Spectral Action Part II

After in part I, I have explained how to go back and forth between spaces with metric information and (C*)-algebras it's now time to add some physics: Let us now explore how to formulate an action principle that will lead to equations of motion when extremised.

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 so we could formulate the requirement that a function should have a gradient which has length less or equal to 1. One could have taken the gradient directly but there was a slight advantage to take the Dirac operator instead since that maps spinors to spinors instead of the gradient mapping scalars to vector fields.

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 . Even better, when we are in d dimensions, taking the d-th power gives us the volume element. Thus, taking these two observations together and using the Clifford algebra, we find that taking powers of gives us . The second term is obviously what we need to integrate to obtain the EH action.

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" which are scalars expressions of the curvature of mass dimension . That is is 1, is basically the Ricci scalar, is a linear combination of scalar contractions of the curvature squared and so on. All those can interpreted as the coefficients of a power series in a formal parameter , i.e. .

The important result now is that the effective action is the integral of over s from 0 to infinity and over all space-time points. Because of the negative powers of , this integral diverges at 0. It turns out, this diverge is nothing but the UV divergence of the one-loop diagrams. Now you can apply your favourite regularisation procedure (dimensinal regularisation, Pauli-Vilars, you name it). Here, for simplicity, we just use a cut-off and start our integration at instead of 0. Connes and Chamseddine do something very similar. Just instead of a sharp cut-off they use a function in the integral that decays exponentially when s approaches 0 (NB has mass-dimension -2 and thus acts as ).

For concreteness, take . Then the term leads to times 1 integrated over the space (i.e. a cosmological constant). The term from gives times the Einstein-Hilbert term and is times an integral of curvature squared. The remaining terms are finite when Lambda is removed. Thus, we find the Einstein action (including a huge cosmological constant) plus a curvature squared correction as the divergence of the one-loop effective action.

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 . Then we have just found that the Einstein Hilbert action can be written as the divergent part of the sum of the log of all those eigenvalues. And this is the spectral action principle: .

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 on spheres can well be approximated by continuous momentum space for momenta large compared to the inverse radius. And there is a final observation detailed in appendix A of the paper: For there are some cancellations of unclear origin which make vanish. This however is a statement about a heat-kernel coefficient and does not have a priori any connection with the non-commutative approach. Furthermore, the physical implications are left in the dark.