A Glimpse of the Big Picture

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 Big Picture

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.  

1. 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.
2. 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.
3. 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.

The Axioms 
   

(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

Going Backward, Going Forward – Part I

In the following 3 posts we will go backward and forward through the seven axiom-sets, on one side to get a better feeling for Q-Orders, on the other to relate Q-orders with classically known concepts. The final post will show that the Hawking-Topology is a Q-Order.

Let’s start in this post with some considerations about the Axioms 1 to 5 and their relation to partial orders.
Axiom 1   Q-Relation

 Lemma F.1   Q-Relation 
  
states that with the proper definitions, any partial order satisfies Axiom I.

Axiom 2 has similar effects for partial orders has it has for Q-Orders:
Axiom 2   Q-reduced

Lemma F.2   Q-Reduced
i.e. we don’t permit isolated points and points that could not be told apart by using the partial order are considered the same. Further on we will consider only reduced partial orders. Please note this condition is weaker than the standard distinguishing conditions.

Axiom 3.1 imposes an additional condition on partial orders, every sequence of three points a<b<c can be completed to have 4, yet Axiom 3.2 follows already from being a partial order.
Axiom 3   Q-regular

Lemma F.3   Q-Regular
  
Please observe that using brute force to prove F.3.2 requires to analyze (3*8)^4=331,776 combinations of binary conditions, which due to internal dependencies may be reduced to less then 6144, but still a substantial quantity, which again for a proof may be further reduced by applying internal symmetries. Instead of wasting three pages with either resulting valid combinations or detailed analysis of symmetries, we just put a small picture and invite the reader to do the latter her- or himself.
 
Suggestions for a 5 lines-proof are obviously welcomed. We let the brute force method being applied by a computer. With respect to F.3.1, we assume henceforth that all partial orders a regular.

Intentionally –it was designed that way- Axiom 4 turns out to be a property of reduced, regular partial orders.
Axiom 4   Q-Order

Lemma F.4  Q-Order
  
Again we will not reproduce the pages of a formal prove, but invite once more to have a look on some pictures.

We break the analysis  of Axiom 5 into two pieces, first the construction of the orientation-set, then the connectivity conditions.
Axiom 5   Q-Orientation

Lemma F 5.1  Q-Orientation

Actually we construct the two orientation-sets, then the F 5.1.3 is direct consequence of the construction, while F 5.1.4 is proven in few steps. This establishes the expected result F 5.1.5: partial orders have an orientation. But please note: the orientation is not the order of partial order itself, as orientation means always oriented cycles.

The final lemma of this post transcribes the connectivity-condition for Q-orders into the language of partial orders.
Lemma F 5.2   Q-Connected

As preliminary result of our comparison Q-Orders versus Partial Orders, we obtain that under relatively weak constraints –Lemmas F.2 and F.3- plus the connectivity-condition –Lemma F.5.2- Partial Orders are models for the Axioms 1-5 for Q-Orders. This will ease the task to establish Hawking-Spaces –alas Causal Sets- as Models for Q-orders, because we can simply rely on the underlying partial-order and are almost done. By default these partial orders satisfy our weak constraints, while –advancing results- the connectivity-condition is one of their key-features.

Yet second this condition points already on a set of minimal constraints that a partial order must comply to become a candidate to be related by whatever structural-knowledge-preserving mechanism to the structure underlying GRT. It turns out that these have to take the form of Second-Order-Predicates, i.e. they can not be expressed as simple statements about relations among points.

Third it appears as if Q-Orders provide a proving-mechanics almost as strong as the transitivity respectively monotony from partial orders, essential not only for proofs but already for constructions like induction or convergence etc., yet without the disadvantage to have to believe that there is a universal beginning and a universal end for all and everything, a non-scientific hypothesis as it can’t be proven nor disproven. (See also Fotini Markopoulou (1)).

As a technical advantage, it will allow us to talk about systems with cyclic behavior –at least during some time-, not only –see Oscillator- a fundamental model in Physics but essential to introduce measurements or without cyclic clocks there is no time and without time there is no measurement at all, as there are no means to measure space as such.

This ends our first considerations about Q-Orders and Partial Orders.

1 Fotini Markopoulou, An insider's guide to quantum causal histories (1999), http://arxiv.org/abs/hep-th/9912137