Vladimir, if there were a diagram of what you wrote I believe it would assist thinking about your ideas-- especially (for me at least) if the diagram were based on a language of objects and arrows.
Georgina-- I now see that the paper I submitted to this contest is too terse. More words are needed to clarify. So I will make the attempt. I also hope the result will be a useful example of a diagram written in the language of objects and arrows. To begin:
By expanding the model of proper time discovered by Minkowski (now basic to General Relativity) the paper shows a mathematical connection between General Relativity and Quantum Mechanics, specifically by linking the Born rule of Quantum Mechanics to proper time of General Relativity.
However this is not a link written in the language of Quantum Field Theory. Rather, it is a link between GR and QM that involves multiple mathematical languages-- non-wellfounded sets, nonstandard analysis, situation theory, channel theory, informationalism, game theory-- and then, the resulting, expanded model of time can be diagrammed using objects and arrows. It turns out to be compatible with something Einstein said:
"The only reason for time is so that everything doesn't happen at once"
Here's how I found this model of time:
Years ago, I used the programming language Smalltalk80 and a diagram based on objects and arrows to simulate a complex and expensive automated manufacturing system-- before it was built. The engineers in charge wanted to know from a simulation whether or not the system would work in all anticipated situations. The answer was not obvious by looking at the layout of machine drawings, because the system had to process a complex schedule of parts.
The objects in the diagram I used to model this system were of two types: (a) places, each drawn with something like a circle, and (b) transitions, each drawn with a rectangle. Hundreds of these objects had to be drawn and connected by arrows in order to model the system.
The result looked like a board for playing a game. There were rules for drawing this game board. There were also rules for moving the tokens on the game board. (The tokens represented the parts and information conveyed between the elements of the system).
Here are the rules for drawing the game board:
(1) Every arrow starting from a place must end at a transition.
(2) Every arrow starting from a transition must end at a place.
Here are the rules for moving tokens on the game board:
(1) When the places "upstream" from a particular transition (as determined by the direction of its attached arrows) become filled with tokens, remove one token from each upstream place.
(2) After the code written for that transition is completed, place a token into each of its downstream places.
Smalltalk (today, the programming language Pharo) allows objects to spawn blocks of parallel code that wait for signals from "semaphores." The block of code will only run when the semaphore signals it. The result is Not lines of code that run one after another exactly as they are written in the source text. Instead the sequence of executing the blocks of code depends on when the semaphores trigger the code-- in "real time." The semaphores look at upstream places from the transition object which, by means of the attached arrows in the diagram, own them. So instead of a sequence following how the source code was written down, in the simulation the sequence of running the blocks of code was determined by the tokens placed on the game board.
This worked-- except when the parts entering the system according to the simulated schedule backed up. I found that in this case, when a transition had "fired" (by means of its underlying semaphore), another token in an upstream place could fire yet another copy of the transition object. So instead of just one of that particular transition existing in the simulation, as a result there could be more than one. The transition was no longer unique. There existed multiple copies of it in the simulation. To anthropomorphize a bit, the transition had no unique self identity. It had multiple identities-- which is an oxymoron except in spy circles and psychotherapy.
In this sense things could "happen all at once." As in Einstein's quote I had to add something to the simulation in order to prevent "everything from happening at once."
The solution is in the attached diagram. I drew one of these kinds of places for just the transition itself for each transition where it was possible for "everything happening at once." To characterize this feature in words, each transition with the possibility for "everything happening at once" needed a place to itself and only for itself, to keep it from firing when it was already firing, and prevent "everything happening at once." I realized that I had to simulate time itself in order to make the simulation work properly.
When running the simulation, these places for each transition's "self" use looked like they were constant-- as if they were always constantly filled with a token. But each of these "self identity" tokens was really, according to the rules, being taken off and put back into its place faster than the eye could see. It was this mechanism, where each transition had a place for itself so that only one instance of itself existed, that kept "everything from happening at once."
Years later at a workshop at Stanford called "The Business Applications of Situation Theory," I learned from Jon Barwise about non-wellfounded sets.
It struck me that the "place for self" I had used years before in the simulation had a structure which looked like a non-wellfounded set:
Eq. 1: unique_constant_identity = (changes, unique_constant_identity).
"Unique_constant_identity" in eq.1 produces the stream of "changes" evident in all of the downstream places from the transition that owns the unique place.
This perspective on proper time is not new. In fact it's ancient. Parmenides, whom Zeno was defending with his story about the race between Achilles and the tortoise, had described the same situation in the fragments left to us of his poem titled "On Nature." The poem tells of a journey from the domain of belief into the domain of knowledge . The person making this journey must be carried in a chariot.
In eq. 1 the chariot is "unique_constant_identity"-- the constant presence of a unique self existing in the present, which is always constantly supporting and evident to a conscious person for as long as that person is alive. The unique constant self that's experienced, and the time in which this self is experienced-- i.e. the present-- are always the same to that person as long as the person is conscious and lives, just as for the place reserved for "self" in the above diagram, and just as for "unique_constant_identity" in eq. 1. The road on which the chariot travels is "changes." Like the old saying, wherever you go, there you are!
Although Minkowski discovered the modern representation of proper time which has become the basis of General Relativity, as far as I'm aware the ancient representation of proper time is due to Parmenides. The ticking of the clocks Einstein imagined to obtain relativity are like the trees and changes along the path travelled by Parmenides' chariot. But Einstein left the constant chariot itself implicit, and not explicit, in his mathematics of relativity. Eq. 1 make this constant chariot, "unique_constant_identity," explicit in the mathematics.
Even if not supportive of numerical calculation, this mathematical connection between General Relativity and Quantum Mechanics supports logical models of dark matter and dark energy.
The diagrams in the following paper show the origin of this idea.
https://docs.google.com/file/d/0B9LMgeIAqlIET3B2NEE2MmxDOWM/edit?usp=docslist_apiAttachment #1: Petri_net.pdf