I am currently a member of American Mathematical Society and when I have the chance I support the activity of this venerable and fundamental association by working as a reviewer for Mathematical Reviews. So, I have the opportunity to read Notices that is a beatiful journal published by AMS and send freely to all members. It can be found here. In the August issue I have read an Opinion by Melvyn B. Nathanson of CUNY that shocked me and mostly my fundations on mathematical truth. The article is here. I think that the core of this article resides on the idea that bosses of the community make truth, mostly when proof implies thousands pages of published material very difficult to be checked carefully. We all know that, for the Wiles’s proof of Fermat theorem, the community was lucky enough to catch such a bug in a long proof. This was conveniently corrected and all agreed that such a demonstration indeed was given. But when, for any reason, we have to rely on authorities to get the truth we are in serious troubles. This means that whatever they said should be taken as granted but in this case does seem that no protestation is indeed possible. As scientists we cannot accept “ipse dixit” position and if mathematics is in such a situation something must be done to correct it.
The next question to be answered is: What is the situation in physics? Physics, differently from mathematics, is an experimental science. This means that experiments should grant our ability to tell where the truth is. Today things do not seem to be that easy for a lot of reasons but I think other sources can give judgments better than mine.
It is a well acquired fact that all the laws of physics are expressed through differential equations and our ability as physicists is to unveil their solutions in a way or another. Indeed, almost all these equations are really difficult to solve in a straightforward way and are very far from the exercises at undergraduate courses. During the centuries people invented several techniques to manage such equations and the most generally known is surely perturbation theory. Perturbation theory applies when a small parameter enters into the equations and a series solution is so allowed. I remember I have seen this method the first time at the third year of my “laurea” course and was Giovanni Jona-Lasinio that showed it to me and other fellow students.
Presently, we see as small perturbation theory has become so pervasive that conclusions derived just at a perturbation level are sometime believed always true. An example of this is the Landau pole or, generally, what implies the renormalization program. It is not generally stated but it is quite common the prejudice that when a large parameter enters into a differential equation we are stuck and nothing can be done than using our physical intuition or numerical computation. This is true despite the fact that the inverse of such a large parameter is indeed a small parameter and most known functions have both a small parameter and a large parameter series as well.
As I said elsewhere this is just a prejudice and I have proved it wrong in a series of papers on Physical Review A (see here, here and here). I have given an overview in a recent paper. With such a great innovation to solve differential equations at hand is really tempting to try to apply it to all fields of physics. Indeed, I have worked for a lot of years in quantum optics testing the approach in a lot of successful ways and I have also found applications to condensed matter physics appeared on Physical Review B and Physica E.
The point is quite clear. How to apply all this to partial differential equations? What is the effect of a large perturbation on such equations? Indeed, I have had this understanding under my nose since the start but I have not been so able to catch it immediately. The reason is that the result is really counterintuitive. When a physical system is strongly perturbed all the terms that imply spatial variation can be neglected. So, a strong perturbation series is a gradient expansion and the converse is true as well. I have proved it numerically in a quite easy way using two or three lines of Maple. These results can be found in my very recent paper on quantum field theory (see here and here). Other results can be found by yourself with similar simple means and are very easy to verify.
As strange as may seem this conclusion, it has obtained a striking confirmation through numerical computations in general relativity. Indeed, I have applied this method also to general relativity (see here and here). Indeed this paper gives a sensible proof of the Belinski-Khalatnikov-Lifshitz or BKL conjecture on the behavior of space-time approaching a singularity. Indeed, BKL conjecture has been analyzed numerically by David Garfinkle with a very beatiful paper published on Physical Review Letters (see here and here). It is seen in a striking way how all the gradient contributions from Einstein equations become increasingly irrelevant as the singularity is approached. This is a clear proof of BKL conjecture and our approach of strong perturbations at work. Since then Prof. Garfinkle has done a lot of other very good work on general relativity (see here).
We hope to show in future posts how this machinery works for pdes. In case of odes we have already posted about (see here).