## Geometrization of the Standard Model

Now we can explain Cones' framework to the Standard Model coupled with (unquantized) general relativity. This requires a bit of a mathematical preliminary: short exact sequences:

$$0\rightarrow A \rightarrow B \rightarrow C \rightarrow 0$$

where the image of each map is the kernel of the next one. The beginning zero means that the A to B map is injective, while the last zero  means that the B to C map is surjective.

Short exact sequences come in several flavors, but we have two cases of interest:

$$0\rightarrow fiber \rightarrow totalspace \rightarrow base \rightarrow 0$$

and

$$0\rightarrow N \rightarrow M \rightarrow M/N \rightarrow 0$$

For the Standard Model coupled with gravity we have the gauge group $$G=U(1)\times SU(2) \times SU(3)$$ and the diffeomorphism group $$Diff(M)$$ of the manifold $$M$$. $$Diff(M)$$ acts on $$G$$ by permutations and the full group of symmetries $$U$$ of the Standard Model and the Hilbert-Einstein action is the semidirect product:

$$U = G\rtimes Diff(M)$$

Now we introduce a toy model: a $$U(n)$$ gauge theory coupled with general relativity. Suppose we have a coordinate algebra $$A$$ on a non-commutative space in the following way:

$$A = C^{\infty}(M) \otimes M_n (C)$$

This means that on the ordinary space-time manifold we attach at each point an $$n \times n$$ complex matrix, which acts as internal degrees of freedom for the generalized notion of coordinate on this (now a) noncommutative space. We are now investigating the automorphisms of the algebra $$A$$: $$Aut(A)$$. One trivial subgrup of $$Aut(A)$$ are the so-called inner automorphisms: $$Inn(A)$$ which is constructed by sandwiching an element of $$A$$ with an invertible element $$u$$ like this:

$$\alpha(x) = u x u^{-1}$$

Moreover this is always a normal subgroup and we have the following short exact sequence:

$$0\rightarrow Inn(A) \rightarrow Aut(A) \rightarrow Aut(A)/Inn(A) \rightarrow 0$$

which is identical with (drum roll please...)

$$0\rightarrow U(n) \rightarrow U(n)\rtimes Diff(M) \rightarrow Diff(M) \rightarrow 0$$

So now the only thing we need to do is find an appropriate algebra $$A$$ such that we get the Standard Model gauge group (subject to the constraint that there is a simple action functional identical with SM+GR action when applied to the noncommutative space). This was easier said than done, and it took Connes and collaborators (Marcolli, Chamseddine) several iterations and many years (I think over 10 years) until they fully recovered the Standard model in its entirety (including 3 generations of mass and the correct value of the Higgs boson mass).

For the Standard Model The algebra $$A$$ is a beast; if I recall correctly it has 96 dimensions. One simple earlier iteration was:

$$A=C\oplus H\oplus M_3 (C)$$

As a side note, Emile Grgin's quantionic quantum mechanics is the same as Connes' toy model when $$A = M_2(C)$$.

To uncover the algebra able to reproduce all the known Standard Model experimental facts, a deep dive into K-theory and spectral triples (a spectral triple is a generalization of a $$Spin^C$$ manifold) was required. As a mathematical curiosity, the non-commutative space in the Standard Model case turned out to be two copies of the manifold M, meaning a product of a continuous manifold with a discrete space. It is interesting to see the physical mapping of the Standard Model in the new language of noncommutative gometry. For example the Higgs boson links the two copies of the manifold and the two copies are a Highs boson Compton length apart, a picture reminiscent of the branes in string theory.

There were attempts to apply Connes' geometrization ideas to string theory, but they were not successful. More interesting facts about the Standard Model were uncovered by Kraimer and Connes: a particular renormalization technique was proven to be the marriage of two well known mathematical areas: Hopf algebras and Birkhoff decomposition (which has direct application in solving solitonic equations)

1. What the hell are you talking about? These Connes et al. papers have always predicted a completely wrong Higgs mass, namely 170 GeV, and it was paradoxically the first value of the Higgs mass that was ruled out by Tevatron in 2008. Check e.g. arXiv

hep-th/0610241
1403.7567

There exists no paper of this kind that would be giving a correct Higgs mass.

I don't really think that the models work at all, but even if they did, there's no virtue in rewriting the field content as some noncommutative algebra. The class of theories one must consider are still given by a field theory with some field content and a specified set of symmetries. Whether something might be written in terms of some noncommutative formalism is unphysical.

It would be fun if one could get a fuzzy compact manifold producing the right low-energy spectrum and nothing else except that I think that the answer is No, there is none. Moreover, at the end, it's clear that such a theory would have a wrong number of degrees of freedom to be compatible with gravity which *does* require 10 or 11 *continuous* dimensions at the end. Having just the SM-like finite field content is like having the size or B-field of the string compactification near the string scale. But when it's so, the excited strings can't be decoupled so there's no meaningful truncation of the compactification down to pure field theory. From a broader perspective, it's pretty clear why one can't find any viable model of this kind.

1. Dear Lubos, it seems to me the second paper 1403.7567 explains on the contrary how the spectral noncommutative standard model extended with a big brother of the Higgs boson, a SM singlet scalar with a vev at a partial unification scale (Pati-Salam like) around 10^12 GeV does the job : to shift the mass of the SM Higgs to its experimental value...
I also think that the most recent articles from Chamseddine, Connes, van Suijlekom and Mukhanov : 1403.271, 1507.08161 and 1606.01189 prove that there has been some virtue working hard on rewriting the field content of the SM as the appropriate noncommutative algebra. It gives a (algebraic operator) geometric insight to understand the nontrivial topology of spacetime beyond the attometer scale. The post at http://noncommutativegeometry.blogspot.fr/2014/11/particles-in-quantum-gravity.html# is quite technical (may be Florin will try to write a better pedagogical summary of its content one day) but it gives a nice summary of the most advance vision of spacetime close to the Planck scale in the spectral noncommutative geometric perspective.
The former formalism lives naturally in a Euclidean context but this does not make it more unphysical than traditional perturbative quantum field investigations about gravitation. The article 1605.032031 by a different trio of researchers explains in detail the prescription for the Wick rotation from the Euclidean theory to the Lorentzian one.
I have no expertise in this field, least of all in string theory but I note in articles like 1409.7574, 1603.01756 some people trained in string theory (and eager to find a sign of new physics at LHC) have different views from you on the recent developments in spectral noncommutative program.
It is probable that the work of Connes et al does not help LHC phenomenologists to foresee exciting discovery at the multi-TeV scale but it opens interesting perspective for cosmologists and astrophysicists with the mimetic dark matter hypothesis (1403.3961) and ultra heavy right-handed Majorana neutrinos compatible with the actual low scale phenomenology and that fit a pretty predictive leptogenesis scenario (1412.4776) inspired by nonsupersymmetric SO(10) model building.

2. Lubos,

A short note as I am about to start on a trip driving about 400 miles today: I don't recall exactly, but Connes was able to obtain the correct Higgs mass a year or two years ago by adding something which appears naturally into his model but he ignored it in the beginning. I'll look it up tomorrow and get back to you on this.

3. Dear Cedric, I don't think it makes any sense to talk about "topology" in the context of noncommutative manifolds. Topology is mathematically defined by saying which subsets of the points in the manifold are "open". But there are no points in a noncommutative manifold - only functions on that manifold - so correspondingly, one can't define any topology.

So I don't know what to do with these things.

You refer to papers by Djordje Minic et al. Fun guy. It would be a controversial discussion whether they're "string theorists". But regardless of the answer, can you - and Florin - understand that most trained people following this portion of physics (I am just an example here) don't consider these papers serious stuff? The Minic 2014 and 2016 papers have 6 and 3 citations right now, respectively. This sort of corresponds to what I thought about these papers *before* I looked at the citation list.

You just can't expect people to think that it's a part of knowledge on par with some established insights such as the heterotic string theory model. The degree of justifiability is smaller by some 3 orders of magnitude here. All the claims you are making are *extraordinary*. Obviously, if one could correctly derive the Standard Model spectrum including masses of the Higgs or other particles from a simple model like that, it would be hugely important. But it almost certainly isn't the case.

The "fuzzy geometry" of the compactification manifold is just a trivial way to rename the fact that the Standard Model contains a finite number of particle species - instead of the infinite tower of Kaluza-Klein (and perhaps excited string) modes. So the idea of a fuzzy geometry here isn't adding anything - it's just renaming the finite number of species which is unavoidable given the Standard Model's being an effective field theory.

But the Connes program claim is that one can get some more interesting QFTs or special QFTs or something like that from demanding some representation in terms of NCG. This is an extraordinary claim, too. I haven't seen any evidence for that. This would change the paradigm of particle physics. In particle physics, its part about effective field theories, there are rather clear rules what theories are allowed, what their set of parameter is, what classes of theories must be considered together, and what assumptions about them may be considered natural.

Connes with NCG clearly want to replace this standard picture by something else except that they have never defined what the rules that should replace the old ones are. It's just some hocus pocus differing from standard QFT analyses in some unreadable ways and claiming to be equally good. But everything that seems to be right about the hocus focus is only right to the extent to which it's exactly equivalent to a quantum field theory from the normal picture and everything that is different is either wrong or ill-defined.

Florin and you are presenting all these weird papers and statements from them as if they were facts or some serious competitors to describe particle physics. Except that they're neither. There could be some ideas along these lines but no one has presented such ideas that would make sense yet.

4. Lubos,

I am back. I got my dates wrong, Connes solved the Higgs mass problem in 2012. This is the relevant paper: http://arxiv.org/pdf/1208.1030v2.pdf

5. Too bad, even the slowest people in the rest of the world learned about the Higgs mass 1 month earlier, in July 2012 when the particle was officially discovered, and we the insiders knew the value since December 2011.

The paper from August 2012 is meaningless fitting of the elephant. The models don't make any sense. It's just pseudoscientific handwaving meant to fool the gullible ones.

Even if one had a model of this kind with a 125 GeV Higgs, one also has a model with the 170 GeV Higgs, so one has a "landscape". Except that Connes and no one else has ever defined what this "landscape" is supposed to be and how it differs from the conventional set of quantum field theories. So the added value is just zero.

6. BTW the August 2012 paper is just silly from a physicist's viewpoint. It says that the fix of the Higgs mass results because of a new scalar singlet field "sigma" in Connes' "Standard Model" which is responsible for the Majorana neutrino masses. But the Standard Model just doesn't have any such additional scalars - it wouldn't be standard. More invariantly, the neutrino masses don't come from *any* physics of the Standard Model. They almost certainly result from physics at a much higher scale close or equal to the GUT scale.

When this stuff is heard by a physicist, e.g. that "Majorana neutrino masses come from a field in the Standard Model", the physicist knows that it sounds nutty. It is an extraordinary claim that requires extraordinary evidence. But people like Connes (let alone you) don't provide the readers with *any* evidence because he (let alone you) doesn't even understand why it's nutty.

So the papers are sequences of notions from the field he knows, rigorous mathematics, pretending to be relevant in physics, except that a big part of the physics "applications" are completely silly and Connes is clearly out of his breadth.

7. Connes blew his shot on correctly predicting the Higgs mass before the actual observation because he insisted for yeas before that it must be 170. The later correction looked like a handwaving salvage of the approach.

In Connes approah the neutrino mass comes from the seasaw mechanism.

8. BTW, I wrote the above before reading your post at your blog. There I have only one question: is "semi anti-quantum zealot" better or worse than "confused"?

And I am not at all "eclectic". This all natural under the category theory which is is the main tool I am using to reconstruct QM. This gave me an idea for a later post.

9. I think that "confused" and "semi-anti-quantum zealot" are basically synonymous in this context. Also, "category theory in physics" is clearly a textbook example of what it means to be "eclectic". The adjective refers to the incorporation of lots of ideas from many corners of human thought even if their usefulness in a given context hasn't been established.

10. The problem is that in category theory you do not yet see the forest because of the trees. There is a cohesive paradigm, and the usefulness of it was rigorously established. I will introduce this paradigm in a series of posts. However for now it is clear to me you do not yet get the point of noncommutative geometry (nothing wrong with this, many other people do not get it either) and next time I will explain the distance formula from last post (which looks a bit of hocus pocus) and start uncovering the depth of noncomutative geometry.

For the SM point of view, Connes approach is simply another equivalent mathematical description of the known physics. Connes is not a physicist and his physics intuition is not very good as his original incorrect prediction of Higgs mass illustrates. However he does understand QM and he is using QM to uncover and organize in a certain way a mathematical landscape.

Noncommutative geometry is geometry though the eyes of quantum mechanics (this is a good title for my next post, maybe I will use it).

11. Cedric,

Thank you for your comment; I had no idea people in NCG area pay any attention to what I am saying. I work in foundations of QM using category theory and the community there does not understand NCG because it is too abstract and too mathematically advanced. I am only trying to bridge the gap.

12. Dear Florin, I do not want to make you or your reader believe that I a work in the NCG area. It just happens that I have been vaguely fascinated in the work of Connes since I was an undergraduate and I got more interested when I had the opportunity to attend some of his lectures at College de France and a few seminars by ALi Chamseddine at IHES. Since the discovery of the Standard-Model like Higgs boson at LHC I have started to keep up with the developments in the NCG-physics potential connections.
I think you make a nice job introducing as simply as possible this subject in your blog. I am curious to see the connection that you could established with your own research program. The continuous deepening of our conceptual understanding of quantum physics is a long and hard work but the Bell theorem, Aspect experiments and the advances in quantum information processing or quantum metrology are worth trying...

2. "I don't think it makes any sense to talk about "topology" in the context of noncommutative manifolds. Topology is mathematically defined by saying which subsets of the points in the manifold are "open". But there are no points in a noncommutative manifold - only functions on that manifold - so correspondingly, one can't define any topology."

What a beautiful display of ignorance. In Connes case he talks about a product of a continuous space with a discrete finite space: MxF which can be understood as a vector bundle. Then one looks at relevant cohomologies to define the topological notion of neighborhood. This is not the topology of the space M by itself and F by itself (which is trivial). See https://en.wikipedia.org/wiki/Topological_K-theory and
http://pages.uoregon.edu/ddugger/kgeom.pdf

As I said in the text, a deep dive into K-theory is needed. There is a lot of topology in KO theory, and there is a notion of a "KO dimension" distinct from the ordinary dimension.

1. One more thing: one can determine the ordinary dimension of a space by looking at the shape of the plot of eigenfrequencies of the geometric object: first mode of oscillation at frequency f1, second mode at f2, etc (for high modes of vibrations). However, there are geometrical objects like quantum groups (quantum spheres) which have the ordinary dimension 0, but they can be distinguished one from another by how they vibrate (again by the shape of the eigenfrequency curve). This is the intuition of the KO dimension. KO dimension tells apart objects of trivial topology and zero ordinary dimension.

2. Your comments are just mathematically wrong. The noncommutative i.e. fuzzy manifolds are in no way "discrete sets". They are not sets at all - only the algebra of functions on these fuzzy manifolds may be identified - which is why there is no topology. Because noncommutative manifolds aren't sets of points, there are no open sets and there are no neighborhoods, either. None of these notions from ordinary manifolds may be applied. Also, one can't define a dimension of a fuzzy space. This is well-known to those who work with D-branes. A collection of D0-branes is equivalent to a noncommutative space - D2-brane or D4-brane etc. with a flux or an instanton number - and those D0,D2,D4 descriptions may be totally equivalent.

3. I do not understand NCG at all. But,I am trying to understand the dispute about prediction of Higgs mass between you (Florin) and Lubos. Does it really matter that much whether it was predicted before or after experimental determination? If it is clean prediction it will remain clean prediction and if it is hand waving it will remain hand waving before or after!!People determined masses of several particles by QCD lattice numerical calculation decades after their masses were experimentally determined!

4. Hi Kashyap,

In this case there is no dispute at all actually. On NCG I will explain its basic spirit in next post. On Higgs mass, the original Connes model predicted 170 and this was before the experimental detection. If that were true, then NCG would have been taken much more seriously, but the prediction was simply wrong. 4 years later Connes found how to obtain the correct mass of the Higgs boson but the damage to his physical credibility was already done. Now the correction to his model does make a new prediction of new physics which is yet to be seen experimentally.

In a nutshell, NCG SM is nothing but a mathematical reformulation of the usual description. If the algebra were simple like the firsts attempts then you can call this reformulation a simplification. However the correct formulation of NCG SM is not simple at all, so what is to be gain?

The jury is out on whether NCG provides genuine physical insights. Yes, the model predicts new physics, but we already know SM is not the correct picture of nature (for example supersymmetry is needed), so nature could throw a curve ball at NCG SM and reveal something different. In this case I guess NCG SM would be fitted again with a correction.

The situation is very different on NCG itself as a mathematical domain. NCG is a new mathematical paradigm extending Riemann geometry using insights from QM.

1. This comment has been removed by the author.

2. (slightly corrected version of the former deleted comment)
I would add that the spectral action principle used by NC geometers to derive particle physics models uses also insight from general relativity (paragraph 7 of 1008.0985)
This fact might explain the extraordinary* NCG claim that one may relate [the very low scale of left-handed neutrinos to the ultra-heavy scale of right ones with the Standard Model Higgs scalar]. To put it differently, there might be some link between Higgs and gravitation as advertised by Martinus Veltman in www.nikhef.nl/pub/theory/academiclectures/Higgs.pdf but the consequences for physics could be different from the ones he envisioned.
@kashyap you might find in the 1008.0985 article a comprehensive answer to the question you asked me in the comment section of http://motls.blogspot.com/2016/07/reality-vs-connes-fantasies-about.html#disqus_thread ;-)

*of course this requires extraordinary evidence as asks legitimately Lubos in his recent post.
On the theoretical side I have already quoted in a comment above the connection between mimetic dark matter and a volume quantization of 3D space.
On the experimental side it will be a very difficult and long endeavour requiring first new Higgs factories (1307.3893) to start the Higgs precision measurement era (1302.3794).

3. Thanks Cedric

4. I would add two things in defense of NCG SM. The bar against which NCG SM is judged is unfairly high. Think of abandoned bootstrap theories of strong force from the 70s which later led to string theory. Just because physicists have a good physics intuition, this is no guarantee they are successful 100% of the time. Second, the math in NCG is rather abstract and requires a lot of effort to understand it. How many physicists took the time to do that? If you want to discover something new, why do you stick with the state of the art of math from 100 years ago? It makes no sense.

5. You are right to insist on the math aspect Florin. Lubos tries to reduce NCG as just a fancy formalism to rewrite quantum field theory in an abstract way but it is much more than that. To quote Thomas Schucker in 1003.5593 "noncommutative spaces are close enough to Riemannian spaces such that Einstein’s derivation of gravity from Riemannian geometry carries over to noncommutative spaces. In Connes’ derivation, the entire Yang-Mills-Higgs action pops up as a companion to the Einstein-Hilbert action, just like the magnetic field pops up as a companion to the electric field, when the latter is generalized to Minkowskian geometry, i.e. special relativity".
I think Lubos completely misses or eludes this point as his paragraph about gravity and noncommutative spaces shows. The understanding (checking by computation) of the Schucker claim requires time and skill. Lubos and all string theorists have skill. The actual paucity of empirical evidence for TeV scale susy/wim-particles may give time to some young(er) bold ones to appreciate the fact that Connes NCG has been extended since 2006 and might have already uncovered at least one really nontrivial fact in our quest to quantum gravity : "...any connected Riemannian Spin 4-manifold with quantized volume >4 (in suitable units) appears as an irreducible representation of a {specific} two-sided commutation relations in dimension 4 and ... th{is} representation give{s} 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" (from 1411.0977). This kind of new commutation relation with the help of the spectral action principle might select the very specific effective quantum field theory beyond the Standard Model viable up to the seesaw scale and further to grand unification.

5. Thanks Florin. So then is NCG Riemannian geometry with non-commuting operators? I can visualize Riemannian geometry, but have hard time picturing geometry with non commutativity!!

1. Yes, you are right and I will show how it works in next post. You cannot quite visualize it in general, although there are particular examples.

6. Latham Boyle (Perimeter Institute) made a good job to explain why NCG formulation of the SM is actually better than usual one (watch the first 35min)
http://pirsa.org/displayFlash.php?id=15050126

Connes/Chamseddine' initial proposal to solve the Higgs mass problem of 170 GeV (with a new scalar) has been picked up and improved by multiple groups working on the field:
- Boyle/Farnsworth: http://arxiv.org/abs/1408.5367
- Devastato/Lizzi/Martinetti: http://arxiv.org/abs/1304.0415
- Connes/Chamseddine/vanSuijlekom: http://arxiv.org/abs/1304.8050

@Lubos, Connes indeed wrongly predicted the higgs mass in 2008. Does it mean the death of NCG as a framework? Shall I remind you how many wrong predictions of the masses of Superpartners Gordon Kane made since the LHC has started?