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Quantization as a Universal Process

On the gnoseological side, the meaning of Bohr’s theory about the nuclear atom is twofold. First, it shows that we can recognize the past in the present simply through the light produced by an atom. Secondly, it shows that, according to Classical Mechanics the light source is not the whole atom or the electron, but mainly the nucleus. Indeed, in a Kepler problem dynamically describing both the electron in the nuclear atom and the solar system, we must have some initial conditions of the motion of electron or planet, referring to the velocity and position that these had at the time when the motion started. In even more general terms, the Mechanics would say that the electron or the planet couldn’t have the motion they show now if this motion did not start in a precisely given place with a precisely given velocity. It is, nevertheless, absurd to even think that we might be able to find these initial parameters for the electron; but the very same goes just as well for the planet in the case of solar system.

However, Mathematics tells us something very interesting: if in the present we observe that the motion of the electron or planet is done along closed orbits of an elliptical form, we can read, at least partially, those initial conditions in the eccentricity of the orbit, i.e. in the ratio between its two diameters. This is the precise meaning of our statement that we can recognize the past in the present. Now, going a little further, the eccentricity of an elliptical orbit can be represented as a vector in space which is physically limited to a region closely surrounding the attractive central material point of the Kepler problem: nucleus in the atom or the Sun in the planetary model. This means that, mathematically, the parameters of the orbit of the electron or of the planet are dictated by the position in space of the vector giving the eccentricity, within the central region mentioned above. Thus, if we think that the set of all possible initial conditions is covered by a certain past motion which, obviously, cannot be presently attested because it’s long gone, then we can at least recognize the trace of this motion in our present. The present form of spiral nebulae and the behavior of the atoms according to Bohr’s rules, bear then witness for this very fact. Thus, the structure of the solar system is just as “accidental” as that of the atom. From this point of view the analogy between the two is as perfect as it should be, and goes both ways as it should go, according to the space and time scale invariance of Newtonian equations of motion (Mariwalla, 1982).

The geometrical form of physical limitation of the eccentricity vector is a Cayleyan or Absolute Geometry equivalent to Hyperbolic Geometry of Lobatchevsky. Based on the ideas initiated by Nottale (Nottale, 1993), Jean-Claude Pissondes (Pissondes, 1999) insisted lately upon the fact that the natural geometry of Quantum Mechanics must be a Hyperbolic Geometry. Pissondes’ idea is, to our perception, possible by the generalization of the metric form, which can be traced back to the fact that the metric form of the Absolute Geometry is not a quadratic form but is given by the general Laguerre logarithmic formula. It seems to us that this idea of Pissondes is indeed the only natural one if considered from a certain historical perspective. However it is natural even from quantization point of view. Indeed, the quantization is a natural consequence of the applicability of Hyperbolic Absolute Geometry. This is how the man perceives the world, and this is how the matter and light can be structured.

In view of this, it is however no less impressing to see in certain scientific circles an “inverse analogy” if we may use this word, whereby the ideas of Bohr quantization are brought in the realm of planetary systems. One can say that, for once, the science follows its natural way: an analogy works not only by the structuring one single scale of contemplation of Nature, but also in reverse, by re-structuring the original scale of contemplation. This is, in our opinion what happened with macroquantization, and the message it carries is then that the quantization in general is the mark of a Newtonian type of forces, inversely proportional with the square of distance between material points, no matter of the scale of contemplation of Nature we consider those forces.

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