I was always confused of what Connes' non commutative geometry was all about, since it did not resemble, at least esthetically, the usual papers on non commutative geometries. Specially, I was always daunted of why something so extremely abstract, that I'd lose myself on the first line, could yield the Standard Model, or close to it. Now, it seems there is the initial string of reasoning with what I could begin to understand.
"S is called an "ideal" and the stickiness of it is summed up by saying for any r, the set rS is contained in S.
That means that once you are in S you cannot get out by multiplying. (By adding yes OK but multiplying by any other element in the ring will not get you out of S.)
This is an extremely simple, idiotically simple, idea and it just happens to also be extremely powerful. As I will illustrate, and that will be enough for a start.
A good example of a ring is C(M) where M is some compact manifold and C(M) is all the complexvalued functions on M. Or realvalued, would be just as good for now, but say complexvalued. So f(m) is a complex number.
You can add two elements f and g of C(M), and you can multiply them in the obvious way.
(f+g)(m) = f(m) + g(m) and (fg)(m) = f(m)g(m)
It turns out that if you pick a point m o in the manifold, and consider the set of all functions f which are zero at that point, that set is an ideal. It is sticky. If a function is in that set then f(mo) = 0 and no matter what other function you multiply it by the result will always be zero at mo.
This also is a very obvious idea. And that ideal {f: f(m o) = 0} is maximal in the sense that there is no larger ideal subset of C(M) which contains it.
It turns out that this describes all the maximal ideals in C(M) and that there is a one-to-one correspondence between the maximal ideals of the ring C(M) and the points of M.
Indeed if somebody secretly chooses a manifold (say a torus) and constructs the ring of functions on it as an algebraic object, and then, without telling you that M is a torus, they describe the ring of functions on it---by some algebraic catalog proceedure. Then you could deduce what the manifold is. All the information about the manifold is contained already in the ring of functions defined on the manifold. As algebraic relations.
This is the step that is hard to accomodate. It is somebody's theorem, a Russian perhaps. ( d Gel'fand-Naimark-Segal theorem)
But once you believe that given any manifold you can have a ring R = C(M) which contains all the information, algebraically somehow, about the manifold, then you have an obvious way to generalize what a manifold is.
You can take any ring at all (even one where the multiplication is not commutative! as it is in the ring C(M) of functions on a manifold) and you can consider the maximal ideals of that ring as a new kind of manifold.
Now that is a bit too much, too general. Connes only wants to consider NICE rings, with a few extra tools and structure, as accessories to make working with the ring a good experience.
There are some axioms. The ring should have a norm, some absolute value or size type measure looking like ||f||. The ring should have a special operator defined on it. Things like that.
But the basic tactic is to generalize the idea of a manifold by studying the maximal ideals of a ring that might or might not be the ring of functions defined on a manifold. Perhaps the manifold was never there! Perhaps the ring just came into existence but you treat it as if it were the ring of functions on some unknown manifold.
Why is this interesting?
Because Bernie Riemann invented the diff manifold in 1850 and we haven't had a new concept of a continuum for over 150 years.
What is string theory defined on? A vintage-1850 manifold. What are branes? Vintage-1850 manifolds. In whatever branch of physics, our usual idea of a continuum is this very ancient idea of Riemann.
Connes comes along and says suppose that spacetime is the set of maximal ideals of a certain ring, and suppose the damned ring is not even entirely commutative! Maybe it is mostly commutative but it has a small discrete noncommutative piece. He calls this "nearly commutative" or "almost commutative". I forget which.
The attraction of this kind of things is that it is the first fundamentally new model of a continuum in a long long time.
Of course Loop uses networks, and simplicial complexes---combinatorial objects---but these are not a continuum, although they might represent a quantum state of geometry.
Because it is such a new idea of the continuum, people are attracted to study it and try to learn what the new possibilities are.
This is a very brief introduction, only enough to take you past what you refer to as the "buzzword" level. You will gradually absorb the rest, just from the atmosphere, if you are patient. :-D"
*****
This is Alain Connes' homepage for downloads:
Note the 2 books on this subject:
Books
- Noncommutative Geometry [PDF] 4.1 MB
- Noncommutative Geometry, Quantum Fields and Motives [PDF] 6.4 MB
With Matilde Marcolli
