Classification of manifolds and smoothness structures

In summary, the conversation discusses the various theories and approaches to quantum gravity, which either use or construct manifolds and smoothness structures. It also mentions the difficulty in defining a "sum over all manifolds" due to the non-decidability and uncountability of manifolds in higher dimensions. The question of whether this sum is a sensible concept is also raised, along with the possibility of a restricted subset of 4-manifolds being the basis for a physical theory of quantum gravity.
  • #1
tom.stoer
Science Advisor
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Except for the work of Torsten I am not aware of any paper which discusses these topics.

Some ideas:
1) all theories for QG I am aware of do either use manifolds and smootheness (string theory, geometrodynamics, shape dynamics, ...) or are constructed from them (LQG, CDT, ...)
2) in some theories the number of dimensions is fixed, whereas in other theories they become dynamical (spectral dimension)
3) we know that piecewise-wise linear and smooth manifolds are not necessarily the same
4) we know that for dim > 3 the classification of manifolds is not decidable, that there is no algorithm to distinguish between arbitrary manifolds, and that manifolds cannot be "listed"
5) we know that in dim = 4 there are in general uncountably many smoothness structures for a given (non-compact) manifold
5') however if we restrict the domain to manifolds which allow for a global foliation M4 = R * M3 then (3-5) become trivial
6) due to (4) we are not able to define a "sum over all manifolds" and due to (5) we are not able to define a "sum over all smoothness structures for a given manifold"
7) in all approaches I am aware of a kind of gauge fixing is required; therefore we need to know the diffeomorphism group / mapping class group and the fibre bundle topology (Gribov ambiguities) for local Lorentz invariance (for the connections and n-Beins)

Is there any paper discussing these topics and their relevance in the contex of QG?
 
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  • #2
287 views but no answer?
 
  • #3
I have no answers, but I think its a great question. Presumably we will realize (or already know?) that summing over manifolds is a bit too naive. For example, even in ordinary quantum mechanics in the path integral language most paths are not smooth.

It should be lattice models all the way down :)
 
  • #4
tom.stoer said:
4) we know that for dim > 3 the classification of manifolds is not decidable, that there is no algorithm to distinguish between arbitrary manifolds, and that manifolds cannot be "listed"

tom.stoer said:
6) due to (4) we are not able to define a "sum over all manifolds" and

Physics Monkey said:
It should be lattice models all the way down :)

http://arxiv.org/abs/1112.5104
"If we try to join the two main messages of Einstein and Feynman we get the equation

Quantum Gravity = Random Geometry. (2)

There is quite a wide agreement on this equation among all the main schools on quantum gravity, although neither string theory nor LQG take it as their starting point. The problem is that geometry is rich. Especially in three or four dimensions there does not seem to be a unique way to put a canonical probability theory on it.

We can take inspiration from another outstanding idea of Feynman, namely to represent quantum histories by graphs. We feel that it is perhaps not sufficiently emphasized in the graph theory community that quantum field theory and Wick's theorem provide a canonical measure ..."
 
  • #5
atty, I understand the basic idea: let geometry emerge from some per-geometric structure; but there are approaches like asymptotic safety relying on smooth manifolds all the way down; so what I am asking for are the mathematical structures we expect to be present in

##\sum_\text{top}\;\int_\text{diff}\;\ldots##

where "diff" are diffential structures for a specific topology "top"

All questions refer to this idea
 
  • #6
hey Tom.

It's a very good question, but ultimately that's the central problem of quantum gravity and the snag where everyone hits the proverbial brick wall for many of the same reasons you list. Namely it's not even obvious how to set up and partition the sum much less solve it.

The state of the art basically doesn't even address the question directly.
For instance a recent paper by Maloney and Castro sets up the sum in 2+1 Ads space, and you can already see the complications they run into. For instance, do you include nontrivial diffeomorphisms in the sum? How do you weigh the geometries etc. The authors make plausible choices, but it's still not obvious if those correspond to the real physics and there are multiple potential technical snags lurking. http://arxiv.org/abs/1111.1987

In a different vein, this type of calculation is easier to write down in perturbative string theory (sum of Riemann structures) and topological string theories but even there you quickly run into calculational complexities.

Even the masters of these types of game (eg Witten) doesn't have much to say in 4d.

I believe a lot of people are skeptical regarding whether it's even a sensible question. Eg This sum over geometries might only be a semi classical artifact valid only in very specific circumstances.
 
  • #7
Haelfix said:
... but ultimately that's the central problem of quantum gravity and ... it's not even obvious how to set up and partition the sum much less solve it.
yes, of course

Haelfix said:
... do you include nontrivial diffeomorphisms in the sum?
yes, why not?

Haelfix said:
I believe a lot of people are skeptical regarding whether it's even a sensible question. Eg This sum over geometries might only be a semi classical artifact valid only in very specific circumstances.
I know; nevertheless it may be interesting to take it seriously.

The interesting thing is that if you define the counting AND sum over ALL dimensions (!) then the sum is naturally peaked at dim=4 due to the uncountably many smoothness structures for non-compact 4-manifolds.
 
  • #8
Does the question remain if we ask for a canonical formulation, instead of a path integral?
 
  • #9
atyy said:
Does the question remain if we ask for a canonical formulation, instead of a path integral?
I am not sure.

The problem with the canonical formalism is that it requires a global foliation M4 = R * M3. Doing this changes (or restricts) the allows 4-manifolds. For the remaining 3-manifolds some questions (3-5) become trivial or are at least solvable, as I said in (5')

However the key question is whether this restriction is physical (and therefore QG must be based on this restricted subset of 4-manifolds) or whether the restriction is unphysical (and the resulting theory is incomplete or wrong).
 
  • #10
tom.stoer said:
I am not sure.

The problem with the canonical formalism is that it requires a global foliation M4 = R * M3. Doing this changes (or restricts) the allows 4-manifolds. For the remaining 3-manifolds some questions (3-5) become trivial or are at least solvable, as I said in (5')

However the key question is whether this restriction is physical (and therefore QG must be based on this restricted subset of 4-manifolds) or whether the restriction is unphysical (and the resulting theory is incomplete or wrong).

I guess I wasn't thinking necessarily of canonical LQG - but more generally - like if the path integral form of the theory summing over dimensions and smoothness structures existed, presumably one could get from there to a Hilbert space and unitary evolution or some sort of canonical-like formulation?
 
  • #11
I don't believe in the PI as a method to construct the canonical formalism rigorously. And I don't believe in the PI as fully equivalent formalism in general.

In the restricted domain the canonical formulation is the better starting point. In the unrestricted domain no canonical formulation will exist - neither constructed directly, nor indirectly via the PI. I think topology and smoothness structures a the weak point of all approaches to QG I know - except for thorsten's work which I still do not really understand.

Afaik in 4-dim. even piecewise linear structures (related to discrete approaches like CDT) can be devided into inequivalent equivalence classes.
 
  • #12
Tom,
these a really important questions.
I know only of one paper (unpublished) which adress them:

Quantum general relativity and the classification of smooth manifolds
by H. Pfeiffer
http://arxiv.org/abs/gr-qc/0404088

But let me also discuss some of your points:

Point 3) Is only true for manifolds of dimension 8 or higher. All manifolds of dimension 7 or smaller have equivalent Piecewise-linear (PL) and smooth structure. So the sum over PL is equivalent to a sum over smooth manifolds

Point 5') Is misleading: Is the global foliation smooth or only continuous? For instance (3-space) cross Line has always uncountable many smooth structures.

Point 4) the decidability depends only on the fundamental group (word problem). But if you assume causality then you have at least demand that spacetime is simply connected (trivial fundamental group). We try to calculate the "sum over smoothness structures" with partly success.

Point 7) I would add to gauge fixing: the group of large diffeomorphisms (not connected to the identity) can be very complicated beginning wth dim 3. In the "sum over structures" you have to include them.

If you want to read a non-technical summary of our approach. I took part of the FQXi essay contest this year. Here is the link:
http://fqxi.org/community/forum/topic/1780
 
  • #13
tom.stoer said:
I don't believe in the PI as a method to construct the canonical formalism rigorously. And I don't believe in the PI as fully equivalent formalism in general.

In the restricted domain the canonical formulation is the better starting point. In the unrestricted domain no canonical formulation will exist - neither constructed directly, nor indirectly via the PI. I think topology and smoothness structures a the weak point of all approaches to QG I know - except for thorsten's work which I still do not really understand.

Afaik in 4-dim. even piecewise linear structures (related to discrete approaches like CDT) can be devided into inequivalent equivalence classes.

Here I completely agree with. In the last two years we analyzed more concrete examples like S^3xR and found some interesting features.
In a canonical approach, one starts with NxR with N a 3-manifold. This spacetime has to be choosen also smoothly, otherwise one contradicts the strong causality. But strong causality means that one has a unique Cauchy surface (some N for a time) and one has unique geodesics starting from the Cauchy surface and going to the future and to the past. These conditions are enough to show that one needs NxR smoothly. Every canonical formulation starts with these assumptions. But what does it mean? If I assume an open future (at least for a quantum system) then why I have to assume unique geodesics? Instead I would assume that there saddle points where I have branching geodesics. Surprisingly we found such structures in exotic smooth 4-manifolds.
 
  • #14
Torsten, I do really appreciate the information you provide!
 
  • #15
tom.stoer said:
6) due to (4) we are not able to define a "sum over all manifolds" and due to (5) we are not able to define a "sum over all smoothness structures for a given manifold"
Good point, this should give us some clue.
And then there are those approaches that point towards getting rid of formulations in terms of Lorentzian manifolds (a lá Connes non-commutative algebra) because they seem too limiting but I think those are not introducing gravity yet..
 
  • #16
Connes still relies on a differentiable (spin-) manifold structure, otherwise he would not be able to introduce a Dirac operator; in that sense his approach does not solve any of the above mentioned problems, as far as I can see
 
  • #17
tom.stoer said:
Connes still relies on a differentiable (spin-) manifold structure, otherwise he would not be able to introduce a Dirac operator; in that sense his approach does not solve any of the above mentioned problems, as far as I can see

You are right it doesn't solve the OP problems.
 
  • #18
But honestly, I didn't think about the (differential) topological properties of non-commutative geometry.
 
  • #19
tom.stoer said:
Connes still relies on a differentiable (spin-) manifold structure, otherwise he would not be able to introduce a Dirac operator; in that sense his approach does not solve any of the above mentioned problems, as far as I can see

Maybe I'm confused, but does Connes really require this? I thought one was, at some level, just working with an algebra of non-commutative degrees of freedom plus extra data specifying the Dirac operator. This algebra indirectly determines a "non-commutative manifold" in the sense that the usual commutative algebra of functions plus extra data also determines a conventional commutative manifold.
 
  • #20
Physics Monkey said:
Maybe I'm confused, but does Connes really require this? I thought one was, at some level, just working with an algebra of non-commutative degrees of freedom plus extra data specifying the Dirac operator. This algebra indirectly determines a "non-commutative manifold" in the sense that the usual commutative algebra of functions plus extra data also determines a conventional commutative manifold.

In what I've read there is no smooth manifold required---instead there is a bunch of axioms that say (among other things) how the Dirac operator should behave.

In special cases/examples he can start with an algebra of functions on a manifold, so there is a manifold in the picture. But there does not have to be.

Tom may be thinking of a special case relevant to physics where the algebra is "almost commutative" in the sense that a big piece of it is constructed by taking the functions on a conventional manifold (and that part of the algebra is commutative) but then a little non-commutative piece is tacked on. The direct sum of the two is non-commutative. this construction comes up when they want to realize the standard particle model.
 
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  • #21
I may not be thinking of ... I simply lack understanding :-]
 
  • #22
tom.stoer said:
I may not be thinking of ... I simply lack understanding :-]
Well if you do get interested in "Spectral Geometry" (Connes stuff) then I reckon you'll pick the basic ideas up pretty fast.

BTW there is a young Hollander working in it who has an appointment at Radboud U (Nijmegen) where Renate Loll is. they also have an Ashtekar PhD named Nelson, who does QG cosmology. They are putting together a strong QG group at Nijmegen. This guy's name is Suijlekom---I wouldn't know how to pronounce it. He has coauthored with Matilde Marcolli (the Caltech prof who does Spectral Geometry)
 
  • #23
In Connes model there is a noncommutative part and also a usual smooth 4-manifold. But Connes need a Riemannian spin manifold. One reason is the definition of the Dirac operator and especially the relation between the spectrum of the Dirac operator and the metric. Thsi relation is problematic for pseudo-riemannian manifolds. The strong relation (which Connes really need) si only true for Riemannian manifolds. Secondly, Connes used the heat kernel method to evaluate the action and obtained the combined Einetin-Hilbert-YM action. But for the heat kernel, one has to assume that the square of the Dirac operator s an elliptic operator, so one assumed a Riemanian spin manifold again.
 
  • #24
Torsten, thanks for clarification; this was my impression as well

tom.stoer said:
Connes still relies on a differentiable (spin-) manifold structure, otherwise he would not be able to introduce a Dirac operator
 
  • #25
Tom,
but let get back to your original question.
Maybe I do not have the full overview but except of our work and the Pfeiffer paper (which was never published!) I know of no other paper discussing your questions.
I remember to read a proceeding article of Penrose (around 1984) where he discussed the Donaldson work. He stated at the end, that Donaldson theory should be important for quantum gravity. But no one followed...
But all of your question point to main problem. I don't think that the current problems in spin foam models as well in LQG can be repaired without adressing your question. The main problem is the usage of discrete models and anyboday is sure that the relation between discrete (or PL) spacetimes and the topological structure is unique. Then one has only to deal with the limit discrete -> continuous and one is done. But as you also discussed, it is wrong.
Maybe I should apply my methods to spin foam models.
Tom or Marcus, did you know a crash course to learn the current state of spin foam models?
 
  • #26
tom.stoer said:
Connes still relies on a differentiable (spin-) manifold structure, otherwise he would not be able to introduce a Dirac operator; in that sense his approach does not solve any of the above mentioned problems, as far as I can see

Tom, I think this is mistaken. Connes NCG setup does not rely on a manifold of any kind. In a spectral triple (H, A, D) the A is a * algebra of operators on a hilbertspace H and the D operator satisfies certain axioms. Nowhere is there a manifold.

It may seem as if Torsten thinks that a manifold is required, but his statement seems to me to be unclear. I would urge not to take his post as confirmation of the mistaken idea that a manifold is required in the definition of NCG.

Manifold IS involved in the definition of the special "almost commutative" case, where functions on a manifold are used in the construction. But we don't know that is is the necessarily the right way to go. There can be other ways to apply NCG. I doubt that "almost commutative" setup is the final word.
 
  • #27
marcus, how would you define D, its spectrum and its heat kernel w/o a manifold
 
  • #28
tom.stoer said:
marcus, how would you define D, its spectrum and its heat kernel w/o a manifold
You have to ask Connes. He defines it on stuff that does not consist simply of functions on manifold (that would be a *commutative*, he defines D on noncommutative structures too, sometimes on finite dimensional vectorspaces). D does not have to be anything you are used to. for him, D can just be something he spells out that satisfies certain axioms. The axioms that D satisfies make it perform the services that he needs, and allow in some commutative cases to RECOVER or reconstruct an algebra of functions defined on a manifold. So it is a generalization.

I'm not an expert, Tom. I just attended a Marc Rieffel seminar on this stuff for several weeks a few years back until his grad students began presenting papers I couldn't follow on obscure side topics, their own research etc etc. I don't remember much.

Maybe if this Marcolli Suijlekom thing develops I will go back and review one of the Connes papers defining NCG that gives the Spectral Triple axioms.
 
  • #29
I suppose the point of calling it "noncommutative" is that it is NON-MANIFOLD BASED. If you take a manifold and take the set of functions defined on it, that is an ALGEBRA because it is a vectorspace and you can multiply two functions together point wise. And multiplication commutes. So with minimal assumptions you have a commutative * algebra. And by looking at the structure of that algebra (the maximal ideals in it) you can recover something about the geometry.

So you are just doing geometry (but instead of dealing with the manifold itself you are "seeing" the manifold thru the algebraic structure of the * algebra of functions defined on it. Basic stuff from over 60 years ago--gelfand, naimark, segal?---I forget)

You are doing geometry but THRU a *commutative* algebra of functions that contain all the information about the manifold they are defined on.

So I think what Connes did was to axiomatize this and say "what if the algebra is not commutative?" What if the algebra is NOT the algebra of functions defined on a manifold, which would make it commutative, but something else? A more general idea of "geometry".

In the commutative case the functions defined on M tell you essentially all about M so you can do geometry without touching the manifold but only studying the algebraic structure. But if the algebra is not commutative then it does not come from a manifold and you can still study the "geometry" implied by the algebraic structure, but what is it the geometry OF? Very strange. This was his creative idea. To generalize differential geometry to something else besides differential manifolds.

I have a lot of family things to do and don't have time to review and be sure about details. some of the others like brian or someone else would be better at explaining NCG anyway. I'll try again tomorrow if no one else volunteers.
 
  • #30
marcus said:
I suppose the point of calling it "noncommutative" is that it is NON-MANIFOLD BASED.

Well, sometimes you use NCG only because you want to manipulate a quotient of a manifold by a group action in a very generic way, including both manifold and non-manifold quotients. Or you want work out a groupoid equivalence relation. I am not sure of how useful NCG is if you only want to work with usual manifolds, surely it provides some new tools when it comes to calculate suspensions, borders, cohomologies, K-theory, and all these high math stuff.
 
  • #31
tom.stoer said:
Except for the work of Torsten I am not aware of any paper which discusses these topics.

Some ideas:

5) we know that in dim = 4 there are in general uncountably many smoothness structures for a given (non-compact) manifold


5'') 11= 7 + 4

Exotic structures in dim=7 should be also of some interest.
 
  • #32
tom.stoer said:
In the restricted domain the canonical formulation is the better starting point. In the unrestricted domain no canonical formulation will exist - neither constructed directly, nor indirectly via the PI. I think topology and smoothness structures a the weak point of all approaches to QG I know - except for thorsten's work which I still do not really understand.

There is Thiemann's, Giesel's `Algebraic quantum gravity' [arXiv:gr-qc/0607099]: "While AQG is inspired by Loop Quantum Gravity (LQG), it differs drastically from it because in AQG there is fundamentally no topology or differential structure...The missing information about the topology and differential structure of the spacetime manifold as well as about the background metric to be approximated is supplied by coherent states."
 
  • #33
marcus said:
Tom, I think this is mistaken. Connes NCG setup does not rely on a manifold of any kind. In a spectral triple (H, A, D) the A is a * algebra of operators on a hilbertspace H and the D operator satisfies certain axioms. Nowhere is there a manifold.

It may seem as if Torsten thinks that a manifold is required, but his statement seems to me to be unclear. I would urge not to take his post as confirmation of the mistaken idea that a manifold is required in the definition of NCG.

Manifold IS involved in the definition of the special "almost commutative" case, where functions on a manifold are used in the construction. But we don't know that is is the necessarily the right way to go. There can be other ways to apply NCG. I doubt that "almost commutative" setup is the final word.

Marcus, there is a fundamental misunderstanding. I spoke about Connes model to obtain the standard model of elementary particle physics (including the Higgs sector). There, he used a manifold (with a cross product of a discrete space), which must be spin and Riemannian.
You are of course right, in the original definition of a spectral triple, there is no manifold. But many examples of noncommutative spaces are derived from or directly related to manifolds. Take the leaf space of a foliation: one is unable to obtain any results of foliation theory by considering the funtion space over the leaf space. So, the introduction of an operator algebra instead of a (commutative) function algebra was an important step.
In the spectral triple (A,H,D), the Dirac operator D (usually called a self-adjoint operator) acts on the Hilbertspace H and A is the space of operators over H. A is a * algebra and the norm of the commutators [a,D] for alle elements a of A must be bounded.
 
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  • #35
TrickyDicky said:
Good point, this should give us some clue.
And then there are those approaches that point towards getting rid of formulations in terms of Lorentzian manifolds (a lá Connes non-commutative algebra) because they seem too limiting but I think those are not introducing gravity yet..

tom.stoer said:
Connes still relies on a differentiable (spin-) manifold structure, otherwise he would not be able to introduce a Dirac operator; in that sense his approach does not solve any of the above mentioned problems, as far as I can see

TrickyDicky said:
You are right it doesn't solve the OP problems.

Physics Monkey said:
Maybe I'm confused, but does Connes really require this? I thought one was, at some level, just working with an algebra of non-commutative degrees of freedom plus extra data specifying the Dirac operator. This algebra indirectly determines a "non-commutative manifold" in the sense that the usual commutative algebra of functions plus extra data also determines a conventional commutative manifold.

marcus said:
In what I've read there is no smooth manifold required---instead there is a bunch of axioms that say (among other things) how the Dirac operator should behave.

In special cases/examples he can start with an algebra of functions on a manifold, so there is a manifold in the picture. But there does not have to be.

Tom may be thinking of a special case relevant to physics where the algebra is "almost commutative" in the sense that a big piece of it is constructed by taking the functions on a conventional manifold (and that part of the algebra is commutative) but then a little non-commutative piece is tacked on. The direct sum of the two is non-commutative. this construction comes up when they want to realize the standard particle model.

torsten said:
Marcus, there is a fundamental misunderstanding. I spoke about Connes model to obtain the standard model of elementary particle physics ...

I understand, Torsten. I was the one who brought up the special application of NCG which DOES involve a conventional 4d manifold in constructing the commutative part and which "obtains the standard model of elementary particle physics ..."

If you look back to the earlier posts you will see that the earlier references to NCG were not about the special application to standard model, but about NCG in general. It is important to make very clear that NCG does not require a manifold. It is merely one of the options to include a commutative (manifold-based) piece.

There was no "fundamental misunderstanding". Unless you thought that the discussion was just about the NCG standard model construction. In your post you did not make clear that you were talking about a specialized application of NCG, so Tom seems to have read what you said as applying to NCG in general.
 

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