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Is anybody here familiar with Asselmeyer's work on exotic smoothness and quantum gravity?
tom.stoer said:Is anybody here familiar with Asselmeyer's work on exotic smoothness and quantum gravity?
They do, thanks a lot!torsten said:Hallo Tom,
Now I try to answer or better react on your question:
...
I hope these answers help a little bit.
tom.stoer said: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.
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?torsten said: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.
torsten said: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.
Fra said: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
czes said: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 said: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)?
mhob said:Why you like to incorporate exotic smoothness with string theory not loop quantum gravity or others?
torsten said: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.
Exotic smoothness refers to the existence of smooth structures on topological manifolds that are not compatible with the standard smooth structure. In other words, it describes a different way of smoothly manipulating the points and curves on a surface. This concept is important in the study of quantum gravity because it can lead to new insights and potential solutions to the long-standing problems in the field.
Exotic smoothness is closely related to the concept of spacetime as it allows for the possibility of non-standard smooth structures on manifolds that represent spacetime. This can have implications for our understanding of gravity and the fundamental laws of the universe.
There are several theories and ongoing research surrounding exotic smoothness and quantum gravity. One prominent theory is the theory of loop quantum gravity, which incorporates the idea of exotic smoothness into its framework. Additionally, there are ongoing studies on the relationship between exotic smoothness and black hole entropy, as well as investigations into potential experimental tests for exotic smoothness.
The concept of exotic smoothness challenges traditional notions of smoothness by introducing the idea that there can be multiple smooth structures on a given manifold. This challenges the idea that there is only one "correct" way to smoothly manipulate points and curves on a surface. It also raises questions about the nature of smoothness and its role in understanding the fundamental laws of the universe.
Exotic smoothness has the potential to revolutionize our understanding of the universe by providing new perspectives and insights into the nature of spacetime and gravity. It could potentially lead to the development of a unified theory that combines quantum mechanics and general relativity, as well as shed light on other fundamental mysteries of the universe such as dark matter and dark energy.