem. A lower bound of the impact parameter assures the applicability of the method.     

On Relativistic Generalization of Gravitational Force

In relativistic theories, the assumption of proper mass constancy generally holds. We study gravitational relativistic mechanics of point particle in the novel approach of proper mass varying under Minkowski force action. The motivation and objective of this work are twofold: first, to show how the gravitational force can be included in the Special Relativity Mechanics framework, and, second, to investigate possible consequences of the revision of conventional proper mass concept (in particular, to clarify a proper mass role in the divergence problem). It is shown that photon motion in the gravitational field can be treated in terms of massless refracting medium, what makes the gravity phenomenon compatible with SR Mechanics framework in the variable proper mass approach. Specifically, the problem of point particle in the spherical symmetric stationary gravitational field is studied in SR-based Mechanics, and equations of motion in the Lorentz covariant form are obtained in the relativistic Lagrangean problem formulation. The dependence of proper mass on potential field strength is derived from the Euler-Lagrange equations as well. One of new results is the elimination of conventional 1/r divergence, which is known to be not removable in Schwarzschild gravitomechanics. Predictions of particle and photon gravitational properties are in agreement with GR classical tests under weak-field conditions; however, deviations rise with potential field strength. The conclusion is made that the approach of field-dependent proper mass is perspective for development of SR gravitational mechanics and further studies of gravitational problems.     

Space-time trajectories of quantum particles

The concept of trajectory is extended theoretically from classical mechanics through nonrelativistic and relativistic quantum mechanics. Forced motion of the particle as might be caused by an electromagnetic field is included in the equations. A new interpretation of the electromagnetic potential and the gauge transformation is presented. Using this formal structure, the problem of collecting particles into packets using trajectories is studied for both quantum mechanics and classical mechanics. Quantum mechanical trajectories are found to be significantly more restricted than those allowed by classical physics. The uncertainty principle comes from the second-order nature of the field equation without recourse to statistical arguments. The trajectories of particles in a quantum state can be calculated explicitly from the wave function (also see article in Volume 20, Number 6).     

Relativity from Light Front and Bethe–Salpeter Equations

We present some results of two independent relativistic approaches to the few-body problem: light-front dynamics and Bethe–Salpeter equation. We show that implementing relativistic invariance leads to new qualitative properties, and that, even driven by the same interaction Lagrangian, both approaches provide different quantitative results, especially in three-body systems. The case of Bethe–Salpeter equation for computing electromagnetic form factors is discussed.     

Classical limit of relativistic quantum mechanical equations in the Foldy-Wouthuysen representation

It is shown that, under the Wentzel-Kramers-Brillouin approximation conditions, using the Foldy-Wouthuysen (FW) representation allows the problem of finding a classical limit of relativistic quantum mechanical equations to be reduced to the replacement of operators in the Hamiltonian and quantum mechanical equations of motion by the respective classical quantities.     

Two Mathematically Equivalent Versions of Maxwell’s Equations

This paper is a review of the canonical proper-time approach to relativistic mechanics and classical electrodynamics. The purpose is to provide a physically complete classical background for a new approach to relativistic quantum theory. Here, we first show that there are two versions of Maxwell’s equations. The new version fixes the clock of the field source for all inertial observers. However now, the (natural definition of the effective) speed of light is no longer an invariant for all observers, but depends on the motion of the source. This approach allows us to account for radiation reaction without the Lorentz-Dirac equation, self-energy (divergence), advanced potentials or any assumptions about the structure of the source. The theory provides a new invariance group which, in general, is a nonlinear and nonlocal representation of the Lorentz group. This approach also provides a natural (and unique) definition of simultaneity for all observers.     

Some novel plane trajectories for carbon atoms and fullerenes captured by two fixed parallel carbon nanotubes

The movement of atoms and molecules at the nanoscale constitutes a fundamental problem in physics, especially following the motion of atoms in many-body systems condensing together to form molecular structures. A number of simplified nanoscale dynamical problems have been analyzed and here we investigate the classical orbiting problem around two centers of attraction at the nanoscale. An example of such a system would be a carbon atom or a fullerene orbiting in a plane which is perpendicular to two fixed parallel carbon nanotubes. We model the van der Waals forces between the molecules by the Lennard-Jones potential. In particular, the total pairwise potential energies between carbon atoms on the fullerene and the carbon nanotubes are approximated by the continuous approach, so that the total molecular energy can be determined analytically. Since we assume that such interactions occur at a sufficiently large distance, the classical two center problem analysis is legitimate to determine various novel trajectories of the atom and fullerene numerically. It is clear that the oscillatory period of the atom for some bounded trajectories reaches terahertz frequencies. We comment that the continuous approach adopted here has the merit of a very fast computational time and can be extended to more complicated structures, in contrast to quantum mechanical calculations and molecular dynamics simulations.     

Instability of the Y string baryon model within classical dynamics

For the three-string baryon model (Y configuration), the known exact solution to the classical equations of motion that describes the rotational motion of the system at a constant speed is investigated for stability. In the spectrum of small perturbations of this solution, modes growing exponentially with time are found, whereby the instability of rotational motion is proven for the Y configuration. This result is confirmed within an alternative approach that makes it possible to determine the classical motion of the system from a specific initial position and initial velocities of string points. A comparison of the Y configuration with the model of a relativistic string with massive ends, in which case rotational motion is stable in the linear approximation, aids in revealing the most adequate string model from the point of view of describing baryon excitations on Regge trajectories.     

Electromagnetic Field Theory Without Divergence Problems 1. The Born Legacy

Born's quest for the elusive divergence problem-free quantum theory of electromagnetism led to the important discovery of the nonlinear Maxwell–Born–Infeld equations for the classical electromagnetic fields, the sources of which are classical point charges in motion. The law of motion for these point charges has however been missing, because the Lorentz self-force in the relativistic Newtonian (formal) law of motion is ill-defined in magnitude and direction. In the present paper it is shown that a relativistic Hamilton–Jacobi type law of point charge motion can be consistently coupled with the nonlinear Maxwell–Born–Infeld field equations to obtain a well-defined relativistic classical electrodynamics with point charges. Curiously, while the point charges are spinless, the Pauli principle for bosons can be incorporated. Born's reasoning for calculating the value of his aether constant is re-assessed and found to be inconclusive.     

On the Electromagnetic Origin of Inertia and Inertial Mass

We address the problem of inertial property of matter through analysis of the motion of an extended charged particle. Our approach is based on the continuity equation for momentum (Newton’s second law) taking due account of the vector potential and its convective derivative. We obtain a development in terms of retarded potentials allowing an intuitive physical interpretation of its main terms. The inertial property of matter is then discussed in terms of a kind of induction law related to the extended charged particle’s own vector potential. Moreover, it is obtained a force term that represents a drag force acting on the charged particle when in motion relatively to its own vector potential field lines. The time rate of variation of the particle’s vector potential leads to the acceleration inertia reaction force, equivalent to the Schott term responsible for the source of the radiation field. We also show that the velocity dependent term of the particle’s vector potential is connected with the relativistic increase of mass with velocity and generates a longitudinal stress force that is the source of electric field lines deformation. In the framework of classical electrodynamics, we have shown that the electron mass has possibly a complete electromagnetic origin and the obtained covariant equation solves the “4/3 mass paradox” for a spherical charge distribution.     

Relativistic Equations of Motion from Poisson Brackets

The inverse problem of Poisson dynamics isreviewed as well as a derivation of the Maxwellequations from a postulated set of Poisson brackets. Theformalism is extended to the relativistic case bypostulating Poisson brackets, as in the nonrelativisticcase, and using the relativistic Hamiltonian. A systemof relativistic equations of motion is obtained, and itis indicated that a system of consistency conditions remains valid in this limit.     

Logical inference approach to relativistic quantum mechanics: Derivation of the Klein–Gordon equation

The logical inference approach to quantum theory, proposed earlier De Raedt et al. (2014), is considered in a relativistic setting. It is shown that the Klein–Gordon equation for a massive, charged, and spinless particle derives from the combination of the requirements that the space–time data collected by probing the particle is obtained from the most robust experiment and that on average, the classical relativistic equation of motion of a particle holds.     

Post-Newtonian gravitational radiation from binary stars

A method is presented for evaluating post-Newtonian orbital corrections to the orbital-period decrease of a relativistic binary, by using previous results on the gravitational-radiation luminosity of a bounded source and the binary's relative motion. The method is based on an energy-balance equation, whose validity to post-Newtonian accuracy is not proved here, but, in view of recent theoretical results on the classical energy-balance equation, seems to be quite reasonable, and, in view of the available observational data, is practically correct. In the case of the binary pulsar PSR 1913 + 16 the proposed relative orbital correction amounts to 10^–6 of the current observational uncertainty, thus greatly favoring the predictions of the classical quadrupole formula.     

Few-Body Physics in a Many-Body World

The study of quantum mechanical few-body systems is a century old pursuit relevant to countless subfields of physics. While the two-body problem is generally considered to be well-understood theoretically and numerically, venturing to three or more bodies brings about complications but also a host of interesting phenomena. In recent years, the cooling and trapping of atoms and molecules has shown great promise to provide a highly controllable environment to study few-body physics. However, as is true for many systems where few-body effects play an important role the few-body states are not isolated from their many-body environment. An interesting question then becomes if or (more precisely) when we should consider few-body states as effectively isolated and when we have to take the coupling to the environment into account. Using some simple, yet non-trivial, examples I will try to suggest possible approaches to this line of research.     

Quantum mechanics as relativistic statistics. I: The two-particle case

It is shown that non-relativistic quantum mechanics can be treated as a kind of relativistic statistical theory, which describes the indeterministic motion of classical particles. The theory is relativistic in the sense that the relativistic notion of the state and two-time equations of motion are used. The principles and relations of quantum mechanics are obtained from the principles of statistics and those of classical mechanics.     

Introducing Relativistic Quantum Mechanics to Energy Students

A method for introducing relativistic quantum mechanics to energy students is described. The method complements existing modern physics courses and relies on Feynman’s relativistic path integral approach to display a relationship between classical dynamics, quantum theory, and relativistic quantum theory.     

Properties of nonautonomous classical systems and quantum mechanics

We show that the relativistic equation of the Brownian motion of a classical particle near a trajectory, stable in Lyapunov's sense, is identical with the Klein-Gordon equation. The conditions which make motions of this type possible are connected with cosmology.