The Einstein’s program forms a consistent system for universe description, beside the standard model of particles. It is founded upon a scalar field propagating at speed of light c, which constitutes a common relativist framework for classical and quantum properties of matter and interactions. Matter corresponds to standing waves. Classical domain corresponds to geometrical optics approximation, when frequencies are infinitely high, and then hidden. Quantum domain corresponds to wave optics approximation. Adiabatic variations of frequencies yield electromagnetic interaction. They lead also to Classical and Quantum Mechanics equations, with unification of first and second quantifications for interactions and matter, and to the wave-particle duality, by space reduction of the introduced space-like amplitude function u(r,t), which completes the usual time-like function ψ(r,t).
The Einstein’s program forms a consistent system for universe description, beside the standard model of particles. It is founded upon a scalar field propagating at speed of light c, which constitutes a common relativist framework for classical and quantum properties of matter and interactions. Matter corresponds to standing waves. Classical domain corresponds to geometrical optics approximation, when frequencies are infinitely high, and then hidden. Quantum domain corresponds to wave optics approximation. Adiabatic variations of frequencies yield electromagnetic interaction. They lead also to Classical and Quantum Mechanics equations, with unification of first and second quantifications for interactions and matter, and to the wave-particle duality, by space reduction of the introduced space-like amplitude function u(r,t), which completes the usual time-like function ψ(r,t).
The Einstein's program forms a consistent system for universe description, beside the standard model of particles. It is founded upon a scalar field propagating at speed of light c, which constitutes a common relativist framework for classical and quantum properties of matter and interactions. Matter corresponds to standing waves. Classical domain corresponds to geometrical optics approximation, when frequencies are infinitely high, and then hidden. Quantum domain corresponds to wave optics approximation. Adiabatic variations of frequencies lead to electromagnetic interaction constituted by progressive waves. It leads to theoretical economy for Quantum Theory, with unification of first and second quantifications for interactions and matter, to the wave-particle duality by reduction of the introduced amplitude space-like function ( ), , t r u which completes the usual time-like function ( ), , t r ψ with hidden variables.
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.
Adiabatic variations of frequencies lead to electromagnetic interaction constituted by progressive waves. By comparison, the consistent quantum theory, based upon time-like equations, of Dirac for fermions and of Klein- Gordon for bosons, is incomplete since it does not take account of a spacelike function u0(k0r0) to describe massive particle extension. It leads to Dirac’s distribution δ(r0) in the geometrical optics approximation, in order to handle a particle as a singularity within a continuous field, and to energymomentum conservation laws, and to least action law, with determination of the Lagrangian.
In this paper, the quantum dynamics was obtained in the framework of the general theory of relativity, where a quantum particle is described by a distribution of matter, with amplitude functions of the matter density, in the two conjugate spaces of the spatial coordinates and of the momentum, called wave functions. For a free particle, these wave functions are conjugate wave packets in the coordinate and momentum spaces, with time-dependent phases proportional to the relativistic Lagrangian, as the wave velocities in the coordinate space are equal to the distribution velocity described by the wave packet in this space. From the wave velocities of the particle wave functions, Lorentz’s force and the Maxwell equations were obtained. From the wave/group equation in the momentum space describing the Lorentz force, the expressions of the electric and magnetic fields as functions of the electric potential conjugated to time and of the vector potential conjugated to the coordinates in the particle-field Lagrangian were obtained. With these expressions, the electric and magnetic fields that satisfy the Faraday-Maxwell law of electromagnetic induction and the two Gauss-Maxwell laws of these fields were obtained. The Ampère-Maxwell law is obtained only by taking into account the physical consistency of the matter-field interaction of the equality of the propagation field velocity with the maximum relativistic velocity c. For a quantum particle in the electromagnetic field, dynamic equations in the coordinate and momentum spaces and the particle and antiparticle wave functions were obtained. It was shown that the electromagnetic potentials as functions of the coordinates describing the matter distribution of the quantum particle do not alter this distribution – under the action of an electromagnetic a quantum particle moves as a whole. The scattering or tunneling rate in an electromagnetic field, for the two possible cases, with the spin conservation, or inversion, were obtained. This description of a quantum particle as a distribution of matter with a density amplitude/wavefunction of the form of a wave packet, with the time-dependent phase proportional to the relativistic Lagrangian as a function of the metric tensor including also the gravitational field, enables the application of this theory in quantum gravity and quantum field theory in agreement with general relativity.
At the beginning of last century, the Universe was considered to be constituted of matter and radiation. The second component of the Universe according to classical physics—radiation—is nothing but propagating electromagnetic waves. The dynamics of a particle was supposed to obey Newton's equation. For a given a field, that is, a physical quantity defined at each point of a region of space, it might be useful to have physical quantities describing the spatial variations of the field. Stationary Schrodinger Equation also known as time-independent Schrodinger equation, and its solutions give the allowed energy values of the system. Energy quantization is a consequence of the space confinement of the particle within a limited volume or, more precisely, due to the fact that the wavefunction probability density tends to zero as the spatial coordinates tend to infinity.
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.