Feynman diagrams for eigenvalue perturbation theory

Question

I posted this question in MathOverflow but was not lucky with the answers, so wil try here.

Suppose I have a matrix given by a sum

$$A=D+\epsilon B$$

where $D$ is diagonal and $\epsilon$ is small, and I want the eigenvalues of $A$ as power series in $\epsilon$. The first two orders in perturbation theory are well known. Third and higher orders are briefly discussed here. However, the equations become horrible.

I hear that Feynman diagrams are an efficient way to formulate perturbation theory, but I can't find an accessible exposition of this approach. Note that I have in mind the simple matrix setting. I don't want vacuum states, quantum field theory, path integrals, many body, etc. Can someone help?

Answer 1

If you want to find the eigenvalues of a finite dimensional matrix $A$ as a Taylor series on $\epsilon$ there are well known procedures to do that. If your object $A$ is infinite dimensional things become more complicated but can be carried out in principle. (Almost) Everything you want to know on the subject can be found in Kato’s book perturbation theory for linear operators.

The Feynman diagrams are essentially simply a convenient way to write a particular term of the perturbation expansion when your objects are (quantum) field theories. For example, it turns out that, because of the form of the interaction term (your $B$) you can already say that some terms in the perturbation expansion are going to be zero. In general you have a diagrammatic way to write such terms which is useful because it provides an easy way to write down these terms. At the same time it also provides a physical intuition for what these terms do and this is probably even more important.

In a simple matrix setting Feynman diagrams are of no use as they cannot even be defined.