Scale Transitions as Foundations of Physics

The following sections are included:

• Contents
• Foreword

At the end of a previous book (Mazilu et al., 2019), we have summarized the points of its continuation, ‘in no particular order’, as we expressed it there. Here, however, we need to specify the choice of a certain order, and the manner to follow it in this part of the work for, obviously, otherwise we cannot properly accomplish the work. And in order to achieve this goal we need to expound its very basis. This is the point where the planetary model with space extended physical components comes in handy for an intuitive regulation of our line of thoughts and ideas…

The idea of hyperbolic geometry of the second kind, mentioned above, brings about another important moment of human knowledge centered around the personality of Albert Einstein, which further clarifies the position and the profound implications of his results from Cosmological Considerations. In hindsight, this even adds to the elucidation of position and implications of the very general relativity theory with respect to classical physics, from the point of view of a general natural philosophy, as we already signaled before. Due to its tremendous importance for our knowledge, we feel like having to dedicate to this moment a substantial portion of this presentation, involving this and the next chapters, for some fairly detailed explanations and proper conclusions. On its historical side, this moment can be safely related to the name of Kurt Gödel, and was, in fact, conceived on the basis of classical natural philosophy (Gödel, 1949, 1952). It had — and actually still has! — a broad impact on the human intellect at large, reflected in a vast literature that covers the many aspects of the problem, and ranges from highly specialized theoretical works down to science fiction subjects. In our opinion though, this moment has a basic natural philosophical significance for human knowledge, originating in the fact mentioned in passing by us before, namely that the space needs to be defined as a state by a manifold of positions. This will be succinctly explained in what follows, using that concept of reverse interpretation briefly touched by us here, just a few pages before. On this note, it should appear as no surprise that it triggered a fundamental revision of the very classical Newtonian mechanics and cosmology alike (Heckmann and Schücking, 1955, 1956). This is actually, by and large, the major idea we are following in our presentation…

In our times, the uniqueness of the universe of natural philosophy of any persuasion, shapes each and everyone of the theories built to explain this universe. Some of these theories, however, carry the imprints of the necessary scale transitions, incidentally imposed by classical dynamics along its way of explaining the physical structures of the universe. As we have shown before [Chapter 1, equations (1.1) ff., and discussion] even the classical dynamics of the Kepler problem itself requires the mixed Mariwalla scale invariance: infrafinite ranges in time transition and finite, or even transfinite ranges, in space transition. This is, in fact, the very nature of the original Newtonian calculus of fluxions. However, one cannot say that, while not recognizing the idea of scale transition of any kind, and therefore categorizing the universe as a unique extant world, our spirit has not encountered similar warning situations in its journey: occasionally it has been forced to remarkable, though sometimes peculiar, attitudes. Such a notorious case, quite rich in lessons for us here, is the theory of regularization of the Kepler problem, whereby the harmonic oscillator plays, again, a central part. This is, indeed, a case of intellectual productions specifically related to scale transitions in the unique universe of the classical natural philosophy. Involving intimately the harmonic oscillator, these productions help us in further elucidate the nature of results of the Hamiltonian theory of Berry and Klein which, as we have seen, are connected to this physical structure…

As we have seen before, the Kustaanheimo–Stiefel regularization could not be accomplished directly, but via a four-dimensional parameter space. This fact is due to the very definition of analyticity: according to Stiefel’s theorem, if the analyticity is expressed by the Cauchy–Riemann linear relations, there is no possibility of regularizing but for dimensions 1, 2, 4 and 8 (Stiefel, 1952). As long as we think of regularization through the property of analyticity, and keep up with the linearity as a property sine qua non of the Cauchy–Riemann conditions, this fact is pathologic: the three-dimensional case of regularization has to be considered indirectly, for it is not among those of the Stiefel’s list. In fact, as long as it does not concern the theory of fluids, the theoretical physics always used, preferentially, some other conditions, than the Cauchy–Riemann ones, in order to deal with forces in the three-dimensional case. These conditions are closer to the spirit of the potential theory of forces…

Kurt Gödel insists on the suggestive concept of a compass, in order to emphasize that the general relativity is physically built upon the observation of motion, and as long as we give up the idea of force, the motion is accomplished in a setting like the one of a ship on the waves. In this context, Einstein’s natural philosophy is based on the idea that the compass of inertia has the same indications as the compass of gravitation. The Newtonian principles of classical dynamics may be taken, indeed, as suggesting this identity (Einstein, 2004) but, in fact, the concept of force as defined by Newton according to Kepler laws, certainly does not suggest it. In order to sustain this point of view, and to give alongside an example of the difference between the two compasses, we shall try to cast into analytic form the Corollary 3 of the Proposition VII from Book I of Newton’s Principia, which we reproduce here for the benefit of our reader: The force by which the body P in any orbit revolves about the center of force S, is to the force by which the same body may revolve in the same orbit, and the same periodic time, about any other center of force R, as the solid SPRP 2, contained under the distance of the body from the first center of forces, and the square of its distance from the second center of force R, to the cube of the right line SG, drawn from the first center of force S parallel to the distance RP of the body from the second center of force R, meeting the tangent PG of the orbit in G. For the force in this orbit at any point P is the same as in the circle of the same curvature. [(Newton, 1974, p. 51); our Italics]…

Before anything else, let us assess the Planck’s fundamental hypothesis represented by equation (5.41) in a little more detail. We take it as representing the statement: the energy of the fundamental physical structure of a universe — whatever this may be — should be itself a statistical variable, characterized by an exponential distribution with quadratic variance function. Insofar as that fundamental physical structure of a universe has to be used in the process of counting for establishing the matter density, we need to know what is its possible physical structure, and how is it related to the concept of scale. In Planck’s original case, it was the harmonic oscillator, on which we need to dwell a little longer, for it has the essential properties needed in characterization of that concept…

It is, we think, the proper moment to concern ourselves with the issue raised by Henri Poincaré, and detailed by us in the Chapter 1 of this work: the problem of gravitation. As it was conceived by Poincaré [see equation (1.20), and discussion], if it is to draw a parallel between gravitation and light as two forms of action at a distance, the problem of gravitation is connected to that of propagation. In Chapter 1 we deepened this connection, considering the classical property of gravitation of acting radially — a property valid as such for both the classical physics and general relativity — which leads to a groupal description involving the SL(2,R) action in two variables with three parameters [see equation (1.21) and the discussion following it]. The ‘proper moment’ here, to treat this important issue, comes with the observation that the absolute geometry may be able to provide a proper way of treatment, by a property of matter, ‘dual’, as it were, to that of light [see equations (6.70)–(6.71), and discussion]: formally, the matter can also be described by a limit velocity, like the light itself…

We uphold the idea that from the point of view of physics, the best approach of the theory of surfaces is the one in which we use the Cartan’s ideas, exploiting the properties of differential forms (Mazilu and Porumbreanu, 2018) [see also Mazilu et al. (2019, Chapters 7 and 8)]. This approach of the theory in question is, in our opinion, the closest to any physical point of view that might be related to the existence of surfaces. The present chapter is an illustration of this statement, taken to the ‘next level’ so to speak, of the affine theory of surfaces. We reserved this place for such an illustration, expressly in order to have at our disposal the Barbilian theory of the Riemann spaces associated to families of cubic equations, previously elaborated here [see equation (6.7) in Chapter 6]. As we see it, this item of knowledge is the main piece in the puzzle of interpretation. It is, in fact, its pivot, allowing mainly for what we have called in this work the reverse interpretation: the interpretation of an ensemble of ‘detached structures’, as a continuum. This reverse interpretation involves, as we said, a further step in the theory of surfaces, namely the affine theory. Its clearest concept, expressed in differential forms of course — otherwise a physical theory of surfaces could not exist! — was provided, according to our views, by Cheng and Yau (1986). The following version is a reproduction of that material [see also the classical work of Shiing-Shen Chern and Chuu-Lian Terng, which inspired, in fact, many other valuable works in the spirit of the one just cited (Chern and Terng, 1980)]…

According to Berry–Klein invariance theory of forces, the central forces of Newtonian type play an essential part in the understanding, through scale transitions, of the construction of the universe we inhabit. The first example of scale transition is implicit to Newton himself, and is referring to the time scales: the validation of central forces, acting permanently, by collision forces, acting instantaneously. Here the action of force is time scaled: permanent action means action in finite or transfinite time intervals, while instantaneous action means action in infrafinite time intervals. The notorious case of transition from finite to infrafinite is represented in mathematics by differential calculus, invented by Newton specifically to serve the purpose of physics. Such a transition is based on continuity [see especially Newton (1974), Volume I, Book II, Lemma II], even though the calculus may not depend on this condition, as in the contemporary fractal calculus…

The following section is included:

• Bibliography