Let me stress however, that although it may look differently, this is entirely classical physics, there is no quantum physics to be found anywhere here even though having possibly non-commutative algebras might remind you of quantum mechanics. This, however, is only a way of writing strange spaces and has nothing to do with the quantisation of a physical theory. We will even compute divergences of one loop Feynman diagrams, but still this only a technical trick to write the classical Einstein action (integral of the Ricci scalar over the manifold) and has nothing to do with quantum gravity! Our final result will be the classical equations of motion of gravity coupled to a gauge theory and a scalar field (the Higgs)!
The trick goes as follows: Last time, we saw that in order to encode metric information we had to introduce a differentiation operator
Another advantage of the Dirac operator is that when it is squared it gives the Laplacian plus the Ricci scalar (which we want to integrate to obtain the Einstein Hilbert action) divided by some number which I vaguely remember to be 12 but which is not essential. In formulas, we have
But how to get the integration? Here the important observation is that we can pretend that this operator is the kinetic term for some field. Then, we can compute the one loop divergence of this field. On one hand, we know that this is the functional trace of the log of this operator. On the other hand, we can compute the divergence of this expression either diagrammatically or with slightly advanced technology in terms of the heat-kernel.
I will explain the heat-kernel formalism at some other time. However, the result of that treatment is a series of "heat-kernel coefficients"
The important result now is that the effective action is the integral of
For concreteness, take
But the effective action can also be written as the functional trace of the log of the operator. For this, we don't need any field for which the operator is the kinetic operator of. Imagine we happen to know all the eigenvalues of
As it happens, Connes and Chamseddine really know the eigenvalues of the Dirac operator on spheres. So, they can really do this sum (which turns out to be expressible in terms of zeta-functions). The spheres are of course compact and thus the eigenvalues are discrete. In our field theory argument above, however, we implicitly used the usual continuous momentum space arguments for the effective action. In the limit of large momentum (which is relevant for the divergence) corresponding to short distances this should not really matter, one can pretend that momentum space is actually continuous. However, with a cut-off, this is not precisely true and the discreteness comes in in sub-leading orders. The difference between the continuous computation and the discrete one is of course tiny for a large cut-off. And it is exactly this difference that lets the two authors find agreement to "astronomical precision" (p. 15).
OK, up to now, we have reformulated the gravitational action in terms of the spectrum of the Dirac operator. But what about gauge interactions. But every physicist should now now how to proceed: Do gravity in higher dimensions and perform a Kaluza-Klein reduction. If the compact space has a symmetry group G then besides some scalars you will find YM-theory with gauge group G.
In the non-commutative setting, one can as well take a non-commutative space for the compact directions. Here, Connes and Chamseddine argue for a minimal example (given some conditions of unclear origin, one might be suspicions that those are tuned to give the correct result). It is minimal in terms of irreducibility. However, the total space being the product (by definition reducible) of this compact space by a classical commutative 4d space time (again reducible) make the naturalness of this requirement a but questionable.
For the concrete specification of the compact space, some Clifford magic is employed (including Bott periodicity) but this is standard material. You end up with a non-commutative description of two points for the two spinor chiralities. The symmetry then determines the gauge group. Here I am not completely sure but it seems to me that they employ the usual representation theory arithmetic from GUT-theories to sort all standard model particles in nice representations.
The non-commutative formulation allows the KK-gauge boson A_mu to also have a leg in the compact direction between the two points. From the 4d perspective this is of course a scalar. This of course will be the Higgs. The two points have a finite distance (see the Landi notes for details) and give a mass term inversely proportional to the distance (that is opposite to a superficially similar D-brane construction as noted by Michael Douglas some years ago).
That's it. We have an algebraic formulation of the classical action for the standard model. Let's recap what went in: The NCG version of a space with metric information in terms of a Dirac operator. Some heat-kernel material to write the gravitational action in terms of eigenvalues, GUT-type representations theory and KK-theory. What kind of 'surprises' were found? GUT type relations are rediscovered, treating the discrete spectrum of
5 comments:
Naively, isn't log(D^(d+2)) just proportional to log(D)? Or am I missing something?
Of course it is. However, to get the field theory computation going you want the operator to have something Laplace-like as highest derivative part. This gives you a nice propagator.
Hi Robert,
nice post, quite clear. I had a cursory look at the paper too, and I have a couple of points I'd like to ask you: first, do you also have in this case the tower of KK states, with the usual masses? And second, in the spectral action, if I understand correctly, the cutoff is not removed at the end but instead it is related to Newton's constant, and that leaves them with a big cosmological constant? Ok a third one: one of the assumptions needed to find the SM group is that the internal space has KO-dimension 6 (whatever that is), but there is no reason for this other than wanting to find the SM, right?
I am not sure about the KK states. My gut feeling is that they don't exists since the compact space really only consists of a finite number of points (with some non-comutative structure to make it behave correctly topologically, that is in K-Theory). Thus I would guess there is no room for higher harmonics.
You are right about the cut-off. That one is left in at a finite value which appears at different powers in all three terms (cosmological constant, EH and higher curvature).
I didn't look really into the part that discusses the choice of the particular compact space. I think that was done in previous papers of the two authors. But already those left me (after a glance at the formulas and a colloquium talk by Chamseddine) with the impression that the specific conditions (like KO-dim=6) were tailored towards the known result.
It is very interesting for me to read the article. Thanx for it. I like such topics and everything connected to them. I would like to read a bit more on that blog soon.
Alex
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