The existence of the so-called ‘dark’ issues of mechanics implies that our present accounting for mass and energy is incorrect in terms of applicability on a cosmological scale, and the question arises as to where the difficulty might lie. The phenomenon of quantum entanglement indicates that systems of particles exist that individually display certain characteristics, while collectively the same characteristic is absent simply because it has cancelled out between individual particles. It may therefore be necessary to develop theoretical frameworks in which long-held conservation beliefs do not necessarily always apply. The present paper summarises the formulation described in earlier papers (Hill, JM. On the formal origin of dark energy. Z Angew Math Phys 2018; 69:133-145; Hill, JM. Some further comments on special relativity and dark energy. Z Angew Math Phys 2019; 70: 5–14; Hill, JM. Special relativity, de Broglie waves, dark energy and quantum mechanics. Z Angew Math Phys 2019; 70: 131–153.), which provides a framework that allows exceptions to the law that matter cannot be created or destroyed. In these papers, it is proposed that dark energy arises from conventional mechanical theory, neglecting the work done in the direction of time and consequently neglecting the de Broglie wave energy [Formula: see text]. These papers develop expressions for the de Broglie wave energy [Formula: see text] by making a distinction between particle energy [Formula: see text] and the total work done by the particle [Formula: see text], that which accumulates from both a spatial physical force [Formula: see text] and a force [Formula: see text] in the direction of time. In any experiment, either particles or de Broglie waves are reported, so that only one of [Formula: see text] or [Formula: see text] is physically measured, and particles appear for [Formula: see text] and de Broglie waves occur for [Formula: see text], but in either event both a measurable and an immeasurable energy exists. Conventional quantum mechanics operates under circumstances such that [Formula: see text] vanishes and [Formula: see text] becomes purely imaginary. If both [Formula: see text] and [Formula: see text] are generated as the gradient of a potential, the total particle energy is necessarily conserved in the conventional manner.
The existence of the so-called ‘dark’ issues of mechanics implies that our present accounting for mass and energy is incorrect in terms of applicability on a cosmological scale, and the question arises as to where the difficulty might lie. The phenomenon of quantum entanglement indicates that systems of particles exist that individually display certain characteristics, while collectively the same characteristic is absent simply because it has cancelled out between individual particles. It may therefore be necessary to develop theoretical frameworks in which long-held conservation beliefs do not necessarily always apply. The present paper summarises the formulation described in earlier papers (Hill, JM. On the formal origin of dark energy. Z Angew Math Phys 2018; 69:133-145; Hill, JM. Some further comments on special relativity and dark energy. Z Angew Math Phys 2019; 70: 5–14; Hill, JM. Special relativity, de Broglie waves, dark energy and quantum mechanics. Z Angew Math Phys 2019; 70: 131–153.), which provides a framework that allows exceptions to the law that matter cannot be created or destroyed. In these papers, it is proposed that dark energy arises from conventional mechanical theory, neglecting the work done in the direction of time and consequently neglecting the de Broglie wave energy [Formula: see text]. These papers develop expressions for the de Broglie wave energy [Formula: see text] by making a distinction between particle energy [Formula: see text] and the total work done by the particle [Formula: see text], that which accumulates from both a spatial physical force [Formula: see text] and a force [Formula: see text] in the direction of time. In any experiment, either particles or de Broglie waves are reported, so that only one of [Formula: see text] or [Formula: see text] is physically measured, and particles appear for [Formula: see text] and de Broglie waves occur for [Formula: see text], but in either event both a measurable and an immeasurable energy exists. Conventional quantum mechanics operates under circumstances such that [Formula: see text] vanishes and [Formula: see text] becomes purely imaginary. If both [Formula: see text] and [Formula: see text] are generated as the gradient of a potential, the total particle energy is necessarily conserved in the conventional manner.
Louis-Victor de Broglie secured his place in the history of quantum physics by proposing the existence of matter waves. Following the concept accepted early in the twentieth century that light waves could display properties characteristic of particles (photons), de Broglie suggested the complementary principle that particles could display properties characteristic of waves. Within a few years, this far-reaching proposal was experimentally verified. The well-known formula for the wavelength λ of the wave associated with a particle of momentum p, namely λ=h/p, does not appear in de Broglie’s original paper, which was published in this journal in 1924. What is derived from the starting assumption that the frequency of the wave is the rest energy of the particle divided by Planck’s constant h, is the wave's velocity, which turns out to be c2/v, where v is the particle’s velocity. This velocity, being superluminal, means that the wave cannot transport energy. De Broglie called it a phase wave and postulated a group of such waves having virtually, but not exactly, the same velocity and wavelength. The group velocity of such a large number of almost identical phase waves is then shown, very satisfactorily, to be equal to the velocity of the particle. The proposal by de Broglie inspired Schrödinger to develop the mathematical foundations of wave mechanics. Whereas in conventional quantum theory, phase waves are considered fictional, de Broglie considered them to have a certain physical reality, renaming them pilot or guiding waves – a concept developed later by Bohm in 1952.
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.
If I could to do this properly, and know this as what is proper (to, of, the thesis), I could say that this is to have balanced the (energy) books – in sum. That a framework of being (multiple) books, I could say text, I could say space, is already in place to have then the balance. Overall – in sum, average, to conserve. I could say, what is already in place? I could say, the atom, rather than the (text)book, if I did not know better, that is, that I can add to the atom in its division: it could (also) be a particle or a wave. So it might be that I adumbrate how it is that de Broglie first thought matter waves, and by so doing, I might observe that the velocity of the phase wave associated to a particle is excessive to the system which produces it. Or rather that to carry energy requires a modification of this phase wave; no specifics, but rather a group (velocity). And in observing this, think through what is at stake in the claims to observers observing that which can(not) be seen. I could say, that this is key (words); how to (re)order the infinite to (re)produce itself? And this in relation to energy. I might (probably) be questioning frames, the mathematics, mechanics; how and why this is invested in as supplement and proof of what is. What would be unity? What is proper (for there to (probably) be unity)? This in relation to eigenvalues, and Schrödinger’s Wave Equation – what is properly characteristic of the being particle, wave? I can only gesture towards this, the field –
The de Broglie wave phenomenon was studied in the framework of Lagrangean formulation of Relativistic Mechanics. The conclusion was made that the de Broglie wave of a particle is a relativistic phenomenon related to the field-dependent proper mass. The wave develops in a process of proper-to-kinetic mass transformation when the particle is given a momentum. The time-part of the 4-wave vector is related to the proper mass oscillation while the spacial part is due to the momentum; both parts can be treated in terms of virtual (longitudinal and scalar) photons characterizing excitation states of a mediating scalar field. It is shown that in Relativistic Mechanics with the variable proper mass both electromagnetic and gravitational force can be treated on the same footing: both are due to the same source that is, the proper mass. Consequently, the scalar mediating field provides a mechanism of unification of gravitational and electromagnetic forces. The concept of the unified divergence-free field is suggested, in which the real (transverse) photon represents a pure space-like particle while the neutrino is a pure time-like particle. In this connection, the parity problem is discussed.
Based on the law of conservation of energy, we have shown that, the steady state electronic motion around a given nucleus in a non-circular orbit depicts a rest mass variation, though the overall relativistic energy remains constant. This is, in no way, conflicting with the usual quantum mechanical approach. On the contrary, it provides us with the possibility of bridging the special theory of relativity and quantum mechanics, to finally achieve a natural symbiosis between these two disciplines, and furthermore, elucidating the “quantum mechanical weirdness”, simply based on the mere law of relativistic conservation of energy. Our theory was developed originally, vis-a-vis gravitational bodies in motion with regard to each other; hence, it is comforting to have both the atomic scale and the celestial scale described on just the same conceptual basis. One way to conceive the rest mass variation of concern is to consider a “jet effect”. Accordingly, a particle on a given orbit through its journey must eject a net mass from its back to accelerate, or must pile up a net mass from its front to decelerate, while its overall relativistic energy stays constant throughout. In other words, a minimal rest mass is transformed into extra kinetic energy through an acceleration process, and kinetic energy condenses into extra rest mass through a deceleration process. The tangential speed of the jet in question, strikingly points to the de Broglie wavelength. This, on the whole, makes the jet speed ‘’superluminal”, yet excluding any transport of energy. This conjecture coincides with the recent measurements (Kholmetskii et al., 2007; Salart et al., 2008), and provides a clue for the wave-particle duality, also insures their concurrent coexistence. An important consequence of the approach presented herein is that, either gravitationally interacting macroscopic bodies, or electrically interacting microscopic objects, sense each other, with a speed much greater than that of light, and this, in exactly the same way. In which case, though, the interaction coming into play, excludes any energy exchange; thus, we would like to call it, “wave-like interaction”. The present approach ends up the existing schizophrenia, between different conceptions painting separately the micro world and the macro world, and unites these two worlds with an unequal, single and simple conception, based on just the law of relativistic energy conservation.
Key words: Special theory of relativity, electric interaction, tachyons, de Broglie relationship, superluminal or wave-like interaction, general theory of relativity, gravitation.
In this paper, we have considered that the particle velocity in the matter wave concept of de Broglie is relative. It reveals that matter wave associated with a moving particle is a photon wave which has longitudinal and transverse values. The conventional de-Broglie’s wavelength-momentum relation corresponds to the longitudinal wave. As a consequence, we have found the longitudinal and transverse relative values of the mass, time, length and velocity including the missing equations of Einstein’s theory of special relativity. It also reveals that momentum and energy are relative quantities and the corresponding expressions for the longitudinal and transverse values for both of them have been presented. Moreover, the addition rules, with a general expression, for all the relative quantities based on Lorentz-Einstein transformation law for the addition of velocities have been given. Finally, the maximum and minimum values of the relative quantities have been expressed in terms of Plank’s constant and the speed of light. Their numerical values have also been calculated and presented in this paper.