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

Revised: The Design Goals for Q-Order Axioms

There have been month of silence in this BLOG, as I was running into to serious trouble beyond the elementary Axioms for Q-Orders, i.e. those that equip Q-Spaces with a suitable topology. More over it turned out that –at least for the moment- an additional axiomatic relation may be needed: while the relation Q clearly models the conformal invariant structure –the Topology- of Space-Time, it misses the projective invariant part, which only jointly as shown by Weyl long time ago fix also the metric. The work isn’t done yet. Nevertheless I would like to present the current stage with its stabilized Axioms for Q-Orders and the first intents to manage the additional relation.

We are looking for a combinatorial framework that, in an essential way, includes the structure of Space-Time as a continuous model on one side and the structure of of Petri-Nets as a finite (countable) model on the other.

Essential means that physically different Space-Times and logically different Petri-Nets shall have different models and that different models produce different Space-Times and different Petri-Nets respectively.

For Space-Times a seminal contribution of S. W. Hawking¹ introduced a unique combinatorial structure –a partial-order– attached to Lorentzian Manifolds with some additional restrictions, that up to conformal mappings defines the manifold (review² on Causal Space-times). David Malament³ showed how this combinatorial structure alone, under suitable conditions, is sufficient to reconstruct Space-time up to conformalty.

For Petri-Nets since 1973 there as been a systematic effort⁴ to detect the underlying combinatorial structures, specifically in the theory of Concurrency or causal structures defined by event-occurrence systems.

The problems

1. the mentioned Space-Time models in their definition make a heavy use of concepts typical for the continuous world, like Hausdorff-spaces as basic model-domain or using properties borrowed from Linear Algebra, all which as such can not be transported into the finite/countable domain.

2. the mentioned Petri-Net models -namely concurrency-theory- require countable models to work and therefore as such a not suited to express all the technical concepts as used in continuous models. On structural level, there is no Linear Algebra, hence appears on first sight impossible to express concepts like convex let alone tensors or more complicated constructs.

3. Both models depend on Global Partial Orders even when expressing purely local concepts, a slight contradiction with the basic idea of General Relativity as something locally defined.

The ideas for solution

1. Both model domains use Paths respectively Curves as a basic building block, where Curves in both domains model trajectories -world-lines- of particles (more precisely potential trajectories see⁵). All expressed relations and properties can be re-written using only curves and the relations among points as defined by curves.

2. As Petri pointed out quite early⁶, on partial orders there exists a generalization for the concept of Dedekind-continuity and -completeness that allows for countable models, yet if applied to full-orders produces the known results. Crucial are two types of points, closed and open, while retaining the idea that the emerging topologies should be connected.

3. A little bit later Petri proposed the separation relation {{a,b},{c,d}} -an unordered pair of unordered pairs- as the basic order-producing relation. This relation expresses the separation of 4 points on a line, and is well defined on any Jordan-Curve, open or closed, i.e. there is no difference between a line, may be with suitable compactification, and a circle.

4. A careful analysis of the original article from Hawking, specifically analyzing the relation between local time-like cones, which form the base for the topology, the definition of regular paths in that topology and their relation to time-like curves, allowed to eliminate the reference to linear concepts like convex and to define local time-like cones and their properties using only combinatorial concepts.

5. This revision in turn demanded a revision of concepts in Petri-Nets. While in the original model the open elements are conditions and the closed elements are the events, and likewise sets border by conditions considered open while sets bordered by events closed, we need exactly the dual: events and event-bordered sets are open, conditions and condition-bordered sets are closed. It should be noted that for countable structures -Petri-nets are normally assumed to be countable- both sets -open and closed- define the same dual Alexandrov⁷ Topology. However already the comparison of Dedekind continuity between total orders and half-orders alas Occurrence-Nets shows that the common type of elements in both -the non-branching conditions- must be closed.

6. In a Hawking-space all points are closed. Therefore it was necessary to overcome the initial interpretation of Einstein as if world points would correspond to physical events. They do not! If the Hawking-space models the loci -the geometry-, then a physical event can not have an exact place as Quantum-Mechanics tells us. A similar observation made decades ago Pauli⁸. Curiously enough, in this interpretation nothing ever happens in Hawking-Space as there are no events. To have events we must coarse grain first.
   

7. Likewise a too naive interpretation by Net-Theory of GRT had to be abandoned, as if each world-point branches into infinite many world-lines. Actually a world-point summarizes the whole time-like pre- respectively post-cones as such and not individual lines. This is the essence of the construction of regular paths by Hawking and the distinguishing conditions from Malament.

Based on the above I obtained the Axiom-sets shown in the picture below.

Some models for Q-Orders are:

(1) Occurrence-Nets (with the above change and some additional requirements) as subclass of Petri-Nets
(2) The Real Numbers (but not Rationals nor Integers) (Q-order is derived from classical order)
(3) The Unit-Circle S1 (and the Circle Group) (but not n-cyclic Groups) and the Real Line (Q-order is derived from the relation among four points)
(4) The Minkowski-Space and the Quaterions (Q-order is derived from Q-Topology)
(5) The Causal structure of a Lorentzian manifold as defined by Hawking and others (Q-order is derived from the relation among four points on a time-like curve)

1 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
2Alfonso García-Parrado, José M. M. Senovilla, Causal structures and causal boundaries, arXiv:gr-qc/0501069v2
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 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
5 David B. Malament, Classical Relativity Theory, arxiv.org/abs/gr-qc/0506065v2
6 Petri, C.A., Concurrency. Lecture Notes in Computer Science Vol. 84: Net Theory and Applications, Proc. of the Advanced Course on General Net Theory of Processes and Systems, Hamburg, 1979 / Brauer, W. (ed.) --- Berlin, Heidelberg, New York: Springer-Verlag, 1980, Pages: 251-260
7Not to be confused with the Alexandrov Topology used by Hawking
8Pauli, Vorlesungen in Turin über nichtlokale Feldtheorien in Google-Books http://books.google.com/books?id=NU9OUj-f8cYC&hl=es Page 34 ff.

The Challenge

Albert Einstein

(1) Die allgemeinen Naturgesetze sind durch Gleichungen auszudrücken, die für alle Koordinatensysteme gelten, d.h. die beliebigen Substitutionen gegenüber kovariant (allgemein kovariant) sind.

Es ist klar, daß eine Physik, welche diesem Postulat genügt, dem allgemeinen Relativitätspostulat gerecht wird. Denn in allen Substitutionen sind jedenfalls auch diejenigen enthalten, welche allen Relativbewegungen der (dreidimensionalen) Koordinatensysteme entsprechen. Daß diese Forderung der allgemeinen Kovarianz, welche dem Raum und der Zeit den letzten Rest physikalischer Gegenständlichkeit nehmen, eine natürliche Forderungen ist, geht aus folgender Überlegung hervor. Alle unsere zeiträumlichen Konstatierungen laufen stets auf die Bestimmung zeiträumlicher Koinzidenzen hinaus. Bestände beispielsweise das Geschehen nur in der Bewegung materieller Punkte, so wäre letzten Endes nichts beobachtbar als die Begegnungen zweier oder mehrerer dieser Punkte. Auch die Ergebnisse unserer Messungen sind nichts anderes als die Konstatierung derartiger Bewegungen materieller Punkte unserer Maßstäbe mit anderen materiellen Punkten bzw. Koinzidenzen zwischen Uhrzeigern, Zitterblattpunkten und ins Auge gefassten, am gleichen Orte und zur gleichen Zeit stattfindenden Punktereignissen.

Die Einführung einen Bezugssystems dient zu nichts anderem als zur leichteren Beschreibung der Gesamtheit solcher Koinzidenzen. Man ordnet der Welt vier zeiträumliche Variable zu, derart, dass jedem Punktereignis ein Wertesystem der Variablen entspricht. Zwei koinzidierenden Punktereignissen entspricht dasselbe Wertesystem der Variablen ; d.h. die Koinzidenz ist durch die Übereinstimmung der Koordinaten charakterisiert. Führt man statt der Variablen beliebige Funktionen derselben, als neues Koordinatensystem ein, so dass die Wertesysteme einander eindeutig zugeordnet sind, so ist die Gleichheit aller vier Koordinaten auch im neuen System der Ausdruck für die raumzeitliche Koinzidenz zweiter Punktereignisse. Da sich alle unsere physikalischen Erfahrungen letzten Endes auf solche Koinzidenzen zurückführen lassen, ist zunächst kein Grund vorhanden, gewisse Koordinatensysteme vor anderen zu bevorzugen, d.h. wir gelangen zu der Forderung der allgemeinen Kovarianz.

The general laws of nature are to be expressed by equations, which are valid for all coordinate systems, ie. are covariant (generally covariant) in relation to the arbitrary substitutions.

It is clear that a physics, which meets this postulate, satisfies the general relativity postulate. Because in all substitutions anyhow are also those contained, which correspond to all relative motions (three-dimensional) of the coordinate systems. The fact that this demand of the general covariance, which takes away the last remainder of physical objectivity of space and time, is a natural demand, results from the following consideration. All our time-spatial stating always end in the determination of time-spatial coincidences. Would for example all events consist only in the motion of material points, then at least nothing would be observable but the meetings of two or several of these points. Also the results of our measurements are nothing else but stating such movements of material points on our yardsticks together with other material points respectively coincidences between watch-hands, points on clock-faces and observed point-events, taking place at the same time and at the same spot.

The introduction of a frame of reference serves for nothing else but for easier description of the totality of such coincidences. One assigns four time-spatial variables to the world such that the value-system of the variables corresponds to every point event. Two coinciding point events correspond to the same value-system of their variables; ie. the coincidence is characterized by the equality of the coordinates. If one uses instead of the variables any function of them as new coordinate-system, so that the values are unambiguously assigned to each other, the equality of all four coordinates also in the new system is the expression for the time-spatial coincidence of the two point events. Because all our physical experiences can be traced back after all to such coincidences, at first glance no reason exists to prefer certain coordinate systems over others, ie. we obtain the demand of general covariance.¹

(2) Man kann gute Argumente dafür anführen, dass die Realität überhaupt nicht durch ein kontinuierliches Feld dargestellt werden könne. Aus den Quanten-Phänomenen scheint nämlich mit Sicherheit hervor zugehen, dass ein endliches System von endlicher Energie durch eine endliche Zahl von Zahlen (Quanten-Zahlen) vollständig beschrieben werden kann. Dies scheint zu einer Kontinuums-Theorie nicht zu passen und muss zu einem Versuch führen, die Realität durch eine rein algebraische Theorie zu beschreiben.

Niemand sieht aber, wie die Basis einer solchen Theorie gewonnen werden könnte.

There are good reasons to suggest that nature cannot be represented at all by a continuous field. From quantum phenomena, it could be inferred with certainty that a finite system with finite energy should be described completely by a finite set of numbers (quantum numbers). This seems not in accordance with a continuum theory and obliges to attempt to describe reality by a purely algebraic theory.

But nobody has any idea of how to obtain the basis for such a theory.²

Carl Adam Petri established the ideas for a theory that may satisfy both Einstein postulates³.⁴

1 Albert Einstein, Die Grundlage der allgemeinen Relativitätstheorie, Annalen der Physik, 49 (1916), Pages 776,777

2 Albert Einstein, Grundzüge der Relativitätstheorie (1924), Reprint Springer Verlag, Page 163

3 Carl Adam Petri State-Transition Structures in Physics and in Computation. Int. Journal of Theoretical Physics, Vol. 21, No. 12, 1982, Pages: 979-992

4 Carl Adam Petri: On the Physical Basics of Information Flow. Petri Nets 2008: 12