Mesoscopic quantum mechanics

• “Classical” thermodynamics may be split in three levels : macroscopic, mesoscopic and microscopic [1]. Let us imagine for a moment that, totally ignorant of the very existence of the microscopic level, Boltzmann would have discovered half empirically the statistic laws of the mesoscopic level, as so as some methods to take avantage of them within calculations. Would we had to believe that the maxwellian distribution was unavoidable ? Would one had been able to imagine the existence of an underlying (microscopic) level where the very notion of temperature may be non-existent (non maxwellian systems).

In a certain manner, it seems rather probable that this is more or less the kind of situation which occurs for quantum mechanics. When one becomes nearer to the quantum (mesoscopic) level, one is inclined to extend here the schemas assimilated at the macroscopic scale, particularly the existence of the “classical” notion of space-time (the “dressed” space-time, which can be named by analogy “maxwellian vacuum”). One therefore extends to hypothetical “naked” elementary interactions occuring in a space-time which maintains a large amount of these aspects, without considering to question back about the very existence of some of them.

• At the microscopic level, “to be at some point in the space” simply means “interact in a particular manner with the surroundings” ; there is probably no space-time “between interactions” (no more than there is some water between the water molecules) : the notion of metric, such as those considered in general relativity, has a meaning only at the macroscopic or mesoscopic level.

In this sense, I disagree with the argument used by Weinberg [2] against Mach's principle. He reasons upon a possible space anisotropy caused at the level of an atomic nucleus, by considering that it is logically the weakless imaginable according to the very small distances involved in this case. Whereas it is true that Mach's still imprecise formulation may allow to think so, nothing in it indicates the manner according to which the different surrounding masses participate in inertia. On one side, in the relativistic interpretation, it is now known that the energies must be considered in place of masses ; on the other side, from the point of vue of relativistic instantaneousness, it appears that all energies in Universe may as much contribuate whatever their distance may be since at this level of reasoning the notion of distance does not exist yet (as a matter of fact the distant ones are very more numerous). This is not equivalent to consider that there exists a privileged reference framework for space description, in so far as it is considered at this level of reasoning that there is no space-time in the “metrical” meaning (this one being only a tool convenient to “statistically” describe some effects of the whole interactions).

• From another point of vue, this kind of interpretation leads to consider the Higgs boson (connected to the spontaneous symmetry breaking in renormalized jauge theories [3]) as a “phonon” in the network of corresponding Feynmann's diagrams ; but is it not the same case for the graviton ?

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References :

1. see as example :
       “les affres d'un passage à la limite”, L. Desvillettes and F. Golse, La Recherche, n° 346, october 2001 ;
       “le comportement des gaz : d'une limite à l'autre”, L. Saint-Raymond, Pour la Science, n° 324, october 2004.

2. “Gravitation and Cosmology”, S. Weinberg, ed. Wiley, § 1.3 and 3.7, 1972.

3. this may be possibly described by the interaction with the Higgs boson ; see as example (the phenomenon is quoted in many articles) : “Qui attrapera le Higgs”, La Recherche n° 364, may 2003.



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