Triadic Logic of Spacetime (Part Two): Quantum logic of orientation

Triadic Logic of Spacetime (Part Two): Quantum logic of orientation1 Let’s attempt to trace the emergence of a model spacetime from the threefold essence of light underwriting the Special Theory of Relativity [for further background on this approach, see The Proximity of Light (Rogers, 2004)]. We consider three in proximity: the Same, the Other and the Third Party, each of which is another to the others and none of which is the same to another. 2 Differance 1 Identity 3 Each of the three is in immediate proximity with its neighbor—one-for-another in a continual process of substitution. There are three distinct indices, or origins, labeled “l”, “2” and “3” in the diagram. These three indices correspond to three distinct instantiations. 1 Appendix to A Physicist’s Guide to [Hegel’s] Phenomenology of Spirit: Resonance, disambiguation and the genesis of spatial orientation, (Rogers, 2013). Available: https://www.academia.edu/3547178/A_physicists_guide_to_Hegels_Phenomenology_of_Spirit_Resonance _disambiguation_and_the_genesis_of_spatial_orientation Timothy Rogers 2013 Page 1 of 9 To see the working of this trope, suppose we establish “1” as the instantiated index, the origin. Then “1” is in a relation of proximity with the two others (here called “2” and “3”). This relation of proximity we call identity. “1” is identical to “2” and “3”, substituting itself for each. However, between “2” and “3”, there is a proximity that is inaccessible to “1” and we can call this difference. In the distinct instantiation of the three, there is both identity and difference. This trope frames interiority through a process of return. In return, “1” substitutes for “2” which in turn substitutes for “3” which finally substitutes for “1”. In return there is a traversing of the inaccessible difference that is the proximity of “2” and “3” according to “1”. This gap becomes the gap or clearing in which creation manifests—the synchronicity of light and word. Moreover, because “2” and “3” can substitute one-for-the-other in the inaccessible gap, there is an indeterminateness at the core of this threefold relatedness. In The Proximity of Light (Rogers, 2004), we explored how this threefold trope manifests temporality through iteration. Iteration is the circular movement around the three, which returns through difference to a different same. Iteration is particular. There is change in the loss and return of proximity, which we postulate as temporal. Timothy Rogers 2013 Page 2 of 9 time ? Resonance is equalizing, the identity in difference through back-and-forth. Unlike the particularity of iteration, resonance is "whole" or “at-once”, the same difference, which we postulate as spatial. space ? From the infinite iterability of return we defined a temporal element (Dt)2 , which we represented as an open circle Timothy Rogers 2013 Page 3 of 9 Likewise, the infinite back-and-forth of resonance brings a connector between two places, which we will call (Dx)2 and represent by a solid line. There are three different resonances corresponding to proximity between the Same, the Other and the Third Party, which we labelled as (Dx)2, (Dy)2 and (Dz)2. (Dx)2 (Dy)2 (Dz)2 Together these connectors form a 3-space. By postulating an equivalence between the resonance of 3-space and the iteration of temporality, we arrived at the figuration of a lightcone. Timothy Rogers 2013 Page 4 of 9 2 1 3 We considered the following two ways to complete a loop, which we postulated as the same difference. In the first way we pause at circle 1 (infinite iterability), then jump (in a finite number of back-and-forth motions) to circle 2 where we pause, then we jump to circle 3 where we pause, and finally back to circle 1. Another way is to infinitely resonate between 1 and 2, then, without pausing, infinitely resonate between 2 and 3 and finally, without pausing, infinitely resonate between 3 and 1. This is a method of combining connectors (resonances, space?) with pauses (return, time?). We represent this equivalence as the fundamental equation of the light cone: (Dx)2 + (Dy)2 + (Dz)2 = (Dt)2 Of course we have been anticipating the solution from the beginning so this is not a derivation as much as a heuristic argument (previously I have used the term étude). Yet something quite interesting has happened. We have postulated 3-space as a system of connectors (or line elements) but these connectors do not have directionality. So we have measure but we have no orientation. This is captured by representing the finite element as Timothy Rogers 2013 Page 5 of 9 a squared quantity. What is missing is the spontaneous symmetry breaking that might establish direction (and therefore orientation). If we are to arrive at a model spacetime we need to disambiguate the temporal and spatial elements. Notice, however, that this disambiguation is prefigured within an overarching threefold relatedness. And the form of that relatedness is each-for-the-other and not each-for-itself. The disclosure of orientation requires surfacing a richer understanding of non-causal association—specifically symmetry creation—as the basis of spatiality. Orientation comes from the disambiguation of pair symmetry. Pair symmetry is already prefigured in the circularity of the temporal element which has two-fold equivalence – clockwise and counterclockwise traversal. Let’s represent the breaking or disambiguation of pair symmetry by two oppositely pointing arrows2: This is the fundamental connector, which brings about an emergent twofold relatedness. 2 We might have chosen something less visually leading, such as “+” and “-“, except that the relatedness becomes visually derivative which is not the intention here. The arrows and the connector are one. Additionally, the arrow will become helpful once we define the concept of spin. However, at this point there is no spatial “direction” intended by the arrows. We are only trying to represent a difference operation that is binary or two-fold. Timothy Rogers 2013 Page 6 of 9 The synchronization of orientation comes about through return. However, with the emergence of pair symmetry, return is now frustrated such that two cycles are needed to recapture identity. The structure of return has the form of a spinor or knot. The figure folds in on itself, as it were, to disclose a deeper interior structure to origin. The inward folding generates the fundamental unit of action – a single rotation or cycle of return – which we call spin. Spin is an emergent property of spacetime that comes from the disambiguation of light. Suppose, now, that we synchronize an origin with a specific spin orientation. This will simultaneously index one of the three axes; let’s call the indexed axis the z-axis. There is a two-fold symmetry which is also broken in choosing a specific spin orientation and this enables us to label the direction of spin along the z-axis. For example, we might call it spin up and label it as | ↑ >z Timothy Rogers 2013 Page 7 of 9 It is important to recognize that the broken symmetry, namely orientation, comes from a three-body process of synchronization. We—the observers as it were—are imagined to be in phase with the specific spin orientation that has been indexed. Of course, instead of we-as-observers, it may be some other origin which is brought into phase with the indexed spin orientation – this process is one of interaction not cognition. What is interesting now is that, for the other two axes, the spin is mixed. | ↑ >x and | ↓ >y | ↓ >x and | ↑ >y Synchronizing spin-up along the z-axis at the same time causes mixing of spin along the remaining two axes (let’s call them x and y). Synchronization of spin along another index axis (eg. x or y) would at the same time de-synchronize the spin along the z axis. The apparent collapse of spin along the x-axis might rather be seen as re-phrasing of threebody interactions. Because of its threefold essence, the disambiguation of light automatically generates the quantum properties of spin and this can be seen to come from the ontology of relativistic spacetime rather than a property of an individual or isolated particle. My hypothesis is that spin is an emergent interiorization of light from which the fundamental unit of in-formation is created. A new form of difference operator, intrinsic to the “geometry” of relativity, which is absent from Euclidean geometry. The archetypal process through which orientation (an interior structure) is externalized as rotational symmetry is the creation of correlated photon pairs. Correlated photons provide a connector between two origins in spacetime which brings their orientations into Timothy Rogers 2013 Page 8 of 9 instantaneous relationship. When the correlated photons (which are a single entity) interact with the external “world”, symmetry is created. This symmetry brings two origins (potential centres of action) in spacetime into a particular, instantaneous relatedness of space-like resonance. This relatedness manifests, for example, in measurement. Timothy Rogers, 2013 [email protected] Timothy Rogers 2013 Page 9 of 9