A Chicken-and-Egg Problem: Which Came First, the Quantum State or Spacetime? by Torsten Asselmeyer-Maluga

Torsten Asselmeyer-Maluga

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.

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[deleted]

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.

Sincerely,

Armin

Torsten Asselmeyer-Maluga

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.

Best

Torsten

[deleted]

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,

Jerzy

Torsten Asselmeyer-Maluga

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.

Torsten

Benjamin Dribus

Torsten,

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 [math]S\times_\theta\mathbb{R}[/math]. What do the leaves [math]S_i\times\mathbb{R}[/math] 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

Torsten Asselmeyer-Maluga

Benjamin,

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 (GR), causality is implemented by the property of 'global hyperbolicity' leading to the model [math]\Sigma\times\mathbb{R}[/math] with the 3-manifold [math]\Sigma[\equation]. But then one obtains the "Block universe of Parmenides": a purely deterministic world. Exotic smoothness enforces me to rethink the causality question. Global hyperbolicity fails, i.e. there are naked singularities (saddle points of a Morse function). For me, it reflects the openess of the future. In a saddle point of this kind, two independent trajectories went ion and two indpendent trajectories went out but there is no a priori relation between them.

Furthermore There is no trivial foliation and if one impose such a foliation then the 3-manifold is wildly embedded. But there is another possibility: one foliates the 3-manifold by surfaces. I have only a glimpse of an interpretation: In one of my last papers (arXiv:1006.2230), we considered the appearance of metter by using exotic smoothness. In this approach, embedded surfaces played a central role, so I would expect this surfaces are the same, i.e. this foliation represents the conservation of matter. But the thoughts are in the flow....

ad 2. Currently I cannot prefer a specific smoothness structure. I know that all exotic smoothness structures have the described properties.

ad 3. Yes that is a good expression: we use rather 2-groups than groups. My coauthor (and friend) Jerzy is also a specialist in topos theory. I think he will go further and change the logic too.

More comments to your essay in your discussion forum.

Best

Torsten

Jayakar Joseph

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,

Jayakar

Torsten Asselmeyer-Maluga

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.

Torsten

Cristinel Stoica

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

Torsten Asselmeyer-Maluga

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.

Best

Torsten

Benjamin Dribus

Torsten,

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.

Torsten Asselmeyer-Maluga

Ben,

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.

Torsten

Lawrence Crowell

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

Torsten Asselmeyer-Maluga

Dear Lawrence,

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

Cheers Torsten

[deleted]

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

http://fqxi.org/community/forum/topic/1413

Torsten Asselmeyer-Maluga

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 Doz

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 dimensions, ie: six areas that touch each one, the seventh center.

This seven spheres, directly involving the seven dimensions compacted, and the confinement of quarks, in its lowest energy state possible: the pi meson, zero.

It is not my intention to describe in detail the whole process, but is reduced to the following result consistent, to a large degree, with the experimental data. Let the difference in mass between quarks, u-anti_u (smaller mass, the state-anti_d d), belonging to the less massive hadron, the pion zero and zero pion mass.

For a compactification length in two dimensions, with the simplest model of Kaluza-Klein compactification, we have that this length is:

[math]l(2d)=(4\pi)^{1/4}=1.882792527

[/math]

For maximum compactness of spheres in two dimensions, there are six areas that touch a central, the seventh. The prime number seven is not factorizable in the Gaussian integers, which means no breakable entanglement.

The confinement of quarks is an intrinsic property of the mutual connectivity of these seven spheres/torus.

When trying to separate the quarks, this entanglement and repetitive the structure itself, the space-time-mass, causes the energy communicated to separate quarks becomes two-dimensional other areas that fill the gaps, the attempted separation.

Summarizing a bit: the ratio between the mass difference between the zero pion mass, and the mass contributed by the quarks, which produce the confinement, and the mass of the electron, is exactly the area of these six areas, with a radius equal to l(2d), that is:

[math][m_{\pi^{0}}-(2m_{u})]/m_{e}=6\cdot[4\pi\cdot l^{2}(2d)]\cdot\sin^{2}(\pi/2\:-2[2\pi/60])=255.7259697

[/math]

[math]60=dim[SO(6)]\cdot dim[SO(4)]\;;\:6d+4d=10d\;;\:6\cdot4=dim[SU(5)]

[/math]

[math]24=Kissing\: number\:4d=(Kissing\: number\:2d)\cdot4d

[/math]

[math]m_{\pi^{0}}=134.9766\: Mev/c^{2}\:;\; m_{u}=2.15\: Mev/c^{2}

[/math]

[math]m_{e}=0,510998928\: Mev/c^{2}[/math]

Thank you very much

Torsten Asselmeyer-Maluga

Interestig coincidences, I have to think about it.

Thanks for your interest.

Best

Torsten

Angel Doz

Yes, other coincidences, dear Armin :

[math]m_{p}/m_{e}=1836.15267245[/math]

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

lenght larger radius torus, compactification in 26 dimensions:

[math]\Biggl(\frac{2\cdot(2\pi)^{26}}{2\pi^{26/2}/\Gamma[26/2]}\Biggr)^{\frac{1}{26+2}}=6.61240539117564

[/math]

[math]60=dim[SO(6)]\cdot4d={\displaystyle \sum_{d/24\: d\neq24}d}\;;\:6d+4d=10d\;;\:6\cdot4=dim[SU(5)]

[/math]

[math]60=dim[SO(5)]\cdot dim[SO(4)]=2(1^{2}+2^{2}+3^{2}+4^{2})

[/math]

[math]6\cdot[4\pi\cdot l^{2}(2d)]\cdot\cos^{2}(2\beta)=255.7259697=[m_{\pi^{0}}-(2m_{u})]/m_{e}

[/math]

[math]\pi/2\:-[2\pi/60]=beta\: angle\: supersymmetry=\beta

[/math]

[math]6\cdot[4\pi\cdot l^{2}(2d)]\cdot\cos^{2}(2\beta)\cdot[\sin(2\pi/l_{26d})+3]\cdot l(2d)+\:\alpha_{s}^{2}(m_{Z})=m_{p}/m_{e}

[/math]

[math]\alpha_{s}(m_{Z})\simeq0.1184

[/math]

[math]Cabibbo\: angle=\theta_{c}=360^{\circ}/(\ln(3^{2})\cdot4\pi)=13.0382^{\circ}

[/math]

[math]\ln(3^{2})\cdot4\pi\simeq26/(\varphi-1)^{1/8}\;;\:\varphi=Golden\; ratio

[/math]

Thank you very much

[deleted]

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.