Feynman Diagrams for Pedestrians and Mathematicians

1.1. About these lecture notes. For centuries physics was a potent source providing mathematics with interesting ideas and problems. In the last decades something new started to happen: physicists started to provide mathematicians also with technical tools, methods, and solutions. This process seem to be especially strong in geometry and low-dimensional topology. It is enough to mention the mirror conjecture, Seiberg-Witten invariants, quantum knot invariants, etc. Mathematicians, however, en masse failed to learn modern physics. There seem to be two main obstructions. Firstly, there are few textbooks in modern physics written in terms accessible for mathematicians. Mathematicians and physicists speak two different languages, and a good “physical-mathematical dictionary” is missing. Thus, to learn something from a physical textbook, a mathematician should start from a hard and time-consuming process of learning the physical jargon. Secondly, mathematicians consider (and often rightly so) many physical methods and results to be non-rigorous and do not consider them seriously. In particular, path integrals still remain quite problematic from a mathematical point of view (due to some usually unclear measure aspects), so mathematicians are reluctant to accept any results obtained by using path integrals. Yet, this technique may be put to good use, if at least as a tool to guess an answer to a mathematical problem. In these notes I will focus on perturbative expansions of path integrals near a critical point of the action. This can be done by a standard physical technique of Feynman diagrams expansion, which is a useful book-keeping device for keeping track of all terms in such perturbative series. I will give a rigorous mathematical treatment of this technique in a finite dimensional case (when it actually belongs more to a course of multivariable calculus than to physics), and then use a simple “dictionary” to translate these results to a general infinite dimensional case. As a result, we will obtain a recipe how to write Feynman diagram expansions for various physical theories. While in general an input of such a recipe includes path integrals, and thus is not well-defined mathematically, it may be used purely formally for producing Feynman diagram series with certain expected properties. A usual trick is then to “sweep under the carpet” all references to the underlying