A post from Lubos triggered me to write a post on non-commutative geometry models for gravity plus the standard model as for example promoted by Chamseddine and Connes in a paper today.
Before I start I would like to point out that I have not studied this paper in any detail but only read over it quickly. Therefore, there are probably a number of misunderstandings on my side and you should read this as a report of my thoughts when scanning over that paper rather than a fair representation of the work of Chamseddine and Connes.
Most of what I know about Connes' version of non-commutative geometry (rather than the *-product stuff which has only a small overlap with this) I know from the excellent lecture notes by Landi. If you want to know more about non-commutative geometry beyond the *-product this is a must read (except maybe for the parts on POSETs which are a hobby horse of the author and which can be safely ignored).
But enough of these preliminary remarks. Let's try to understand the spectral action principle!
Every child in kindergarden knows that if you have a compact space you get a commutative C*-algebra for free: You just have to take the continuous functions and add and multiply complex conjugate them point-wise. As norm you can take the supremum/maximum norm (here the compactness helps). This is what is presented in every introduction section of a talk on non-commutative geometry, but onfortunately, this is completely trivial.
The non-trivial part (due to Gelfand, Naimark and Segal) is that it works also the other way round: Given a (unital) commutative C*-algebra, one can construct a compact space such that this C*-algebra is the algebra of the functions on it. Furthermore, if one has got the algebra from the functions on a space, the new space is homeomorphic to (i.e. the same as) the original one.
How can this work? Of course, anybody with some knowledge in algebraic geometry (a similar endeavour but there one deals with polynomials rather than continuous functions) knows how: First, we have to find the space as a set of points. Let's assume that we started from a space and we know what the points are. Then for each point x we get a map from the functions on the manifold to the numbers, we simply map f to f(x). A short reflection reveals that this is in fact a representation of the algebra which is one-dimensional and thus irreducible. It turns out that all irreducible representations of the algebra are of this form. Thus, we can identify the points of the space with the irreducible representations of the algebra.
I could have told an equivalent story in therms of maximal ideals which arise as kernels of the above maps, i.e. the ideals of functions that vanish at x.
Next, we have to turn this set into a topological space. One way to do this is to come up with a collection of all open or all closed sets. In this case, however, it is simpler to define the topology in terms of a closure map, that is a map that maps a set of points to the closure of this set. Such a map has to obey a number of obvious properties (for example if I apply the closure a second time it doesn't do anything or a set is always contained in its closure). In order to find this map we have to make use of the fact that if a continuous functions vanishes on a set of points then by continuity it vanishes as well on the limit points of that set, that is on its closure. Therefore, I can define the closure of a set A of points as the vanishing points of all continious functions that vanish on A. This definition then has an obvious reformulation in terms of irreducible representations instead of points. Think about it, as a homework!
Now that we have a topological space, we want to endow it with a metric structure. But instead of giving a second rank symmetric tensor, we specify a measure for the distance between two points x and y as this is closer to our formalism so far. How can we do this in terms of functions?
The trick is now to introduce a derivative. This allows us to restrict our attention to functions which are differentiable and whose gradient is nowhere greater then 1, i.e. the supremum norm of the gradient is bounded by one (this is where the metric enters implicitly since we use it to compute the length of the gradient). Amongst all such functions f we maximize l=|f(x)-f(y)|. Take now the shortest path between x and y (a geodesic). Since the derivative of f along this paths is bounded by one l cannot be bigger than the distance between x and y. And taking the supremum over all possible f we find that l becomes the distance.
Again, I leave it as a homework to reformulate this construction in the algebraic setting in terms of irreducible representations and the distance between them. There is only a slight technical complication: We started with the algebra of scalar functions on the manifold but the gradient maps those to vector fields which is a different set of sections. This makes life a bit more complicated. Connes' solution is here to forget about scalar functions and take spinors (sections of a spinor bundle precisely) instead. If those exist, all the previous constructions work equally well. But now we can use the Dirac operator and this maps spinors to spinors (with another slight complication for Weyl spinors in even dimensions).
In the algebraic setting, the Dirac operator D is just some abstract linear (unbounded) operator D which fulfills a number of properties listed on p2 of the Chamseddine_Connes paper and once that is given by some supernatural being in addition to the algebra, you can actually reconstruct a Riemannian manifold from an abstract commutative C*-algebra and D.
Next, we have to write down actions. Unfortunately, I now have to run to get to a seminar on the status of LHC. This will be continued, so stay tuned!