that appeared on hep-th tonight. So far, it looks fascinating. I plan to study it in detail, and let me admit that at this moment I remain unable to tell you any details why it should be wrong and/or entirely vacuous. Believe me that the previous sentence is written not only because Alain Connes is an amazing thinker and a genius. It is also partly written because most of the page 10 and one third of the page 11 is dedicated to a "formula" that looks much like the Standard Model Lagrangian. No octopi and no unworldliness are involved. ;-)
The basic paradigm is that one assumes something like extra dimensions but their structure is not described by a metric tensor but rather by a Dirac operator. The possibilities for its structure are more general than in the case of the metric tensor with its corresponding box operator that would describe a Riemannian geometry. The Dirac operator is capable to match a certain kind of non-commutative geometry - a generalized manifold with a finite-dimensional space of possible functions on it. We will discuss whether the notion of a "geometry" is justified or whether it is just a fancy word.
The hidden discrete or noncommutative Connes manifold "F" has the KO-dimension equal to 6, just like in perturbative string theory, making the total KO-dimension of spacetime equal to 10. This dimension is only defined modulo 8, so you can also imagine that the dimension of spacetime equals 26 as implied by the left-movers of the heterotic string. Recall that in string theory, the chiral bosons must be added in groups of 8 for modular invariance. The dimension 10 mod 8 is probably also necessary from a spacetime perspective, to get the right violation of the C, P, and CP symmetries. I will mention a few more details about it later.
In previous papers co-authored by Alain Connes, it was argued that the weak SU(2) gauge group follows from a non-commutative solution of certain constraints for the Dirac operator. The colors had to be added by hand and the model suffered from a fermion doubling problem. Alain Connes now claims that these problems have been resolved. His proposed theory is supposed to include
- the Standard Model and gravity
- right-handed neutrinos with the seesaw mechanism
- the same gauge coupling unification as in SU(5)+ GUT theories
- possibly some other relations between the parameters.
Believe me that I have not yet seen a paper with a theory of everything that I couldn't deconstruct and reliably debunk in 15 minutes - which may be partly because I did not read the previous Connes' physics papers too carefully. This is the first one in my life but I hope to be able to return with a more meaningful analysis what's nontrivial about this proposal later. ;-)
Some things were however clear after the first minute. Connes can't explain the number of generations that is put in. Also, he only discusses the classical Lagrangian and his treatment has nothing to say about the UV problems of gravity and the Standard Model itself.
After an hour or so
It seems clear that the main task for a physicist trying to fully understand the paper is to be able to separate things that are just a translation of the usual laws of physics to a particular mathematician's language from things that are non-trivial and different from any kind of translation.
In equation (2), Connes defines an algebra whose parts seem to be in one-to-one correspondence with the factors of the Standard Model gauge group, up to having two copies of the quaternions that is probably needed to match the fact that the fundamental representation of SU(2) is pseudoreal. Some involutions are added. If I summarize, my feeling is that Connes postulates some data that are equivalent to the choice of the gauge group, so he doesn't derive the gauge group.
What seems to be the most conceivable non-trivial task is to derive the 90-dimensional representation "H_F" that encodes the quarks and leptons. The number 90 is mentioned at the bottom of page 2. It makes it clear that his spectrum of fermions, or at least the part called "H_F", does NOT contain right-handed neutrinos. 90 is a multiple of 10+5, using the dimensionalities of the SU(5) representations, not a multiple of 16 that would be needed to construct the full representation of SO(10) including the right-handed neutrinos. So I am already a bit suspicious how the see-saw mechanism is gonna work without adding the right-handed neutrinos by hand (which is clearly not Connes' discovery or prediction).
The rest of the page 4 is, in my opinion, dedicated to introducing the features that are needed for the right reality properties of the various representations. Three generations are put in by hand. As I indicated above, I am actually rather unimpressed by the statement that he can formally derive that the KO-dimension of his compact quasi-manifold F equals 6 modulo 8.
Imagine that you start with something like heterotic string theory but in D dimensions where D is not necessarily ten. D must obviously be even to get a chiral theory from any non-singular compactification (G2 manifolds for M-theory must be singular in order to be realistic) - and his construction is a counterpart of a non-singular although arguably highly non-standard compactification. Moreover, the spinors of SO(D-4) must have complex representations in order to be tensor-multiplied with the complex Weyl spinors of SO(3,1). The groups SO(8k+6) are the only even-dimensional Euclidean orthogonal group that have complex spinor representations, inequivalent to their complex conjugates, and that simultaneously lead to real spinor representations in the full spacetime whose dimension is higher than 4, namely the spacetime with the Lorentz group SO(8k+9,1). If you changed the dimension of "F" by four, you would get pseudoreal spinor representations in the full spacetime, which would lead to a doubling of fermionic degrees of freedom in 4D.
I think that no string theorist would thus be really surprised that in this kind of generalized construction, the dimension of the compact manifold must be 6 mod 8 (which is the periodicity of the qualitative properties of the spinors) because Connes just analyzes elementary properties of spinors in various dimensions. The correct choice, of course, follows from string theory. The string theorists have never emphasized these things but with extra non-singular dimensions, 10 mod 8 is really the only possible correct dimensionality that admits realistic physics. Those people who say that any number would be equally good as 10 clearly misunderstand not only the purely stringy reasons why 10 is unique, but also the spacetime (spinor) arguments why 10 mod 8 is necessary.
So I am sufficiently confident that the page 4 only contains things I know about the duality and conjugation properties of different representations that we have known before. Various pieces of discrete information are rearranged a bit, but I feel that the set of assumptions about his structures is equivalent to what we normally assume. Because of the Standard-Model-like algebra assumed in (2), it is not shocking that the Standard Model gauge group is derived on page 5.
At this moment, we have already the right gauge group because an equivalent piece of information was assumed. On page 6, the Dirac operators start to be discussed. We know very well physically what Dirac operators we can have - and his definition of the operator seems conventional: we face the choice between the usual Dirac operators with the covariant derivatives corresponding to different representations for the elementary fields.
Connes can't derive three generations, as we mentioned, so the question is whether he can say something non-trivial about the representations of the gauge group in which one generation appears.
I am reasonably confident that on page 6 in theorem 2.7, the representations for the fermions are already assumed, not derived from anything. It is only shown that the usual Dirac operators for the fermions satisfy some rules that more or less agree with the physics rules for the Dirac operator anyway. Note that the theorem 2.7 is an existence theorem. Of course, the classical Lagrangian of the Standard Model exists and the Dirac operators satisfy the elementary rules.
It seems almost certain at this moment that more complicated representations - and maybe not just anomaly-free representations - of the Standard Model gauge group could be assumed at this point, and a corresponding existence theorem 2.7 for the Dirac operators could be proved. This fact makes me reasonably certain that there is nothing important we didn't know up to page 6. Note that the right-handed neutrinos are added by hand, via "M_R", with symmetric mass matrices, and an introduction to the see-saw mechanism is offered on the top of page 7.
The rest of page 7 seems to confirm some trivial facts about gamma matrices in 4 dimensions and the reality conditions of the spinors in different dimensions, as discussed above. Various involutions resulting from the reality and other discrete conditions are listed. The fermion doubling problem is resolved, by using the correct dimension mod 8 that normally follows from string theory. That's probably just a correction of a simple error from the older papers.
Deriving Higgs, and the whole SM from a dozen of letters
When you start to think that everything is trivial, things change and explode on the top of page 8. It is argued that all bosons of the Standard Model, including the Higgs, are derived from some unimodularity of certain fluctuations. ;-) Moreover, the Standard Model action is written on half a line - see the first displayed formula on page 8.
This compact formula is supposed to give you the whole Standard Model, as derived later, with the usual gauge coupling unification - although some of the compactness has a similar origin to the compactness of the "unwordliness theory of everything". ;-) I was trying to figure out whether the Higgs mass is determined because "alpha_h" seems to be determined on page 11. Later I thought that the Higgs mass was adjustable because of an extra parameter "M". UPDATE: However, recently, Prof. Connes has pointed to me that "M" stands for the mass of the W-boson.
The conclusions say that the Higgs mass at the unification scale could be fixed and the running puts it below the big desert hypothesis but the precise numbers don't seem to be available.
ANOTHER UPDATE: In equation 39 and below it, a derived relation between Yukawa couplings is written down which effectively predicts the top quark mass to be 198 GeV in the zeroth approximation. (The top mass was already argued to be predicted by previous papers.) Connes argues that the large discrepancy may be fixed by the tau neutrino Yukawa couplings which I find very unlikely.
The calculations and their discussion continue up to the page 12.
- the parameters of the Standard Model are geometrized, assuming that the word "geometry" can also mean "non-commutative geometry" of Connes' type with a finite-dimensional space of functions on it - a geometry that is determined by the Dirac operator
- it happens that the particular "geometry" F - which we can start to call the Connes manifold - predicts gauge coupling unification
- as far as I see right now, the construction has enough arbitrary assumptions to set (not derive) the gauge group, the representation for the fermions, and all parameters except for two of the coupling constants that are determined by the gauge coupling unification
- it is still not clear to me whether the origin of the gauge coupling unification is based on some hidden respect to the grand unified group or whether Connes' argument for the coupling unification is entirely independent
- it is not yet clear to me whether the construction actually implies a natural argument why there should be one Higgs doublet and no triplets or other representations, or whether the Higgs representations are effectively inserted by some of these convoluted enough pieces of Connes' formalism
- as far as I understand, the unification of gravity only means that gravity is used for 4 dimensions, while the other forces are associated with the Connes manifold and interpreted as the rest of the same gravity, but there is no actual unification because the internal Connes manifold F follows different rules than the four-dimensional Minkowski space; note that the unification in string theory is much less vacuous because the forces morally arise from geometry of the extra dimensions that follow the same laws as the large dimensions - they're the same fields on the worldsheet, to use a perturbative example
I would say that there seem to be at least two general features that are predicted rather naturally from the Connes framework:
- KO-dimension of "F" is 6 mod 8; I have already discussed that this seems to be trivially required for any non-singular compactification of anything that shares qualitative features with string theory
- gauge coupling unification; recall that gauge coupling unification holds in many vacua of string theory, even those that break the grand unified symmetry at the string scale (including e.g. some intersecting branewolds); the Connes manifold could be a more general example suggesting that the prediction is rather general
Connes doesn't predict SUSY but I don't quite see what is better about his fermion-only model as opposed to a model in which he would put the full supermultiplets. After all, he needs SUSY not to be falsified by the high-precision gauge coupling unification whose validity he predicts. A SUSY Connes model would have to predict two Higgs doublets.
To summarize: Connes seems to describe some common features of a class of compactifications. You could imagine that all the Connes-like semi-realistic compactifications of string theory must formally reduce, at energies below the Calabi-Yau compactification scale (using the heterotic language), to the discrete Connes manifold. This Connes manifold, in some sense, only knows about the Dirac zero modes on the Calabi-Yau manifold (or its generalization). My main problem is, of course, that I have not quite understood in what sense the Connes manifold is a geometry rather than the same old physics of the covariant derivatives inside Dirac operators pretending to be "a new kind of geometry".
The deepest wisdom about the real world is, in some sense, generalized geometry. Many of us believe this statement. However, one shouldn't be using the notion of "generalized geometry" for random concepts without a good reason - for example, chess in SWF is not a version of non-commutative geometry, I guess, and the closest thing in the real life that could deserve to be called generalized geometry is Toyota Matrix M-theory - and I am so far uncertain whether there is a really good reason to call the Connes construction "geometry" or "Connes manifold" or whether it is just a misleading label that suggests a non-existent unification.
From a purely predictive vantage point, Connes seems to predict that the gauge coupling unification is generic for all string vacua that satisfy certain conditions whose character I can't describe independently of Connes' formalism. I would like to know whether the gauge coupling unification in his picture is a non-trivial result or a consequence of a secret SU(5) or other broken GUT structure.
He also seems to derive the existence of the Higgs doublet simultaneously with the derivation of the gauge bosons. I would love to know whether this prediction is really natural - whether the Higgs bosons are naturally "additional components" of the gauge fields - or whether the Higgs doublet is effectively assumed. In other words, I want to know whether Connes could write a similar paper with a different algebra and a different geometry that would predict the Standard Model with different representations - or even any representations - of the Higgs bosons.
New eyes and unification
Finally, Connes' paper may turn out to be nothing else than a translation of physics of the Standard Model to an unusual language. Even if it is so, it could still be true that the notion of the Connes manifold - which seems to be a truncation of the full stringy manifold below the Calabi-Yau compactification scale, i.e. the truncation down to the Dirac zero modes - could be useful. But if there is no way how to connect the Connes manifolds with usual manifolds, then I would be skeptical about the usefulness of the notion of the Connes manifold, too.
You know, the unification of geometry (gravity) and other forces (and matter) in string theory is real. These things are all made out of the same "stuff". The Connes unification seems fake to me. Until you show that the local or other detailed physical properties of the Connes manifold "F" are analogous or physically equivalent to the local properties of the four-dimensional spacetime, your calling both of them "geometry" is just a trick to suggest unification of something that is not really unified.
Imagine that aliens study the terrestrial buildings. They can already draw plans of each floor but they can't capture three-dimensional images. Someone proposes that there is a unification of the horizontal dimensions and the vertical dimension - but he still describes the horizontal directions by continuous numbers while the floors are labeled by integers. Is it unification? I don't think so.
You only unify the dimensions of the skyscraper once you start to build on the three-dimensional continuous geometry - or, speculatively, something that satisfies the same equations of motion. I deliberately used the term "equations of motion" instead of more vague "mathematical constraints" because it is not enough to find some common general mathematical characteristics of two structures in order to claim that you have unified them. In physics, you must show that these two structures both follow from a set of equations that completely determines the dynamics. A unification built on calling two different things by the same name is fake, and it can't imply any predictions.
Half a day later
Although it's been a great fun, I feel almost certain now that there is no new physics in the paper. For example, the Higgs fields are effectively added by hand already in equation (2) that contains two copies of the quaternions. One combination of these two quaternionic spaces gives you the core of the SU(2) weak gauge group, after a projection. Another combination gives you a Higgs doublet. Using the words of Nima, the Higgs doublet arises according to the logic of deconstruction from the bi--fundamental matter whose equivalent is inserted at the beginning.
One could construct similar algebras with other projections that would give different Higgses - different numbers or different representations. Also, the gauge coupling unification seems to be a consequence of an underlying SU(5)-like structure that is partly explicitly broken by Connes' formalism.
As I mentioned above, the representations for the quarks and leptons and the number of generations were added by hand, too.
Everything is just a derivation of the Standard Model from an unfamiliar set of algebras employed as the starting point - algebras that, however, contain the very same information as the normal information that we use to specify the Standard Model - the gauge group, representations, and the data about the couplings. Someone might think that string theory is doing the same thing - namely that it derives the familiar phenomenological data from some other starting point that has a comparable number of parameters - but it is not. The starting point in string theory is truly unified and follows other fundamental laws (that manifest themselves as conformal symmetry in the perturbative approximation). The unification in string theory and its uniqueness is real; Connes' unification is fake and his construction is extremely far from being unique.
It's been fun but it's good to be finished with it. Also, I am happy that Alain Connes now agrees with us that a unifying theory with fermions in a non-singular bulk must have 10 dimensions (at least mod 8). And because I still believe that he could really move with theoretical physics, I hope that it will take less time to agree about hundreds of other things that are probably pre-requisites for all serious attempts to unify the fundamental theories in 2006. The dimensionality mod 8 is a nice result but it would probably occupy 3 lines in Polchinski's book if it were mentioned explicitly, and there are many other important things in physics...