The forum discussion centers on the classification of manifolds and smoothness structures in the context of quantum gravity (QG). Key points include the undecidability of manifold classification in dimensions greater than three, the existence of uncountably many smoothness structures for non-compact four-manifolds, and the necessity of gauge fixing related to the diffeomorphism group for local Lorentz invariance. The participants express skepticism about the feasibility of summing over all manifolds and smoothness structures, highlighting the complexities involved in defining such sums in quantum gravity theories.
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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
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.
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.
arivero said:5'') 11= 7 + 4
Exotic structures in dim=7 should be also of some interest.
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 ...
torsten said:The whole discussion was confusing, one spoke about gravity, Tom spoke about Dirac operator (as the self-adjoint operator) and all the others spoke about the general NCG.
My post was a direct reaction to Toms. I wanted to clarify that in Connes model using the Dirac operator (which is Connes version of the standard model, no other model was proposed by him) he assumed a Riemannian spin manifold.
I'm deeply sorry for the confusion. Next time I will retain myself in posting such things.
marcus said:I'm very excited by the Suijlekom-Marcolli paper which introduces the concept of a "gauge network", have you looked at the paper?
They are aiming at the Connes standard model without any commutative part of the algebra! All the spectral triples are purely non-manifold and also they are finite dimensional i.e. in A,H,D the algebra A is finite dimensional and so is the hilbertspace.
torsten said:... to simplify a triangulatio of a 3-manifold: use a 2-complex (a so-called spine of a 3-manifold). It is known that every 3-manifold can be constructed by using spines but nothing less (including no 1-complex like a spin network).
But I have to think more carefully about it.
Hi Julian, it is not directly but related to Kirby calculus etc. q-deformed spin networks are more related to skein theory (as far as I can remember Kauffman proved that there is a strong relation). Skeins are related to knot theory and the knot polynomials. The relation to Kirby calculus was proved by Lickorish, he obtained 3-manifold invariants (of Witten type).julian said:You can't classify 3 manifolds with 1d spin networks but what about q-deformed spin networks? Here vertices are replaced with discs and links with ribbons which can have twists in them...isn't this related to Dehn surgery and Kirby calculus or something? What is the additional information in q-deformed spin network states? Just the twists?
tom.stoer said:yes, of course
yes, why not?
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.