Last week Steven Weinberg gave a Lee Historical Lecture at Harvard, entitled Glimpses of a World Within. There’s a report on the talk at the Harvard Gazette.

In essence, Weinberg argues in the talk for an idea that first started to dominate thinking among HEP theorists nearly forty years ago, one that is sometimes called the “Desert Hypothesis”. The idea is that by looking at what we know of the SM and gravity, you can find indications that the next level of unification takes place around the Planck scale, with no new physics over the many orders of magnitude between the scales we can observe and that scale, at least no new physics that will affect running of coupling constants for instance. The evidence Weinberg gives for this is three-fold (and very old by now):

- He describes listening to Politzer’s first talk on asymptotic freedom in 1973, and quickly realizing that if the strong coupling decreases at short distances, at some scale it would become similar to the coupling for the other fundamental forces. In a 1974 paper with Georgi and Quinn this was made explicit, and he argues this is evidence for a GUT scale a bit below or around the Planck scale.
- He explains about the Planck scale, where gravity should be of similar strength to the other interactions. This idea is even older, well-known in the fifties I would guess.
- He refers to arguments (which he attributes to himself, Wilczek and Zee in 1977) for a Majorana neutrino mass that invoke a non-renormalizable term in the Lagrangian that would come from the GUT scale.

Weinberg sees these three hints as “strongly suggesting” that there is a fundamental GUT/Planck scale, and that’s what will explain unification. Personally though, I don’t see how three weak arguments add up to anything other than a weak argument. GUTs are now a forty-year old idea that never explained very much to start with, with their best feature that they were testable since they generally predicted observable proton decay (which we haven’t seen). We know nothing at all about the source of particle masses and mixing angles, or the reason for their very different scales, and there seems to be zero evidence for the mechanism Weinberg likes for getting small neutrino masses (including zero evidence that the masses are even Majorana). As for quantum gravity and the Planck scale, again, we really have no evidence at all. I just don’t think he has any significant evidence for a desert up to a Planck unification scale, and this is now a very old idea, one that has been unfruitful in the extreme.

Weinberg ended his talk with another very old idea, that cosmology will somehow give us evidence about unification and GUT-scale physics. That also hasn’t worked out, but Weinberg quotes the BICEP2 value of r as providing yet more evidence for the GUT scale (he gives it a 50/50 chance of being correct). Again though, one more weak piece of evidence, even if it holds up (which I’d give less than 50/50 odds for at this point…), is still weak evidence.

For a much more encouraging vision talk, I recommend listening to Nati Seiberg at the recent Breakthrough Prize symposium. Seiberg’s talk was entitled What is QFT?, and to the claim that QFT is something understood, he responds “I really, really disagree”. His point of view is that we are missing some fundamental insights into the subject, that QFT likely needs to be reformulated, that there exists some better and more insightful way of thinking about it than our current conventional wisdom. In particular, there seems to be more to QFT than just picking a Lagrangian and applying standard techniques (for one thing, there are QFTs with no known Lagrangian). Seiberg takes the fact that mathematicians (who he describes a “much smarter than most quantum field theorists”…) have not been able to come up with a satisfactory rigorous version of QFT to indicate not that this is a boring technical problem, but that we don’t have the right definition to work with.

To make things more specific, he describes joint recent work (for another version of this see here) on “Generalized Global Symmetries” that works with global symmetries associated to higher co-dimension spaces than the usual codimension one case of Noether symmetries and Lagrangian field theory. Evidently there’s a forthcoming paper with more details. I’m in complete agreement with him that there must be better ways of thinking about QFT, and I think these will involve some deeper insights into the role of symmetries in the subject.

**Update**: The paper Seiberg mentions is now available here.

I know nothing at all about non-commutative geometry, but martibal’s post seems to imply at least one prediction, which is the existence of three right-handed neutrinos. If the “relation between masses” include predictions of the neutrino masses (sterile or active), that would be even better. Does it make these predictions?

If the new mathematics does not simplify, I would not call it “deep”, but “obtuse”. If the mathematics simplifies, it should postdict everything the old theory did, but why would it need to predict anything new to be considered a step in the right direction?

@Will: there was an older version of the model with massless neutrinos. NCG seems to be dead when neutrinos were discovered to have a mass. Some years after, it came out that some flexibility can be introduced by making the distinction between two notions of dimensions (the metric dimension, and the so called KO-dimension). In the commutative case, i.e. for a Riemannian manifold, both notions coincide with the usual dimension of the manifold. It was implicitly assumed that the same should be true in the noncommutative case.

At the same time, Connes and Barrett noticed independently that there was no reason to identify these two dimensions in the noncommutative case. By fixing the KO-dimension (which can take value from 0 to 7) to 6, one solves at the same time two problems: 1. The number of particles/generation is 2(2 x a)^2, hence the possibility to incorporate a massive neutrino; 2. One solves a problem of over-counting degrees of freedom in the fermonic action (“fermion doubling”).

The point is that NCG does not say anything about the number of generations. So with some contorsion one might built a model where the three generations do not have the same KO-dimension, meaning that one could have massless neutrinos (i.e. KO-dimension 0, as in the older version) for some generations, and massive neutrinos for other (i.e KO-dimension 6) for others. However I do not know if such weird possibilities have been checked.

The relation is between the sum of the mass of the fermions ans the mass of the W. Since the top is much heavier than the other fermions, in pratical the relation is between the mass of the top and the W.

Peter & Anon: a postdiction that the gauge group of the SM is U(1) x SU(2) x SU(3) would be a high level postdiction, or there already exists some justification to it ?

Martibal,

SU(3)xSU(2)xU(1) is already a very simple mathematical structure. If you predicted this before it was known in terms of some other simple mathematical structure, that would be compelling. If you start already knowing this, claiming to derive it from some other “simpler” structure is going to be convincing only if the simpler structure is quite a bit simpler, and what I have understood of the NCG explanations doesn’t to me convincingly do that. In any case, the real test I think is whether you can explain the more complicated aspects of the standard model in terms of the simpler structure you are advocating, it is that which would be convincing.

Peter,

to push a little bit the argument: the U(1) x SU(2) x SU(3) structure is simple, but in usual QFT nothing explains (or am I wrong ?) why it should stop here: why not … x SU(4) x SU(5) x… ?

The whole point of NCG is that (Euclidean) GR + SM are obtained from one single action, which is geometrical. In other terms the SM is viewed as “gravity” on the noncommutative part of space (time). So a postdiction would be that – assuming the geometrical structure of space(time) is not a manifold, but a manifold x noncommutative part – then there are only four interactions: gravity on the commutative part, the SM which has to be a U(1) x SU(2) x SU(3) gauge theory on the noncommutative part. But I agree this is not the most highly convincing postdiction.

Before I stop monopolizing the discussion: I do not think anybody working in the field is pretending that this is fully convincing. But as you said earlier, none of the other approaches to unification or quantum gravity seems much more convincing. Looking at the ratio produced results/attention received, I think NCG scores not too bad.

Martibal,

some months ago I have searched through arxiv, google scholar and the SCI about this topic; I found just one postdiction of U(1)xSU(2)xSU(3) from a simpler structure. And that postdiction is not taken seriously by anybody in the high energy physics community. Many have tried to relate the three gauge groups to the complex numbers, for U(1), the quaternions, for SU(2) and the octonions, for SU(3), but without success. Many other attempts failed and were not even published – ask your favourite high energy physicist. So the present situation is: There is no justification, as you call it, of the interaction gauge groups from a simpler structure.

Many are trying to deduce them uniquely from other, more general structures, dropping the requirement of simplicity, but still without success. (Uniqueness is the key here, of course.) Also Peter’s work can be seen as an attempt in this direction, I guess. But no such postdiction or justification is available yet. Finding a postdiction seems to be easy at first sight, since the result is known, but in fact it is a surprisingly hard problem.

martibal,

I definitely agree that NCG ideas are worthy of attention, especially because of the lack of good ideas about unification.

In your description though of unification by NCG, the problem is with the claim of deriving SU(3)xSU(2)xU(1) by just doing geometry on the “non-commutative part”. The danger is that among many possibilities for the “non-commutative part”, you have just picked the one that gives SU(3)xSU(2)xU(1). When I’ve been trying to read the NCG literature, this is a point at which I’ve found things unconvincing. Possibly I’m not reading the right explanation…

Didn’t I read somewhere that NCG predicted the wrong mass for the Higgs? Did they do something about that?

Peter, well the hope is that precisely, on the noncommutative part, one does not have so much way to do geometry but the one that gives you the standard model. All depends on what you call “doing geometry” of course.

The starting point is the observation that all the information of a compact spin manifold M is encoded within a triple given by the commutative algebra C^infty(M) of smooth functions on M, the Hilbert space of spinors and the Dirac operator. Conversely, Connes worked out the conditions that must satisfy a commutative algebra A, an Hilbert space H and an operator D such that – given any triple (A, H, D) with A commutative – then there exists a spin manifold such that A = C^infty(M). Then one drops out the commutativity requirement, and define a “noncommutative geometry” as a spectral triple (A, H, D) where A does not need to be commutative.

Now the result is, assuming three things:

– the algebra is the product of C^infty(M) by a finite dimensional algebra A_F,

– the KO-dimension (see a post above) of the finite dimensional part is 6,

– there is an ad-hoc “symplectic hypothesis” to define a real form on A_F,

then A_F = C + H + M_3(C) (complex, quaternion, 3×3 matrices), which is precisely the algebra the spectral triple of the SM is built on. All this is in the paper “Why the standard model” by Chamseddine and Connes (arXiv: 0706.3688 [hep-th]). In their most recent work you have already mentionned, it seems they can get rid of the ad-hoc symplectic hypothesis.

So once you buy the definition of a spectral triple (which has a strong justification, since in the commutative case this is equivalent to the definition of a Riemannian spin manifold) one basically deduces the gauge group of the SM from the first two assumptions above.

Hans-Peter: thanks for the information.

David: from the beginning of the model (early 90′) there was a prediction for the Higgs mass around 170GeV. It has been ruled out in 2008 by Tevatron. After the discovery of the Higgs, Chamseddine and Connes noticed that if one takes into account an extra-scalar field (proposed in a completely different context by particle physicist to avoid the instability of the electroweak vacuum due to the “low mass” of the Higgs), then one can obtain the correct Higgs mass.

The question is then: why this extra field was not present before, since the claim is that the SM is almost uniquely derived from the NCG principles ?

It turns out that to justify this new field inside the NCG framework, one has to allow a little but of flexibility, regarding one of the conditions that defines a spectral triple (namely the one that tells you the operator D is a first-order differential operator). The present state of the art is that the extra-field (which violates the first-order condition) can be viewed as a small excitation around a vacuum (which does satisfies the first-order condition).

Martibal,

the key assumption in what you write about NCG is the KO-dimension 6 of the finite dimensional part. It indeed implies the three gauge groups – but does not imply them uniquely: if you took 7 or 8 or 9 as an assumption, the argument dissolves. Besides, the assumption is not simpler than the result, as required; it is just a hidden way to introduce the result in the argument right from the start. So it should not be counted as a justification of the standard model.

Seiberg’s argument is very interesting and deserves closer attention. We don’t understand QFT and only have a “practical” or pragmatic understanding of how to calculate with it in certain cases. That’s not satisfactory. The situation becomes acute when gravity is brought into the picture and the spacetime background moves to the front of the stage and becomes part of the show. We need something background-independent — indeed, spacetime-independent — to make progress. Higher groups? Maybe. My pet cause is starting with field fluxes as fundamental. You can do that already, although it’s merely a curiosity in QFT as we know it now.

Hans Peter: as I said in a precedent post, the KO-dimension 6 also has a justification regarding the fermonic action. In fact it is justified in this way in Connes and al paper. Then one “noticed” that it allows to have massive neutrino. In Barret paper the KO-dimension 6 is justified because it corresponds to the Dirac operator with Minkovski signature. So the choice is far from arbitrary.

Even if you consider it as a free parameter, that one single parameter with only 8 possible values (from 0 to 7), fixes the gauge group of the SM, the number of particles per generation, their representation, the form of the Bosonic Lagrangian – including the Higgs – obtained together with Einstein-Hilbert action… well, I would say this is quite an interesting step towards an understanding of the mathematical structure of the SM.

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Martibal,

Connes indeed achieved an interesting step. He discovered a structure that allows to tie the gauge groups together in a different way from that of GUTs. He showed that the three gauge groups are deeply related somehow. On the other hand, he also states that these ideas do not explain most parameters of the standard model, nor the number of generations. So we have to take the truth in these ideas and continue the search. But in which direction …?

Well, recently there have been several works by various groups on a new scalar field who comes out by relaxing the first-order condition (see a precedent post). This opens some possibility towards some Pati-Salam models, whose SM would be the vacuum. Hopefully some phenomenology could be done there.

At more fundamental level, there is also the idea to get rid of the assumption that space (time) is described by the product of a manifold by a finite dimensional space. In Connes and all last work there are very interesting things in that direction. The idea would be NOT to postulate from the beginning that there is a manifold (or say differently, NOT to start with an algebra with an infinite dimensional center and only a finite dimensional noncommutative part) but to be able to reconstruct everything from some “abstract” noncommutative relations.

The problem is not to find research projects, the problem is to have the possibility to work on them with a little bit of stability (meaning contracts longer than 1 or 2 years). But that is another story.