# Eigenvalue perturbation theory via Feynman diagrams

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 a 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, Dirac equation, etc. Can someone help?

• If $D$ has multiple eigenvalues, the power series will involve fractional exponents (Puiseux series). By the way, mathematicians rarely go beyond the second order (first and second derivatives at $\epsilon=0$). – Denis Serre Oct 17 '17 at 13:54
• @RodrigodeAzevedo these are diagrams for computing gaussian integrals. I want diagrams for computing eigenvalue perturbation theory. Are they the same? Because this is definitely not clear to me. – thedude Oct 17 '17 at 23:54

I think the simplest way is to use the very simple and very useful resolvant formula $$(A+B)^{-1}=A^{-1}-A^{-1}B(A+B)^{-1}.$$ For perturbative theory, we just iterate this formula $$(A+B)^{-1}=A^{-1}-A^{-1}BA^{-1}+A^{-1}BA^{-1}B(A+B)^{-1}$$ and obtain a power series in $B$. $$(A+B)^{-1}=A^{-1}-A^{-1}BA^{-1}+A^{-1}BA^{-1}BA^{-1}-A^{-1}BA^{-1}BA^{-1}BA^{-1} +\cdots$$ With your notation it gives $$A^{-1}=D^{-1}-\epsilon D^{-1}BD^{-1}+\epsilon^2 D^{-1}BD^{-1}BD^{-1}-\epsilon^3 D^{-1}BD^{-1}BD^{-1}BD^{-1} +\cdots$$ For the eigenvalues there is then the Cauchy formula $$z_0 = \frac{1}{2i\pi}\oint \frac{z}{z-z_0} dz$$ which for matrices gives
$$\lambda_0 = \frac{1}{2i\pi}\oint \lambda Tr(A-\lambda)^{-1}d\lambda$$ where the integral is a closed loop around an eigenvalue of $D$. And we obtain the perturbative expansion: $$\lambda_0 =\frac{1}{2i\pi} \oint \lambda Tr(D-\lambda)^{-1} d\lambda \\-\epsilon \frac{1}{2i\pi} \oint \lambda Tr (D-\lambda)^{-1}B(D-\lambda)^{-1} \\+\epsilon^2 \frac{1}{2i\pi} \oint \lambda Tr (D-\lambda)^{-1}B(D-\lambda)^{-1}B(D-\lambda)^{-1}\\-\epsilon^3 \cdots$$

Now take a loop sufficiently close to the initial eigenvalue of $D$ and it will give what you wanted.

• Thanks. But where are the Feynman diagrams? – thedude Oct 17 '17 at 16:53
• @thedude You can see a translation into pictorial language at the beginning of chapter 11 of "Many-body quantum theory in condensed matter physics" by Henrik Bruus and Karsten Flensberg phys.lsu.edu/~jarrell/COURSES/ADV_SOLID_HTML/Other_online_texts/… and probably many other similar textbooks. – j.c. Oct 17 '17 at 18:06
• In the "for matrices gives..." equation, is it the reciprocal of the trace or the trace of the inverse? – Ian Oct 18 '17 at 0:56

Feynman diagrams work really well when considering Green's functions which encapsulate information about eigenvectors and eigenvalues but in an indirect way. If you want these objects themselves, then you need an explicit and therefore necessarily combinatorial/diagrammatic form of the Rayleigh-Schrörindger perturbation series which, in general, is more complicated than the Feynman diagram expansions for Green's functions. Such a formalism is developed in the article "The Rayleigh-Schrödinger perturbation series of quasi-degenerate systems" by Brouder, Duchamp, Patras and Toth. A similar philosophy is followed in the recent work by Imbrie on Anderson localization.

• Thanks. And what reference would you recommend for Feynman diagram expansions of Green's functions? – thedude Oct 17 '17 at 15:54
• Basically any book on quantum field theory or many-body quantum systems should cover that. Try this book by Zinn-Justin global.oup.com/academic/product/… – Abdelmalek Abdesselam Oct 17 '17 at 16:01
• Thanks again, but I would prefer something that involves only linear algebra, no quantum field theory or many-body quantum systems. This should be possible, no? – thedude Oct 17 '17 at 16:33
• Not really. The main feature of the Feynman diagrammatic formalism is that it is insensitive to the dimension of the underlying vector space $V$. The diagrams are a way of encoding index contractions of tensors that live in various tensor products of $V$ and its dual. In QFT, $V=L^2(\mathbb{R}^d)$. In your case $V=\mathbb{R}^n$ if your matrices are $n\times n$. The same diagrammatic series will appear in both cases. It is just the numerical evaluation of (diagram indexed) terms of the series which will differ... – Abdelmalek Abdesselam Oct 17 '17 at 17:18
• Yes. You can use Feynman diagrams without knowing anything about QFT in the linear/multilinear algebra context on finite dimensional spaces. You can have a look at these papers of mine for example: arxiv.org/abs/math/0212121 arxiv.org/abs/math/0208173 arxiv.org/abs/math/0208174 arxiv.org/abs/0904.1734 – Abdelmalek Abdesselam Oct 17 '17 at 18:02