Synchronization of Non-Commutative Unitary Actions

Since reality is characterized by the three different chrono-ontological formats the "time-space of the present'' in statu-nascendi has to be understood in its temporal binding potential not in the sequential event temporal order of the "continuum of the real line'' that models only the factual aspect of reality, but in another chrono-topological form. The sought-after chrono-topological binding process is characterized homotopically and group-theoretically by the ability of irreversible actions to compose and synchronize with respect to a fulcrum that is freely instantiated up to conjugation. This is expressed by the commutator of the free non-Abelian group generated by these actions.

All non-Abelian free groups of actions are isomorphic to each other, and all of them exist as subgroups of the fundamental non-Abelian free group generated by two non-commutative actions. In the quantum theoretic formalism the notion of a temporal action is formalized in terms of an one-parameter unitary group. Since there exists a bijective correspondence between one-parameter unitary groups and quantum observables, which infinitesimally generate the former, the ordering of these unitary action paths is reflected on the ordering of observable actions. An immediate consequence is that, due to the non-commutativity of conjugate quantum observables, like the position and momentum, the ordering of the corresponding unitary actions is irreversible. It is precisely the quality of irreversibility in the homotopic re-presentation of these actions as loops non-deformable to each other that, if based on the same fulcrum state - up to conjugation - can synchronize by linking proportionally. The proportional linking of these non-contractible loops at a fulcrum state - up to conjugation - is what characterizes the synchronization of two corresponding unitary actions generated by conjugate observables giving rise to phenomena of temporal non-locality. All these phenomena exemplify the symmetry of a non-reducible indirect and non-chain-like topological link re-presenting the synchronization independently of distance.

Entangled Symplectic Area

The non-local synchronization link generated by two conjugate unitary actions is represented in quantum theory in terms of a central element of the Heisenberg group. This is what gives rise to the ubiquitous type of non-commutativity expressed by the commutator of the Heisenberg group in the Weyl representation. In other words, the group commutator, expressed in terms of Planck's constant, is the imprint of synchronization of conjugate unitary actions, and thus, the imprint of temporal non-locality and interference.

The notion of a central element should be also interpreted in terms of a based loop - up to continuous deformation - that bears a particular characteristic which allows the measurement of an appropriate attribute qualifying the effect of synchronization and interference. This characteristic is what qualifies the time-space of the present in statu-nascendi, that is, as a pre-conditioning ontophainetic platform through which such an observable attribute can be recorded. The necessary condition is that the based loop corresponding to a central element in the Heisenberg group, which encodes the pertinent synchronization link, is transcribed to an area-bounding cycle in space whose measure translates the effect of temporal non-locality, or equivalently, synchronization, in terms of the entangled symplectic area bounded by this cycle. Therefore, the latter is qualified as a homological boundary in space that encloses the entangled symplectic area invariantly. In this manner, spatial entanglement is the transcription of the pertinent synchronization - pertaining to temporal non-locality - into spatial terms with respect to symplectic area bounding cycles.

Group-Theoretic Modelling of the Transcription from Temporal Non-Locality to the Spatial Measure of Entangled Symplectic Area

The major homological characteristic of the entangled symplectic area, bounded ontophainetically by means of boundary cycles, is that it is integral in units of the quantum of action, expressed by Planck's constant. Equivalently, the measures of the entangled symplectic area instantiated by two conjugate unitary actions differing by integers are homologically equivalent. Thus, they give rise to the same global geometric phase factor that constitutes the measurable spatial measure of temporal non-locality and synchronization with respect to its homological Abelian symplectic shadow instantiated by the joint areal action of any two conjugate observables. The notion of a central element of the Heisenberg group - that specifies the quantum commutation relations - interpreted as the imprint of synchronization according to the proposed schema, indicates that the substance of the Abelian symplectic shadow, where spatial entanglement takes place, is of a homological descent. In this way, "the past'' and "the future'' exist paratactically in their potentia to convey holonomic meaning with respect to the ``time-space of the present'', that is, in terms of a global geometric phase factor, and not hypotactically as in the linear sequential chain model of time pertaining solely to the factual layer of reality. This is in remarkable consonance with the interpretation of the geometric phase, or equivalently its underlying homological measure of entangled symplectic area, as the memory of a quantum system.

From Proto-Time Non-Locality to Proto-Space Entanglement

Proto-time engulfing temporal non-locality (and thus non-sequential) is synchronizing unitary complementary observable actions in quantum mechanics. Proto-space is the spectral Abelian shadow of this synchronization that manifests as non-local entangled area. The proto-space is non-local and of a homological nature, that is, it pertains to the means of comparing and proportioning entangled area with respect to a topological boundary, at each present (time-space of the present, not space-time). This is possible due to the non-squeezing or irreducibility of entangled area, expressed in integer units of Planck’s constant. This quantum of non-irreducibility of entangled area transcribes the memory of the temporal non-locality at each present (what maintains the self-referential core of the constellatory self-unfolding) into proto-space homologically with respect to a boundary (i.e. the setting of the Stern-Gerlach experiment). This is the essential core of ontophainetics – the proof that there always exists such an irresolvable quantum of entangled non-local spectral area (proto-space) as a transcription of temporal non-locality in spatial terms at each present.

Leibniz's Ontophainetic Screen

We have to postulate that there is a screen in this dark room to receive the species, and that it is not uniform but is diversified by folds representing items of innate knowledge; and, what is more, that this screen or membrane, being under tension, has a kind of elasticity... This action would consist in certain vibrations or oscillations, like those we see when a cord under tension is plucked ... (G.W. Leibniz, New Essays on the Human Understanding, H, 12, 1)