I’ve been reading Marcolli & Connes’ new book *Noncommutative Geometry, Quantum Fields, Kitchen Sinks, and Motives* lately. It’s a good read, full of interesting calculations and intriguing ideas. Frankly, I don’t understand half of it (I won’t be writing about motives today), but the bits that I do get look really cool.

For example: Connes & Marcolli (and their collaborator Chamseddine) have found a rather clever way of writing down the Standard Model. Basically, they’re doing “non-commutative Kaluza-Klein theory”. They write down a very simple, essentially gravitational Lagrangian on a spacetime which is a direct product of 4d Minkowski and a “very small” non-commutative space F. Then they ask what this Lagrangian looks like in 4d. The answer is, of course, determined by F, and it turns out that there’s a very simple choice of F that gives exactly the Standard Model. It’s the non-commutative space whose algebra of functions is

(I’m cheating somewhat here: You also have to specify a Hilbert space on which acts and a Dirac operator . These data encode the Standard Model fermions and their couplings.)

The situation isn’t perfect though: This construction hints at some deep underlying structure, but that structure is maddeningly obscure. Their model is explicitly a low-energy approximation to some unknown theory. The Lagrangians they consider all have the form

where is some even function, is the Dirac operator on , and is a cut-off scale, which is used to regularize the theory. Which is to say: they are at the end of the day studying an ordinary effective field theory. We should think of this theory as an approximation to quantum gravity on some weird non-commutative background — remember that the Lagrangian they start with is purely gravitational — but at the moment, we can’t say much of anything about this quantum gravity theory.

The authors realize this, of course, and say so in their introduction. Their hope is that by using non-commutative geometry to exploit formal similarities between quantum gravity and number theory, we can learn something about. (I wonder if there’s any formal similarity between universality in statistical mechanics and the theory of motives in algebraic geometry: Certain properties inevitably emerge when one chooses to ignore enough information, determined only by the sorts of data you choose to relate?) I’m not qualified to make any judgments about this program, but it does look exciting!

I should mention also: The book also contains the most readable explanation I’ve seen of the relationship between Hopf algebras and the renormalization of perturbative field theories. In fact, if you’re a mathematician who wants to learn something about perturbative QFT, you could do a lot worse than reading their survey. As of this writing, it’s got decent explanations of most of the confusing stuff: Wick rotation, bare vs. physical coupling constants, renormalization schemes, and so forth. The only things missing are 1) some discussion of scattering theory, and 2) a good explanation of universality, which explains in exactly what sense all of the computations in renormalized QFT make sense. I was planning to write something about this, but as this post has gotten rather long, I think I’ll leave that for later.

Gee, thanks! I can’t wait to get hold of a copy. So what are the kitchen sinks? I understand that the book probably has everything But the K.S. but then they spoilt that by putting the K.S. on the cover!

In fact, that algebra is known for about twenty years now, I think. The recent progress which apparently makes this NCgGKK-reduction give pretty much exactly the standard model is due to just a tiny modification in one of the gradings that enter the defintion of the representation of this algebra.

I once tried to summarize some elements of this

here.

Indeed. Connes starts with a worldline theory for a super particle, encoded in his spectral triple. He then argues that, by a simple consistency argument, whatever effective target space theory is encoded by this worldline theory, it should to come from an expansion of .

As far as I am aware he and his collaborators make no attempt whatsoever to understand the connection between the worldline theory and the target space theory on more standard grounds. It would be nice to understand whether or not one can argue that the “spectral action” in fact follows from some standard but NCG version of perturbative field theory.

Chamseddine once went throught the trouble of checking (here and here) that the spectral action induced by the Dirac-Ramond operator which governs the superstring does reproduce the super string’s effective target space theory to lowest nontrivial order.

So that suggests some deep significance of the spectral action.

When I wrote my latest post about the fractional quantum hall effect a moment ago, I saw the new tag you’d introduced things I don’t understand. I was going to tag my post with it too, but in the end decided against, because in the end it would just be too much work, having to click an extra checkbox for every single thing I ever write.

Actually, I introduced that tag for a draft which I never published. It may return someday soon, but I wasn’t happy with the version I had. Apparently if you cancel a post it still keeps the tags you added for it. AJ picked up the tag from there, I think.