### Causality in Discrete Models of Spacetime

#### Posted by John Baez

*guest post by Gavin Wraith*

Excuse me pestering you with a query about an article in the Scientific American — ‘The Self-Organizing Quantum Universe’ by Ambjörn, Jurkiewicz and Loll. I found the article interesting but frustrating. It gives hints but no definite description of the mathematics involved. The references given were evidently written for a readership of physicists not mathematicians.

The diagrams of the article suggested, at least to me, that the Causality Principle meant enriching simplicial complexes (no “semi”) with a partial order on the vertices — i.e. one considers finite subsets of a partial order, closed under intersection. From the language of the article, which talks about dimensions “emerging” from some statistical procedure, I take it that no prior restrictions are made on the simplicial dimension. The barycentric subdivision functor on simplicial complexes extends naturally to the partially ordered case once one defines a simplex $a$ to be less-than-or-equal to a simplex $b$ if every vertex of $a$ is less-than-or-equal to every vertex of $b$. I could find no clue from the article what the measure on the set of isomorphism-classes of these things was supposed to be.

My ideas are evidently very hazy and naive. The article gives the impression that starting from as little as possible (but what?), the authors manage to produce (but how?) something very like de Sitter space. Does any of this stuff approach any of your purlieus (or should that be purlieux)? Have simplicial complexes plus partial ordering swung into your ken at any point? They seem a natural enough concept, but I have never seen them used in the literature. There used to be a vogue at one time for an axiomatic approach to spacetime, based entirely on time-ordering. Also papers looking at other topologies than the Euclidean on Lorentz space — e.g. say a set is open if every timelike line intersects it in an open interval — for which propagators appear much better behaved. I do not think anything particular came of them.

When it comes to finite models and ambitious programs to get something that resembles spacetime to drop out of “nothing”, i.e. some simply describable mathematical gadget that has not been forged in the crucible of “phenomenology”, my thoughts turn to random graphs. For any type of structure on finite sets you can count the isomorphism classes of structures with $n$ elements, weight each class by the inverse of the number of automorphisms and come up with the probability that a randomly chosen structure belongs to a given class. Then you can start asking how these probabilities behave as $n$ varies. That is where the fun starts. Let us say that a structure type is ‘interesting’ if asymptotically there are only a finite number of isomorphism classes with nonzero probability. So random graphs are definitely interesting in this sense as there is only one such class. That is the sort of scenario I would love to see develop.

## Re: Causality in Discrete Models of Spacetime

In reply to Gavin Wraith’s questions, I’ll start by reposting a few of my own summaries of what Ambjörn, Loll and Jurkiewicz have been doing. The first is from week206, written on May 10, 2004, back when I still worked on quantum gravity. It’s part of a report from a quantum gravity conference in Marseille.