Feynman diagrams for pedestrians and mathematicians

Abstract

This is a simple mathematical introduction into Feynman diagram technique, which is a standard physical tool to write perturbative expansions of path integrals near a critical point of the action. I start from a rigorous treatment of a finite dimensional case (which actually belongs more to multivariable calculus than to physics), and then use a simple dictionary to translate these results to an infinite dimensional case. The standard methods such as gauge-fixing and Faddeev-Popov ghosts are also included. Resulting Feynman diagram series often may be used rigorously without any references to the initial physical theory (which one may sweep under the carpet). This idea is illustrated on an example of the Chern-Simons theory, which leads to universal finite type invariants of knots and 3-manifolds.