I'm not usually a fan of "Causal Sets" QG, but this is a significant advance in that theory. http://arxiv.org/abs/gr-qc/0604124 On Recovering Continuum Topology from a Causal Set Seth Major, David Rideout, Sumati Surya 31 pages, 5 figs. Dedicated to our friend and teacher Rafael Sorkin, to celebrate his 60th year "An important question that discrete approaches to quantum gravity must address is how continuum features of spacetime can be recovered from the discrete substructure. Here, we examine this question within the causal set approach to quantum gravity, where the substructure replacing the spacetime continuum is a locally finite partial order. A new topology on causal sets using 'thickened antichains' is constructed. This topology is then used to recover the homology of a globally hyperbolic spacetime from a causal set which faithfully embeds into it at sufficiently high sprinkling density. This implies a discrete-continuum correspondence which lends support to the fundamental conjecture or 'Hauptvermutung' of causal set theory." =============== In his paper The Case for Background Independence Smolin calls this the "Inverse Problem" that arises in any Background Independent approach---and he indicates that it is especially severe in the case of Causal Sets. If you start with a continuum, and extract a Causal Set from it by throwing away almost all the information and structure you were initially given, HOW DO YOU know that what you have left came from a continuum and how do you GET THE CONTINUUM BACK? Causal Set model of spacetime is extremely minimalist. All it is is a set of events (points) which may or may not be a finite set and pairs of points may or may not be causally related with the provision (of local finiteness) that if any two points ARE causally related then there's at most a finite number of intermediate steps in a causal chain from one to the other. So if spacetime consists of events, this is the fanatical bare minimum structure you could use to represent it. How can these people expect to accomplish anything when they give themselves so little to start with? Basically they just give themselves a set of some events with some causal arrows between some of the events. So the problem is. You start with a cliché vanilla 4D continuum spacetime vintage 1905, and you "sprinkle" points on it to get a finite set of events in the spacetime continuum and then you look to see what events are in the lightcone of what other. And if B is in the forward lightcone of A, then A could influence B, so you draw an arrow. So now you have a causal set of events which was gotten by SPRINKLING. It came from a continuum by selecting out a set of events randomly. Now you FORGET that it came from a continuum, and you collapse it down into a LIST, and all you have now is a directory that lists the events and says which ones causally precede which others. It is just a list of abstract codenames annotated to show precedence. There is no geometry any more. Now, from this list, how do you recover the original continuum? Major Rideout Surya say that it is the "HAUPTVERMUTUNG" of Causal Sets that you can get something of the original back, that there is a solution to what Smolin called the "inverse problem" of Causal Sets. that means "Principal Conjecture"----haupt means main and vermutung means hunch----hauptvermutung means Basic Hunch (German is probably a more earthy less Latiny language) So MajorRideoutSurya have undertaken to verify the Basic Hunch of Causal Set approach to quantum gravity. It gets rather technical because they give themselves so little to start with that they have to be quite clever just to get anywhere at all. Part of me wants to shoot them for being so absolute stubborn minimalist, and again partly I want to applaud.