The relativistic Schrodinger equation is reinterpreted as describing a classical particle that mutually-interacts with other objects via electromagnetic-like gravity waves. The accompanying derivation equates the usual quantum mechanical energy and momentum operators to the effects of negative or attractive energy. Lorentz-like transformation equations are obtained that yield the uncertainty principle such that quantum uncertainty is ascribed to the disregard of a magnetic-like component of a gravity wave. Finally, quantum-level Maxwell-like equations that involve the above gravity waves are derived.
The relativistic Schrodinger equation is reinterpreted as describing a classical particle that mutually-interacts with other objects via electromagnetic-like gravity waves. The accompanying derivation equates the usual quantum mechanical energy and momentum operators to the effects of negative or attractive energy. Lorentz-like transformation equations are obtained that yield the uncertainty principle such that quantum uncertainty is ascribed to the disregard of a magnetic-like component of a gravity wave. Finally, quantum-level Maxwell-like equations that involve the above gravity waves are derived.
The goal of this paper is an extension of the de Broglie wave mechanics model for a single spinless particle in an electromagnetic field. The analysis indicates that the motion of a particle separates naturally into particle dynamics through the classical Hamilton-Jacobi equation and quantum wave behavior through a pilot or interaction wave equation. The interaction wave equation travels at the classical particle velocity. We study gauge invariance and interpret it in the light of the interaction wave. The Heisenberg uncertainty relations are shown to be implicit in the interaction wave. We also develop a complex quantum-mechanical, relativistic energy-momentum conservation expression using a complex quantum-mechanical four-vector.
In a previous paper, we showed that the Hamilton equations of motion of a quantum particle are obtained as group velocities of the wave packet describing this particle only if the time dependent phase of a particle wave is proportional to the Lagrangian, not to the Hamiltonian as in a solution of the conventional Schrodinger equation. When a Lagrangian of relativistic form is considered, the wave packet of a quantum particle takes a physical form, with a finite spectrum of a cut off velocity (c). Based on a relativistic quantum principle, asserting the invariance of the time dependent phase for an arbitrary change of coordinates, we obtained the relativistic kinematics and dynamics, the electromagnetic field equations, the spin and the electromagnetic and gravitational interactions. When the Lagrangian is considered as a function of the Hamiltonian, we obtain a Schrodinger type equation with an additional term depending on the velocity and the momentum operator. Based on this equation, we investigate the dynamics of a relativistic quantum particle. In this framework, such a particle is described as a continuous distribution of conservative matter, according to the general theory of relativity. In an electromagnetic field, any time dependent phase variation is modified with a term proportional to a vector potential conjugated to the spatial coordinates and a scalar potential conjugated to time. However, this classical description tells us nothing about the structure of the physical world. Only Quantum Mechanics tells us something about the structure of this world. Namely, that this world is composed of species of identical quantum particles. On one hand, experimentally, it has been found that these particles are of a wavy nature. On the other hand, one could find that the simplest way to define a quantum particle is by a wave packet, with momenta conjugated to coordinates and energy conjugated to time Consider a quantum particle as a wave packet, and find that the group velocities in the coordinate and momentum spaces are in agreement with the Hamilton equations only when the Lagrangian is considered in the time dependent phases, instead of the Hamiltonian in the conventional forms of these waves as solutions of the Schrodinger equation. We define a relativistic quantum principle, and derive a wave equation for a relativistic quantum particle, the relativistic kinematics and dynamics of the particle waves, the Maxwell equations and the Lorentz force of a field interacting with the particle waves, the relativistic transform of such a field, and the spin as a characteristic of the particle waves. We consider a quantum particle as a distribution of conservative matter propagating according to the General Theory of Relativity. We obtain the dynamics of this matter in a gravitational field, the propagation in plane waves perpendicular to geodesic tracks, and equations of conservation. We describe the matter dynamics as a positively defined density and show that, according to the general theory of relativity, such a distribution can be conceive only as of a fragment of matter with a finite mass equal to a mass , as a characteristic of the matter dynamics, – the matter quantization. The group velocities of the Fourier conjugate representations in the coordinate and momentum spaces describe the dynamics of a quantum particle in agreement with the Hamiltonian equations. Under the action of an external (non-gravitational) field, the acceleration of the quantum matter has two components: 1) A component perpendicular to the velocity, given by the relativistic mechanical component of the time dependent phase, and 2) A component given by the additional field terms of this phase. A free quantum particle is described by a non-dispersing wave function , contrary to the solution of the Schrodinger equation. A coherent electromagnetic field, in resonance with a system of active quantum particles in a Fabry-Perot cavity, has a wave vector approximately proportional to the metric elements, as the resonance frequency is approximately constant – a gravitational wave can be detected by the transmission characteristics of an active Fabry-Perot cavity. In a constant gravitational field, a quantum particle undertakes a velocity and an acceleration, which, at the boundary of a black hole are null – absorption and evaporation processes at the boundary of a black hole arise only by gravitational perturbations. Generally, a quantum particle is described by a time-space volume, called graviton, with a spin 2, and a distribution of a specific matter in this volume, with a half-integer spin for Fermions and an integer spin for Bosons. A graviton Lagrangian is obtained as a curvature integral on a graviton volume, and a Hamiltonian tensor is obtained for the gravitational coordinates and velocities. In a gravitational field, the time space coordinates are deformed. In such a field, any plane wave remains perpendicular geodesic, while an additional acceleration is possible in the wave plane.
We consider a quantum particle as a wave packet in the coordinate space. When the conjugate wave packet in the momentum space is considered, we find that the group velocities of these two wave packets, which describe the particle dynamics, are in agreement with the Hamilton equations only if in the time dependent phases one considers the Lagrangian instead of the Hamiltonian which leads to the conventional Schrodinger equation. We define a relativistic quantum principle asserting that a quantum particle has a finite frequency spectrum, with a cutoff propagation velocity c as a universal constant not depending on the coordinate system, and that any time dependent phase variation is the same in any system of coordinates. From the time dependent phase invariance, the relativistic kinematics is obtained. We consider two types of possible interactions: 1) An interaction with an external field, by a modification of the time dependent phase differential with the terms proportional to the differentials of the space-time coordinates multiplied with the components of this field four-potential, and 2) an interaction by a deformation of the space-time coordinates, due to a gravitational field. From the invariance of the time dependent phase with field components, we obtain a mechanical force of the form of Lorentz’s force, and three Maxwell equations: The Gauss-Maxwell equations for the electric and magnetic fluxes, and the Faraday-Maxwell equation for the electromagnetic induction. When the fourth equation, Ampere-Maxwell, is considered, the interaction field takes the form of the electromagnetic field. For a low propagation velocity of the particle waves, we get a packet of waves with the time dependent phases proportional to the relativistic Hamiltonian, as in Dirac’s famous theory of spin, and a slowly-varying amplitude with a phase proportional to the momentum and this velocity. In the framework of our theory, the spin is obtained as an all quantum effect, without any additional assumption to the quantum theory. When a space-time deformation is considered in the time dependent phase of a quantum particle, from the group velocity we get the particle dynamics according to the general theory of relativity. In this way, the relativistic dynamics, the electromagnetic field, and the spin of a quantum particle are obtained only from the invariance of the time dependent phases of the particle wave functions.
We consider a quantum particle as a wave packet, and find that the group velocities in the coordinate and momentum spaces are in agreement with the Hamilton equations only when the Lagrangian is considered in the time dependent phases, instead of the Hamiltonian in the conventional forms of these waves as solutions of the Schrödinger equation.We define a relativistic quantum principle, and derive a wave equation for a relativistic quantum particle, the relativistic kinematics and dynamics of the particle waves, the Maxwell equations and the Lorentz force of a field interacting with the particle waves, the relativistic transform of such a field, and the spin as a characteristic of the particle waves.We consider a quantum particle as a distribution of conservative matter propagating according to the General Theory of Relativity.We obtain the dynamics of this matter in a gravitational field, the propagation in plane waves perpendicular to geodesic tracks, and equations of conservation.
A wave equation complying with Galilean transformations is proposed for electromagnetic and matter waves. A fundamental feature entirely different from the principle of relativity is that the wave equation is referred specifically to the proposed local-ether frame. For electromagnetic wave, the local-ether wave equation accounts for a wide variety of experiments of propagation or interference. For matter wave, the wave equation leads to modifications of Schrodinger's equation which in turn lead to a unified quantum theory of electromagnetic and gravitational forces in conjunction with the identity of inertial and gravitational mass. Further, it leads to modifications of the Lorentz force law and of Maxwell's equations. Moreover, the wave equation leads to dispersion of matter wave, from which the speed-dependences in the mass of a particle and in the wavelength, angular frequency, and quantum energy of a matter wave are derived. The consequences of the wave equation are in accord with a wide variety of experiments which are commonly ascribed to the special relativity, the general relativity, the Lorentz mass-variation law, or to the de Broglie matter wavelength. Thereby, the local-ether wave equation unifies quantum mechanics, electromagnetics, and gravitation.
The Schrödinger equation: the quantum description of one massive, slow-moving particle To establish a probability wave equation for one photon, it was useful to note that probability density and electromagnetic energy density were proportional. The governing equation for EM radiation fields is the Maxwell wave equation, which is also the governing equation for photon probability wave functions. Converting from EM radiation to photon information is made plausible by identifying energy and momentum operators with time- and space-derivatives, respectively. Thus, the Maxwell wave equation can be interpreted as being equivalent to the energy-momentum relation for photons. Though we don’t have a “Maxwell wave equation” to start with for massive particles, we do have an energy-momentum relation, so borrowing the energy and momentum operators from photons leads to an analogous wave equation for massive particles.