RESEARCH PROGRAM

Research Program

The Concept and Formalization of Constellatory Unfolding as a Cross-Cutting Principle

of an Autogenetic Universe

Albrecht von Müller and Elias Zafiris

Third Book Volume

(In Preparation to be Submitted for Publication to Springer)

On the Foundations of Ontophainetics and the Role of Primes in the Present

Main Idea in Relation to Quantum Ontophainetics

Ontophainetics of the Quantum of Action: Whither the Double Slit Interference Pattern?

An Autogenetic Perspective on the Symplectic Nature of the Weyl-Heisenberg Group

The philosophical theory of an "Autogenetic Universe", formulated by A. von Müller, proposes new "categorial foundations'' for science aiming to overcome the inherent limitations, incompatibilities and structural pitfalls of the current scientific paradigm. In order to overcome these structural pitfalls, we need (a) to unearth the categorial foundations of our present theories, (b) to re-think our notions of time and reality, and that, in turn, allows us (c) to re-conceptualize some of the big, unresolved issues in modern science in a fundamentally novel perspective. The basic conceptual underpinning of the "Autogenetic Universe" theory is based on the notions of "constellatory self-unfolding" and "strong self-referentiality". Autogenesis means precisely that something unfolds out of itself, within itself and towards itself.

According to Richard Feynman, one of the leading figures in the quantum theoretic description of the physical word, the central mystery of quantum mechanics revolves around the double slit experiment. The particular challenge posed by this experiment necessitates the appropriate deciphering of the interference pattern observed on a screen when no measurement takes place beforehand at any of these slits.

We would like to take this challenge seriously, and qualify precisely the origin and constitution of the type of the quantum interference pattern recorded on a screen. The proposed approach leads to a fundamentally different categorial framework of conceptualizing time and reality. For this purpose, it turns out that it is essential to move beyond the standard functional analytic frameworks employed in quantum theory, and make essential use of synthetic tools and methods from harmonic analysis and algebraic topology, admitting a precise conceptual interpretation, without modifying any of the principles of quantum mechanics.

The unveiling of the invariants, literally making them appear in light, called "epiphaneia'' in ancient Greek, in its philosophical and technical meaning as a spectrum of distinguishable appearances following the analysis of distinctions incident to the architectonics of adjoining and modular substitution, requires for its articulation a well-defined framework of ontophainetics, called "chroma'', pre-conditioning the process of spectral resolution and qualifying the "ana-phenomenon" in the time-space of the present. In particular, the process of constellatory unfolding of the chromatic invariants requires tempering within uniformly-partitioned chromatic intervals, underlying in turn the notion of an objective uniform probability distribution in relation to a directly inaccessible domain, like the quantum one. We will show that the progressive unveiling of the chromatic invariants in the time-space of the present - characterizing genuine novelty in this manner - is subordinate to the appropriate qualification of the "primitive roots of unity" in the complex arithmetic domain, and thus congruent with the novel features unveiled through the relative primes.

At a first stage, one of the major turning points in our work - in relation to the standard probabilistic or path integral approaches - is a careful scrutinization of the temporal aspects involved in the quantum theoretic framework. These aspects pertain to the role and function of a global irreducible geometric phase - interpreted in terms of a holonomic and chromatic "symplectic area" of unveiling in the present- with respect to a maximal commutative modular algebra of quantum observables. It is the manifestation of such a global geometric phase through the Weyl-Heisenberg group of unitary transformations that gives rise to the canonical commutation relations. In this manner, it is precisely this pertinent algebraic group - including its continuous and discrete articulation - that encodes the cipher of the proposed ontophainetics in the quantum domain.

Henceforth, we show that the nilpotent non-commutative Weyl-Heisenberg group via its representation in the quantum state space holds the key for deciphering the interference pattern. In particular, it plays the role of the "golden mean mediator'' between pure "apeiron non-commutativity'' and "pre-factual commutativity'' as recorded at the interference pattern of the screen.

"Apeiron non-commutativity" pertains to the non-reversible ordering of unitary actions capable of giving rise to a temporal entanglement bond in the not-directly experiential quantum domain where this bond is established. It can be expressed in topological terms through the non-local linkage formed by the "Borromean rings'', and its concomitant re-presentation in terms of the commutator of the non-Abelian free group in two generators by means of unitary group actions. We prove that the reflection of this bond in the quantum state space is faithfully re-presented by the group commutator of conjugate unitary actions in the Weyl-Heisenberg group, thus unveiling the origin of non-commutativity in the quantum domain.

In this manner, we deduce that "pre-factual commutativity in statu-nascendi" - a pre-requisite for quantum measurement - pertains to the "epiphaneia of the present'', modelled as an Abelian Galois covering cross-section of the Weyl-Heisenberg group reflection of this entanglement bond. In turn, this specifies objectively the "time-space of the present'' underlying the interference pattern imprinted on the screen, qualified thus chromatically as an invariant holonomic ana-phenomenon with respect to the established ontophainetic platform.

We demonstrate explicitly that this epiphaneia does not bear any sequential ordering of events, and is characterized by a pattern of double periodicity, emerging by the commutativity of modular conjugate quantum observables subject to a topological integrality condition of the Dirac type. In particular, the interference pattern is derived as the recorded homological boundary of the joint symplectic action of two one-parameter unitary groups infinitesimally generated by two conjugate observables, i.e. the position and the momentum of an electron beam.

The ubiquity and difficulty of deciphering the quantum interference pattern requires overthrowing all classical preconceptions referring to particles or waves and stems from the intertwining of non-classically displayed aspects of time, like non-sequential ordering, constellatory unfolding, joint periodicities and self-referentiality, being at the root of the objective indistinguishability depicted by the non-classical probabilistic framework of quantum mechanics. Nevertheless, such a conceptual stance is indispensable since it probes the actual taking place of reality occurring in a primordial form of time, the not yet sequentially structured through facts epiphaneia of the ``time-space of the present''. Concomitantly, both the sequentially ordered aspect of time and the factual aspect of reality appear as emergent traces that come into being only after reality has actually taken place.

Visualization of the Distribution of Relative Primes and the Emergence of Patterns of Novelty in the Ontophainetics of the Complex-Valued Disk of Harmonic Analysis

Concept of the Visualization

The basic concept is the autogenetic emergence of relative primes as roots of the unit circle on the complex plane-epiphaneia. The constellatory self-unfolding of this process as we ascend the ladder of roots, or equivalently, going deeper and deeper in unveiling the resolution spectrum of this chromatic ontophainetic platform, gives rise to the pertinent patterns of genuine novelty out of the emergence of new primes. The video aims also in capturing the subtle bidirectional relation between strong self-referentiality and autogenesis through the visualization of constellatory self-unfolding in its threefold ekphrasis, namely, in, out, and towards the identity:

(i) “in” targets the emergence of genuine ontological novelty, through iterations of the process of unfolding bearing the power to delve into deeper and deeper manifestations of this strongly self-referential structural identity/unity;

(ii) “out” targets the absence of an external cause triggering the multi-faceted unfolding of the strongly self-referential identity/unity; and

(iii) “towards” targets the reverse, vertically encompassing internalization of all these emergent ontological manifestations, explicated through unfolding, as energetic qualifications of the strongly self-referential identity/unity.

The video displays the process up to the 30^th roots, where the pattern of unfolding can be easily grasped. Then it utilizes the rotational symmetries of the layers in the process of unfolding to explicate the whole diversity of possible visual patterns. In this manner, the progression in time through the autogenetic emergence of new relative primes is visually associated with the genuine novelty arising from the novel patterns.

The towards the unity aspect of the autogenesis, which seems to be the most challenging conceptually aspect to grasp, is also attempted to be visually represented in the video. After the utilization of the rotational symmetries, the process is visually reversed as a way of zooming in self-referentially towards the identity, encompassing the unfoldment manifested up to the 30^th roots. This takes place through the polygonal articulation of the relative primes starting from the top layer and going backwards, meaning internalizing self-referentially. This brings new light on the old Archimedean motif of articulating the transcendental nature of the circumference of the unity circle through a chain of polygons, the inner meaning of the method of exhaustion. It also provides a bridge, regarding the role of the relative primes in the present articulation, with the rudiments of the Galois theory of groups and prime Galois fields.

At a further stage, the latter can be utilized in their function as models of Abelian Galois covering cross-sections of the Weyl-Heisenberg group, explicating and manifesting the process of reflective appearance of a quantum entanglement bond on the chromatic ontophainetic platform in the shape of an interference pattern.