Wave-like interaction, occurring at superluminal speeds, or the same, de Broglie relationship, as imposed by the law of energy conservation: Gravitationally bound particles (Part II)
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.
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.
Up to this point, our study of quantum mechanics has concerned itself with the behaviour of particles that inhabit a Galilean spacetime. For many purposes, in atomic, molecular and condensed matter physics, this theory is quite adequate. We saw in earlier chapters, however, that our actual spacetime has a structure which is much closer to that of the Minkowski spacetime of special relativity and that more general structures must be considered when gravitational phenomena are signicant. From a purely theoretical point of view, it is therefore important to formulate quantum theory in a way which is consistent with these more general spacetimes. The benets of constructing a relativistic quantum theory actually go far beyond the aesthetic satisfaction of making our geometrical and quantum-mechanical reasoning compatible. For one thing, we shall discover that the relativistic theory provides a deeper understanding of spin and the distinction between fermions and bosons, which in the non-relativistic theory appear simply as facts of life that we must strive to accommodate. Also, of course, there are many situations in which relativistic eects become observable, for which non-relativistic theory provides no explanation. The most obvious are high-energy scattering experiments, in which particles acquire kinetic energies comparable with or greater than their rest energies mc2, and the correct 4-momentum (3.33) must be used. There are, however, more subtle eects, such as the spin-orbit coupling that is essential for interpreting atomic spectra, which are also of relativistic origin.
Up to this point, our study of quantum mechanics has concerned itself with the behaviour of particles that inhabit a Galilean spacetime. For many purposes, in atomic, molecular and condensed matter physics, this theory is quite adequate. We saw in earlier chapters, however, that our actual spacetime has a structure which is much closer to that of the Minkowski spacetime of special relativity and that more general structures must be considered when gravitational phenomena are significant. From a purely theoretical point of view, it is therefore important to formulate quantum theory in a way which is consistent with these more general spacetimes. The benefits of constructing a relativistic quantum theory actually go far beyond the aesthetic satisfaction of making our geometrical and quantummechanical reasoning compatible. For one thing, we shall discover that the relativistic theory provides a deeper understanding of spin and the distinction between fermions and bosons, which in the non-relativistic theory appear simply as facts of life that we must strive to accommodate. Also, of course, there are many situations in which relativistic effects become observable, for which nonrelativistic theory provides no explanation. The most obvious are high-energy scattering experiments, in which particles acquire kinetic energies comparable with or greater than their rest energies mc2, and the correct 4-momentum (3.34) must be used. There are, however, more subtle effects, such as the spin-orbit coupling that is essential for interpreting atomic spectra, which are also of relativistic origin.
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 idea that all changes in nature must be smooth and continuous pervades classical physics, and indeed Einstein’s theory of relativity as well. There are several kinds of ‘discontinuity’ and their implications differ. Descartes’ laws of collision had stated that if a small body hits a larger one then it rebounds with equal speed, whereas if a large body hits a smaller one they move together in such a way that the total quantity of ‘motion’ is conserved. Field theory promised to get round these conceptual difficulties in classical particle mechanics, by surrounding such particles by fields whose effect would be to prevent ‘sharp’ collisions and to replace them by rapid but smooth accelerations and decelerations. In classical physics ‘energy’ is regarded as being carried both by material particles and by fields.
Special relativity seems to be associated with linking physical relations as seen in one frame with those as seen in a frame moving at a speed relative to such a frame. A constant speed is often used for the moving frame. This suggests that classically there is a special frame for a particle with rest mass called a rest frame. In such a case, one simply introduces a parameter for rest mass and does not worry about the complicated motion and interactions which actually create the rest mass. It seems that even quantum mechanics, which considers a particle to be moving as one does not know its location, one uses the idea of a potential V(x) which in principle is associated with a very large rest mass, presumably at rest as no variables of such a rest mass appear in the Klein-Gordon or Dirac equations. It seems that one uses a preferred frame when solving relativistic quantum mechanical equations. Furthermore, given V(x) and a bound state with exp(-i E bound t), E applies to the overall bound system, not to a possible exp(ipx) constituent. The special relativistic invariant equation seems to exist at the system level, like with the rest mass and one may apparently calculate E in the special rest frame of the large mass and hence find a rest mass of the bound system, i..e large rest mass*cc + mo cc + E, mo is the rest mass of the single bound particle. As a result of the use of the special rest mass frame for the V(x) producing large rest mass, it seems that one does not have to worry about special relativistic constraints on the interaction with V(x), as only one frame is involved. In other words, a bound calculation is about finding a new rest mass for an overall new system. Then this system may be used in Lorentz boosts. One simplifies by working in the rest frame of the mass creating V(x). Perhaps surprisingly then, we note that exp(-iEt+ipx) for a free particle is Lorentz invariant and in fact must be Lorentz invariant for the following reason. Imagine that one wishes to create a probability for free particles which is linked with conservation of momentum. Any p must have the same real value weight because for free particles there is no bias for one p or the other. There is, however, a bias for a p1 and p2. It should have the same probability as a p3 and p4, if p1+p2 = p3+p4. (We use one dimension for simplicity.) As a result, exp(ip) seems to be a probability which respects these conditions. The problem is that one may have p=m1v1=m2v2 hence (E1,p) and (E2,p). With respect to momentum, it is irrelevant if one particle has E1 and the other E2, but not if one does a Lorentz boost. Then, in one case: p→ g(v) p + g(v) v E1 and in the second, p→ g(v) p + g(v)vE2. Here g(v) =1/sqrt(1-vv/cc). In other words, p is the same for each particle in one frame and not in the other which violates conservation of momentum because in frame 1 both particles can complete the same conservation of momentum equation, but in the second frame this is not the case, because each has a different momentum. Thus, exp(ip) cannot hold. One requires a linear form in p which is Lorentz invariant, i.e. exp(-iEt+ipx), we argue (and have done so in previous notes). We see now that momentum probability is linked with space in exp(ipx). The idea seems to be that for free colliding particles there is no preferred frame and the probabilities used must reflect this Lorentz invariance. As a result, we argue that a free particle probability in E,p arises because of Lorentz invariance, but then apply it to a V(x) problem and no longer worry about Lorentz invariance at the interaction level. V(x) with no associated rest mass variables assumes a rest frame for the V(x) producing mass. A Lorentz boost which then invalidates exp(ip), would only apply to the total rest mass of the bound particle system which has no overall momentum. It seems that one does apply the Lorentz transformation to the interaction of exp(ipx)’s with V(x). One is only interested in the overall bound system as a single entity for special relativity. As a result, one may only impose momentum considerations as V(x) is a spatial function and one must link it with exp(ipx). In fact, one may write V(x) = Sum over k Vk exp(ikx) to associate it with momentum hits linked with Vk energies, which already moves away from the idea of special relativity as Vk and k are not linked as a 4-vector. The idea of exp(ip1x)exp(i k1x) = exp(ip2x)exp(ik2x), however, seems to still hold (as it does for free particles), linking these two values (i.e. Vk1 and Vk2) with (p1+k)(p1+k) exp(i (p1+k)x). Lorentz invariance for a particle with rest mass seems to be linked with the idea that one ultimately has a rest frame for the particle. One then establishes a relation between the rest mass in one frame and its new E and p as seen by another frame. One does not, however, need to impose Lorentz invariance conditions on the interaction which creates the rest mass it seems. For example, exp(-iEt+ipx) is the Lorentz form used to describe a free particle, but exp(ipx) is the only portion used to describe its interaction with V(x) in the rest frame of the large mass creating V(x). Furthermore, exp(-iEt+ipx) ensures conservation of energy and p for a free particle, but in an interaction case, that does not happen as the potential may absorb energy.
Based on the law of energy conservation, the author shows 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. Quite on the contrary it provides one 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 energy conservation. The author developed his theory, originally, vis-a-vis gravitational bodies in motion with regards to each other; hence, it is comforting to have both the atomic scale and the celestial scale described on just the same conceptual basis. The approach leads to the de Broglie relationship, and superluminal speeds, yet excluding any exchange of energy, making the basis of quantum mechanics.