Non Standard stuff

I am not sure I really know how to make use of a "blog" like this one. Recently I had to write a sollicited paper describing the perspective on the structure of space-time obtained from the point of view of noncommutative geometry. At first I thought that I could just be lazy and after the paper was written (it is available here) just use pieces of it to keep this blog alive during the summer vacations. However, when trying to do that, I realized that it was better (partly because of the impractical use of latex in the blog) to first make the paper available and then tell in the blog the additional things one would not "normally" write in a paper (even a non-technical general public paper such as the above). I am not keen on turning the blog into a place for controversies since it is unclear to me that one gains a lot in such discussions. The rule seems to be that, most often, people have prejudices against new stuff mostly because they dont know enough and take the lazy attitude that it is easier to denigrate a theory than to try and appreciate it. I am no exception and have certainly adopted that attitude with respect to supersymmetry or string theory. A debate will usually exhibit the strong opinions of the various sides and it is rare that one witnesses a real change taking place. So much for the "controversy" side. However I do believe that there are some points that can be quite useful to know and which, provided they are presented in a non-polemic manner can help a lot to avoid some pitfalls. I will discuss as an example the two notions of "infinitesimals" that I know and try to explain the relevance of both. This is not a "math paper" but rather an informal discussion.

When I was a student in Ecole Normale about 40 years ago, I fell in love with a new math topic called "nonstandard analysis" which was advocated by A. Robinson. Being a student of Gustave Choquet at that time, I knew a lot about ultrafilters. These maximal filters were (correct me if I am wrong) discovered by H. Cartan during a Bourbaki workshop. At that time Cartan had no name for the new objects but he had found the remarkable efficiency they had in any proof where a compactness and choice arguments were needed. So (this I heard from Cartan) the name he was using was "boum" !!! Of course he knew that it gave a one line proof of the existence of Haar measure (boum...). And also that because of uniqueness of the latter it was in fact proving a rather strong convergence statement on the counting functions that approximate the Haar measure. He wanted to make sure, and wrote in a Compte-Rendu note the full details of a direct geometric argument proving the expected convergence. From ultrafilters to ultraproducts is an easy step. And I got completely bought by ultraproducts when I learnt (around that time) about the Ax-Cochen theorem: the ultraproduct of p-adic fields is isomorphic to the ultraproduct of local function fields with the same residue fields. Thus I started trying to work in that subject and obtained, using a specific class of ultrafilters called "selective", a construction of minimal models in nonstandard analysis. They are obtained as ultraproducts but the ultrafilters used are so special that, for instance, in order to know the element of the ultrapower of a set X, one does not need to care about the labels: the image ultrafilter in X is all that is needed. I wrote a paper explaining how to use ultraproducts and always kept that tool ready for use later on. I used it in an essential manner in my work on the classification of factors. So much for the positive side of the coin. However, quite early on I had tried in vain to implement one of the "selling adds" of nonstandard analysis, namely that it was finally giving the promised land for "infinitesimals". In fact the adds came with a specific example: a purported answer to the naive question "what is the probability "p" that a dart will land at a given point x of the target" in playing a game of darts. This was followed by 1) the simple argument why that positive number "p" was smaller than epsilon for any positive real epsilon 2) one hundred pages of logic 3) the identification of "p" with a "non-standard" number...
At first I attributed my inability to concretely get "p" to my lack of knowledge in logics, but after realizing that the models could be constructed as ultraproducts this excuse no longer applied. At this point I realized that there is some fundamental reason why one will never be able to actually "pin down" this "p" among non-standard numbers: from a non-standard number (non-trivial of course) one canonically deduces a non-measurable character of the infinte product of two element groups (the argument is simpler using a non-standard infinite integer "n", just take the map which to the sequence a_n (of 0 and 1) assigns its value for the index "n"). Now a character of a compact group is either continuous or non-measurable. Thus a non-standard number gives us canonically a non-measurable subset of [0,1]. This is the end of the rope for being "explicit" since (from another side of logics) one knows that it is just impossible to construct explicitely a non-measurable subset of [0,1]!

It took me many years to find a good answer to the above naive question about "p". The answer is explained in details here. It is given by the formalism of quantum mechanics, which as explained in the previous post on "infinitesimal variables" gives a framework where continuous variables can coexist with infinitesimal ones, at the only price of having more subtle algebraic rules where commutativity no longer holds. The new infinistesimals have an "order" (an infinitesimal of order one is a compact operator whose characteristic values \mu_n are a big O of 1/n). The novel point is that they have an integral, which in physics terms is given by the coefficient of the logarithmic divergence of the trace. Thus one obtains a new stage for the "calculus" and it is at the core of noncommutative differential geometry.

In Riemannian geometry the natural datum is the square of the line element, so that when computing the distance d(A,B) between two points one has to minimize the integral from A to B along a continuous path of the square root of g_\mu\nu dx\mu dx\nu. Now it is often true that "taking a square root" in a brutal manner as in the above equation is hiding a deeper level of understanding. In fact this issue of taking the square root led Dirac to his famous analogue of the Schrodinger equation for the electron and the theoretical discovery of the positron. Dirac was looking for a relativistic invariant form of the Schrodinger equation. One basic property of that equation is that it is of first order in the time variable. The Klein-Gordon equation which is the relativistic form of the Laplace equation, is relativistic invariant but is of second order in time. Dirac found away to take the square root of the Klein-Gordon operator using Clifford algebra. In fact (as pointed out to me by Atiyah) Hamilton had already written the magic combination of partial derivatives using his quaternions as coefficients and noted that this gave a square root of the Laplacian. When I was in St. Petersburg for Euler's 300'th, I noticed that Euler could share the credit for quaternions since he had explicitly written their multiplication rule in order to show that the product of two sums of 4 squares is a sum of 4 squares.

So what is the relation between Dirac's square root of the Laplacian and the above issue of taking the square root in the formula for the distance d(A,B). The point is that one can use Dirac's solution and rewrite the same geodesic distance d(A,B) in the following manner: one no longer measures the minimal length of a continuous path but one measures the maximal variation of a function: ie the absolute value of the difference f(A)-f(B). Of course without a restriction on f this would give infinity, but one requires that the commutator [D,f] of f with the Dirac operator is bounded by one. Here we are in our "quantized calculus" stage, so that both the functions on our geometric space as well as the Dirac operator are all concretely represented in the same Hilbert space H. H is the Hilbert space of square integrable spinors and the functions act by pointwise multiplication. The commutator [D,f] is the Clifford mulltiplication by the gradient of f so that when the function f is real, its norm is just the sup norm of the gradient. Then saying that the norm of [D,f] is less than one is the same as asking that f be a Lipschitz function of constant one ie that the absolute value of f(A)-f(B) is less than d(A,B) where the latter is the geodesic distance. For complex valued functions one only gets an inequality, but it suffices to show that the maximum variation of such f gives exactly the geodesic distance: ie we recover the geodesic distance d(A,B) as Sup f(A)-f(B) for norm of [D,f] less than one.

Note that D has the dimension of the inverse of a length, ie of a mass. In fact in the above formula for distances in terms of a supremum the product of "f" by D is dimensionless and "f" has the dimension of a length since f(A) - f(B) is a distance.

Now what is the intuitive meaning of D? Note that the above formula measuring the distance d(A,B) as a supremum is based on the lack of commutativity between D and the coordinates "f" on our space. Thus there should be a tension that prevents D from commuting with the coordinates. This tension is provided by the following key hypothesis "the inverse of D is an infinitesimal".

Indeed we saw in a previous post that variables with continuous range cannot commute with infinitesimals, which gives the needed tension. But there is more, because of the fundamental equation ds = 1/D which gives to the inverse of D the heuristic meaning of the line element. This change of paradigm from the g_\mu\nu to this operator theoretic ds is the exact parallel of the change of the unit of length in the metric system to a spectral paradigm.
Thus one can think of a geometry as a concrete Hilbert space representation not only of the algebra of coordinates on the space X we are interested in, but also of its infinitesimal line element ds. In the usual Riemannian case this representation is moreover irreducible. Thus in many ways this is analogous to thinking of a particle as Wigner taught us, ie as an irreducible representation (of the Poincaré group).