I’ve been asked whether there is a single text, that comprises the most essential of Q-Orders. Well – there is not, or not yet. To get an idea of what is and what not yet, may be the below picture helps.
The left side shows in a very simplified manner the tower of mathematics beneath contemporary, classical General Relativity Theory. The right side, as far as I’m aware less solidified and standardized yet, the tower of mathematics beneath contemporary usage of Petri-Nets in Informatics.
I had this picture already in my office about 30 years ago. Now, I thought, if one would like to relate seriously the truly interesting part on top of the left tower with something may be interesting on top of the right tower –meaning by proofs and not by analogies- then one would need a mathematical bridge between both towers, starting already on some quite low-level of both towers. This is, where the work on Q-orders started. They should permit both types of domains –Real and Countable- and should produce one single category of a topology to relate both towers.
To complicate the issue, I discarded partial orders as the funding concepts, as both from Physics and Net-Theory we knew that it are the cycles, that produce basic invariants, on the very end even enable measurement: while we can measure our time as cycles, we can’t measure space without using time. [I know Carlo Rovelli will most strongly disagree].
I did know already the red elements towards the center, they were developed while I was still a GMD. 10 years later, Olaf Kummer and Mark-Oliver Stehr (1) give a quite complete résumé of what has been found out. Yet –though published already in 1976- I was not aware of the proposal of Hawking et. al (1) for a New Topology for Space-Time, i.e. the blue elements towards the center.
May be if I’d known, my life’s history would have been different. Yet I did know already then that the basic invariant of embedding Petri-Nets into (1+n)-Vector Spaces seemed to be the group of conformal transformations. So after leaving GMD in 1985 I accumulated notes and proofs on predecessors for Q-Orders, yet without any serious break-through, still I succeeded giving Axioms 1-5 for Q-orders their current form. (See Going in Circles Part I to III).
After getting back more seriously, about in 2007 or so, and using the resource Internet (+some additional dollars, unfortunately many seminal papers are still sold, while they should be free for humanity), I got across the cited paper from Hawking and a later companion by David Malament (3). Suddenly there was a correspondence already worked-out: a structure on both sides, whose geometrical invariant is the conformal group.
So the only task remaining was to find an axiomatic definition for the Q-Topology, that covered both sides –the Hawking Topology and the Net Topology-. Yet Hawking and Malament use for definitions and proofs many features intrinsically related with lower parts of their tower, starting with the standard definition of paths, which carries automatically the Hausdorff-properties of Reals into the Topology to be defined, over concepts like locally convex, which make sense in a Real-Linear-Vector-Space setting, yet not in the right tower etc. etc. etc. So it took some time to get to the current axioms of Q-Topology (Axiom 6) and Q-Loop-Topology (Axiom 7), which both do only rely on concepts available on both sides of the Big Picture. (The trapped arrow of Time Part I-III).
So what is finished –at least in my electronic scrap-book- is the basic bridge. And I will continue to present its definition and related results during the next weeks. Specifically I will introduce a rich set of models, all by themselves important, for Q-Orders, which will construct something may be close to the fundaments for middle-tower that might be of some use by itself. Done, I’ll proceed to recompile the essays into one single paper, to be published may be through my arXiv account.
Yet I’m fully aware that there is still neither Informatics nor Physics in the picture, which both start on top of their respective towers. The fundamental problem: we have no tools yet to formulate equations or even quantify invariants. Though it’s known that the Hawking-Topology allows to reconstruct the metric and it’s known that the conformal group of transformation corresponds to a single central source of gravity, but as far as I know –and found googleing the Internet- nobody has investigate yet the full way back: i.e. given a Einstein (or Einstein-Cartan) field-equation and posing some reasonable constraints on its right –Energy-Tensor- side, what are the effects on the underlying Casual Structure? (Though Alfonso García-Parrado and Miguel Sánchez (4) may give some hints).
As these tools are still missing, it doesn’t make much of sense either to speculate about the formal relations between the big tower underlying contemporary Quantum-Mechanics and the modest elements presented so far.
There is however the sketch of a work-program to complete and solidify the middle-tower, once finished the above presentation.
- Going downwards, it appears attractive to introduce the concept of a Q-Manifold. The Q-Manifold will be defined using the Complex or Quaternion (alas 2 and 4 dimensional Minkowski-Spaces) as base-space, using q-continuous functions instead of the usual Euclidian ones. The advantage: it appears as if a Q-Manifold is automatically smooth.
- As defining measures on S1 is quite standard and the q-continuous images of S1 correspond to Q-Loops, it should be possible to get some notion of distance for Q-Loop-Spaces.
- Similar remembering that the Tangent-Space may be defined as local equivalence classes of paths (alas Q-loops in our model), it should be possible to have a sort of Tangent-Space for Q-Loop-Spaces.
My only hope: my conditions of work will permit to continue … and the readers of this BLOG don’t get too impatient too soon as truly seminal posts are still month away.
(1) Olaf Kummer, Mark-Oliver Stehr: Petri's Axioms of Concurrency - A Selection of Recent Results , Proceedings of the 18th International Conference on Application and Theory of Petri Nets, Toulouse, June 23-27, 1997, Lecture Notes in Computer Science 1248, © Springer-Verlag , 1997
(2) S. W. Hawking A.R. King and P. J. McCarthy, A new topology for curved space-time which incorporates the causal, differential and conformal structures Journal of Mathematical Physics Vol. 17, No 2, February 1976
(3) D. Malament, The class of continuous timelike curves determines the topology of spacetime Journal of Mathematical Physics, July 1977, Volume 18, Issue 7, pp. 1399-1404
(4) Alfonso García-Parrado, Miguel Sánchez, Further properties of causal relationship: causal structure stability, new criteria for isocausality and counterexamples, arXiv:math-ph/0507014v2