Tuesday, December 02, 2008

Heat expansion of the spectral action and loop diagrams

Ali Chamseddine and Alain Connes have a new paper,
The uncanny precision of the spectral action,
which argues that under some assumptions they find worth considering, two terms in an expansion of an action give the correct initial 6820 decimal places of an action.

It's a typical paper where I feel that there is something extremely cool and exciting going on, but once I spend ten minutes with each such thing, it seems to evaporate. Recall that they want to write a theory of everything as a simple system in noncommutative geometry. The action is
S = Tr [ f(D/Lambda) ]
It's the so-called spectral action. The trace should probably be taken over a first-quantized (one-particle) Hilbert space only, generating the classical spacetime action of a field theory in some way. Here, "D" is the Dirac operator - including all dimensions they consider.

The letter "Lambda" stands for a cosmological constant. You would think that it is extremely small but they actually study "Lambda goes to infinity" limit. That's bizarre and I am afraid that you won't find any justification of such an approach in the paper but some other things are fun so you shouldn't stop reading yet.

The letter "f" is a function of a real variable "x" that drops to zero if "x" goes to plus minus infinity, something like "exp(-x^2)" except that the authors want to prefer a slightly different function with the same asymptotics but with derivatives of all orders vanishing at x=0. Again, not sure what justifies such a strange and arbitrary function. 

Clearly, for some contrived choices of functions and parameters, certain approximations to some quantities may be very accurate, but another question is whether such an observation under certain conditions is interesting, non-trivial, or relevant for anything in science or just an artificially created curiosity from the realm of recreational mathematics. If the accurate values of both sides of the (approximate) equation are physically irrelevant, so is the (approximate) identity!

But what's funny and possibly non-trivial is a "heat expansion" of the spectral action, "Tr [f(D/Lambda)]", for large values of Lambda. You might think that it is just a Taylor expansion except that it is apparently not. I feel that in their non-commutative geometrical context, they have an alternative calculation scheme for loop diagrams in field theory but all the links seem very obscure so far.

For example, I think that the numerator "11" in the equation below (27), obtained from zeta functions, could be equivalent to the "11" in the one-loop beta functions of gauge theories. Analogously, the calculation of the curvature-cubed terms around page 27 could be a reorganization of the calculation of two-loop diagrams in general relativity except that they don't get the desired (or at least expected) non-zero result. 

Note that the Goroff-Sagnotti two-loop result, proving non-renormalizability of general relativity, is a cubic function of the Weyl tensor only, and the rational coefficient is 209/2880. The number 209 could be related to 208 in some of their formulae.

If a reader knows something about the relationship between the heat expansions and loop diagrams, or why it's not there if it's not there, I would like to know! Thanks. Another topic worth research is the conceptual origin of the cancellations of the "a4" and "a6" terms in their expansion and their possible applications in the solution to various hierarchy problems.

1 comment:

  1. Triggered by your post I had a look at the paper. Since answering your questions would be too much for a comment I have started writing my own post. Yesterday, I wrote a first installment on some non-commutate geometry background


    I hope to have time tonight to write to write a part II actually covering the spectral action. As a spoiler: I don't think there is much surprising going on: The increadible precission expresses that you make only a small error when you replace the discrete spectrum of the Dirac (or any other differential operator) on S^3xS^1 of the size of the universe with a continuous momentum space.

    The cancelation is about ordinary Sealy-deWit heat-kernel coefficients on S3 and has nothing to do with NCG. I have no explanation but I also cannot see a significance.