Combinatorial QFT on graphs: First quantization formalism
Iván A. Contreras, Santosh Kandel, Pavel Mnëv, Konstantin Wernli · Annales de l’Institut Henri Poincaré D Combinatorics Physics and their Interactions · 2024
We study a combinatorial model of the quantum scalar field with polynomial potential on a graph. In the first quantization formalism, the value of a Feynman graph is given by a sum over maps from the Feynman graph to the spacetime graph (mapping edges to paths). This picture interacts naturally with Atiyah–Segal-like cutting-gluing of spacetime graphs. In particular, one has combinatorial counterparts of the known gluing formulae for Green’s functions and (zeta-regularized) determinants of Laplacians.