Archimedean Synchronization of the Quantum Phase Space in the Present

The quantum transition probability bears the internal symmetry of a process, which resolves a moment of factual time in terms of a symplectic area-bounding loop in the projective Hilbert space. This symplectic loop constitutes the root leading to the time-space of the present as categorially distinct from the sequential ordering of factual time, and thus, requiring an ontophainetic platform for its manifestation. The basic idea is that this symmetry is elucidated through the Archimedean synchronization method applied to the quantum phase space, which requires a categorially distinct ontophainetic domain for its articulation in the present independent of the measurement of events. Otherwise, the symplectic area-bounding loop symmetry remains only implicit, and essentially ignored, due to the insistence of interpreting the temporal variable as a classical one, overlooking the subtle interrelation of the complex with the symplectic structure of the projective Hilbert space of quantum states. This leads to the qualification of quantum amplitudes in terms of symplectic spinors which doubly cover the corresponding complex state vectors obtained through squaring in accordance to the Born rule of transition probability between quantum states. Their interrelation can be simply formulated in terms of null vectors of a 3-d Minkowski space, which proves to be instrumental for the Hermitian and unitary representations of the symplectic group.

Main Theorems of Quantum Ontophainetic Synchronization

A. The quantum transition probability assignment (Born Rule) constitutes an equiareal transformation from the annulus of symplectic spinors to the disk of complex quantum state vectors, which makes it equivalent to the equiareal measure-preserving synchronization transformation of Archimedes.

B. In view of the Archimedean areal rectification method of the synchronization disk in terms of spirals applied to the quantum phase space (De Broglie's Energy/Frequency relation) the probability simplex pertaining to quantum transition probabilities should be integral in units of the quantum of action (Planck's constant).

Harmonic Parallel Transport at the Centre of Synchronization in the Present

and the Equiareal Transformation on the Unit Synchronization Disk

The disk of all synchronized momenta records synchronically and invariantly the latent area of the spinorial annulus of quantum amplitudes with respect to a center that is not explicit in advance - to be identified with the present with respect to an ontophainetic screen. Since the tangential component in the direction of motion following the ordering aspect of time does not have any impact on the rate of change of the spinorial annulus area, it all depends mechanically upon the phase of unitary rotation with respect to this center, as well as upon the frequency rate of this rotation, that is, how fast the winding takes place.

Spinorial Area Measure of Time in the Present

At the categorial level of the time-space of the present, time is conceptualized as a helix unfolding continuously and orthogonally to the complex pane, such that its projection is the unit synchronization disk on the complex plane. The measure of time in this context becomes spinorial area inherited by the symplectic quantum conjugate variables equal to the area of the rectifying Archimedean triangle modulo ℤ . But, since spinorial area is interpreted as the squared modulus of the transition probability segment of the corresponding state vector being doubly covered by the underlying symplectic spinor, we conclude that the measure of time on the helix should be counted modulo 2ℤ . Equivalently, for each winding, time amounts to a spinorial-area preserving transformation in the symplectic variables. This is precisely the ontophainetic quality that is encoded in the structure of the Heisenberg group and the concomitant uncertainty relations.

Double Helix and Helical Standing Wave of Moment Resolution in Prime Harmonics - De Broglie Quantum Phase Wave

The physical idea is that a single phase of the unit circle on the complex plane is resolvable by an integer spectrum of harmonic frequencies, or equivalently, energies, giving a precise meaning to the poly-strophic quality of the Archimedean spiral. The effect of this manoeuvre through the integer domain of harmonic frequencies is that we can now qualify the integer windings as a quantum spectrum recorded on the imaginary axis of the complex plane via branches of the complex logarithm, corresponding to the observables. In this manner, the criterion of equivalence, or quantum indistinguishability, can be imprinted on the imaginary axis in terms of the angular temporal interval of one whole period 2π times the integer harmonics. Relative prime harmonics lead to patterns of genuine novelty in the present.

Quantum Effect of Synchronization: Non-Locality and Area Entanglement

Since the area of the synchronization disk in the categorial domain of the present is spontaneously and synchronically covered, it exemplifies the main characteristic of non-locality and spatial entanglement in the symplectic conjugate variables. In other words, it is objectively indistinguishable in terms of separate spatial points or parts, and as an effect it can be accessed only probabilistically. In this sense, the symplectic quantum geometry exemplifies a global characteristic that is not analysable into separable parts, thus the entangled area is symplectically rigid. The Archimedean measure-preserving equiareal bijection of the qubit sphere on the synchronization disk in the present gives the information that the entangled area of this disk remains invariant under proportionate re-scaling of the symplectic conjugate variables with any positive real number.

The areas under the hyperbola show that by proportionate re-scaling of both symplectic conjugate variables x and y through a positive real factor λ, the invariance remains intact. Thus, product areas arecalculated under the rectangular hyperbola and preserved under reciprocally co-related extension/contraction actions in the symplectic variables. In the limit that the range of values of one of these variables becomes sharp the range of the other becomes totally uncertain, but their product remains constant. In effect, this is precisely the transcription of the invariance of the area of the synchronization disk of Archimedes leading to the uncertainty relations.

From Synchronization to Areal Entanglement: The Non-Squeezing Theorem

There does not exist any symplectic transformation that can squeeze the symplectic ball of radius R through the circle in the plane of the conjugate symplectic variables of radius smaller than R. The theorem applies in the case of a general 2n-dim symplectic phase space, and is known as Gromov's theorem, or theorem of symplectic rigidity.

The physical idea is to qualify the non-squeezing theorem via the notion of the synchronization disk emerging through Archimedes’ measure-preserving bijection in relation to any 2-real dimensional symplectic plane of conjugate variables. In other words, the symplectic rigidity of any symplectic plane is the result of the synchronization effected through harmonic parallel transport to the center of the disk in the present that makes the enclosed area entangled, and thus physically objectively indistinguishable in terms of separable parts. This leads naturally to the interpretation of Planck’s constant in terms of a minimal invariant non-analysable and non-squeezable symplectic area giving rise to the quantum of action under the condition of modularity and integrality of the transition probability simplex.From this viewpoint, we qualify the uniqueness theorem of Stone and von Neumann pertaining to the Schrödinger model, and in particular, realize the possibility of unitary equivalence between the Schrödinger model, and the direct sum of copies of this model in relation to the continuous group action of R ⊕ R. Although the first requires infinite dimensionality of the pertinent Hilbert space, this is not the case pertaining to the second due to the integrality condition, which above all leads to a precise qualification of Planck’s constant in terms of a minimal invariant symplectic area scale tracing its roots to the measure-preserving Archimedean projection to the synchronization disk in the present qualifying the Born rule.