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CATEGORY: Questioning the Foundations Essay Contest (2012) [back]
TOPIC: A Chicken-and-Egg Problem: Which Came First, the Quantum State or Spacetime? by Torsten Asselmeyer-Maluga [refresh]
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Author Torsten Asselmeyer-Maluga wrote on Aug. 24, 2012 @ 17:04 GMT
Essay Abstract

In this essay I will discuss the question: Is spacetime quantized, as in quantum geometry, or is it possible to derive the quantization procedure from the structure of spacetime? All proposals of quantum gravity try to quantize spacetime or derive it as an emergent phenomenon. In this essay, all major approaches are analyzed to find an alternative to a discrete structure on spacetime or to the emergence of spacetime. Here I will present the idea that spacetime defines the quantum state by using new developments in the differential topology of 3- and 4-manifolds. In particular the plethora of exotic smoothness structures in dimension 4 could be the corner stone of quantum gravity.

Author Bio

I'm a post-doc worker at the German Aerospace Center. I received my PhD at Humboldt university. My research interests are wide-spreaded from evolutionary algorithms and quantum computing to quantum gravity. Since more than 15 years I try to uncover the role of exotic smoothness in general relativity and quantum gravity.

Download Essay PDF File

Armin Nikkhah Shirazi wrote on Aug. 26, 2012 @ 02:02 GMT
Dear Torsten,

You have written a sophisticated essay which I must admit I only partially understood, but from what I can tell, it is different from other mainstream approaches to quantum gravity.

My ability to provide constructive criticism is limited, but I would like to mention that the idea of constructing a Hilbert space from "wild embeddings" is intriguing, I would have liked to see a fuller conceptual discussion of how this comes about (it seems hard to visualize). Also, does this approach account for the fact that after a "collapse" the state spreads out again, in accordance with the Schroedinger equation? Does it account for entanglement?

Finally, let me remark that I do not actually believe that a quantum theory of gravity is the way to resolve the current situation, but rather that in my view, quantum theory and general relativity have in an as yet unrecognized manner separate domains of validity. I have outlined this idea in my essay, and given that we hold opposite views on this, and you are obviously very knowledgeable in this field, I would appreciate your perspective and criticism.



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Author Torsten Asselmeyer-Maluga replied on Aug. 27, 2012 @ 09:10 GMT
Dear Armin,

thanks a lot for your interest. Yes, this approach is different from the mainstream. My main motivation came from the question of naturalness: what is the next natural step after general relativity? For me, it was the variation of the smoothness structure. Otherwise one has to change too much (variation of topology, dimension etc.). The idea to use wild embeddings is caused by exotic smoothness.

A wild embedding is a hierarchical, infinite structure (like a fractal). So, the Hilbert space is direct consequence of the infinity: every level is a basis vector of the Hilbert space. Then a wild embedding is a linear combination of this vectors.

What about state reduction etc.? In dimension 4, there is a common structure to describe exotic smoothness: the Casson handle. The details are too complicated but in short: a Casson handle is a tree consisting of immersed disks (a disk with finite self-intersections). The wilderness of this object came from the embedding of the disk. The original motivation to construct the Casson handle came from the problem to find an embedded disk inside of the 4-manifold. Technically, an infite number of immersed disks are needed but one obtains a disk after three stages. That means: there is a state reduction after a finite time but one obtains afterwards a full state again (stage four to infinity). The details of this view will be worked by us (Jerzy and me) in the next paper.

I'm looking forward to read your essay.



Jerzy Krol wrote on Aug. 26, 2012 @ 11:59 GMT
Dear Torsten,

Great essay! Even though we work together on some quantum aspects of exotic 4-smoothness your essay is the work which inspires me much and which I red as a very fresh and well argued new work. I have to say that I did not think about QG state as a wild embedding nor I assigned precisely this universal meaning to it as you propose in the essay. But now, I start thinking like that, especially you have collected conviencing arguments for. And this wild embedding is indeed realized within smooth structures in dimension 4. Thus, topology in dim. 3 and 4 is complicated enough to generate a state for QG (if all works well). Certainly, further work is needed but this only makes the approach even more interesting. Especially, when we start with the smoothness, we are not looking for the deeper (quantum) level, does it mean that physical world is rather classical? Or it is just the potential of smoothings without clear further implications? Another thing: if the smoothness in dimension 4 were only standard we would not have had a QG state. So, if we lived in the world with only standard smoothness in dim. 4, gravity would not be quantized. So, maybe, this is some reason behind the difficulty with quantizing gravity - we do it, in general, as if the smooth structure were only standard?

Certainly, I wish you success in the contest,


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Author Torsten Asselmeyer-Maluga replied on Aug. 27, 2012 @ 10:13 GMT
Dear Jerzy,

thanks for your words. Similar ideas were a motivation for me.

Your question is very interesting. A wild embedding is a hierarchical object which includes every scale. Therefore the quantum level must be included.

Our previous work about foliations, exotic smoothness and non-commutative geomery gives a quantum structure. I think the wild embedding is the expression for the quantum level. So, gravity must be also quantized but more work is needed to uncover it.

Yes, right if there is only standard smoothness then there is no quantization (in this model). I agree with you, that can be reason for the problems in quantum gravity.

I'm looking forward to read your essay, Good luck too.


Member Benjamin F. Dribus replied on Aug. 27, 2012 @ 16:26 GMT

I really enjoyed your essay; it is uncommon to see simple motivating ideas combined with mathematical maturity. I have a few questions.

1. I am not quite sure how to think about time in your exotic
. What do the leaves
represent physically?

2. I assume you are thinking of using all the exotic smoothness structures, rather than suggesting there is a preferred one?

3. If so, would it be accurate to say that you are changing the covariance principle from a local group symmetry to a family of local group symmetries parametrized by exotic smoothness structures?

The reason I am interested in your qualitative view of the covariance principle is because I suspect myself that covariance is not best thought of as a symmetry. I prefer to think of it in order theoretic terms. My essay

On the Foundational Assumptions of Modern Physics

briefly explains this. Personally, I think manifolds are too good to be true, but I do find your approach very intriguing. Take care,

Ben Dribus

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Author Torsten Asselmeyer-Maluga replied on Aug. 28, 2012 @ 11:32 GMT

I also really enjoyed your essay with many deep thoughts; I will write more on the discussion area of your essay.

Now to your questions.

At first, my interest in exotic smoothness of 4-manifolds was the driving force for the direction of my work. I was forced by the theory to change some concepts and keep some other ones.

ad 1. In usual general relativity...

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Jayakar Johnson Joseph wrote on Aug. 28, 2012 @ 05:00 GMT
Dear Torsten Asselmeyer,

Spacetime is also applicable in Coherently-cyclic cluster-matter universe model with modifications, in that quantization of eigen-rotational strings is expressional instead of quantum state of particles.

With best wishes,


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Author Torsten Asselmeyer-Maluga replied on Aug. 28, 2012 @ 11:35 GMT
Dear Jaykar,

I do not understand your essay fully. But quantum states are really important in my opinion. Otherwise, you have a problem to understand uncertainity etc.


Cristinel Stoica wrote on Aug. 28, 2012 @ 20:33 GMT
Dear Torsten,

I am happy to find your essay here. I read it with enthusiasm, being interested in your work and exotic smoothness in general (and having so much to catch up on). I like how you managed to take the reader from simpler facts from differential topology and geometry, to very advanced topics and results. There are so many things we don't yet understand about the four-dimensional spacetime, that I am amazed why so many physicists consider that we already know everything worth knowing about them. My initial investigations and hopes were towards getting particles and quantumness from topology, but apparently this is not rich enough. Exotic smoothness definitely is richer, and deserves far much more attention than it presently receives. As much as I understand, the idea of using wild embedding to get quantum gravity is novel, and I think it is brilliant. Further developments may reveal that it indeed answers the question.

Best regards,

Cristi Stoica

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Author Torsten Asselmeyer-Maluga replied on Aug. 29, 2012 @ 08:42 GMT
Dear Chris,

thanks for your words. It is great if the very small exotic smoothness community will growing. The real power of the idea is the connection between wild embeddings and exotic smoothness. In arXiv:1105.1557 we used the wild embeddings to understand quantum D-branes. This approach was a little bit artificial. The connecetion to exotic smoothness is much better.

I will read your essay too.



Member Benjamin F. Dribus wrote on Aug. 28, 2012 @ 20:51 GMT

I appreciate your answers. A couple of other questions came to mind, if you have time for them.

1. Do you regard what is conventionally understood as matter-energy to be part of the quantum state given by a wild embedding, arising together with the quantum spacetime, or do you regard it as auxiliary? I understand how you derive an operator algebra and Hilbert space from such an embedding, but are "particle states," for instance, actually generated somehow, or are they merely constrained by the noncommutative structure, as the Poincare group constrains possible particle types in ordinary QFT?

2. A related question is about representation theory. This is partly why I asked about your view on covariance. In ordinary Minkowski spacetime, the symmetries define what covariance means and also constrain the particle types (representations). Obviously representation theory is absolutely central, appearing also with gauge groups, etc. I am wondering if the "representation theory of 2-groups" (if such a thing has been developed) might play a role, possibly even leading toward predictions of "wild embedding particles" not appearing in the standard model.

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Author Torsten Asselmeyer-Maluga replied on Aug. 29, 2012 @ 09:00 GMT

Of course I have time, I appreciate your interest.

ad 1. I expect that the wild embedding determines also the matter-energy part. The wild embedding is a direct consequence of exotic smoothness but as shown in arXiv:1006.2230, one obtains the Dirac and YM action from it (Fermions are knot complements and bosons are torus bundles). But maybe more is true. In arXiv:1112.4885 we speculated about a relation to the Connes-Kreime renormalization schema.

ad 2. Jerzy is the expert in higher-categories. For me, the 2-groups is an expression to understand the diffeomorphism group. It is a pseudogroup, especially there is strong division between global and local diffeomorphism. The 2-group can help to express this difference: local diffeomorphisms (or coordinate transformations in phyiscs) are the usual morphism but global diffeomorphisms (like the Dehn twist in my essay) are 2-morphisms. Particle representations are equivalence classes with respect to loacl diffeomorphisms. We used this concept in the arXiv:1006.2230 article to obtain the correct gauge group SU(3)xS(2)xU(1) (by considering the connecting component of the isometry group of the torus bundles). I think we are closer with our ideas then we think. I also look for more general symmetries away from the usual Lie groups.


Lawrence B Crowell wrote on Aug. 31, 2012 @ 18:48 GMT
Dear Torsten,

I read your essay last week or so with the intention of reading it again in more depth. I just again gave your essay a review type of reading. I studied exotic manifolds some time ago. I even pondered their role in quantum gravity. I will need to pull down my copy of Donaldson & Kronheimer to refresh myself on the subject of exotic 4-manifolds. I respond in greater detail in my essay blog area to your response. I think potentially the connection between what you have done is more than just boring, as you put it, but there might be richer connections in the structure of Yangians (which I did not break out in my essay) there exists a dual description, one with spacetime configuration variables possible and one without. One might be the chicken and the other the egg, where I am not sure which came before the other.

Your essay seems to point to a map between spacetime structure and quantum states. This might not be that far off from the duality I just mentioned. In order to comment more fully I really need to review these mathematical topics. I commented more fully in my area, and we can maybe dialogue on possible connections between Yangians in supergravity, quantum groups and exotic manifolds.

Cheers LC

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Author Torsten Asselmeyer-Maluga replied on Sep. 3, 2012 @ 09:09 GMT
Dear Lawrence,

for an answer I will move to your essay discussion page.

Cheers Torsten

Yuri Danoyan wrote on Sep. 3, 2012 @ 10:29 GMT
Dear Torsten,

first of quote from your essay:

"For example consider the foliation of an

exotic spacetime like S3  R can be very complicated."

I did kind of foliation in my essay

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Author Torsten Asselmeyer-Maluga replied on Sep. 4, 2012 @ 18:53 GMT
Dear Yuri,

I see that you use codimension-1 foliations but of a kind which I cannot use.

My foliations are really complicated: the obvious foliation contains a serie of wild embeddings.

Cheers Torsten

Angel Garcés Doz wrote on Sep. 3, 2012 @ 22:51 GMT
His essay I consider excellent.

The problem of the mass gap in physics can be easily calculated precisely, if quantized space-time-mass, as a single entity

The key is in reducing, by holography, space-time-mass, information contained in surfaces.

Exactly: the space-time-mass, is a geometric entity, whose starting point is the compactification of spheres (or torus) in two...

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Author Torsten Asselmeyer-Maluga replied on Sep. 4, 2012 @ 18:56 GMT
Interestig coincidences, I have to think about it.

Thanks for your interest.



Angel Garcés Doz wrote on Sep. 4, 2012 @ 23:09 GMT
Yes, other coincidences, dear Armin :

(six quarks x three colors) + eight gluons = 26

lenght larger radius torus, compactification in 26 dimensions:

Thank you very much

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Yuri Danoyan replied on Sep. 10, 2012 @ 08:12 GMT
Mp/Me= 1836

For example Mp/Me=1836 is a true dimensionless constant. I found that it is a beautiful symmetric number because 1+8=3+6=9, after it is converted to numerological addition. In the binary code 1001 present mirror symmetry.

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Yuri Danoyan wrote on Sep. 9, 2012 @ 11:01 GMT
A Chicken-and-Egg Problem: Which Came First, the Quantum State or Spacetime?

To my opinion best solution is Bootstrap model in the modern level.

I suggest that 3:1 ( examples #1,#2,#3) is enclosed in a total interaction of Bose and Fermi particles or fields, and it is a bootstrap relationship between mentioned evidences.

Surprisingly, the container(space-time), QM content(fermions-bosons), SR content (energy-matter) obey the same law 3:1.

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Vasily Kletushkin wrote on Sep. 11, 2012 @ 17:23 GMT
English I don't know. Read esse can't. Tried to understand the meaning of the annotations and comments. The chicken and the egg are odnovrmenno. Quantum state and space odnovrmenno without time exist. This metaphysics. I wish the best of luck.

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Eckard Blumschein replied on Sep. 11, 2012 @ 20:37 GMT
odnowremenno = at the same time

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Author Torsten Asselmeyer-Maluga replied on Sep. 14, 2012 @ 08:31 GMT
Thanks for the wishes. Unfortunately my russian is very bad to answer (I never practice it).

Right, the chicken and the egg is at the same level. I use the famous philosophical problem to give the hint that spacetime and the quantum state are the same.


Frank Martin DiMeglio wrote on Sep. 12, 2012 @ 00:51 GMT
What comes first is us and our [natural/ordinary/typical] experience taken TOGETHER. Nature, truth, us, and the fullness of reality together as one -- this is what the ultimate order fundamentally and truly is, and it is what the physicists should be thinking/considering. Indeed, especially these days, they need to be. The truth is simple and deep. It involves order and variability and randomness and order.

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Author Torsten Asselmeyer-Maluga replied on Sep. 14, 2012 @ 08:34 GMT
Interesting. The space of the essay is limited to also discuss questions like: what is time? (I suppose you say order) what is dynamics? (variability) what is the free will? (randomness)


Constantinos Ragazas wrote on Sep. 14, 2012 @ 00:30 GMT
Hello Torsten,

I will not comment on your deep and sophisticated essay. But I do wonder. Is Nature so complicated? Possibly. But I keep believing probably not! And have many reasons to think so. Some of which I try to argue in my essay, “The Metaphysics of Physics”. At the Endnotes of my essay you will find a very simple and elegant proof of Planck's Formula for blackbody radiation. As...

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Author Torsten Asselmeyer-Maluga replied on Sep. 14, 2012 @ 08:40 GMT
Hello Constantinos,

I will read your essay in the next days.

But I will comment on your question: Is Nature so complicated?

I believe that Nature is not deterministic but if this assumption is true then one needs really complicated processes rpoducing a kind of randomness in Nature. Nonlinear interactions are a possible explaination (producing bifurcations in the solution space or chaos as expression of randomness in the initial conditions).

That is my motivation.

Cheers Torsten

Constantinos Ragazas replied on Sep. 14, 2012 @ 21:31 GMT
Dear Torsten,

I agree Nature is not deterministic. A 'chicken/egg' interaction has no deterministic 'cause and effect'. Our experience of Nature is as random as is not. And only we can make it less random by imposing order and predictability. All these conclusions are crystal clear were we to consider our 'human being'. But we lose sight of that when we seek physics. Thinking physics somehow is different. Were we to mathematically model a cloud to describe a cloud, wont an artist's rendition be more truthful?



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Patrick Alan Hutchinson wrote on Sep. 14, 2012 @ 23:06 GMT
Hello Torsten

I have just read the introduction to your essay. It is coherent and bold. Is it necessary to be so bold?

Your stated assumptions 1 and 2 appear attractive. Like you, I want to keep them. In fact, it would be nice to go further: unless there is a really compelling cause, I see no reason to consider quantizing space-time at all. One can develop lots of good...

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Author Torsten Asselmeyer-Maluga replied on Sep. 17, 2012 @ 14:33 GMT
Hello Alan,

I think we agree more than you thought. You start with a smooth manifold and consider Riemnnian geometry to explain our world. In my opinion, your asymmetric metric is not new (and was already considered by Einstein). I start at the same place (a 4-manifold) but make one step further. There is no canonical way to choose a differential structure on a 4-manifold. Therefore I studied the influence of the smoothness on our (field) theories and found amazing things.

The motivation is the same as the motivation of Eddington. I looked into his book "Fundamental theory". His relational theory reminds me on the mathematical manifold theory where one uses equivalence classes of coverings (or atlases) to define a differential structure. Every concrete realization of a differential structure is a concrete physical situation (a collection of reference systems, with overlapp to express the information exchange between these systems).

I'm driven by the question: what is the physically natural structure to describe quantum gravity? For me, it is the exotic smoothness.

(The "exotic" is misleading: all differential structures except one are exotic on a 4-manifold)

Best wishes


Alan Hutchinson replied on Sep. 17, 2012 @ 23:18 GMT
Hello again Torsten

Thanks for looking at my essay. I must read the rest of yours. I suspect it will be hard going. I have a copy of Donaldson & Kronheimer, but it is in one of a few big boxes in the attic (put there when the family told me to give up my study to make way for a new child's bedroom), and I can't remember which box.

You are quite right about Einstein: he did consider skew metrics. My copy of his book is still on an open shelf. The big relevant changes in understanding since he wrote it are Hannabuss' discovery of an extension to Clifford's construction which works for nonsingular asymmetric metrics, and the Dubois-Violette Launer construction. They both stem from the metric. From the little I remember of the D&K book, exotic smooth structures stem from properties of connections instead. Have I got that right?

You seem to accept that there is such a thing as quantum gravity. For all I know, there may be. It promises a way to reconcile GR with quantum theory, but the idea doesn't sound right to me. My point is that things might work the other way: it is conceivable that gravity may not be quantized, and instead the right quantum theory may not be linear. This approach provides enough orthodox mathematics to make constructions which could potentially model physics.

Best wishes


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Anonymous wrote on Sep. 18, 2012 @ 22:54 GMT
Hello Torsten

Your essay was easier for me than I expected. I wrote a thesis on topology of 4-manifolds in the 1970s, and many of the ideas you touch on are familiar. I enjoyed it, and I admire your hard work.

We are both trying to reconcile GR and QM, but our approaches are quite different.

Best wishes, Alan H.

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Author Torsten Asselmeyer-Maluga replied on Sep. 23, 2012 @ 20:44 GMT
Hello Alan,

our approaches are far more related.

You start with a metric and a connection (with respect to the tangent bundle) and demand some stationary YM functional. I also assume also the first two (metric and connection). The Yang-Mills functional is derived but I have to assume the Einstein-Hilbert action. You mention the Clifford, the Hannabuss algebra and some Hopf algebra. We found a strong relation between smoothness structures and codimension-1 foliations used to construct the Clifford algebra (of the Hilbert space, i.e. the Fock space of a free fermion) also related to Tomitas modular theory (an example of a Hannabuss algebra, Example 1.6). But codimension-1 foliations are also strongly related to Hopf algebras (as shown by Connes and Moskovic). Interestingly this Hopf algebra is isomorphic to the Connes-Kreimer Hopf algebra of renormalization in QFT.

So, my theory also fulfills your criteria.

Best Torsten

Patrick Alan Hutchinson replied on Sep. 24, 2012 @ 08:16 GMT
Thanks Torsten.

This looks like closely related work, as you say. I shall try to understand it.

bw, Alan H.

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Patrick Alan Hutchinson replied on Sep. 27, 2012 @ 22:25 GMT
Hello Torsten

After a scamper through a few references, here is an ill-formed picture. Please forgive me for the mistakes.

1. Your topological constructions.

An exotic space-time is formed by choosing a rather unusual 3-manifold K and a normal bundle for it whose boundary, a double cover of K, is a 3-torus; and choosing an embedding of a 2-torus in ordinary space-time, with a tubular neighbourhood whose boundary is also a 3-torus; and replacing the 2-torus with K by surgery, so constructing a 4-manifold M. One then has to prove two theorems: a/ that M is homeomorphic to flat 4-space, and b/ that M is not diffeomorphic to flat 4-space. None of this is easy.

2. Properties of the resulting M.

c/ On M, Einstein's equation has an extra term because the Einstein tensor of any connection on M cannot vanish on the neck where the surgery was performed - otherwise the connection could be flat (I suppose) which would contradict theorem b/. d/ ... lots of wonderful results about function spaces related to M ...

3. Consequences for my conjectures:

While writing my essay, I did not get as far as imagining that space-time might be anything more complicated than ordinary flat 4-space. Perhaps the only interesting solutions of the Yang-Mills problem occur on the sort of exotic 4-spaces which you discuss.

Thank you for your guidance.

Alan H.

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Hoang cao Hai wrote on Sep. 19, 2012 @ 13:37 GMT

Very interesting to see your essay.

Perhaps all of us are convinced that: the choice of yourself is right!That of course is reasonable.

So may be we should work together to let's the consider clearly defined for the basis foundations theoretical as the most challenging with intellectual of all of us.

Why we do not try to start with a real challenge is very close and are the focus of interest of the human science: it is a matter of mass and grain Higg boson of the standard model.

Knowledge and belief reasoning of you will to express an opinion on this matter:

You have think that: the Mass is the expression of the impact force to material - so no impact force, we do not feel the Higg boson - similar to the case of no weight outside the Earth's atmosphere.

Does there need to be a particle with mass for everything have volume? If so, then why the mass of everything change when moving from the Earth to the Moon? Higg boson is lighter by the Moon's gravity is weaker than of Earth?

The LHC particle accelerator used to "Smashed" until "Ejected" Higg boson, but why only when the "Smashed" can see it,and when off then not see it ?

Can be "locked" Higg particles? so when "released" if we do not force to it by any the Force, how to know that it is "out" or not?

You are should be boldly to give a definition of weight that you think is right for us to enjoy, or oppose my opinion.

Because in the process of research, the value of "failure" or "success" is the similar with science. The purpose of a correct theory be must is without any a wrong point ?

Glad to see from you comments soon,because still have too many of the same problems.

Regard !


August 23, 2012 - 11:51 GMT on this essay contest.

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James Lee Hoover wrote on Sep. 20, 2012 @ 21:25 GMT

Quite interesting.

"In general relativity, gravity is treated as a massless spin-2 field is not one that can be given precise meaning outside the context of the linear approximation."

Is this indicative of the continuing mystery of gravity? Observations I mention in my essay might show a different character than we model.


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Author Torsten Asselmeyer-Maluga replied on Sep. 23, 2012 @ 20:49 GMT
Hello Jim,

Yes right, it is the origin of the mysteries for gravity. I remember on a paper where ist was shown that spin-2 graviton is not enough to derive Einsteins equation.

I have to read your essay.



Lawrence B. Crowell wrote on Sep. 28, 2012 @ 01:52 GMT

At the risk of sounding like a crank, I have done some thinking about this matter of exotic manifolds, which I thought I might relay. If this is wrong that is fine; if this is wrong I’d rather know this is flawed than to think otherwise. My essay discusses noncommutative geometry, which implies manifold structure exhibits quantum fluctuations on a very small scale. As one...

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Anonymous replied on Sep. 28, 2012 @ 13:13 GMT
Dear Lawrence,

Jerzy answers some of the questions. (Thanks, Jerzy) The relation to modular forms is fascinating. In Seiberg-Witten theory one has the modular curve which expresses the properties of the TQFT. From that point of view, the relation is not fully unexpected but the results of our last two papers are go further.

But now some words to your objection (BTW I like it, our...

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Author Torsten Asselmeyer-Maluga replied on Sep. 28, 2012 @ 13:16 GMT
My asnwer was to long for FQXI.

It was me


Jerzy Krol wrote on Sep. 28, 2012 @ 09:05 GMT
Dear Lawrence,

certainly Torsten will comment more thorough your very accurate observations, but let me comment here on one very interesting aspect. You mentioned T-duality as a way for changing the scale on manifold/noncommutative realms. Many expressions generated from QFT on such R4, and GR, in the so called foliated topological limit are represented by Eisenstein 2nd series, which are quasi-modular. That would indicate that under modular transformations (or T-duality) changing the scale on exotic R4 we would have quasi-modular invariants. Exotic diffeomerphisms probably are based on quasi-modular expressions. This is based on Connes-Moscovici construction representing the codimension-one foliations with non-zero GV. Given the noncommutative algebras assigned to exotic R4, we face the strange pattern of scale changing alone with the manifold/noncommutativity duality, and quasi-modular diffeomorphisms. This seems very close to your picture, but more settled in the smoothness alone. Torsten and me, we are just at the beginning of this story, but the subject seems very promising and fascinating indeed. So, maybe, this T-duality thread you propose, has more to do with exotic R4 alone that one could suspect at the beginning.



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Author Torsten Asselmeyer-Maluga replied on Sep. 28, 2012 @ 13:16 GMT
Thanks Jerzy, we have to understand this relation much further.


Lawrence B. Crowell wrote on Sep. 29, 2012 @ 01:59 GMT
Torsten and Jerzy,

I just finished a rather long post on Ben Dribus’ essay page, where his is a good essay to read. So I am a bit tired of writing. I will write more explicitly on your issues tomorrow or early next week. I did write something of relevance, so I will relay the core part of this that has to do with an uncertainty between time and spatial coordinate.

When it...

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Peter Jackson wrote on Oct. 2, 2012 @ 15:19 GMT

Have you ever wandered lost around a strange complex city, then are shocked to find a place identical to one you know, then find it IS that same place! I've just been there in your essay. It was at once an exhilarating and bizarre experience. I can't say I can find any fault or disagreement (I'd have to understand it better anyway for that!).

But I DID "start with a concrete action" so I can only justify the mechanism I used to arrive there with a naive ontological construction, not the thorough theoretical and mathematical approach you use, expressing the logic in the current language of physics. I commend you on the theory and the essay.

I hope you may also speak literal formalisms and will read and understand mine. It's more prosaic and poetic (well..a sonnet anyway) compared to yours but with a little theatrical entertainment to aid kinetic visualization. Eight assumptions are identified and more consistent re-interpretations form a model which turns out to have the hierarchical structure of Truth propositional Logic, first deriving the STR postulates from an evolving quantum mechanism, then finding spacetime also emergent.

It astonished me how central the torus became, as that is central to my astronomical work and at all scales (i.e. fractal), barely touched on here, but in the end notes and last years essay. Also momentum and indeed frame transitions, both of which I also show "...must be seen in the context of a measurement device.." producing noncommutative geometry but conserving the wave equation and causality.

Very well done for your work, and showing me another route to the place where we both seem to find truth lies. I hope you can follow my own, and perhaps we can meet there to discuss further.

Best of luck in the final placings.


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Lawrence B. Crowell wrote on Oct. 3, 2012 @ 01:23 GMT
Torsten & Jerzy,

It took me a while to get back to this. I have been working some on how Yangians work in this. The Chern-Simons Lagrangian and the appearance of knot crossing equations seems to suggest there is a dual gauge form of the Yang-Baxter equation underlying this.

A Lie algebra gl(N) is a set of NxN matrices with a bracket structure [A, B] that is in general nonzero. The...

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Sergey G Fedosin wrote on Oct. 4, 2012 @ 06:18 GMT
If you do not understand why your rating dropped down. As I found ratings in the contest are calculated in the next way. Suppose your rating is
was the quantity of people which gave you ratings. Then you have
of points. After it anyone give you
of points so you have
of points and
is the common quantity of the people which gave you ratings. At the same time you will have
of points. From here, if you want to be R2 > R1 there must be:
In other words if you want to increase rating of anyone you must give him more points
then the participant`s rating
was at the moment you rated him. From here it is seen that in the contest are special rules for ratings. And from here there are misunderstanding of some participants what is happened with their ratings. Moreover since community ratings are hided some participants do not sure how increase ratings of others and gives them maximum 10 points. But in the case the scale from 1 to 10 of points do not work, and some essays are overestimated and some essays are drop down. In my opinion it is a bad problem with this Contest rating process. I hope the FQXI community will change the rating process.

Sergey Fedosin

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