Derivation of the postulates of quantum mechanics from the first principles of scale relativity

Abstract

phone:1 (0)1 45 07 74 03 2 (0)1 45 07 75 38 February 2, 2008Abstract Quantum mechanics is based on a series of postulates which lead to a very good description of the microphysical realm but which have, up to now, not been derived from first principles. In the present work, we suggest such a derivation in the framework of the theory of scale relativity. After having analyzed the actual status of the various postulates, rules and principles that underlie the present axiomatic foundation of quantum mechanics (in terms of main postulates, secondary rules and derived 'principles'), we attempt to provide the reader with an exhaustive view of the matter, by both gathering here results which are already available in the literature, and deriving new ones which complete the postulate list.

Scale Relativity and Fractal Space-Time: Theory and Applications

“…Therefore, in case of simple reversal of the sign of this potential energy, the self-organization properties of this quantum-like behavior would be immediately turned, not only into a weakly organized classical system, but even into an increasing entropy diffusing and desorganized system. We tentatively suggest [96] that such a view may provide a renewed way of approach to the understanding of tumors, which are characterized, among many other features, by both energy affinity and morphological desorganization.…”

“…This relation (which can actually be proved instead of simply being set as a simplifying choice, see [94,96]) is actually a generalization of the Compton relation, since the geometric parameter D =<dξ 2 > /2dt can be written in terms of a length scale as D = λc/2, so that, when S 0 = , it becomes λ = /mc. But a geometric meaning is now given to the Compton length (and therefore to the inertial mass of the particle) in the fractal space-time framework.…”

“…Therefore, just after the measurement, the system is in the state given by the measurement result, which is precisely the von Neumann postulate of quantum mechanics. The Born postulate can also be inferred from the scale-relativity construction [17,94,96]. Indeed, the probability for the particle to be found at a given position must be proportional to the density of the geodesics fluid at this point.…”

“…One may consider several applications to biology of the various tools and methods of the scale relativity theory, namely, generalized scale laws, macroscopic quantum-type theory and Schrödinger equation in position space then in scale space and emergence of gaugetype fields and their associated charges from fractal geometry [69,85,89,6,96]. One knows that biology is founded on biochemistry, which is itself based on thermodynamics, to which we contemplate the future possibility to apply the macroquantization tools described in the theoretical part of this article.…”

Application of Scale Relativity to the Problem of a Particle in a Simple Harmonic Oscillator Potential

“…The derivation of the postulates of quantum mechanics from basic principle of SR [11], is basis of the present work. It shows that quantum mechanical behavior appears without any use of the Schrodinger equation, but as a consequence of the fractality of space-time.…”