Major Rideout Surya recapture continuum

[marcus]

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.

 

[marcus]

looks like causal sets people are on the move

it is like the "Volkenwanderung" period around the breakup of the Roman Empire
last year it was Loll's people, now it is Sorkin'shttp://arxiv.org/abs/gr-qc/0605006
Discreteness without symmetry breaking: a theorem
Luca Bombelli, Joe Henson, Rafael D. Sorkin
7 pages
"This paper concerns sprinklings into Minkowski space (Poisson processes). It proves that there exists no equivariant measurable map from sprinklings to spacetime directions (even locally). Therefore, if a discrete structure is associated to a sprinkling in an intrinsic manner, then the structure will not pick out a preferred frame, locally or globally. This implies that the discreteness of a sprinkled causal set will not give rise to 'Lorentz breaking'' effects like modified dispersion relations. Another consequence is that there is no way to associate a finite-valency graph to a sprinkling consistently with Lorentz invariance."

 

[Chronos]

I agree this is an interesting idea, despite being a bit off the trodden path, marcus. I am a fan of Sorkin and Smolin. Both men have a history of writing high quality papers that equitably consider all the issues, IMO. I like the causal set approach [Kea has brainwashed me].

 

[Careful]

Just one tiny note : this paper is still millions of lightyears removed from a proof of the Hauptvermutung.

 

[marcus]

Just one tiny note : this paper is still millions of lightyears removed from a proof of the Hauptvermutung.

thanks for amplifying. Their abstract says that what they have found LENDS SUPPORT to the basic conjecture----they don't say they have a PROOF.

It is a hard problem, perhaps one can never reach a satisfactory conclusion with Causal Sets (my suspicion as outside observer). They have to go through a lot of technical labor and be very clever just to make any progress at all.

 

[Careful]

thanks for amplifying. Their abstract says that what they have found LENDS SUPPORT to the basic conjecture----they don't say they have a PROOF.

It is a hard problem, perhaps one can never reach a satisfactory conclusion with Causal Sets (my suspicion as outside observer). They have to go through a lot of technical labor and be very clever just to make any progress at all.

I did not amplify but merely provided a realistic picture. This paper is about the coarsest of all possible TOPOLOGICAL invariants (the obtained result is fairly obvious) - the real difficulty resides in constructing (robust ??) metric invariants without reference to the Poisson process.

 

[marcus]

I did not amplify but merely provided a realistic picture...

well thanks for whatever you provided
I found both your comments helpful.

Causal sets is not a big interest of mine. Please enlarge on it, if you wish. We have two new Causal Sets papers here to discuss, if you want.

 

[Careful]

well thanks for whatever you provided
I found both your comments helpful.

Causal sets is not a big interest of mine. Please enlarge on it, if you wish. We have two new Causal Sets papers here to discuss, if you want.

Nah, I will leave that to Kea - I am interested to hear how she thinks about causal sets.

 

[Kea]

Hi Careful and Marcus

I am busy writing and have no time to read this paper right now. Of course, I think causal sets are a nice stepping stone to the more sophisticated, but amazingly simple, category theoretic point of view. Generally speaking, there are a lot of reasons for thinking they can't get a proper hauptvermutung without additional ideas. A modern String theorist might point out that since strings are arrows (as Schreiber and Baez like to say) then one at least needs causal sites, meaning some sort of categorification of causal sets, which are based on very ordinary sorts of points. A GR aficionado might come to the same conclusion, based on the knowledge that points are not as physical as fields, the collection of which would naturally form a topos-like structure. And from the point of view of a study of the logic of QM, it's quite clear that sets in any guise won't do the job. All this assumes that we won't find the proper continuum limit until we have a pretty good understanding of how unification comes about. And from my perspective, for what it's worth, this is the case.

Causal set theory is largely motivated, as I'm sure you know, by the result that one can recover a Lorentzian metric up to a conformal factor from an Alexandrov space, being a strongly causal spacetime in the GR lingo. There is lots of research on classical causal boundaries and such things, and no doubt hopes that holography can be understood using such technology. But one has to get the whole Standard Model out as well, and therein lies the question!

Cheers Kea

P.S. Well, maybe I do have time for a chat, so long as no one else is actually reading the paper either...

 

[marcus]

Hi Kea, you say you are BUSY WRITING. That is interesting. What?
What general topic? Hope it's going well.

you mentioned Causal SITES, we had a discussion of this Causal Sites paper here at PF:

http://arxiv.org/abs/gr-qc/0410104
Causal sites as quantum geometry
J. Daniel Christensen, Louis Crane
21 pages, 3 figures; to appear in JMP

I remember starting a thread about it. Must have been back in 2004 when it came out.

More recently, IIRC, I started a thread on
http://arxiv.org/abs/gr-qc/0602120
Categorical Geometry and the Mathematical Foundations of Quantum General Relativity
Louis Crane
Comments: Contribution to the Cambridge University Press volume on Quantum Gravity

"We explore the possibility of replacing point set topology by higher category theory and topos theory as the foundation for quantum general relativity. We discuss the BC model and problems of its interpretation, and connect with the construction of causal sites."

this did not get as much comment as the earlier thread about the paper with Dan Christensen----the Crane/Christensen had some detailed proofs which were fun to follow.

If you or anybody ever feels like chatting about Causal SITES, or sets, or categorified versions of them, I'd be happy to hunt down the earlier discussion of the Crane/Christensen.

===============
You say
"All this assumes that we won't find the proper continuum limit until we have a pretty good understanding of how unification comes about. And from my perspective, for what it's worth, this is the case."

To me it looks like Freidel is achieving this with spinfoams. So far one cannot say anything too conclusive, but I see his approach as the QG closest to finding matter.

this tends to diminish my interest in other approaches like Loll Triangulations and Crane CausalSites which (correctly or not) I see as not as far along in contacting matter.

 

[Kea]

What general topic?

Topic? Er, quantum topos theory, of course. No claims to have solved any big problems, mind you. But the way the logic of vector spaces gets defined really is pretty, I think. More later.

 

[marcus]

...quantum topos theory, of course...

of course

--------------------------------
sounds like would be an eminently good choice of topic for a PhD thesis
(don't know if that applies to you, I forget details and you may be way past that stage----but it does have the ring of good thesis material to it)

I think "Careful" would like a new research direction. Maybe we can encourage him to look in the direction of quantum topos theory. Or do you prefer to shoo off the interlopers?

 

[Careful]

** Generally speaking, there are a lot of reasons for thinking they can't get a proper hauptvermutung without additional ideas. **

Hmm, are you thinking here in terms of fibre bundle like ideas such as Raptis and Mallios are persuing (that is adding vector spaces (arrows) to the points of the causal sets)? Technically speaking, I do think you can get out the hauptvermutung without any additional structure. The downside of additional structure is the possibility of compatibility problems with the information given already in the causet itself - something most people seem to forget. Moreover, a causet person can only speak in terms of points and relations between them since he believes matter to be encoded in geometrical patterns. Concerning the QM of causets, you know I agree extra relational information is needed unless you can reformulate QM as a deterministic theory. Actually, causet QM changes the latter anyway since only manifold like causets are incoorporated (a wise thing to do) - in contrast to the well known fact that almost smooth structures have zero measure in the path integral formulation of the free particle.

Cheers,

Careful

 

[Kea]

Hmm, are you thinking here in terms of fibre bundle like ideas such as Raptis and Mallios are persuing (that is adding vector spaces (arrows) to the points of the causal sets)?

Except that we would never talk about additional structure without respecting higher dimensional algebra. That is, there is no plain 'adding spaces' to points, because as you say this is much too ad hoc and will clash with the information that's already there.

Technically speaking, I do think you can get out the hauptvermutung without any additional structure.

That would be interesting. We might actually be sort of in agreement. When I talk about 'extra structure' it really comes down to understanding the topology in a more subtle way, so although there is 'extra stuff' the basic principle is not all that different.

 

[Kea]

sounds like would be an eminently good choice of topic for a PhD thesis...

Yes, indeed, Marcus!

 

[Careful]

**Except that we would never talk about additional structure without respecting higher dimensional algebra. **

Ohw, could you give a simple example of this ?

**
That would be interesting. We might actually be sort of in agreement. When I talk about 'extra structure' it really comes down to understanding the topology in a more subtle way, so although there is 'extra stuff' the basic principle is not all that different.
**

Ah well, causets were my first love (with respect to QG), it is just that more urgent questions are pushing themselves IMO.


Cheers,

Careful

 

[Careful]

I think "Careful" would like a new research direction. Maybe we can encourage him to look in the direction of quantum topos theory. Or do you prefer to shoo off the interlopers?

Ah marcus, I think my research direction is pretty clear and very demanding - I have no desire of changing whatsoever. As I said before, causet QM changes the latter too. At least I can sympathize to some extend with people who recognize that QM needs a change.

 

[Kea]

**Except that we would never talk about additional structure without respecting higher dimensional algebra.**

Ohw, could you give a simple example of this?

I just meant that in category theory 'extra structure' often means higher dimension. Conversely, higher dimension always means interesting extra structure. For example, if one wants to talk about tensor products then one demands they be put in the language of 2-dimensional categories, because that's just what they are.

The problem is that 'simple', although simple from a certain point of view, is not necessarily illustrative, or simple, from another point of view. But think about the fibre bundle idea. Sticking to manifolds, this puts sets on top of other sets, point by point. But in topos theory, neither the sets nor the points nor the underlying number field is the fundamental thing. In fact, a manifold is a really difficult thing to understand. What is fundamental is the almost functorial nature of the assignation of fibres. Now I say almost because claiming to understand a functor means having a solid grip on the nature of the category at its source, and if one thinks a manifold is, well, just a manifold, then one does not have such an understanding. So when I say higher dimensional algebra it is to make the point that anything at all to do with fibre bundles must at least be 2-categorical, because one can't even say what they are without that language if one wishes to keep all the topological information intact since the topological information is an interesting sort of category in itself and taking limits of such categories is a 2-categorical thing.

 

[Careful]

**I just meant that in category theory 'extra structure' often means higher dimension. Conversely, higher dimension always means interesting extra structure. For example, if one wants to talk about tensor products then one demands they be put in the language of 2-dimensional categories, because that's just what they are. **

Let us vulgarize : suppose M and N be manifolds and f,g : TM -> TN be morphisms, then we can construct f * g : TM * TM -> TN * TN (with * any of your favorite associative products) - I presume this would be an example of a specific bi-morphism in a 2-dimensional category, no? Phew... I was looking for a good paper on examples of two categories, I found a recent one http://arxiv.org/PS_cache/math/pdf/0602/0602510.pdf (can't say I read it all :-) ). You said before that 2-dimensional categories are there to extract more subtle topological information, but the paper under discussion (surya, Rideout) manages to recover (scale dependently) all homotopy groups from the basics - I am sorry to be ignorant about this (I think almost any physicist is) but what is there to be gained more from going to derived structures (products etc.) of these groups (I mean, one would expect the gain in information to stop at some *early* point, no) ? Concering the a priori meaning of taking tensor products of `vectorspace´structures to be derived from the (say) causal set itself, I am much more worried. That is: I do not regard the derived structures from the (co)tangent bundle to be the fundamental juice of a manifold at all. All this information is way too unstable when you are dealing with structures (such as causal sets) which have no appropriate notion of smoothness. Even the beautiful work done by Alexandrov, Bishop, Gromov et al to generalize curvature notions to the more rough framework of Alexandrov spaces seems to be too `fine grained´ for *direct* application in physics (and actually there is a very nice theorem about `being almost manifold´ in this context). Even within the more restricted class of triangulated manifolds (where such notions are all much easier to obtain), and where tensor product constructions and alike have been used for example by Cheeger and Schrader to construct higher curvature invariants (analogues of the higher Euler, Gauss Bonnet invariants), there is a generic instability unless the triangulated structure is *properly* embedded in the underlying manifold it is supposed to approximate.

Bottom line is, first probe the primary structures and learn then how useful (useless) it is to import the usual algebraic notions of direct products and so on to construct higher curvature `tensors´. Prior to achieving this, I would not try tro promote my field by attributing to it `the deeper content´ of a manifold.

Cheers,

Careful