#feynmangraph — Public Fediverse posts

The #paperOfTheDay is "Exponents for the excluded volume problem as derived by the Wilson Method" from 1972. This one-page letter makes one elementary, but far-reaching observation: The O(N)-invariant scalar #quantumFieldTheory , which had a few months before been treated by Wilson in terms of dimensional regularization, has a special interpretation when N=0. Namely, one assigns to every #FeynmanGraph a "symmetry factor", which is a polynomial in N. The coefficient of N^k in this polynomial counts how many ways there are to decompose the 4-valent vertices of the graph such that one obtains exactly k cycles. If one sets N=0, all that remains is the constant term: It counts the ways of decomposing the graph without forming any cycle.
One is interested in the statistical behaviour of non-crossing paths on a lattice, called a self-excluding walk. This can be studied with the methods of statistical #physics . One introduces a Boltzmann-type weight exp(-n*p), where p is a parameter (analogous to the inverse temperature or the Planck constant), and n is the length of a self-excluding walk. Let N_n be the number of different such walks (for a fixed size of the lattice, or counted relative to the lattice size), then the sum of N_n*exp(-n*p) is analogous to a partition function, or path integral. Hence, it can be analyzed perturbatively with Feynman integrals, namely those mentioned above of the O(N) theory at N=0. This way, one obtains, for example, the critical exponents.
https://www.sciencedirect.com/science/article/pii/0375960172901491

I recently discovered an excellent #math article by Krajewski and Martinetti that I had overlooked so far: https://arxiv.org/abs/0806.4309
Basically, renormalization of #Feynmangraph s in #qft is organized in terms of rooted trees, and so are solutions to differential equations, concatenation of differential operators, numerical integration schemes, and the Hopf algebra of power series. It is intuitively clear that all these things must be closely related, but I wasn't aware that there is this article where the relations are actually spelled out in detail. They also include a derivation of Wigner's semicircle law for Gaussian random matrices in the rooted-tree formalism, which is something I didn't think about at all. Learned another unexpected connection 😀 .