A Quantum-Like Description of the Planetary Systems

Abstract

The Titius-Bode law for planetary distances is reviewed. A model describing the basic features of this rule in the "quantum-like" language of a wave equation is proposed. Some considerations about the 't Hooft idea on the quantum behavior of deterministic systems with dissipation are discussed.

Nambu at Work

“…(4) which for large n and e 2λ > 1 produces a Titius -Bode type law. Comparing with experimental data on various planetary or satellite systems, the parameter s, unlike Planck's constant which it replaces, does not take a universal value, but changes from system to system, However the parameter λ remains essentially the same [25].…”

“…The relevance of a Schrödinger-type equation for such systems has been observed by Albeverio, Blanchard , Høegh-Krohn [24] and others. These issues are reviewed and expanded in a paper by Scardigli [25]. In the simplest way, the appearance of a Schrödinger-type equation can be seen as follows [25].…”

“…These issues are reviewed and expanded in a paper by Scardigli [25]. In the simplest way, the appearance of a Schrödinger-type equation can be seen as follows [25]. In a Bohr-like model let us require the usual equality of the gravitational attraction and centrifugal forces, GMm/r 2 = mv 2 /r (1) and then impose the new type of "quantization" condition J/m = vr = se λn (2) instead of the Bohr quantization…”

Order out of Randomness: Self-Organization Processes in Astrophysics

“…Alternative mechanisms besides gravitational N-body resonance self-organization have been proposed also, such as: (i) Hierarchical self-organization processes based on sequential resonance accretion (starting with the accretion of massive objects first) and 2-body resonance capture of planetesimals in the primordial solar nebula (Patterson 1987); (ii) plasma self-organization driven by the development to minimum energy states of the generic solar plasma during protostar formation (Wells 1989a(Wells , 1989b(Wells , 1990; (iii) susequent mass ejections into planetary rings around a central rotating body with magnetic field properties predicted by stochastic electrodynamics (Surdin et al 1980), (iv) retarded gravitational 2-body resonance, i.e., macroscopic quantization of orbital parameters due to finite gravitational propagation speed (Gine 2007); or (v) quantization of orbital periods in terms of the quantum-mechanical Schrödinger equation (Perinov et al 2007;De Neto et al 2004;Scardigli 2007;Chang 2013).…”

On the canonical quantization of the electromagnetic field and the emergence of gauge invariance

“…One possibility in such a framework, is represented by loss of information: it may happen that a deterministic theory has a dissipative character, so that, on a given time scale, the dynamics leads to a reduction of the number of degrees of freedom of the system, and the result is the emergence of ordinary quantum states with non-local character. Several examples of such a scheme have been discussed [1]- [5], in particular the system of two coupled damped-amplified harmonic oscillators (Bateman system) [6]- [10], turned out to be a paradigmatic one, in the sense that the quantum harmonic oscillator here emerges out of the dissipative dynamics of the original (deterministic) system [2,4]. A common feature of these models, is the fact that the Hamiltonian of the (primordial) deterministic system is not positive-definite implying a spectrum unbounded from below: by imposing the constraint of positive definiteness, one then obtains a genuine quantum system with unitary evolution and a stable groundstate.…”