Adventures in math of a quantum thinker

...once one took String Theory seriously, one soon found a lot of reasons that physicists would have to pay attention to previously unfamiliar topics in more or less modern mathematics. For example, at a basic level, a string moving in spacetime sweeps out a two-dimensional surface with the property of what mathematicians call a Riemann surface. Riemann surfaces are an important topic in the mathematics of the last century, and they became important for physicists primarily because of String Theory. 

From my vantage point, all this made the interaction of physics with more or less contemporary mathematics far more robust and significant. Opportunities to apply physics-based insights to “purely mathematical” problems stopped seeming like exceptions. 

In my own work in the years just after 1984, the development that is most worth mentioning here involves topological quantum field theory. This was partly motivated by hints and suggestions by the mathematician Michael Atiyah, who pointed to mathematical developments that he suggested should be better understood using physical insight. Other hints came from developments in physics.

Each problem here involved applying physics ideas to a problem that traditionally would have been viewed as a math problem, not a physics problem. These were all problems that I would not have seriously considered working on until String Theory broadened our horizons concerning the relations between mathematics and physics. In each case, the aim of my work was to try to show how a problem that naively is “purely mathematical” could be approached by methods of physicists. 

I will just tell you about one of these problems. It involved knots in ordinary three-dimensional space. A tangled piece of string is a familiar thing in everyday life, but probably most of us are not aware that in the 1900's, mathematicians built a deep and subtle theory of knots. By the time I became involved, which was in 1987-8, there was a puzzle, which Atiyah helped me appreciate. The mathematician Vaughn Jones had discovered a marvelous new way of studying knots – for which he later received the Fields Medal. Vaughn Jones had proved that his formulas worked, but “why they worked was mysterious. 

It may be hard for someone who does not work in mathematics or science to fully appreciate the difference between understanding “what” is true and understanding “why” it is true. But this difference is an important part of the fascination of physics and mathematics, and I guess all of science. I will say, however, that the difference between “what” and “why” depends on the level of understanding one has at a given level of time. One generation may be satisfied with the understanding of “why” something is true, and the next generation may take a closer look.  

Anyway, getting back to knots, I was able to get a new explanation of Vaughn Jones's formulas by thinking of a knot as the trajectory followed by an elementary particle in a three-dimensional spacetime. There were a few tricks involved, but many of the ideas were standard ideas of physicists. Much of the novelty was just to apply the techniques of physicists to a problem that physicists were not accustomed to thinking about. 

This work became one of my best-known contributions, among both mathematicians and physicists. But it is also an excellent illustration of something I said in my acceptance speech the other night. No matter how clever we are, what we can accomplish depends on the achievements of our predecessors and our contemporaries and the input we get from our colleagues. My ability to do this work depended very much on clues I got from work of other scientists. In several cases, I knew of these clues because colleagues pointed out the right papers to me or because the work was being done right around the corner from me by colleagues at the Institute for Advanced Study in Princeton. It also helped at a certain point to remember some of what I had learned from Sidney Coleman back when I had been at Harvard, involving yet another insight of Albert Schwarz.

ADVENTURES IN PHYSICS AND MATH 

Edward Witten

En 1979, j'étais à une conférence sur les théorie de jauge à Cargèse où l'un des vedettes était ... Ed Witten. Il se trouve qu'on était dans le même hôtel et que j'ai eu donc souvent l'occasion de l'écouter ; et il y a une chose qui m'a beaucoup frappé. Tandis que les autres physiciens ... parlaient toujours en termes de phénomènes, éventuellement en termes de modèles concrets testés sur ordinateurs, Witten jonglait tout le temps avec les théories ellees-mêmes, leur manière d'intéragir, de se compléter ou tout simplement d'exploser. Il me semble qu'il y a un lien direct entre cette manière de penser et la M-théorie d'aujourd'hui. 

In 1979, I was at a lecture on gauge theory at Cargèse where one of the stars was ... Ed Witten. It turns out we were in the same hotel and so I often had the opportunity to listen to him ; and there is one thing that struck me a lot. While other physicists ... always spoke in terms of phenomena, possibly in terms of concrete models tested on computers, Witten juggled all the time with theories themselves, their way of interacting, of complementing each other or simply of exploding. It seems to me that there is a direct link between this way of thinking and today's M-theory. (blogger's translation)

Autour de l'hypothèse de Poincaré

Valentin Poénaru