Christensen Crane: Causal Sites paper

[marcus]

Dan Christensen, Louis Crane
Causal sites as quantum geometry
20 pages, 3 figures
http://arxiv.org/abs/gr-qc/0410104

Starting a separate thread about the Causal Sites paper in case there's interest. I will quote some exerpts in the next post.

Here's my comment where I flagged the paper earlier, just to introduce things:

ordinary (loop) QG is done on a set of points called a differentiable manifold-----a continuum---analog of ordinary 3D space but without
a precommittment to some particular geomety---a floppy continuum

that was the basis for classic 1915 GR too.

Now Christensen and Crane want to replace the diff-manif.
they want to get rid of the point set continuum and replace it with a new mathematical arena called a Site.

Grothendieck made up Sites. A site is a category with a "Grothendieck topolopy"

you consider your old pointset topological space and you notice that the subsets A of X form a partially ordered-by-inclusion structure and you abstract this notion. Now you have a bunch of "subsets" but they don't have points they are just abstract entities with an ordering relation (taken from the old "order-by-inclusion")

that's not all, these things (A, B,...) are also ordered by causality. One of them can precede another, sometimes.

Grotend. made up a topology to put on this kind of thing, and various
superstructure---presheaf, gerbe, bundle, gadgetry---which he and his friends always enjoyed doing.

Along come christensen crane and notice it would be a neat thing to do QG on instead of doing it on a manifold.

Einstein always said that the points of the manifold had no physical existence. So maybe christensen crane are purifying. and sometimes
when you purify it is like throwing overboard the balast and the ship or balloon can get off the ground.

so i want to call attention to this paper. it has the beginnings of a new approach. mostlikely one that will fail! of course. that is the game. one must try anyway. good luck to them.

[edit: it might succeed too, might be a really good idea---can't tell at this point]

 

[marcus]

Mike had a question

in the link reference-library thread where I flagged this paper, Mike2 had a question:

Is it possible that the sets of a causal site can be shrunk down to points? Are causal site a generalization of point sets?

anyone want to respond. I have my plate full right now, have to go to an appointment and also want to post some sample quotes from the paper before i go.

the basic answer, Mike, is yes and no. The causal site is a generalization of a key feature of a pointset situation-----the system of subsets.

But a causal site does not have points in the ordinary sense.
so you cannot "shrink the objects down" to points

on the other hand some of the objects are analogous to points, so it is not that simple, certain objects are called "absolute points" but that does not mean they are points in a conventional sense

a causal site is not a set of points

a causal site is a category---a set of objects connected by arrows.

these objects do not correspond intuitively to points in pointset topology----they correspond instead to the subsets of a topological space

you just have to read the article I think

 

[selfAdjoint]

Causal sites are sets of regions, modeled on but not a genralization of Causal Sets, which are sets of points. It would be pointless (no pun) to shrink them down because you would lose precisely the property that makes them interesting. The authors want to get away from points and manifolds made out of points.

 

[marcus]

Here are Louis Crane's 23 articles since he got his degree in 1985.
http://arxiv.org/find/grp_physics,grp_math/1/au:+Crane_Louis/0/1/0/all/0/1
You can learn about him from their titles. his degree is in math but his drive is clearly to create math enabling quantum gravity----quantum general relativity.

Most people are likely to have heard of Crane, if at all, in connection with the Barrett-Crane spin-foam model.
Here's a picture of John Barrett, 1998 when giving a seminar at Penn State on the barrett-crane model.
http://cgpg.gravity.psu.edu/online/Html/Seminars/Fall1998/Barrett/

Last year John Baez and Dan Christensen co-authored some papers about computer calculations using the Barrett-Crane model.

In February this year Louis Crane gave a talk at Perimeter called
The point of pointless topology
http://www.perimeterinstitute.ca/activities/scientific/seminarseries/alltalks.cfm?CurrentPage=7&SeminarID=277

Why do topology without points? Why is this especially appropropriate for modeling quantum gravity? In 1915 Gen Rel points do not have a physical existence---famous quote of Einstein about this---how to do topology (or diffl geometry) without sets of points. In LQG you start with a continuum of points but then you factor them out, get rid of them, at some convenient stage in constructing the hilbert space of quantum states. points are "gauge"----i.e. bogus----entities.
Still how about never having points in the first place? never having to squeeze out the bogus slop. A way of getting the mathematical model to be closer to the physical reality---eliminating a layer of "gauge" entities right at the start. Sounds good, wish I had slides and audio for Crane's talk.
I will keep looking.

 

[Mike2]

Why do topology without points? Why is this especially appropropriate for modeling quantum gravity? In 1915 Gen Rel points do not have a physical existence---famous quote of Einstein about this---how to do topology (or diffl geometry) without sets of points. In LQG you start with a continuum of points but then you factor them out, get rid of them, at some convenient stage in constructing the hilbert space of quantum states. points are "gauge"----i.e. bogus----entities.

It sounds like if you imposed a coordinate system throughout the causal site, then you'd have a manifold.

It still seems strange that you could get a metric out of causal sites without a point set.

 

[marcus]

It sounds like if you imposed a coordinate system throughout the causal site, then you'd have a manifold.

It still seems strange that you could get a metric out of causal sites without a point set.

Causal sites would be a good topic for Baez to bring up in TWF #208, wouldn't it? I have been reading the paper this evening but can't say anything helpful as yet. I will try it again when I'm fresh tomorrow.
In the meanwhile maybe sA or someone else may contribute insight.
the main thing that impresses me is that I don't know enough to guess about this. I don't know if the causal site idea is promising or not. I can only
tell it's an innovative representation of spacetime.

typically you couldn't coordinatize one of the things----not with familiar 3D or 4D coordinates---for one thing typically it is the wrong cardinality (too many objects)

have to get back to this L8R

 

[selfAdjoint]

The point is that instead of trying to get back to familiar manifold math, you follow them into their category math (John Baez's 2-categories!) to see how they do it. "He who takes his hand from the plow to look back is not worthy of the kingdom", as Jesus said on another subject.

Marcus, you know an awful lot of work on quantization and nonlinear diff eqs is done topologically, and cell complexes with their topology come out of the darndest places in this work. The authors of this paper define a topology, similar but not identical to the Grothendiek topology, on their category, and that gives them the opportunity to reconstruct spacetime.

Actually a lot of their paper is sheer speculation or work they plan to do but haven't got enough results to publish yet. An awful lot of the paper is just definitions - vocabulary - and references to the existing work they're incorporating. That's not intended as a knock because I'm very interested in their approach, but we shouldn't go overboard just yet.

 

[marcus]

so i want to call attention to this paper. it has the beginnings of a new approach. mostlikely one that will fail! of course. that is the game. one must try anyway. good luck to them.

[edit: it might succeed too, might be a really good idea---can't tell at this point]

I don't know if the causal site idea is promising or not. I can only
tell it's an innovative representation of spacetime.

Actually a lot of their paper is sheer speculation or work they plan to do but haven't got enough results to publish yet. An awful lot of the paper is just definitions - vocabulary - and references to the existing work they're incorporating.

I can see that. Maybe some other readers didn't notice it, so good that you pointed it out.

That's not intended as a knock because I'm very interested in their approach,

I thought you might be ----one reason I flagged it.

but we shouldn't go overboard just yet.

Thanks for the advice. Personally I'm not overboard but you never know when a fluky wave will sweep the deck, so everybody should hold on.

 

[arivero]

Anyone interested in causal neighbourghoods should chech local quantum field theory, whose main textbook is the one of Haag.

 

[marcus]

Anyone interested in causal neighbourghoods should chech local quantum field theory, whose main textbook is the one of Haag.

Alejandro,
Can you explain the connection between a causal site (as Christensen and Crane define it) and a causal neighborhood?

C and C say their idea is new, and different from the causal structure on a manifold. A causal site, as they define, can exist without an underlying manifold and may embody a certain "graininess".

they say on page 7 at the start of section 3. The Intrinsic Geometry of Causal Sites

---quote---
It turns out that a causal site can contain more information than just the causal structure of a manifold. The reason is that a causal site may have a fundamental graininess which sets a length and time scale. Physically, this graininess is expected to occur at the Planck scale, and serves as a measuring rod or clock. Heuristically, a measurement at a smaller scale would result in the formation of black hole, so the maximum possible number of successive measurements along a timelike path gives its duration in Planck units.
---end quote---

So they are distinguishing between causal site and "causal structure on a manifold".

I do not know Haag textbook that you mention and do not have immediate access, but I would be interested to know if Haag has really anticipated them. C and C think they have hold of something new. On page 2 (fourth paragraph) they say:

---quote---
Now, since up to this point nobody has tried to use sites as a foundation for relativity, the natural structure that occurs when we combine the two ideas has never been considered. We will show that it is surprisingly rich and elegant.
---end quote---

If they are mistaken, and the thing has been studied---only under a different name or formalism---it would be good to know.

 

[Mike2]

Causal sites would be a good topic for Baez to bring up in TWF #208, wouldn't it? I have been reading the paper this evening but can't say anything helpful as yet. I will try it again when I'm fresh tomorrow.
In the meanwhile maybe sA or someone else may contribute insight.
the main thing that impresses me is that I don't know enough to guess about this. I don't know if the causal site idea is promising or not. I can only
tell it's an innovative representation of spacetime.

This approach sounds a little like LQG which finds spacetime to be discrete, etc. So the same issues would arise. Perhaps the answer are the same as well. My question to quantized spacetime is how can you still have wave motion or any sense of a direction of change if the spacetime it travels through is not continuous? If you introduce a discontinuity, how does information travel from one section to another? Not only that, but how can you have various invariance of continuous groups with disconnected spacetime? I've not figured that out yet.

 

[selfAdjoint]

This approach sounds a little like LQG which finds spacetime to be discrete, etc. So the same issues would arise. Perhaps the answer are the same as well. My question to quantized spacetime is how can you still have wave motion or any sense of a direction of change if the spacetime it travels through is not continuous? If you introduce a discontinuity, how does information travel from one section to another? Not only that, but how can you have various invariance of continuous groups with disconnected spacetime? I've not figured that out yet.

Have you ever seen "The Wave" done by a discontinuous collection of fans in a stadium? Of course in that case there is preplanning, but in LQG for example, the tetrads are constrained where they meet at their sides and edges, hence they "communicate" there.

 

[Mike2]

Have you ever seen "The Wave" done by a discontinuous collection of fans in a stadium? Of course in that case there is preplanning, but in LQG for example, the tetrads are constrained where they meet at their sides and edges, hence they "communicate" there.

So is it the case that the information passed from one region to the next is quantized? So then under some minimal change that info is not passed along?

 

[marcus]

Have you ever seen "The Wave" done by a discontinuous collection of fans in a stadium? Of course in that case there is preplanning, but in LQG for example, the tetrads are constrained where they meet at their sides and edges, hence they "communicate" there.

beautiful analogy! I know I am supposed to keep my enthusiasm in bounds but this is a literary matter so there should be some allowance

BTW Mike mark well that the Causal Site does not HAVE to have this discreteness. the mathematical richness of the structure is such that it ALLOWS one to define some CS which have this discreteness property;
and which therefore may (or may not) be more successful in modeling the real world. Discreteness is an OPTION which many people think is desirable because they expect we will find it exhibited by spacetime at a fundamental scale. But if we dont, well, Causal Sites can model infinitely divisible situations too.

 

[marcus]

So is it the case that the information passed from one region to the next is quantize? So then under some minimal change that info is not passed along?

Mike we are not arguing about discreteness here. Let's focus on the basics of Causal Sites---they can model either the discrete spacetime or the continuum: that is not the issue.

I for one have a long way to go just understanding the basics of CS.
also am reserving judgement as to the usefulness of this new math tool.

 

[Mike2]

Have you ever seen "The Wave" done by a discontinuous collection of fans in a stadium? Of course in that case there is preplanning, but in LQG for example, the tetrads are constrained where they meet at their sides and edges, hence they "communicate" there.

If the separate quantized regions are rigid, then there would be a preferred reference frame, the frame in which the rigid regions are still. And it would be impossible for some regions to move with respect to other regions; for the regions are rigid and each touches the other. But if the quantized regions are not rigid, and their edges and sides are allowed to deform to allow some regions to move wrt others, then that would be no different from a continuous spacetime. Right?

 

[marcus]

one problem in trying to discuss
is that we haven't said what it means for a C.S. to be discrete

I believe it has something to do with having a finite upperbound on
the length of a sequence of "consecutive" objects. For any A and B, then
no matter how you try you can't get a sequence

$$A \prec A_1 \prec A_2 \prec...\prec A_n \prec B$$

that is any longer than a certain number N

 

[marcus]

I think one way to imagine discreteness of C.S., or tell oneself a story about it, is this. Suppose people of Earth sent a roundtrip probe to AlphaCentauri and we think of two events (A) departure: when and where the main booster fires and (B)arrival: when and where we recover the probe.

we can imagine A and B as two causallyrelated regions of spacetime----we don't assume a spacetime manifold exists, that is just how we picture it intuitively. A and B are two regions and A precedes B causally.

Now the people of Earth play a game of seeing how many separate times they can read a clock between probe departure and probe recovery.

that is, they create a sequence of N distinct causally related actions/events which are mediate between A and B

Now the universe can be one of two different ways. Either it has a maximum number of possible causally ordered events between A and B, or it does not.

If it has a maximum number then we want to model it with a discrete C.S.

If it does not, then in principle it will allow us to make an unlimited number of events where we observe the time on some real physical clock while we are waiting for the probe to return.
No matter how many we arrange to make, it would still be possible for us to have arranged to make some more----to shoe-horn more events into the causal sequence. In that case we will wish to model the universe with a non-discrete C.S.

 

[marcus]

Now suppose someone comes to us who thinks that the word "discrete" has only one meaning. When you say space or spacetime is discrete, she thinks that means only just one thing: that it is like the stars of the American Flag.

She thinks "discrete" means an array of little bitty dots with gaps of "space" in between them.

what shall we do? How shall we talk to her?

First we must say "congratulations! that is one very good meaning of the word discrete! the field of stars on the American Flag is indeed a way to imagine a discrete array or lattice of objects!"

but we don't want to always mean that picture. we want to talk about C.S. where that picture would be irrelevant---simply couldn't apply.
So for the next 5 minutes, when we say discrete we are going to be talking about how you put the idea of discrete into C.S. context.

A typical C.S. is an extremely infinite collection of extremely "overlapping" things-----it is full of overlap and containment and inclusion, because it is basically a substantial chunk of the powerset and overlap and inclusion is what it is all about.

when I say a typical C.S. is extremely infinite I mean it is too infinite to ever think of "coordinatizing" it. You can't impose ordinary coordinates on something that has the cardinality of the set of all subsets of Euclidean 4D space.

So when we say a C.S. is discrete we are going to mean that it has a natural intrinsic idea of the time lapse between two causally related objects---which is the max number of objects you can insert in sequence between.

and if it does not have a maximum in such a case, then, well, it is not a discrete C.S.

We still, like, don't know if the C.S. is a good mathematical idea or if it has any real potential to elucidate Quantum Gravity. But let's try to understand the basics at least. CAVEAT: what I just said is not at all rigorous and is just a first handwavy impression, so not to take as gospel