In a Galilean theory recently developed, the Lorentz force law has been modified in such a way that the electromagnetic force is invariant when observed in different inertial frames, as expected intuitively. In this investigation, based on the local-ether model of wave propagation, a Galilean-invariant wave equation is proposed. This wave equation incorporates a natural frequency and the electric scalar potential. A unique feature is that this potential connects to the augmentation operator which in turn is associated with the momentum operator and the velocity of the source particle. From this local-ether wave equation, a first-order time evolution equation is derived, which presents Galilean modifications of Schrodinger's equation. This modified equation in turn leads to the Galilean-invariant electromagnetic force law given in terms of the augmented potentials.
In a Galilean theory recently developed, the Lorentz force law has been modified in such a way that the electromagnetic force is invariant when observed in different inertial frames, as expected intuitively. In this investigation, based on the local-ether model of wave propagation, a Galilean-invariant wave equation is proposed. This wave equation incorporates a natural frequency and the electric scalar potential. A unique feature is that this potential connects to the augmentation operator which in turn is associated with the momentum operator and the velocity of the source particle. From this local-ether wave equation, a first-order time evolution equation is derived, which presents Galilean modifications of Schrodinger's equation. This modified equation in turn leads to the Galilean-invariant electromagnetic force law given in terms of the augmented potentials.
Recently, we have presented a local-ether wave equation incorporating a nature frequency and the electric scalar potential, from which the speed-dependences in the angular frequency and wavelength of matter wave, in the mass of particle, and in the energy of quantum states are derived. These relations look like the postulates of de Broglie and the Lorentz mass-variation law, except that the particle speed is referred specifically to a geocentric inertial frame and hence incorporates earth's rotation for earthbound particles. Further, the wave equation is extended by connecting the scalar potential to the augmentation operator which is associated with a velocity difference between involved particles. Then the electromagnetic force law is derived, which under some ordinary conditions reduces to the modified Lorentz force law. In this investigation, the interaction of atoms with electromagnetic radiation is explored. Then it is shown that the time evolution equation derived from the wave equation is substantially identical to Schrödinger's equation incorporating the vector potential, if the latter is observed in the atom frame and if the source generating the vector potential is electrically neutralized, as in common practice.
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.
Recently, we have presented the local-ether model of propagation of electromagnetic wave. This new classical model accounts for a wide variety of propagation phenomena, including the gravitational effects associated with the deflection of light by the Sun and the increment in interplanetary radar echo time, which are commonly ascribed to the general relativity. In this investigation, based on the local-ether model, a wave equation incorporating the gravitational potential, the electric scalar potential, and a natural frequency is proposed for the matter wave associated with a charged particle. The local-ether wave equation leads to a time evolution equation similar to Schrödinger's equation. Then both the gravitational and the electrostatic forces are derived in a quantum-mechanical approach. Furthermore, it is found that the gravitational mass and the inertial mass are identical to the natural frequency, aside from a common scaling factor. Thus the identity of gravitational and inertial mass is also derived from the local-ether wave equation.
The theory of Quantum Electromagnetics presents a wave equation which is constructed, in compliance with Galilean transformations, 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 on propagation and 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 origin and 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 the dispersion of matter wave, from which the speed-dependence in the mass of a particle and in the wavelength, angular frequency, and quantum energy of a matter wave can be derived. The consequences of the wave equation are in accord with a wide variety of experiments that are commonly ascribed to the special relativity, the general relativity, the Lorentz mass-variation law, or to the de Broglie matter wave. Thereby, the local-ether wave equation unifies quantum mechanics, electromagnetics, and gravitation.
Textbook treatments of the Galilean covariance of the time-dependent Schrödinger equation for a spinless particle seem invariably to cover the case of a free particle or one in the presence of a scalar potential. The principal objective of this paper is to examine the situation in the case of arbitrary forces, including the velocity-dependent variety resulting from a vector potential. To this end, we revisit the 1964 theorem of Jauch which purports to determine the most general form of the Hamiltonian consistent with “Galilean-invariance,” and argue that the proof is less than compelling. We then show systematically that the Schrödinger equation in the case of a Jauch-type Hamiltonian is Galilean covariant, so long as the vector and scalar potentials transform in a certain way. These transformations, which to our knowledge have appeared very rarely in the literature on quantum mechanics, correspond in the case of electrodynamical forces to the “magnetic” nonrelativistic limit of Maxwell’s equations in the sense of Le Bellac and Lévy-Leblond (1973). Finally, this Galilean covariant theory sheds light on Feynman’s “proof” of Maxwell’s equations, as reported by Dyson in 1990.
A hypothesis is put forward according to which the formulas of Relativity are a consequence of the laws of quantum physics and may be applied only to quantum objects. According to the wave - particle duality quantum bodies are simultaneously the waves and considered in the present research with a use of the wave equation and requirement of its Lorentz‘s invariance. The property of invariance with respect to the Lorentz’s transform is analysed in hyperbolic coordinates in an area of complex numbers. An examination showed that formulas of relativistic kinematics can be obtained based on the condition of equality of the amplitudes and phases of the wave, observed from different inertial frames. There is no need to introduce the concept of a unified entity of space-time, but the return to the idea of aether and matter waves de Broglie occurs. The physical model becomes borrowed partly from quantum field theory. It is assumed that the moving particle binds to a wave of the aether disturbance, and only this wave acts on the other particle like an electromagnetic field pushes an electrical charge. The model of colliding elastic balls becomes inapplicable, but equations of relativity mechanics remain the same, however, with a new interpretation. The mass of a particle in all coordinate systems is constant, in exchange the expressions of energy and momentum alter relative the classical case, include the property of aether – coefficient c of the wave equation, and acquire relativistic forms.