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.