In this paper we develop and study, as the second part of one more general development, the energy transmutation equation for the material singularity, previously obtained through the symmetrisation of a wave packet, that is, we develop the correlation between the terms of this equation, which accounts for the formation of matter from a previous vibrational state, and the different possible energy species. These energetic species are ascribed, in a simplified form, to the equation E ¯ ω = E ¯ k + E ¯ f , which allows us, through its associated phase factor, to gain an insight into the wave character of the kinetic energy and thus to attain the basis of the matter-wave, and all sorts of related phenomenologies, including that concerning quantum entanglement. The formation of the matter was previously identified as an energetic process, analogous to the kinetic one, in which finally the inertial mass is consolidated as a mass in a different phase, now, in addition, the mass of the material singularity is identified as a volumetric density of waves of toroidal geometry created in the process of singularisation or energy transfer between species, which makes it possible to establish the real relation or correspondence between the corpuscular and photonic energy equation ( E=m c 2 =hν ), i.e. to explain through m the intimate sense of the first equivalence, which explains what ν is in the second one.
In this paper we develop and study, as the second part of one more general development, the energy transmutation equation for the material singularity, previously obtained through the symmetrisation of a wave packet, that is, we develop the correlation between the terms of this equation, which accounts for the formation of matter from a previous vibrational state, and the different possible energy species. These energetic species are ascribed, in a simplified form, to the equation E ¯ ω = E ¯ k + E ¯ f , which allows us, through its associated phase factor, to gain an insight into the wave character of the kinetic energy and thus to attain the basis of the matter-wave, and all sorts of related phenomenologies, including that concerning quantum entanglement. The formation of the matter was previously identified as an energetic process, analogous to the kinetic one, in which finally the inertial mass is consolidated as a mass in a different phase, now, in addition, the mass of the material singularity is identified as a volumetric density of waves of toroidal geometry created in the process of singularisation or energy transfer between species, which makes it possible to establish the real relation or correspondence between the corpuscular and photonic energy equation ( E=m c 2 =hν ), i.e. to explain through m the intimate sense of the first equivalence, which explains what ν is in the second one.
The expectation value of the total energy of a wave packet, associated with the speed (moments) space by means of the Lorentz transformation, allows us to obtain by means of quantum wave formulation an energetic expression equivalent, and at the same time less restricted (more versatile and informative), to the one obtained by means of a corpuscular treatment. On the one hand, the energetic formants of the matter appear, on the other the expression in undulatory terms of it, of that which we call mass, and finally an additional phase factor as the last vestige of the undulatory nature after the formation process, that is to say, as an accrediting element of that nature, and mechanism of restitution of it, responsible for the wave-corpuscle duality, of the related creation-annihilation processes, and of the relations among the different generations of elementary particles, as it was shown in https://vixra.org/abs/2003.0001.
This article is a continuation of Part(I) to demonstrate more aspects for a successful merge of classical mechanics to the quantum theory. The article starts first with the deduction of the general form of the complimentary energy, next a robust firm mathematical proof is given indicates that the complimentary energy has a wave nature and it is a potential energy, then the radiation condition has been determined, a vector formula expresses the corresponding force was deduced and this force was called adhesion force. A mathematical convergence process has proved the duality of the electromagnetic wave parameters and particle parameters, which results in the existence of a “wave-particle” object. While finding the relationship between the electromagnetic wave parameters and particle parameters, a mathematical relationship was found indicating that not every moving charged mass generates a magnetic field. This relationship for the mass was linked to the effective mass tensor, The geometric representation of the constant energy surface and its relationship to the energy levels of both the orbital model (Bohr model) and the cloud model for different geometric 3D shapes were investigated as a sample of the possible geometric interpretations of the energy equation, that is by employing the triangle method, it is possible to create a geometric representation of the constant energy surfaces. The relation between the energy levels, quantum numbers, and band width is represented geometrically. As a result of proving the relationship between the motion of an electromagnetic wave and the motion of a particle associated with the wave, two new, unprecedented mathematical formulas were developed. The first formula specifies a moving charged mass and its relationship to the resulting wavelength. Thus, the radius of the charged mass is obtained in terms of its wavelength and mass density. The second formula specifies the kinetic energy of a photon. The grand potentials are the potential energy of the Newton’s Law of Universal Gravitation, Coulomb’s Law, and Inverse Square Law of Electromagnetic Waves Intensity. From the potential energy of Newton’s law of universal gravitation, the wavelength of the gravitational wave, as well as the wavelength of quantum gravity were deduced. A mathematical relationship was then deduced that determines the lengths of the functions of conic sections, and in particular the length of perimeter of ellipses. From the potential energy of Coulomb’s law of attraction, a mathematical formulation of the energy levels in general for any atom of an element was obtained. This mathematical formulation was compared with the results obtained by Bohr and Schrödinger for the hydrogen atom. As for the inverse square law of electromagnetic radiation intensity, several innovative mathematical equations have been deduced that link different parameters together. Following the contents of this article requires to be quite familiar with the contents of the first part with the same title, as well as a complete understanding of the concepts of intermediate parameters derived and used in the various equations. Otherwise, it will be extremely difficult to follow the contents of this article.
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.
A study was carried out to explain a wave propagation of the relativistic energy that comes in the squared form. What looks physically incompatible concept of wave and particle was combined into a general wave of the complex energy. From the fact that the energy of matter is expressed by the squared form, a wave of the massless and massive energies was presented, which plays a role in the formation of matter wave. This study revealed that the energy is separated into the complex value with a imaginary mass that makes the wave different from the plane wave. The wave propagation is described by plugging the complex energy into the wavefunction, in which a modified mass-energy equivalence from the special relativity was used. From the fact that the energy of matter is consist of a complex value, it is also explained that the space around a rest-mass is a space with a imaginary mass. As a result, a general expression of matter and wave is described by the complex wave including an imaginary mass and a real momentum, by which a reasonable explanation is given for the radiation of mass energy, and explains the role of imaginary mass in the propagation of energy.
Since the interest in the quantum phenomena started, there have been two main approaches to the quantum theory, the first approach investigates the motion of a wave-particle object by assuming the wave has the characteristics of a moving particle, momentum and energy, this approach was developed by physicists Max Planck, Albert Einstein, and others. The second approach investigates the motion of a particle-wave object by assuming the particle has the characteristics of a moving wave, wavelength, wave vector, and frequency, this approach was developed by physicists De Broglie, Schrodinger and others which led to the presence of quantum mechanics. This article focuses on the second approach and merging it with classical mechanics. An early mathematical formulation for the quantum theory was proposed by De Broglie’s relations and later on, another formulation was given by Schrodinger’s equation. A successful attempt to bridge the gap between classical mechanics with quantum theory implemented via a mathematical derivation for De Broglie’s relations and Schrodinger formulation, The derivation process overcame a main obstacle was the convergence of the deterministic nature of classical mechanics with the probabilistic nature of the quantum theory by adopting an innovative method based upon replacing constant values in a deterministic relation with probabilistic variables in the same relation, and this led to switch the relation from deterministic representation to probabilistic representation. Both classical and quantum explanation are given for the deduced formulas. An energy approach is instrumented. Since classical mechanics formulas are not sufficient to explain the quantum phenomena, a new concept of complementary energy introduced in order to connect the parameters of a moving particle with the parameters of an associated wave, the value of the complementary energy for each particle adjusted the total energy equation of N moving particles without changing the magnitude of the total energy of a system of N particles. In special cases, the value of the complementary energy for some particles equals the value of the total energy for the particles. The derivation process indicates that complementary energy corresponds to the potential energy on the surface of the material which prevents the particles to escape from the boundary of the matter to the outer space and maintains its coherence, complementary energy concept also manifest the impact of the moving particles on a single particle due to the energy conveyed during direct contact in case of collision or by the remote influence through an existing field in the space, so it represents an energy of dual nature for both matter and radiation wave combined. The complementary energy is not introduced as mathematical assumption in order to manipulate the total energy equation, it’s actually exists in the total energy equation and its validity has not been applied before. A general format for any potential energy is obtained regardless the nature of field it represents (a universal format for potential energy) that is by using mathematical manipulation for the equation of total energy of N moving particles. The derivation process led to a particle-wave formula. The substitution into the particle-wave formula with the speed of light, De Broglie’s relations, revealed. Some additional applications for the particle-wave formula are suggested, also as a well known energy quantization formula obtained by applying all of De Broglie’s relation, a general energy, and speed formulas resulting during the derivation process. Matching time independent equation called Schrodinger’s amplitude equation with classical mechanics formulation demonstrated by deducing both the wave function and the magnitude of the wave vector from the classical formulation after converting it to probabilistic formulation. A dedicated section for wave function solutions for Schrodinger’s time dependent equation was given in terms of the wave functions and the magnitude of the wave vectors obtained from the matching process of Schrodinger’s amplitude equation with classical mechanics.
This work is a revised version (Version 2) of a previous paper by the author, presented in a manner intended to emphasize more clearly the connection between its epistemological postulates and the subsequent development of its formalism. Several modifications have been introduced, and new sections have been added. As in the first version, the study concerns the physical interpretation of the imaginary quantities that arise in the formulation of superluminal relativity (including complex coordinates, velocities, angles, charges, and masses). In this framework, the kinematics of tachyons is analyzed, and the nature of the imaginary velocities appearing therein is examined. Furthermore, it is shown that a physical interpretation of the complex quantities (masses and charges) occurring both in quantum mechanics and in superluminal relativity is indeed possible. In this context, the complex nature of the electric charge is investigated, with particular attention to its divisibility. It is demonstrated that singular fractions of the electric charge exist whose squared values coincide with the observed quark charges. Within the same conceptual framework, the physical nature of a complex mass is also explored. It is shown that the imaginary component of such a mass implies that its gravitational field intensity behaves as a wave, propagating with the velocity of the wave associated with the particle itself. It is further demonstrated that the corresponding wavelength coincides with that given by the De Broglie relation, and that the resulting wave equation—second order in both spatial and temporal coordinates—is compatible with the Klein–Gordon wave equation. This constitutes a significant indication of the physical character of the associated wave and suggests an alternative interpretation to the orthodox probabilistic view of quantum mechanics.