Exotic smoothness and quantum gravity

[tom.stoer]

Is anybody here familiar with Asselmeyer's work on exotic smoothness and quantum gravity?

 

[marcus]

Is anybody here familiar with Asselmeyer's work on exotic smoothness and quantum gravity?

We had some threads on it, in which Torsten (and also sometimes Helge) took part. I was glad they came here and I think people here were welcoming and encouraging. But I can't say I'm familiar with the work. Maybe someone else got a clearer impression.

 

[tom.stoer]

The idea interesting b/c it relies on smooth structures (whereas many other QG approaches favours discrete models) and b/c it somehow singles out 4-dim. space time.

 

[marcus]

Torsten has 32 posts on PF, many discussing his work:
https://www.physicsforums.com/search.php?searchid=2847499

Here's a 2010 thread, both you and Torsten took part.
https://www.physicsforums.com/showthread.php?t=412582

It is definitely interesting for the reason you mentioned.

 

[torsten]

Hallo again,
yes I'm familiar with this work... ;-)
What are the questions? I remember back on a very interesting discussion with you.

Torsten

 

[tom.stoer]

Hi torsten, nice to talk to you!

There are a lot of questions - but most of them may be due to my ignorance in topology and differential geometry :-)

Let's start with some basic questions:

• Do you believe in a spacetime manifold as a fundamental entity, or do you think that the manifold is only an effective description of a more fundamental (discrete) structure?
• Is my understanding correct that you re-interpret the Einstein equation G + Λg = T as G’ = 0 such that T (incl. the Λ-term) are only an "effective description" of some underlying geometrical structure?
• Is my understanding correct that you try to describe matter degrees of freedom using exotic smooth structures?
• Do you think that exotic smoothness singles out dim=4 for some reason? (*)
• Is there a recent review paper?

Regarding (*) some years ago I had the following idea: suppose you could formulate a quantum field theory using a path integral formalism summing over all dimensions, all different manifolds in each dimension and all different smooth structures. Then by a simply counting argument picking a smooth structures randomly, the probability to pick a 4-dim. non-compact manifold is one (b/c for all other dimensions the number is either finite or countable whereas for non-compact dim=4 the number is infinite). Does this makes sense?

 

[czes]

https://www.physicsforums.com/showthread.php?t=412582 :
"Now some words about quantization:
Jerzy Krol and me studied a model where we are able to show that exotic smoothness implied quantization.
see http://arxiv.org/abs/1001.0882
So, our models are possibly quantized...
Torsten"

Riemann is asking what it is about the structure of space that makes it possible to talk about measurable things like distance, area, volume and angles (the "metric relationships"), and he is contrasting the case in which the deep structure of space is continuous with the case in which it is discrete.

What do you think about the causal sets programme of Rafael Sorkin ? Its founding principle is that spacetime is fundamentally discrete and that the spacetime events are related by a partial order. This partial order has the physical meaning of the causality relations between spacetime events.
"Order + Number = Geometry"

 

[torsten]

Hallo Tom,

Now I try to answer or better react on your question:
- Do you believe in a spacetime manifold as a fundamental entity, or do you think that the manifold is only an effective description of a more fundamental (discrete) structure?
I think the spacetime manifold is the fundamental entity. The question whether this manifold is discrete or continuous is also very fundamental.
Usually a discrete structure is associated with a triangulation (or PL structure). If one assumes this PL structure as discrete
counterpart then one can also use the following fact (true up to dimension 7): to every PL structure there is an unique smoothness structure and vice versa.
This sounds amazing but at a second view it is natural: the smoothness structure is continuous but the contained (topological) information is only countable,
encoded into the intersections of the charts. It doesn't matter for topology what happens in between, i.e. if two curves
intersect then only this fact is important not the coordinates of the intersection.
BTW, czes, I know Sorkins proposal and like it. Especially the causal information is most important (see also the dynamical triangulation approach of Loll et.al.)

Is my understanding correct that you re-interpret the Einstein equation G + Λg = T as G’ = 0 such that T (incl. the Λ-term) are only an "effective description" of some underlying geometrical structure?
Yes, your understanding is correct, matter and interaction have a geometrical/topological root. If that is true, then G'=0 is the correct equation
and all the (classical) rest should be derived from it.

Is my understanding correct that you try to describe matter degrees of freedom using exotic smooth structures?
Yes, fermions are given by singular disks and interactions by torus bundles. These structures are not artificial but rather appear naturally
by the decription of exotic smoothness, the Casson handle. The Casson handle is an infinite structure with the potential to contain
also QFT (or quantum mechanics). Currently we are working on this approach.


Do you think that exotic smoothness singles out dim=4 for some reason? (*)
Yes I agree with you including the nice explanation. One year ago I calculated the path integral contribution (arXiv:1003.5506)
of exotic smoothness (for compact manifolds). The contribution is not small.

Is there a recent review paper?
Unfortunately not, currently we try to place these papers in some journals. But I will do it in the near future.

I hope these answers help a little bit.

Torsten

 

[tom.stoer]

Hallo Tom,

Now I try to answer or better react on your question:

...

I hope these answers help a little bit.

They do, thanks a lot!

A last remark: I understand the relation between smoothness and PL. I understand that what essentially counts are the discrete (countable) topological entities. But if that is indeed true, then I do not understand why still the continuous manifold shall be the relevant fundamental object instead of a (to be identified) discrete structure (or e.g. a topological field theory). But if this conclusion (regarding fundamental discreteness) is correct, then my argument which singles out D=4 (via counting of equivalence classes of manifolds w.r.t. to diffeomorphism) does no lonager apply.

 

[torsten]

They do, thanks a lot!

A last remark: I understand the relation between smoothness and PL. I understand that what essentially counts are the discrete (countable) topological entities. But if that is indeed true, then I do not understand why still the continuous manifold shall be the relevant fundamental object instead of a (to be identified) discrete structure (or e.g. a topological field theory). But if this conclusion (regarding fundamental discreteness) is correct, then my argument which singles out D=4 (via counting of equivalence classes of manifolds w.r.t. to diffeomorphism) does no lonager apply.

The answer to this question is more philosophical.
The manifold must be continuous because of quantum mechanics. Sounds like a contradiction...
Quantum fluctuations enforce the spacetime to be continuous. Given a discrete serie of measurement results, each result is caused by a
measurement of the quantum state (as reduction of the state). The randomness of the fluctuations implies the continuous variation
as uncertainity of space and momentum. Then spacetime is the space of all possible events, not more (otherwise you will get a block universe).
So, the discrete values are only a part of the story.
Another lione of argumentation comes from the contradiction between determinism and indeterminism.
I believe in indeterminism, which means, there is no law to forecast uniquely the next measured value. From the point of sequences,
one has to consider so-called random sequences (of nonalgorithmic complexity). But there are uncountable many random sequences
which can be only realized on continuous manifolds.

 

[tom.stoer]

this is indeed philosophical ;-)

 

[Fra]

Unlike Tom, I don't know anything whatsoever about the formalism you work so I'm unable to comment on that but your conceptual argument caught my attention:

I believe in indeterminism, which means, there is no law to forecast uniquely the next measured value. From the point of sequences, one has to consider so-called random sequences (of nonalgorithmic complexity). But there are uncountable many random sequences which can be only realized on continuous manifolds.

If I get the idea right, one has to consider all possible future sequences, in order to construct the action - right so far? And this set is uncountable thus you arrive at continuum structures?

I see the logic but I wonder:

Isn't one here an observer (O)? If so, how about the possibility that the action of O, actually reflects a truncation, where the O, due to limiting information capacity simply can't count all mathematically possible sequences? Maybe the action is rather just reflecting the retained historical sequences belonging to the history of this observer?

A everday analogy here is where the way a person behaves, really depends on wether it's able to mentally reflect over the consequences of certain actions. Ie. kids do things that seem irrational simply because they are unable to make the perfect decision.

/Fredrik

 

[czes]

Quantum fluctuations enforce the spacetime to be continuous. Given a discrete serie of measurement results, each result is caused by a
measurement of the quantum state (as reduction of the state). The randomness of the fluctuations implies the continuous variation
as uncertainity of space and momentum.

The phase coherence of light from extragalactic sources - direct evidence against first order Planck scale fluctuations in time and space.
http://arxiv.org/abs/astro-ph/0301184
Does it mean the spacetime isn't continuous because there aren't the fluctuations ?

 

[tom.stoer]

Hi Torsten,

Unfortunately I am still not able to ask the mathematical questions (and I am afraid I will never be), but the whole approach sounds very interesting, therefore I would like to understand at least the conceptual issues.

First of all can you comment on the equation Ric=0? I mean, w/o matter degrees of freedom I would have expected G=0 (G: Einstein tensor); how is the step towards Ricci flatness justified?

If you rely on a smooth manifold as a fundamental object plus the relation to PL manifolds then there should be some connection to e.g. CDT or Regge theory. Would you say that these are just tools (for certain regimes) but that all essential entities are already provided by the framework “Riemann manifold + smooth structures)?

Regarding quantization: for a smooth manifold with Ric=0 (or G=0) the Asymptotic Safety approach seems to be reasonable, especially as you do not have to worry about matter degrees of freedom. Does that mean that your theory will be something like Asymptotic Safety for f(R) theories? If this is true, then how does exotic smoothness (which in your construction seems to be “localized” and is therefore an UV property) affect renormalization?

How do you define a PI measure? I mean it cannot simply be Dg restricted to one manifold topology and one smooth structure (as in the AS case), but it has to integrate over the smooth structures as well. How does it look like?

Regarding spin: my impression was that spin always introduces extra structures like torsion in Einstein-Cartan theory (or at least that this would be natural). So w/o using tetrads you cannot even describe the coupling of spin to geometry. But we know that Einstein-Cartan-theory is equivalent to the Einstein-Hilbert-theory in the vacuum b/c torsion does not propagate and has to vanish exactly in vacuum. But in your theory there is only vacuum! So you do neither have any non-trivial torsion which could back-react on something, nor do have this “something”, namely spinors. Therefore your theory is entirely described in terms of a metric g from which spin (spin ½) cannot emerge. I could continue with objections regarding chirality, left-right asymmetry, P- and CP-violation etc. (I mean: I don’t expect that all the topics are already worked out in detail, but I would like to understand how you can overcome the problem that in a pure Einstein-Hilbert metric approach they cannot even be formulated – as far as I can see)

Do you rely on M4 ~ M3 * R, global hyperbolicity, foliations or something like that? Or is your approach valid for any compact or non-compact M4? Could a foliation be introduced and would there be a relation to ADM formalism (or perhaps even Ashtekar variables)?

Thinking further into this direction: in Loop Quantum Gravity (prior to the introduction of spin networks) the Ashtekar formulation (and loops) turned out to be completely equivalent to GR (in ADM formalism). Now in LQG you factor away the diffeomorphisms completely (you do that during quantization but the argument would work classically as well), whereas in your approach you need at least the different equivalence classes. So there seems to be a question already at the classical level: does LQG factor away “too much”? Or did they miss to introduce different smooth structures, so is their configuration space “too small” from the very beginning? Or is the configuration space (defined via PL manifolds i.e. Regge-like) large enough simply b/c all PL manifolds are somehow contained? Could it be that LQG perhaps misses some details of the dynamics, but that it could be equivalent to your approach? Which means that it would already contain emergent matter?

Back to gravity: is there an idea what a black hole would be? How do you treat singularities? Are there topology-changes allowed? Or would singularities be replaced by some highly non-trivial, local exotic structure?

Are there any affects regarding long-range or global / topological properties of spacetime? Is your theory equivalent to GR for large-scale physics or do you expect imprints (like LQC effects on CMB)?

 

[torsten]

Unlike Tom, I don't know anything whatsoever about the formalism you work so I'm unable to comment on that but your conceptual argument caught my attention:

If I get the idea right, one has to consider all possible future sequences, in order to construct the action - right so far? And this set is uncountable thus you arrive at continuum structures?

I see the logic but I wonder:

Isn't one here an observer (O)? If so, how about the possibility that the action of O, actually reflects a truncation, where the O, due to limiting information capacity simply can't count all mathematically possible sequences? Maybe the action is rather just reflecting the retained historical sequences belonging to the history of this observer?

A everday analogy here is where the way a person behaves, really depends on wether it's able to mentally reflect over the consequences of certain actions. Ie. kids do things that seem irrational simply because they are unable to make the perfect decision.

/Fredrik

I agree with your claims. But I don't need an observer. QM has a continuous state space inducing a continuous probability theory. For indeterminism I need
random sequences. But for a discrete spacetime we have only finite many possible sequences and get the determinism. If I assume
indeterminism then I need non-algorithmic sequences (or random sequences) but then I have the continuum.

 

[torsten]

The phase coherence of light from extragalactic sources - direct evidence against first order Planck scale fluctuations in time and space.
http://arxiv.org/abs/astro-ph/0301184
Does it mean the spacetime isn't continuous because there aren't the fluctuations ?

Thanks for the paper reference. But see my answer above. To my opinion there is no discrete spacetime. I used the fluctuations only as illustration.

 

[torsten]

Hi Torsten,

Unfortunately I am still not able to ask the mathematical questions (and I am afraid I will never be), but the whole approach sounds very interesting, therefore I would like to understand at least the conceptual issues.

First of all can you comment on the equation Ric=0? I mean, w/o matter degrees of freedom I would have expected G=0 (G: Einstein tensor); how is the step towards Ricci flatness justified?

If you rely on a smooth manifold as a fundamental object plus the relation to PL manifolds then there should be some connection to e.g. CDT or Regge theory. Would you say that these are just tools (for certain regimes) but that all essential entities are already provided by the framework “Riemann manifold + smooth structures)?

Regarding quantization: for a smooth manifold with Ric=0 (or G=0) the Asymptotic Safety approach seems to be reasonable, especially as you do not have to worry about matter degrees of freedom. Does that mean that your theory will be something like Asymptotic Safety for f(R) theories? If this is true, then how does exotic smoothness (which in your construction seems to be “localized” and is therefore an UV property) affect renormalization?

How do you define a PI measure? I mean it cannot simply be Dg restricted to one manifold topology and one smooth structure (as in the AS case), but it has to integrate over the smooth structures as well. How does it look like?

Regarding spin: my impression was that spin always introduces extra structures like torsion in Einstein-Cartan theory (or at least that this would be natural). So w/o using tetrads you cannot even describe the coupling of spin to geometry. But we know that Einstein-Cartan-theory is equivalent to the Einstein-Hilbert-theory in the vacuum b/c torsion does not propagate and has to vanish exactly in vacuum. But in your theory there is only vacuum! So you do neither have any non-trivial torsion which could back-react on something, nor do have this “something”, namely spinors. Therefore your theory is entirely described in terms of a metric g from which spin (spin ½) cannot emerge. I could continue with objections regarding chirality, left-right asymmetry, P- and CP-violation etc. (I mean: I don’t expect that all the topics are already worked out in detail, but I would like to understand how you can overcome the problem that in a pure Einstein-Hilbert metric approach they cannot even be formulated – as far as I can see)

Do you rely on M4 ~ M3 * R, global hyperbolicity, foliations or something like that? Or is your approach valid for any compact or non-compact M4? Could a foliation be introduced and would there be a relation to ADM formalism (or perhaps even Ashtekar variables)?

Thinking further into this direction: in Loop Quantum Gravity (prior to the introduction of spin networks) the Ashtekar formulation (and loops) turned out to be completely equivalent to GR (in ADM formalism). Now in LQG you factor away the diffeomorphisms completely (you do that during quantization but the argument would work classically as well), whereas in your approach you need at least the different equivalence classes. So there seems to be a question already at the classical level: does LQG factor away “too much”? Or did they miss to introduce different smooth structures, so is their configuration space “too small” from the very beginning? Or is the configuration space (defined via PL manifolds i.e. Regge-like) large enough simply b/c all PL manifolds are somehow contained? Could it be that LQG perhaps misses some details of the dynamics, but that it could be equivalent to your approach? Which means that it would already contain emergent matter?

Back to gravity: is there an idea what a black hole would be? How do you treat singularities? Are there topology-changes allowed? Or would singularities be replaced by some highly non-trivial, local exotic structure?

Are there any affects regarding long-range or global / topological properties of spacetime? Is your theory equivalent to GR for large-scale physics or do you expect imprints (like LQC effects on CMB)?

Hi Tom,

I start with Ric=0:
Einsteins equation G=0 means Ric=1/2 g*R or by contraction with respect to the metric R=0 implying Ric=0. So, G=0 is equivalent to Ric=0.

Currently I'm working on the relation to CDT or Regge. I think these theories sum over the essential information contained in the smooth structure: the transition function between patches.
Up to now I never spoke about gravity. The gauge interactions are covered by the torus bundles. There are three possible torus bundles related to U(1), SU(2) and SU(3) groups. But gravity has to be described differently.
Not long ago, I saw how to do it: gravity is described by sphere bundle, i.e. S^2x[0,1] (in space). It explains the universality of gravity: I can always put such a bundle between two fermions and also every other intersection (as gauge interaction) also contains this bundle.

Quantization: we are able to describe exotic smoothness as "localized" but by definition it is extended to the whole space. In our last paper, we relate exotic R^4 to QFT and got a clear sign that the theory is realted to Connes-Kreimer renormalization. Especially the Casson handle (described by a tree) must contain informations about the counter term structure

The PI measure was defined in a previous paper. Usually one divides Dg into a geometric part (integration over the geometries) and an integration over the smoothness structures. For compact manifolds one has counatble many smoothness structures and so we will get a sum. Currently I think about the non-compact case with continuous many smoothness structures.

Spin, chirality etc.: Right, that seems to be a problem. But Einstein-Cartan or Einstein-Hilbert theory don't say anything about the topology of the space. In some sense, matter is concentrated along space with special properties. There is a paper of Sorkin et.al. "spin 1/2 from gravity" which explains how to generate spin. So, our vacuum has a structure.
I will explain it by toy model: Exotic smoothness favours hyperbolic 3-manifolds, i.e. the spatial component contains these negatively curved 3-manifolds. Outside of this 3-manifold we have the empty space (or the vacuum). But hyperbolic 3-manifolds ahev special properties.
For instance: hyperbolic 3-manifolds cannot be scaled, i.e. they are not contractable. They have a preferred orientation (chirality) and there are rotations of the manifold which are not smoothly connected (definition of spin a la Sorkin).

Foliation: You touch a crucial point. Consider S^3xR as model. If S^3xR admits an exotic smoothness structure then you never got a global foliation. Otherwise the S^3 with unique smoothness structure will induce the standard smoothnes structure on S^3xR.
But by foliation theory, you can introduce a codimension-1 foliation which is necessary to get a Lorentz metric. In a concrete example of an exotic S^3xR, there is a topology-change inside of S^3xR, i.e. somewhere S^3 changes to the Poincare sphere P and back.
All topology changes induced by exotic smoothness of S^3xR are of this kind: the S^3 is changed to a homology 3-sphere and changed back. So, the approach is related to ADM but I have to understand how.

LQG: The main problem is the division of the diffeomorphism group into spatial and temporal diffeomorphism. This problem is reflected in the contraints: the Hamilton and the diffeomorphism contraint depend on each other.
Furthermore the restriction to global hyperbolity excluded all exotic structures (see above). Especially one never considers the whole diffeomorphism group.
Only the diffeomorphism connected to identity are used, i.e. one forget the isotopy classes. I give an example: take a torus and cut it to get a cylinder. Then twist one end of the cylinder by a full 2\pi twist and sew both ends together (the procedure is called a Dehn twist). You will get a different looking torus. But this torus is diffeomorphic to the original one. The diffeomorphism is not given by coordinate transformation (small diffeomorphism) but by a global(or large) diffeomorphism.
All large diffeomorphisms are forgotten by LQG but I think it is an error.

More later
Torsten

 

[tom.stoer]

Hi Torsten,

Regarding Ric=0: thanks for the reminder 

Regarding gravity you seem to have something different in mind. My impression was that you can simply use GR, even for exotic smoothness structures. But you seem to have something different in mind.

Regarding Connes-Kreimer renormalization I have no idea 

Regarding spin: Sorkin’s paper „spin 1/2 from gravity“ sounds interesting – but I can’t find it; I’ll check „Spin and Statistics in Quantum Gravity“.

Regarding loops and missing of large diffeomorphisms: in the end they may turn out to be relevant [but there are similar problems in many treatments of QCD like ignorance regarding Gribov copies, missing large gauge transformations … but nevertheless the main results seem to be rather robust with respect to such errors] I am asking b/c my impression is that LQG has nearly the same potential as your approach regarding emergent matter (stable configurations in spin networks, noiseless subsystems, graphs w/o a dual triangulation …) but nobody seems to study these topics. They simply add matter on top of LQG.

Thanks for your time
Tom

 

[John86]

http://arxiv.org/abs/1109.1973
The modification of the energy spectrum of charged particles by exotic open 4-smoothness via superstring theory
Torsten Asselmeyer-Maluga, PawełGusin, Jerzy Król
(Submitted on 9 Sep 2011)
In this paper we present a model where the modified Landau-like levels of charged particles in a magnetic field are determined due to the modified smoothness of $\mathbb{R}^{4}$ as underlying structure of the Minkowski spacetime. Then the standard smoothness of $\mathbb{R}^{4}$ is shifted to the exotic $\mathbb{R}_{k}^{4}$, $k=2p$, $p=1,2...$. This is achieved by superstring theory using gravitational backreaction induced from a strong, almost constant magnetic field on standard $\mathbb{R}^{4}$. The exact string background containing flat $\mathbb{R}^{4}$ is replaced consistently by the curved geometry of $SU(2)_{k}\times\mathbb{R}$ as part of the modified exact backgrounds. This corresponds to the change of smoothness on $\mathbb{R}^{4}$ from the standard $\mathbb{R}^{4}$ to some exotic $\mathbb{R}_{k}^{4}$. The calculations of the spectra are using the CFT marginal deformations and Wess-Zumino-Witten (WZW) models. The marginal deformations capture the effects of the magnetic field as well as its gravitational backreactions. The spectra depend on even level $k$ of WZW on SU(2). At the same time the WZ term as element of $H^{3}(SU(2),\mathbb{R})$ determines also the exotic smooth $\mathbb{R}_{k}^{4}$. As the consequence we obtain a non-zero mass-gap emerges in the spectrum induced from the presence of an exotic $\mathbb{R}_{k}^{4}$.

 

[mhob]

Why you like to incorporate exotic smoothness with string theory not loop quantum gravity or others?

 

[tom.stoer]

Why you like to incorporate exotic smoothness with string theory not loop quantum gravity or others?

See torsten's answer in post #17:

LQG: The main problem is the division of the diffeomorphism group into spatial and temporal diffeomorphism. ... Furthermore the restriction to global hyperbolity excluded all exotic structures. Especially one never considers the whole diffeomorphism group. ... All large diffeomorphisms are forgotten by LQG but I think it is an error.