Finite-dimensional Feynman Diagrams

The Stanford Linear Accelerator Virtual Visitor Center website has a Theory section including a page on Feynman diagrams and the ``Feynman rules''. The focus is exclusively on the phenomenological interpretations of the diagrams (electrons in, electrons out). Web references on Gaussian integrals include lecture notes for Math 221A from Berkeley and for Chem 461 at Michigan.

1. What every Freshman should know.

``If I were in charge of the world, all physics students would learn how to do Feynman diagram calculations as college freshmen, while their brains are still fully functioning.'' -John Baez

Feynman diagrams are a fundamental tool for the investigation and explanation of phenomena in quantum field theory. Their origin, however, is purely mathematical: they give a convenient way of organizing and encoding certain important calculations.

In this column we will look at a finite-dimensional calculation that shares many of the formal properties of the calculation of interest.

The mathematics involved is more technical than usual in this series of columns, but it is quite concrete. With the restriction to finite-dimensionality we will see exactly where the ``diagrams'' come into the picture, using no more than fairly elementary procedures from calculus and linear algebra. So any third or fourth-year undergraduate should be able to follow the details, while a broader audience should be able to gain from the examples an accurate picture of the whole procedure.

This column is an attempt to reconstruct the first lecture of Misha Polyak's minicourse Quantum Field Theory and Topology at the ``Graphs and Patterns in Mathematics and Theoretical Physics'' conference in Stony Brook, June 2001.

--Tony Phillips
Stony Brook