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Other “Macro” Facts from “Micro” Analogy

Now, that we got a glimpse on this gnoseological message carried by Bohr quantization in the macroscopic scale of contemplation of Nature, let’s signalize some other remarkable theoretical facts from the microscopic world which can be considered as having a macroscopic counterpart.

The geometry of the space region admissible for this vector of eccentricity is a Hyperbolic Geometry which can be represented in complex numbers. Such a complex number represents a system of generalized elliptical coordinates that can be attached to a tensor having precise physical meaning. Indeed, the material point represented by the atomic nucleus, by the Sun or even by the nucleus of a galaxy, reveals to the Universe a variable physical structure depending of the point of view, so to speak. According to Hertz’s philosophy, no matter of the particular form of this structure, it is given by agglomerations of material particles between which forces act. We cannot bring a verdict on these forces, the way a verdict was brought upon Newtonian forces: one cannot say for instance if they are central or not, if they have intensity inversely proportional with the square of distance, etc. What we can say for sure is that they yield some resultants which can be represented by a tensor of tensions, and this is actually all that matters. Indeed, any physical structure of a material point can be characterized by equilibrium of forces between its material particles, and a set of material particles in equilibrium define a tensor of tensions (Williamson, 1894). This tensor varies with the structure of the material point.

The variation in question has a form to which the modern Physics of Elementary Particles often makes reference. Indeed, the above tension tensor determines an orthogonal reference frame by its principal directions. For instance in the case of the solar system this reference frame is given by the plane of planets’ orbits and a perpendicular direction to this plane. The values of the tensor along the three directions of the reference frame are then the generalized elliptic coordinates. The evolution we are talking about here can then be described by the variation of these generalized elliptic coordinates, with respect to a fixed reference frame. Indeed, when we measure the intensity of the tension tensor associated to a structure, we actually measure the so-called orthogonal invariants having physical meaning related to octahedral planes of the fixed reference frame (see the figure). These invariants represent the magnitude of the normal and tangential components to the octahedral plane of the vector of “position” given by the three principal values of the tensions. The evolution is then described by a rotation of the tangential component of the position vector in the octahedral plane, and can be represented by a transformation within the Kinematics generated by the Hyperbolic Geometry associated with the eccentricity of the orbit (the so-called Bäcklund transformation).

There are eight such evolution possibilities, because there are obviously eight octahedral faces. They can reveal themselves, one or another, as the case may be, in situations where the physical structure of an atom, to take the classical case, is exposed to variation, as in the collision experiments which validated the so called “eightfold way” discovered by Murray Gell-Mann and Yuval Ne’eman in 1964 (Gell-Mann, Ne’eman, 1964). The fact that the eight possibilities are considered objective material points in current Particle Physics is of no consequence for our argument, for if it is to talk about physical structure here, this is rather given by the Feynman partons – material particles in the sense of Hertz. It is only the experimental form of these structures which matters here, and this is given by the “eightfold way”.

The very same philosophy can be applied at the level of solar system or galaxy. For instance, in the case of solar system the structure exposed by the Sun to its planets is approximately “constant”, the only variation being that which produces the energy discharged by Sun in the form of light and heat. In general strokes the principle acting at the microscopic scale is therefore also acting at the astrophysical scale.

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