Ali Chamseddine and Alain Connes have a new paper,
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.
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.
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.