Are There « Sequential Descriptions » in Physical Systems Analogous to DNA Ribbons in Living Cells ?

According to recent developments of arithmetical relaters, the « sequential descriptions » of some physical systems might have some analogy with DNA ribbons in living cells. But such descriptions might have a physical existence only in « space-time-imbrication » reference frames. After recalling several properties of arithmetical relaters, we explain in the framework of this formalism why these descriptions disappear when an arbitrary observer tries to observe them. A stabilized arithmetical relater (AR) expresses in a structural way the adaptation of a natural system to its environment. Underlying ideal structures of the model appear as Lie structures. The coupling between a system and its environment involves a notion of time which usually cannot be reduced to classical Galilean or reh1ti vistic time. It evokes the numbering of successive symbols in a narrative. The structure of this narrative that gives rise to anticipatory aspects is destroyed by an arbitrary observation. A sequential description contains objective and non-objective elements and depends of the level of imbrication. At macroscopic scale, the objective part of a sequential description is extracted by means of a «Peaceable Working Observer». He does not perturb the system because he knows and respects the structure of imbrication. Then, in simple cases, roles of system and observer can be permuted. The displacement of a rudimentary system in two-dimensional space or the non quantum harmonic oscillator are typical examples; these models use degenerate AR's. Surprising results appear when the narrative is dealing with a system moving in three-dimensional physical space: the solution involves either the use of spinors or an algebraic extension of the formalism. However, the notion of «Peaceable Working Observer» is inadequate at the quantum level. We propose to extract objective descriptions by means of a «spinorial pilot» who works at the imbrication level K/2 when the physical system is described at the imbrication level K. This approach is applied to the quantum harmonic oscillator. Our starting point is nilpotent Lie algebra of dimension 3. A rule of recurrence is deduced from the sequential description. This rule is applied at the level K to an arbitrary point of a « linearized » one-dimensional lattice. Different « linearized » states, belonging to the same level K, are associated only to one point. Without additional information, this rule cannot give automatically certain states at other level K' > K. The « non deterministic part » of the spreading of states, deduced from this rule, appears to be a sequential description of the physical interaction. The deterministic part is a discrete approximation of a derivative operator. The whole description may be considered as a discretization of the Schrödinger equation. This conclusion results from a tedious analysis using an extension of Morse-Thue sequences. Bases of the formalism are presented in two books Relateurs arithmétiques, volumes 1 and 2 (Editions Belrepere, 1997). Some details are given in a recent book Objectivite et pilotage spinoriel, (same publisher, 2000, 443 pages). But there is no information in this book about relations between nilpotent Lie algebra and our model of quantum harmonic oscillator, nor is a comparison between non quantum and quantum models. These points are new. Following these results, we suggest an analogy between peculiar sequential descriptions at quantum scale and DNA ribbons in living cells. This other point is new too. However, such a suggestion is deduced from arithmetical relators describing the environment by only one variable. The presence of an arbitrary observer increases the number of environment variables and destroys the imbrication structure. On the contrary, if this imbrication structure is preserved by hard chemical and geometrical constraints and if a peculiar observer respects them too, the « sequential description » might be observed. There is no inconsistency with the results of quantum physics because this peculiar observer can observe only what he knew before.