Space-Time Events
 $\Large{k}$ $\large{ \delta _{\hat{m}}}$ $\large{ \delta _{\hat{e}}}$ $\large{ \delta _{\theta} }$ 1 +1 0 +1 2 0 -1 +1 3 -1 0 +1 4 0 +1 +1 5 +1 0 -1 6 0 -1 -1 7 -1 0 -1 8 0 +1 -1

Consider a particle P described by a repetitive chain of events

$\Psi ^{\sf{P}} = \left( \sf{\Omega}_{1}, \sf{\Omega}_{2}, \sf{\Omega}_{3} \; \ldots \; \right)$

where each cycle can be parsed into eight sub-orbital components

$\sf{\Omega} ^{\sf{P}} = \left( \sf{P}_{1}, \sf{P}_{2} \ldots \sf{P}_{\it{k}} \ldots \sf{P}_{8} \right)$

so that there is one component $\sf{P}_{\it{k}}$ for each combination of the phase $\delta _{\theta}$, the magnetic polarity $\delta _{\hat{m}}$ and the electric polarity $\delta _{\hat{e}}$ as shown in the accompanying table. This arrangement ensures that P has a fixed relationship with the electric, magnetic and polar axes. It provides a logically sufficient array of sensation to make an account of events that is fully three-dimensional. We can assign $\bar{r}$, a well-defined position, and $t \,$, the time of occurrence to these events without making further assumptions. Definition: Compound events like $\sf{\Omega}$ are called space-time events. Chains of space-time events like $\Psi$ are called trajectories. Particle trajectories are generically written as $\Psi \left( \bar{r}, t \right)$ to emphasize that their events have space-time coordinates. Orientation of a space-time event in quark space. The red numbers indicate values for k as shown the the accompanying table.

This image illustrates the relationship between spatial axes and P's sub-orbital components. The eight components may be composed from some miscellaneous collection of quarks beyond the bare minimum required to establish a spatial orientation. So sub-orbital events are shown as different pie-shaped wedges. Events $\sf{P}_{5}$ through $\sf{P}_{8}$ are out-of-phase with events $\sf{P}_{1}$ through $\sf{P}_{4}$ so they are depicted in a lower tier on the polar axis. Click here for a movie showing all eight sub-orbital events in an atomic-cycle.

## Coherent Interactions

Consider some generic particles $\mathbb{X}$, $\mathbb{Y}$ and $\mathbb{Z}$ that are objectified from space-time events like the ones defined above. And let these particles interact with each other as $\mathbb{X} + \mathbb{Y} \leftrightarrow \mathbb{Z}$. If this process preserves relationships between sub-orbital events such that

$\sf{P}_{1}^{ \, \mathbb{X}} + \sf{P}_{1}^{ \, \mathbb{Y}} \leftrightarrow \sf{P}_{1}^{ \, \mathbb{Z}}$and$\sf{P}_{2}^{ \, \mathbb{X}} + \sf{P}_{2}^{ \, \mathbb{Y}} \leftrightarrow \sf{P}_{2}^{ \, \mathbb{Z}}$and$\sf{P}_{3}^{ \, \mathbb{X}} + \sf{P}_{3}^{ \, \mathbb{Y}} \leftrightarrow \sf{P}_{3}^{ \, \mathbb{Z}}$$\ldots$$\sf{P}_{8}^{ \, \mathbb{X}} + \sf{P}_{8}^{ \, \mathbb{Y}} \leftrightarrow \sf{P}_{8}^{ \, \mathbb{Z}}$

then we say that the interaction is coherent. That is, relationships that determine the phase and orientation do not get mixed-up when a particle is formed or decomposed. Alternatively, we say that an interaction is incoherent if information about phase and orientation gets scrambled during the process. Next step: spatial homogeneity.
page revision: 502, last edited: 05 Nov 2019 12:43