Dispersion dynamics applies wave-particle duality, together with Maxwell’s electromagnetism, and with quantization E = hν = ħω (symbol definitions in footnote) and p = h/λ = ħk, to special relativity E2 = p2c2 + m2c4. Calculations on a wave-packet, that is symmetric about the normal distribution, are partly conservative and partly responsive. The complex electron wave function is chiefly modelled on the real wave function of an electromagnetic photon; while the former concept of a “point particle” is downgraded to mathematical abstraction. The computations yield conclusions for phase and group velocities, vp⋅vg = c2 with vp ≥ c because vg ≤ c, as in relativity. The condition on the phase velocity is most noticeable when p≪mc. Further consequences in dispersion dynamics are: derivations for ν and λ that are consistently established by one hundred years of experience in electron microscopy and particle accelerators. Values for vp = νλ = ω/k are therefore systematically verified by the products of known multiplicands or divisions by known divisors, even if vp is not independently measured. These consequences are significant in reduction of the wave-packet by resonant response during interactions between photons and electrons, for example, or between particles and particles. Thus the logic of mathematical quantum mechanics is distinguished from experiential physics that is continuous in time, and consistent with uncertainty principles. [Footnote: symbol E = energy; h = Planck’s constant; ν = frequency; ω = angular momentum; p = momentum; λ = wavelength; k = wave vector; c = speed of light; m = particle rest mass; vp = phase velocity; vg = group velocity].
Dispersion dynamics applies wave-particle duality, together with Maxwell’s electromagnetism, and with quantization E = hν = ħω (symbol definitions in footnote) and p = h/λ = ħk, to special relativity E2 = p2c2 + m2c4. Calculations on a wave-packet, that is symmetric about the normal distribution, are partly conservative and partly responsive. The complex electron wave function is chiefly modelled on the real wave function of an electromagnetic photon; while the former concept of a “point particle” is downgraded to mathematical abstraction. The computations yield conclusions for phase and group velocities, vp⋅vg = c2 with vp ≥ c because vg ≤ c, as in relativity. The condition on the phase velocity is most noticeable when p≪mc. Further consequences in dispersion dynamics are: derivations for ν and λ that are consistently established by one hundred years of experience in electron microscopy and particle accelerators. Values for vp = νλ = ω/k are therefore systematically verified by the products of known multiplicands or divisions by known divisors, even if vp is not independently measured. These consequences are significant in reduction of the wave-packet by resonant response during interactions between photons and electrons, for example, or between particles and particles. Thus the logic of mathematical quantum mechanics is distinguished from experiential physics that is continuous in time, and consistent with uncertainty principles. [Footnote: symbol E = energy; h = Planck’s constant; ν = frequency; ω = angular momentum; p = momentum; λ = wavelength; k = wave vector; c = speed of light; m = particle rest mass; vp = phase velocity; vg = group velocity].
From a combination of Maxwell’s electromagnetism with Planck’s law and the de Broglie hypothesis, we arrive at quantized photonic wave groups whose constant phase velocity is equal to the speed of light c = ω/k and to their group velocity dω/dk. When we include special relativity expressed in simplest units, we find that, for particulate matter, the square of rest mass , i.e., angular frequency squared minus wave vector squared. This equation separates into a conservative part and a uniform responsive part. A wave function is derived in manifold rank 4, and from it are derived uncertainties and internal motion. The function solves four anomalies in quantum physics: the point particle with prescribed uncertainties; spooky action at a distance; time dependence that is consistent with the uncertainties; and resonant reduction of the wave packet by localization during measurement. A comparison between contradictory mathematical and physical theories leads to similar empirical conclusions because probability amplitudes express hidden variables. The comparison supplies orthodox postulates that are compared to physical principles that formalize the difference. The method is verified by dual harmonics found in quantized quasi-Bloch waves, where the quantum is physical; not axiomatic.
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.
In massive free particles the square of Relativistic energy, E’2 = p2c2 + mo2c4/h2 divides into two: relativistic momentum p=m’vg and rest mass mo (where c is the speed of light and h is Planck’s constant}. The rest mass is a function of phase velocity vp; the momentum a function of group velocity vg. In each mass component, mo and (m’-mo) the velocity dependence is exclusive. This consequence of their formulations defines their functions in mass, in internal motion, in intrinsic spin, in superconductivity, in wavefunction collapse, in conservation laws such as momentum, spin, etc.
Essentially, we solve the particle-field equations obtained in the matter distribution quantum mechanics. In the framework of this theory, a particle is described by a density amplitude function, we call wave function, as a packet of waves with the time dependent phases proportional to the relativistic Lagrangian. The mass quantization of such a particle, as a distribution of matter, is obtained from the equality of the spatial integral of the density with the mass as a characteristic of the relativistic dynamics of this matter. The particle dynamics, as a motion of a mass spatial distribution, is obtained from the equality of the wave propagation velocities with the distribution coordinate velocities. This description is a physical one, compared to the Schrödinger-Heisenberg description leading to unphysical results as a null velocity for the kinetic energy as a constant of motion, or Zitterbewegung. The description of a particle interaction with an electromagnetic field by additional Lagrangian terms, with a vector potential conjugated to coordinates, as in the Aharonov-Bohm effect, and an electric potential conjugated to time, leads to the Lorentz force, and the Maxwell equations. In the two conjugated spaces, of the coordinates and of the momentum, the dynamics of a quantum particle is described by propagation operators, depending on the coordinate-momentum product, applied to time dependent wave functions satisfying matter-field equations. Unlike the standard Schrödinger or Schrödinger-Dirac equations, depending only on the Hamiltonian operator, instead of the Hamiltonian these equations contain the Lagrangian, which, besides the Hamiltonian, include additional terms with an explicit dependence on velocity, as it is expected in a relativistic theory. We obtain the wave function of a particle-antiparticle system interacting with an electromagnetic field, as a wave packet of spinors propagating in space with finite dimensions, given by the finite definition domain of the density function, in the space of the matter-field momentum, as a constant of motion. We show that an electrically charged quantum particle, rotating in a central electric field, generates a vector potential with a magnetic field perpendicular to the rotational plane and an electric field in this plane. By a quantum transition, a photon is emitted in the particle rotational plane, with the magnetic field perpendicular to this plane, and the electric field contained in this plane. In this framework, the vector potential rotation is correlated to a particle momentum rotation, for the canonical matter-field momentum as a constant of motion, without any electromagnetic field radiated outside.
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.
Toward the end of the nineteenth century, it became clear that some experiments cannot be explained on the basis of known physics, now referred to as classical physics. In 1900, Max Planck was able to correctly describe the distribution of blackbody radiation energy as a function of wavelength λ or frequency ν = c/λ. He had to accept the existence of energy packets (quanta) with energy E = hν. A few years later, Albert Einstein found that this unexpected relation between energy and frequency is also necessary in order to understand photoemission from metallic surfaces. A decade later, Niels Bohr arrived at the correct equation for the electronic spectrum of the hydrogen atom by applying “quantum conditions” on the motion of the electron around the nucleus and the Planck’s relation E = hν on the energy difference between the energy levels (1916). A decade later, Heisenberg, Born, and Jordan introduced “quantum mechanics” and de Broglie and Schrödinger (1925-1926) introduced “wave mechanics.”