# Alain homers with Ali and Slava on base

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I think Alain Connes, Ali Chamseddine, and Slava Mukhanov have scored a big hit with the November 2014 paper titled Geometry and the Quantum: Basics.

I'll try to explain bit by bit what is going on with that paper, and will appreciate additional understanding and explanation others may wish to volunteer.

Connes and Chamseddine are both at Bures-sur-Yvette (basically a French "institute for advanced studies" called IHES) and Mukhanov is based in Munich. Chamseddine also has a professorship at Beirut, so you might hear of seminars on this stuff being held in Beirut.

Basically it was a "waiting for the other shoe to drop" situation. C&C seemed to pull the Standard Model particles out of some algebra, but it wasn't clear where a piece of the algebra came from. And it looked like they might be going to get quantum gravity too in a combined package. Finally, it seems, they did! and the same algebraic elements emerged. So they have a unified QG&M (quantum geometry and matter, if that makes sense). Particles and quanta of geometry turn out to grow from a common root, and be encompassed by the same algebra. Have to do something in the kitchen, back soon.

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wabbit, atyy and ShayanJ

## Answers and Replies

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Here's the paper:
http://arxiv.org/abs/1411.0977
Geometry and the Quantum: Basics
Ali H. Chamseddine, Alain Connes, Viatcheslav Mukhanov
(Submitted on 4 Nov 2014)
Motivated by the construction of spectral manifolds in noncommutative geometry, we introduce a higher degree Heisenberg commutation relation involving the Dirac operator and the Feynman slash of scalar fields. This commutation relation appears in two versions, one sided and two sided. It implies the quantization of the volume. In the one-sided case it implies that the manifold decomposes into a disconnected sum of spheres which will represent quanta of geometry. The two sided version in dimension 4 predicts the two algebras M2(H) and M4(ℂ) which are the algebraic constituents of the Standard Model of particle physics. This taken together with the non-commutative algebra of functions allows one to reconstruct, using the spectral action, the Lagrangian of gravity coupled with the Standard Model. We show that any connected Riemannian Spin 4-manifold with quantized volume >4 (in suitable units) appears as an irreducible representation of the two-sided commutation relations in dimension 4 and that these representations give a seductive model of the "particle picture" for a theory of quantum gravity in which both the Einstein geometric standpoint and the Standard Model emerge from Quantum Mechanics. Physical applications of this quantization scheme will follow in a separate publication.
33 pages, 2 figures

Mukhanov is (by all signs) a world class cosmologist and physics theorist, that's as far as I can tell. Others here will know more of him, and have surely used his textbook, so they can speak more precisely. It seems to me like a fairly big deal for C&C who have hitherto been focused on spectral geometry and high-level algebra to be now working with Mukhanov. Mukhanov recently came out with a paper showing that inflation does not necessarily involve producing a multiverse. At the recent Paris conference reviewing the 2014 Planck results, he was on a four member panel critically examining the issues around inflation. Other panel members were Paul Steinhardt (Princeton), and Robert Brandenberger (McGill).

Check out the Wiki on him:
http://en.wikipedia.org/wiki/Viatcheslav_Mukhanov
"Viatcheslav Mukhanov (...born October 2, 1956) is a theoretical physicistand cosmologist. He is the best known for the theory of Quantum Origin of the Universe Structure. Working in 1980-1981 with Gennady Chibisov in the Lebedev Physical Institute in Moscow he predicted the spectrum of inhomogeneities in the Universe, which are originated from the initial quantum fluctuations. The numerous experiments in which there were measured the temperature fluctuations of the Cosmic Microwave Background Radiation are in excellent agreement with this theoretical prediction, thus confirming that the galaxies and their clusters originated from the initial quantum fluctuations. Later on V. Mukhanov proved that the results, he obtained with G. Chibisov in 1981, are of the generic origin and he has developed the general consistent quantum cosmological perturbation theory"...
Authored a major textbook I've heard a lot of good things about.
The Physical Foundations of Cosmology
https://www.amazon.com/dp/0521563984/?tag=pfamazon01-20&tag=pfamazon01-20
that line of thought and learning is so important! Because cosmology is what we observe (where nature goes to extremes) and so comparing observation with fundamental theory is the primary hope we have to TEST fundamental theory at the extremes (of radiation temperature, density, curvature, particle energies) where ground based laboratories can't go.

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ShayanJ
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Notice in the previous post, where CC&M say "which will represent quanta of geometry. The two sided version in dimension 4 predicts the two algebras M2(H) and M4(ℂ) which are the algebraic constituents of the Standard Model of particle physics."

Here M4(ℂ) simply means the algebra of 4x4 matrices of complex numbers, so that has real dimension 32. A matrix is made of 16 complex numbers and the complex plane ℂ has dimension 2.
And H denotes the "quaternions", fancy gadgets that form a non-commutative 4 dimensional algebra, so M2(H) is simply the 2x2 matrices where the entries are quaternions, instead of complex numbers. you can still multiply matrices and add them together just as before, because you can do those things with the quaternion numbers that comprise them. It has real dimension 16, because each matrix has 4 entries and the individual entries are (4 dimensional) quats.

So one thing that CC&M seem very happy about is this:
Recall when Connes and friends discovered some years ago how to pull the Standard Model particles out of an algebraic hat, they found that to make it work they needed exactly this algebraic gadget: M2(H) ⊕ M4(ℂ)
and it seemed kind of arbitrary, why that?
And now, which is cause for celebration, CC&M have succeeded in implementing quantum GR geometry and it turns out that the quanta of geometry live in the same algebraic setting! To make their QG work it turns out they need exactly the same algebraic gadget: M2(H) ⊕ M4(ℂ) .
Who would have guessed that matter particles and quanta of geometry are enough alike that they would grow from the same algebraic root? Somehow it augurs well. It seems like a bit of luck, that nature is simpler than we might have expected, so Chamseddine Connes and Mukhanov are expressing satisfaction about that, in the paper.

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Jimster41 and ShayanJ
ShayanJ
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This was exciting marcus, thanks!
Recall when Connes and friends discovered some years ago how to pull the Standard Model particles out of an algebraic hat, they found that to make it work they needed exactly this algebraic gadget: M2(H) ⊕ M4(ℂ)
Can you give a link to that paper?(Yeah, I know, I'm not grown up enough for this much advanced stuff but I really wanna see it!)

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That algebra appears for example on page 8, in equation (3.1) in the C&C paper http://arxiv.org/pdf/0706.3688v1.pdf where they defined the algebra A which they then used throughout the rest of the paper. They denote it with a fancy-font A which I copy using one of the PF fonts called "courier" because I don't have their exact font. Sometimes they call it "the algebra (3.1)".
===quote C&C "Why the Standard Model?" page 8===
A conceptual description of the algebra A and its representation in H is then obtained from Remark 2.9. We let V be a 4-dimensional complex vector space. Our algebra is

(3.1).... A = EndH(W ) ⊕ End(V ) ∼ M2(H) ⊕ M4(ℂ) .

It follows from the grading of W that the algebra (3.1) is also...
==endquote==
This paper is the kind of thing I find hopelessly difficult to understand. They are speaking pure high algebra and they are trying to find a explanation within the domain of algebraic art for why the symmetry group of the standard model array of particles should be U(1)xSU(2)xSU(3) . To be fair, physicists have looked at that and asked "where did that come from? what's the reason for that? there's no earthly reason it should be that!" I recall Feynman was quoted saying something frankly disparaging about that particular Lie group. He may have called it ugly, or ad hoc, or words to that effect. Physicists have expressed the feeling that somehow U(1)xSU(2)xSU(3) does not seem to have been "made in Heaven" So it is fair for C&C to have taken up the challenge and try to find some other way of expressing the StdMdl symmetry that would by their lights be more beautiful or natural---less ad hoc.

In the page 8 quote, H is a Hilbert space that that they've already defined, H is the quaternions and ℂ is the complex numbers.

What seems noteworthy is that the same algebraic gizmo that occurs on page 8 of the 2007 paper when they are studying StdMdl particles also shows up in the 2014 paper where they are exploring how to quantize GR geometry.

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Even though I don't understand much of the algebra in this 2007 C&C paper I will post the summary, in case others can get something out of it:
http://arxiv.org/abs/0706.3688
Why the Standard Model
Ali H. Chamseddine, Alain Connes
(Submitted on 25 Jun 2007)
The Standard Model is based on the gauge invariance principle with gauge group U(1)xSU(2)xSU(3) and suitable representations for fermions and bosons, which are begging for a conceptual understanding. We propose a purely gravitational explanation: space-time has a fine structure given as a product of a four dimensional continuum by a finite noncommutative geometry F. The raison d'etre for F is to correct the K-theoretic dimension from four to ten (modulo eight). We classify the irreducible finite noncommutative geometries of K-theoretic dimension six and show that the dimension (per generation) is a square of an integer k. Under an additional hypothesis of quaternion linearity, the geometry which reproduces the Standard Model is singled out (and one gets k=4)with the correct quantum numbers for all fields. The spectral action applied to the product MxF delivers the full Standard Model,with neutrino mixing, coupled to gravity, and makes predictions(the number of generations is still an input).
13 pages

ShayanJ
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So in 2007 they got SM in curved spacetime out of $M_2(\mathbf H) \oplus M_4(\mathbb C)$ and now SM in curved spacetime+QG? So $M_2(\mathbf H) \oplus M_4(\mathbb C)$ is mathematically a TOE waiting for getting a physical interpretation?

EDIT: That $\mathcal A$ is written using latex with \mathcal{A}.

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marcus
Berlin
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Who would have guessed that matter particles and quanta of geometry are enough alike that they would grow from the same algebraic root? Somehow it augurs well. It seems like a bit of luck, that nature is simpler than we might have expected, so Chamseddine Connes and Mukhanov are expressing satisfaction about that, in the paper.
Curious whether this would imply an explanation for all sorts of duality things between the 'two wings of the buttefly', like holography, EPR=ER, AdS/CFT?

Berlin

marcus
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Shyan, Berlin, I like very much these two questions. To derive consequences from the CC&M idea and physical interpretations must surely be ways to test the idea. I hope I hope that young researchers at various places will see the interest in the sort of questions you both immediately raised and will make a concerted effort to derive the consequences of the idea if it should happen to be right. Whether nature thinks it is right or wrong I would like to see this idea catch fire in the community. Quantum gravity is getting to be a lot of fun to watch! (I personally am just a watcher and a gossip :) ) It could be an interesting year or two, ahead.

It's not just CCM, there are also novel initiatives by others that have surfaced in the past year Wolfgang Wieland "new spinfoam action", Bianca Dittrich "flux formulation" of LQG, Lanery and Thiemann "projective limit" LQG.

It may seem like an irrelevant concern but I would be worried about funding problems and growing pains. the field has to grow. where do you find the money to support the various postdocs you need to stay in the race, and to bring up the PhD students prepared to tackle the new approaches. When new prospects open, the senior people in the departments must scramble for grants. Yes, that is an irrelevant concern. It's not my business to wonder, so I won't waste any more words about that.

atyy
Curious whether this would imply an explanation for all sorts of duality things between the 'two wings of the buttefly', like holography, EPR=ER, AdS/CFT?
I have no idea if this is related, but Marcolli (with Lin, Ooguri and Stoica) recently wrote an AdS/CFT paper: http://arxiv.org/abs/1412.1879. Marcolli has worked with Chamseddine and Connes http://arxiv.org/abs/hep-th/0610241, so if there is any link, there is some hope that someone knowledgeable is thinking of it.

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Thanks for the suggestion, and the reminder to keep an eye on Marcolli's work! BTW Garrett Lisi commented briefly on the CCM paper in the current Lisi thread.
... On that CCM paper, I think their approach is great and I'm happy that a whole community of researchers have been working in that direction. With this paper in particular though, it's mightily complex, and I worry that with each step of complexity they're getting further away from good physics. This is often the case with research programs that get popular -- there's a tendency to pile on complications, constructing more and more theory until it becomes a bit of a nightmare. And though I do think this is happening here, at least it hasn't yet gotten as bad as strings.

On the content side, I think I see some of what they're doing, and it is getting closer to my work. Their $M_2(\mathbb{H}) \times M_4(\mathbb{C})$ algebra contains the $su(2) \times su(2) \times su(4)$ Pati-Salam model, and they're presumably getting gravitational degrees of freedom from the rest of $M_2(\mathbb{H})$, though I couldn't see quite how in their paper. If that is what they're doing, then they could embed that algebra in $spin(4,4) \times spin(8)$, add their spinor representation and triality, and be looking squarely at $E_8$.

I do consider them my closest competition. But I'm not especially worried, because my approach is to minimize rather than expand complexity, which is a terrible strategy for building a career but possibly the best for figuring something out.

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I understand Garrett's concern about things getting more complicated. But my impression of this particular paper was that it represented a simplification because gravity was being handled in parallel with particles---almost as if there were "quanta of geometry". Another thing that impressed me as a simplification was what I quoted in this earlier post: Wigner identified particles as irreducible representations of the Poincaré group. And it looks to me, from this passage on page 25, that they are identifying both particles and quanta of geometry as irreducible representation of a generalized Heisenberg uncertainty principle.
...this from page 25 of their [geometry quantum basics] paper. In the following passage think of n = 4:
==quote Chamsedding Connes Mukhanov==
A tentative particle picture in Quantum Gravity

One of the basic conceptual ingredients of Quantum Field Theory is the notion of particle which Wigner formulated as irreducible representations of the Poincaré group. When dealing with general relativity we shall see that (in the Euclidean = imaginary time formulation) there is a natural corresponding particle picture in which the irreducible representations of the two-sided higher Heisenberg relation play the role of “particles”. Thus the role of the Poincaré group is now played by the algebra of relations existing between the line element and the slash of scalar fields.

We shall first explain why it is natural from the point of view of differential geometry also, to consider the two sets of Γ-matrices and then take the
operators Y and Y′ as being the correct variables for a first shot at a theory of quantum gravity. Once we have the Y and Y′ we can use them and get a map (Y,Y′) : M → Sn ×Sn from the manifold M to the product of two n- spheres. The first question which comes in this respect is if, given a compact n-dimensional manifold M one can find a map (Y, Yç) : M → Sn × Sn which embeds M as a submanifold of Sn × Sn. Fortunately this is a known result, the strong embedding theorem of Whitney, [24], which asserts that any smooth real n-dimensional manifold (required also to be Hausdorff and second-countable) can be smoothly embedded in the real 2n-space. Of course R2n = Rn × Rn ⊂ Sn × Sn so that one gets the required embedding. This result shows that there is no restriction by viewing the pair (Y,Y′) as the correct “coordinate” variables. ...
==endquote==
On page 27 they have the result that in their new version of Quantized GR Geometry the volume is quantized which would seem very good to a LQG researcher and something one would like to understand more about. And then on page 29 there are the conclusions:
==quote==
Conclusions
In this paper we have uncovered a higher analogue of the Heisenberg commutation relation whose irreducible representations provide a tentative picture for quanta of geometry. We have shown that 4-dimensional Spin geometries with quantized volume give such irreducible representations of the two-sided relation involving the Dirac operator and the Feynman slash of scalar fields and the two possibilities for the Clifford algebras which provide the gamma matrices with which the scalar fields are contracted. These instantonic fields provide maps Y, Y ′ from the four-dimensional manifold M4 to S4. The intuitive picture using the two maps from M4 to S4 is that the four-manifold is built out of a very large number of the two kinds of spheres of Planckian volume. The volume of space-time is quantized in terms of the sum of the two winding numbers of the two maps. More suggestively the Euclidean space-time history unfolds to macroscopic dimension from the product of two 4-spheres of Planckian volume as a butterfly unfolds from its chrysalis. Moreover, amazingly, in dimension 4 the algebras of Clifford valued functions which appear naturally from the Feynman slash of scalar fields coincide exactly with the algebras that were singled out in our algebraic understanding of the standard model using noncommutative geometry thus yielding the natural guess that the spectral action will give the unification of gravity with the Standard Model (more precisely of its asymptotically free extension as a Pati-Salam model as explained in [5]).
Having established the mathematical foundation for the quantization of geometry, we shall present consequences and physical applications of these results in a forthcoming publication [6].
==endquote==

Publication [6] is something [CC&M] have in the works...
A lot may depend on what success they have with their presentation of "consequences and physical applications of these results" in the work-in-progress [6].
Given that Mukhanov is a cosmologist he must be asking himself if this CCM approach to quantum GR geometry has what it takes to resolve or remove the cosmological singularity at the start of expansion. That's too much to ask, I realize, so early in the development of this new model. But it must have occurred to him to wonder. Might the theory, for example, predict a bounce?

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I just saw this post Marcus. Kindof goosebumps. Looking forward to catching up. Also, I have been watching your "Intuitive content..." super thread.

I wish I had a better understanding of what the Special Unitary Groups (n=1-3) sort of "mean". Right now I only have this cartoon that interactions between sets that can be described by them obey a set of rules that we sort of expect: Commutation, Identitification, Inversion, Closure (no interactions step off to infinity)... they do something something invariant under something something, Cartesian something something... I'm just spitting stuff out that I saw on wiki... I don't have any sense of what the identification of theses algebras as theories of the crime really carry - in terms of descriptive elegance, or completeness. Even though I can kindof take the blue highlights in quote on faith.

Frustrating.

Still digging the Unger-Smolin tome, but biting it off a bit at a time.

Also, now reading a really nicely written (slim) book on "Quantum Chance" (Nicholas Gisin - winner of the Bell prize). When I dangerously mix these two things...

If our space-time geometry and all the bits in it are emergent products of some evolutionary process, doesn't that imply some minimal notion of "environment", the fitness landscape or game board or "payoff matrix", the rules and incentives of the game, the something that can't be nothing that is driving emergence of the structure that defines our experience?

I can't stand the "infinite number of universes" notion (it seems like an utterly structure-less notion). However, if we are trying to integrate the problem of "non-local coordination" (Bell's game, Spooklike Action, Quantum non-locality) with the emergence of space-time... it seems unavoidable to me to hypothesize that "space-time" as we mean it (it is where we are hopelessly stuck), is not singular but is joined with something that is pretty much not space-time (at least as we know it), this environment that provides the evolutionary gradient and the stepping (because "clock" is a loaded word) of our "emergence index or iterator". Currently I'm hoping that this is where Unger is heading (expanding the notion of time to be more of an all inclusive iterator - rather than a clock as we think of it).

In this Gisin book, to prove (at least for the layperson) that quantum entangled systems cannot be cloned Gisin argues ad absurd um to conclude that cloning of an entangled system would predict "communication without transmission" which by "common sense" is impossible. My common sense doesn't quite get it. I guess I object to the notion of defining "transmission" narrowly to include only transmission via our assemblage (our reef) of emerged space-time with it's transmission distance measuring stick. I mean what is the difference between a "non-local whole" and "transmission" if you withhold a-prior assuming some measuring stick for the transmitting medium in question. Seems like the question becomes "what's the difference between the measuring stick that seems to be zero length, and the one that transmission obviously travels some distance by". How do we know it's zero? We just know it looks like zero to us. In other words it doesn't seem obvious to me that something about the "non-locally coordinated becoming" couldn't be divined locally, if you could arrange the right pinch of whatever medium with it's weird measuring stick (that looks to us like it has zero length), our space-time is becoming in.

This is just the kind of thing that sends me off into a fugue of wondering whether aspects of our emergent space-time actually reflect or indirectly show, the existence of this (apparently from our point of view) zero-ish space-time length container. Maybe things that seem unlikely to have emerged in an "uncoordinated" way like extremely massive black holes, synchronized pulsars or quasars or whatever?, or dark energy/matter, or ubiquitous interference-like patterns, things that are unlikely to be explainable by causal superposition of space-time histories of more mundane things,or at least have other possible explanations, like non-local coordination. The idea that the measuring stick of our container isn't entirely orthogonal (we can see it at least as the source of our index of iteration and environment of emergence) and extreme forms, or periodic forms (or extreme periodic forms) are literally showing us that there is something there, right up against us. Our space time rolls down it, but we can feel/see the reflection/refraction of our timeless, or maybe just thus-far emerged history, vibrating through it.

What kind of algebra are fractal sets describable in? Is it "closed" or "compact"?. Seems confusing to think of how the notion of closed and compact relate when any sense of specific scale vanishes periodically under iteration as long as you can't see the index of iteration.

Sorry, it just get's me excited to ponder this stuff...

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I realize now re-reading your set of posts for like the fifth time, that the new algebra is a replacement for the U and SU(n) Unitary and Special Unitary Groups. So now what I wish is that I understood what this new algebra sort of looks like, if there is some way to appreciate it's descriptive elegance and completeness w/respect to space-time geometry and the particles of the SM.

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So in 2007 they got SM in curved spacetime out of $M_2(\mathbf H) \oplus M_4(\mathbb C)$ and now SM in curved spacetime+QG? So $M_2(\mathbf H) \oplus M_4(\mathbb C)$ is mathematically a TOE waiting for getting a physical interpretation?
So is there any sense in which $M_4(\mathbb C)$ being an algebra on the complex plane implies fractal-like, self-similar, or discrete scale invariant properties for the space? If so is that sort of a "duh" w/respect to what we already understand as the standard model, which as I understand this is consistent with? Or am I trying to project too much intuitive meaning on one specific term of encoding.

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You may be interested in this talk:

wabbit
wabbit
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Thanks. I just came accross this thread and this is very helpful - I've been hoping to get some understanding of this CCM paper for a while as it does seem very suggestive and intriguing, but I can barely struggle through the abstract. Even though Connes' use of the words "obvious" and "of course" can be slightly at odds with mine, this talk at least starts in plain language so I'll try to see if I can pretend to understand a tenth of it : )

I'm too lazy to read this thread or the paper, did they actually make a quantitative QG prediction?

If not... yaaaaaaawn.

EDIT: That's a). New, and b). testable in the foreseeable future.

I'm too lazy to read this thread or the paper, did they actually make a quantitative QG prediction?

If not... yaaaaaaawn.

EDIT: That's a). New, and b). testable in the foreseeable future.
Then why are you even here? This doesn't contribute to the thread in any way, and I've noticed this kind of responses several times before. usually in the form of destructive criticism regarding the field of high energy theoretical physics.

My rebuttal is the following;
Work has to be done in order to verify that the model that is apparently similar to the standard model and the QG part that they expect to follow are exactly that.
Once they have confirmed this, then they can look for different predictions/explanations for physical processes.

Did every single theory ever invented stay appropriate? Think Ether Theories (I know forbidden) which as far as I know can be doctored to coincide with special relativity.
Heck why even do experiments that might lead to new physics? If it doesn't pan out its just a waste of grant money.

wabbit
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From an earlier post about the CCM paper:
On page 27 they have the result that in their new version of Quantized GR Geometry the volume is quantized which would seem very good to a LQG researcher and something one would like to understand more about. And then on page 29 there are the conclusions:
==quote==
Conclusions
In this paper we have uncovered a higher analogue of the Heisenberg commutation relation whose irreducible representations provide a tentative picture for quanta of geometry. We have shown that 4-dimensional Spin geometries with quantized volume give such irreducible representations of the two-sided relation involving the Dirac operator and the Feynman slash of scalar fields and the two possibilities for the Clifford algebras which provide the gamma matrices with which the scalar fields are contracted. These instantonic fields provide maps Y, Y ′ from the four-dimensional manifold M4 to S4. The intuitive picture using the two maps from M4 to S4 is that the four-manifold is built out of a very large number of the two kinds of spheres of Planckian volume. The volume of space-time is quantized in terms of the sum of the two winding numbers of the two maps. More suggestively the Euclidean space-time history unfolds to macroscopic dimension from the product of two 4-spheres of Planckian volume as a butterfly unfolds from its chrysalis. Moreover, amazingly, in dimension 4 the algebras of Clifford valued functions which appear naturally from the Feynman slash of scalar fields coincide exactly with the algebras that were singled out in our algebraic understanding of the standard model using noncommutative geometry thus yielding the natural guess that the spectral action will give the unification of gravity with the Standard Model (more precisely of its asymptotically free extension as a Pati-Salam model as explained in [5]).
Having established the mathematical foundation for the quantization of geometry, we shall present consequences and physical applications of these results in a forthcoming publication [6].
==endquote==

Publication [6] is something that Chamseddine and Connes have in the works with Slava Mukhanov...
This echoes what was already quoted, in the paper's abstract:
http://arxiv.org/abs/1411.0977
Geometry and the Quantum: Basics
Ali H. Chamseddine, Alain Connes, Viatcheslav Mukhanov
... model of the "particle picture" for a theory of quantum gravity in which both the Einstein geometric standpoint and the Standard Model emerge from Quantum Mechanics. Physical applications of this quantization scheme will follow in a separate publication.
33 pages, 2 figures

So we should be on the lookout for that follow-up publication

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With a potentially important paper like this it's good to have the Inspire link handy
http://inspirehep.net/record/1325968?ln=en

and we should check the SHORT version (4 or 5 pages) that was published in March 2015 Physical Review Letters
http://inspirehep.net/record/1325968?ln=en
Quanta of Geometry: Noncommutative Aspects
Ali H. Chamseddine (American U. of Beirut & IHES, Bures-sur-Yvette) , Alain Connes (College de France & IHES, Bures-sur-Yvette & Ohio State U., Dept. Math.) , Viatcheslav Mukhanov (Munich U. & Munich, Max Planck Inst.)
Sep 8, 2014 - 5 pages

Abstract (APS)

In the construction of spectral manifolds in noncommutative geometry, a higher degree Heisenberg commutation relation involving the Dirac operator and the Feynman slash of real scalar fields naturally appears and implies, by equality with the index formula, the quantization of the volume. We first show that this condition implies that the manifold decomposes into disconnected spheres, which will represent quanta of geometry. We then refine the condition by involving the real structure and two types of geometric quanta, and show that connected spin manifolds with large quantized volume are then obtained as solutions. The two algebras M2(H) and M4(C) are obtained, which are the exact constituents of the standard model. Using the two maps from M4 to S4 the four-manifold is built out of a very large number of the two kinds of spheres of Planckian volume. We give several physical applications of this scheme such as quantization of the cosmological constant, mimetic dark matter, and area quantization of black holes.

The short paper in PRL has been cited 5 times so far
http://inspirehep.net/record/1315431/citations
We may have to wait until CCM bring out the promised follow-up, their reference [6], before we can gauge the significance of this one.
that's where they said at the conclusion on page 30 of the "Basics" paper:
"Having established the mathematical foundation for the quantization of geometry, we shall present consequences and physical applications of these results in a forthcoming publication [6]. "
[6] A. H. Chamseddine, Alain Connes and V. Mukhanov, in preparation.

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Are the prereqs for this paper just differential geometry, group theory and QFT?

arivero
Gold Member
A new one with extra dimensions only in the "tangent space"

arXiv:1602.02295 On Unification of Gravity and Gauge Interactions
Ali H. Chamseddine, Viatcheslav Mukhanov
(Submitted on 6 Feb 2016)
The tangent group of the four dimensional space-time does not need to have the same number of dimensions as the base manifold. Considering a higher dimensional Lorentz group as the symmetry of the tangent space, we unify gravity and gauge interactions in a natural way. The spin connection of the gauged Lorentz group is then responsible for both gravity and gauge fields, and the action for the gauged fields becomes part of the spin curvature squared. The realistic group which unifies all known particles and interactions is the SO(1,13) Lorentz group whose gauge part leads to SO(10) grand unified theory and contains double the number of required fermions in the fundamental spinor representation. We briefly discuss the Brout-Englert-Higgs mechanism which breaks the SO(1,13) symmetry first to SO(1,3)×SU(3)×SU(2)×U(1) and further to SO(1,3)×SU(3)×U(1) and gives very heavy masses to half of the fermions leaving the others with light masses

marcus