Louis Crane's great paper on categorical geometry

[marcus]

Louis Crane has contributed a great paper to Dan Oriti's "QG Approaches" book.

I started a thread on Crane's paper when it made arxiv in February, but to get a better focus on it I want to start a fresh thread.

Around 1 March, Crane corrected an error in the abstract and reposted on arxiv. Earlier he said the book's publisher was Oxford, instead of Cambridge U. P. This was confusing because Oriti's book is the obvious place for the essay.

I recommend this paper to anyone interested in Quantum Gravity. Why? Because of something Carlo Rovelli said in his book. He said that physics can get along fine for extended periods of time----sometimes for many decades-----ignoring philosophical issues (like "what is time?" what is "space?") and working physicists will often act like they think considering basic questions are a waste of effort-----but sometimes you reach a point where you don't make progress unless you ask deep philosophical questions and check to see if your math foundations are actually right (or are they just what you inherited from the previous occupant of your office or picked up in grad school?)

And Rovelli said that both Newton and Einstein were people who asked philosophical questions and changed the framework, or altered the foundations. They came at a time when there was a philosophical river to cross----so to make progress they had to be MORE than just routine physicists.

Rovelli suggests that if people are having an extraordinary protracted struggle to make a GENERAL RELATIVISTIC QUANTUM FIELD THEORY that is may be because they are philosophically lax. They neglect to tackle philosophical questions and get an up-to-date mathematical model of spacetime to work with.

So here is Crane ( of the Barrett-Crane spinfoam model) and HE MAY BE WRONG or he may be right but at least he grapples the foundation issues.

http://arxiv.org/abs/gr-qc/0602120
Categorical Geometry and the Mathematical Foundations of Quantum General Relativity
Louis Crane
"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."

In this paper he cites a paper with Dan Christensen that we had a thread about over a year ago IIRC
http://arxiv.org/abs/gr-qc/0410104
Causal sites as quantum geometry
J. Daniel Christensen, Louis Crane
21 pages, 3 figures; v2: added references; to appear in JMP
J.Math.Phys. 46 (2005) 122502
"We propose a structure called a causal site to use as a setting for quantum geometry, replacing the underlying point set. The structure has an interesting categorical form, and a natural 'tangent 2-bundle', analogous to the tangent bundle of a smooth manifold. Examples with reasonable finiteness conditions have an intrinsic geometry, which can approximate classical solutions to general relativity. We propose an approach to quantization of causal sites as well."

Even if Crane is wrong this 2006 paper is a good place to start thinking about QG and I'm inviting people to discuss it.

My thought just at the moment is that Georg Riemann invented the continuum (around 1850) that Einstein used (around 1915) to build Gen Rel.
The continuum (a diff manif) was good because you could have one WITHOUT GEOMETRY. it didnt have a rigid metric structure, Riemann showed how to put any number of metrics on it. So Einstein could make the geometry itself, the metric, be a dynamical variable thing----the X that was not given to start with but that you solved his equation to get, and interacted with matter.

So that is already very good, and using Riemann's continuum allowed Einstein to make a BACKGROUND INDEPENDENT theory of spacetime that didnt have a prior-chosen background metric.

But Crane says THIS IS NOT GOOD ENOUGH and he says that working relativists do not believe that the Riemann continuum (the diff manif) is realistic. It is not a realistic model: it has arbitrarily small distances and uncountably many points arbitrarily close together and the same integer dimension all the way down-----a lot of stuff that has no operational meaning and doesn't correspond to realworld measurements.

Crane suggests that we will never succeed in getting a good QG----a good quantum physics of spacetime and matter-----until we THROW OUT RIEMANN!

And he suggests to replace Riemann with GROTHENDIECK!

Alexander Grothendiek was the greated mathematician circa 1950 in the same way that Georg Riemann was the greatest circa 1850. Grothendiek is the immortal and awesome eminence of mid-20th century math.

To me this sounds both crazy and remarkably reasonable. If, at this point, an idea of how to do QG has to be crazy enough to be right then this has a chance.

Also this Crane paper is very well written and gives lots of intuitive/conceptual verbal description. It is not all formulas and technical stuff.
Crane has thought about what the difficulties are and come to some general conclusions. So I hope a bunch of us read it and can discuss it a little.

 

[marcus]

here is the earlier thread I started on this Crane paper
https://www.physicsforums.com/showthread.php?t=112567
I want to refocus and make a fresh start.Here is a key quote right at the start of the paper

====quote from Crane=======

General Relativity is a classical theory. Its mathematical foundation is a smooth manifold with a pseudometric on it. This entails the following assumptions:

1. Spacetime contains a continuously infinite set of pointlike events which is independent of the observer.

2. Arbitrarily small intervals and durations are welldefined quantities. They are either simultaneously measurable or must be treated as existing in principle, even if unmeasurable.

3. At very short distances, special relativity becomes extremely accurate, because spacetime is nearly flat.

4. Physical effects from the infinite set of past events can all affect an event in their future, consequently they must all be integrated over. The problem of the infinities in quantum general relativity is intimately connected to the consequences of these assumptions.

In my experience, most relativists do not actually believe these assumptions to be reasonable.

Nevertheless, any attempt to quantize relativity which begins with a metric on a three or four dimensional manifold, a connection on a manifold, or strings moving in a geometric background metric on a manifold, is in effect making them. ======endquote======

the current Crane paper cites one he did with Christensen
here is the PF thread, from October 2004, about that paper
https://www.physicsforums.com/showthread.php?t=49108

several people commented, including selfAdjoint, but it was not a long thread

 

[selfAdjoint]

I was kind of dumbfounded by my contributions then. I have never really followed up on 2-categories, but I am getting into Grothendieck topology in another context. Anything further will have to wait till I get home and can consult my printoffs.

 

[Kea]

And he suggests to replace Riemann with GROTHENDIECK! ...To me this sounds both crazy and remarkably reasonable.

Hi Marcus

Sigh. Does this mean you like it better than CDT now?

 

[marcus]

I was kind of dumbfounded by my contributions then. I have never really followed up on 2-categories, but I am getting into Grothendyke topology in another context. Anything further will have to wait till I get home and can consult my printoffs.

I actually didnt like the "Causal Sites" paper with Christensen very much. It seemed to me that it introduced a lot of elaborate gear, and then you had to jump thru hoops to make it work

You had to sweat to get things that are taken for granted in a diff manif situation-----infact what you could do was never quite as good. it was always a little bit awkward or inadequate

But now with this paper there is a comprehensible rationale. All the slickness of the smooth manifold continuum is not realistic, and so one WANTS to be a bit lame and clumsy because one wants the mathematical foundations to be closer to realworld-----and one is exploring the limitations that result.

So I don't want to go back to that 2004 paper. I feel like following the chain of thought of this 2006 one. After the philosophy is done there are bound to be several different ways to go.

 

[marcus]

No Kea

I don't like putting one thing above another. If I like them both I will hold both in mind (even if they seem contradictory) and imagine that they are two aspects of something yet to be discovered.

the thing about CDT is it gives you an example of a space which is not a diff manif with a definite dimension

it is a space which is a limit of finer and finer simplicial complexes

now Crane in this paper is TALKING ABOUT JUST THAT

See on page two:
"One often hears from quantum field theorists that the continuum is the limit of the lattice as the spacing parameter goes to 0. It is not possible to obtain an uncountable infinite point set as a limit of finite sets of vertices, but categorical approaches to topology do allow us to make sense of that statement, in the sense that topoi of categories of simplicial complexes are limits of them. "Sigh. Is Kea being just a little "see I told you so"?

 

[Kea]

...but I am getting into Grothendyke topology in another context.

What context is that, selfAdjoint?

 

[Kea]

Sigh. Is Kea being just a little "see I told you so"?

No, I'm being a LOT like that!

 

[Hurkyl]

Oh bleh. Just what I need, more reasons to figure out Topoi, 2-categories, and simplicial complexes!

 

[marcus]

Oh bleh. Just what I need, more reasons to figure out Topoi, 2-categories, and simplicial complexes!

nya, nya, we told you so!

=======================

at christine dantas blog,
http://christinedantas.blogspot.com/2006/03/matters-of-gravity-and-categorical.html
she posted about Crane's paper and one of the regulars (a student at Marseille where Rovelli is) replied:

I read the Crane paper. Madness.

 

[marcus]

actually Hurkyl, a possible alternative way to approach this is to look at Renate Loll
The Universe from Scratch
http://arxiv.org/abs/hep-th/0509010
where she makes a path integral for spacetime using
simplicial complexes

and then it turns out that DIMENSIONALITY at some definite scale of magnification (looking at the microstructure) is an OBSERVABLE
and the expectation value of it turns out to closer than 2D than to 3D at very small scale but in any case not to be an integer

so the limit as the triangulation gets fine cannot very well be a manifold.

and these are spacetimes you can SIMULATE IN A COMPUTER

simplicial complexes are not necessarily all that complicated, they are pretty visual and intuitive, and this whole thing is not very abstract----it is hands on----the only thing is that it describes a spacetime which is not a manifold.

So then you just say I WILL DO WHATEVER IT TAKES to model this kind of thing, this kind of spacetime (and matter), that is not a manifold.

and then if it turns out to require a TOPOS well then at least the topos is WELL MOTIVATED. I would never take something merely on the word of Grothendieck or Ross Street or whoever, or Alain Connes. I would not approach it top-down by learning all about some abstract mathematics castle-in-the-air first and then trusting I could apply it. Start with something concrete that defies conventional treatment and see what is needed.

But I agree that you are talking about doing the conventional grad school thing: learn N-categories and Topoi and all that stuff. Probably it will pay off. You will need it later in life. The Voice of Wisdom.

it's a dilemma, personally can't advise but can sympathise

 

[selfAdjoint]

I was struck by this quote from the Crane paper: it goes to an old concern I had about path interals:

One often hears from quantum field theorists that the continuum is the limit of the lattice as the spacing parameter goes to 0. It is not possible to obtain an uncountable infinite point set as a limit of finite sets of vertices, but categorical approaches to topology do allow us to make sense of that statement, in the sense that topoi of categories of simplicial complexes are limits of them.

My problem is that the motivation for Feynmann's technique in textbooks make use of this same invalid limit of countable -> uncountable as parameters approach 0. In topology we learn that the space of paths over a manifold is not paracompact. Paracompact means that any arbitrary open cover has a locally finite refinement. That means that (1) every upen set in the refinement cover is a subset of some open set in the original cover, and (2) any point of the manifold is in only finitely many of the open sets of the refinement. Essentially what this means is that you can take limits in a paracompact space and your limits will have meaning.

So by definition the set of "all paths" that they want to integrate amplitudes over should not on the face of it support meaningful limits. When I exposed this concern on sci.physics.research, John Baez rather brushed me off with a reference to his book, but that didn't answer my question. Does anybody here have anything to say on the subject?

 

[Kea]

Does anybody here have anything to say on the subject?

Hi selfAdjoint

When Crane says categories of simplicial complexes one should think of a simplicial complex internal to some category, ie. as a functor. Functor categories are naturally toposes in the Grothendieck sense (we get to sheaves when we start worrying about topologies). Now it was precisely because Grothendieck was trying to do Algebraic Geometry, and wanted to study categories like functors from topological spaces to sets that he had to worry about the size of categories. Now initially categories were always discussed in terms of homsets, which were actually sets, and the collection of all maps from one topological space to another (such as the interval into a spacetime manifold) simply wasn't a space in the right sense. Russell's paradox etc. This means that the category Top of topological spaces is what is known as large. That's OK so long as one refrains from discussing functor categories. The collection of all functors between large categories is just too big. However, by thinking of spaces as lattices of open sets we seem to have gained a great advantage, so we would like to stick with this approach. And, of course, the spaces need not be paracompact.

The solution was to invent the Grothendieck universe, an add-on to ZF which basically says that a category of sets is really the category of sets contained in a universe U. If one needs to add other levels of universe, fine, that works OK. This idea has since been refined further, but that is an aside.

The point is, one simply cannot do algebraic geometry, or rigorous QFT, without facing such dilemmas. Sorry if I was just rambling.

 

[Kea]

The point is, one simply cannot do algebraic geometry, or rigorous QFT, without facing such dilemmas.

And even doing 'QFT' Careful's way should end up facing similar questions.

 

[Chronos]

I see no good reason to assume the universe is 4 dimensional at all scales. There are many observations, IMO, that suggest the classical notion of dimensionality breaks down as you approach the Planck scale. QFT, as Kea noted, is a good example.

 

[Careful]

And even doing 'QFT' Careful's way should end up facing similar questions.

What kind of questions ? There are pleanty left but I would like to know whether you have the same in mind as I do

Cheers,

Careful

 

[Kea]

What kind of questions?

Ah, Careful, as much as I would like to , I regret to say that I have put absolutely no effort into understanding the details of the papers to which you refer.

But I was imagining that backreaction effects should act analogously to renormalisation processes in QFT, and hence encounter size issues. (BTW, this is a thread on Crane's paper).

 

[Careful]

Ah, Careful, as much as I would like to , I regret to say that I have put absolutely no effort into understanding the details of the papers to which you refer.

But I was imagining that backreaction effects should act analogously to renormalisation processes in QFT, and hence encounter size issues. (BTW, this is a thread on Crane's paper).

radiation backreaction effects are perfectly well defined for point particles, it is the coulomb self energy which is troublesome in that case. If the particle has a finite size (determined by some GR-EM model), no such problem arises at all. ADDENDUM: you might be saying that I need a cutoff for my radiation spectrum (sure I do). However, since this approach is entirely classical I have the liberty to put that in the initial data: I am not stuck with a Feynman path integral sum which tells me I have to go to arbitrary small distances. Anyway, this is indeed a thread on Cane's paper.

 

[selfAdjoint]

What context is that, selfAdjoint?

I bought Lieven le Bruyn's book Noncommutative Geometry@n, Volume 1, The Tools, and I have been trying to read it. I have to admit that with all the other calls on my hard thinking time (like Smolin's videos), I haven't made an awful lot of headway.

 

[Careful]

I bought Lieven le Bruyn's book Noncommutative Geometry@n, Volume 1, The Tools, and I have been trying to read it. I have to admit that with all the other calls on my hard thinking time (like Smolin's videos), I haven't made an awful lot of headway.

Good, I have read a part of his course (long time ago) which was available on his web page before he started to work on the book. Good stuff from an excellent mathematician, although not ``much´´ physics in it