Combinatorial QFT on graphs: first quantization formalism

TL;DR

This paper introduces a combinatorial quantum field theory model on graphs using a first quantization approach, connecting graph mappings with Feynman diagrams and gluing formulas for Green's functions and determinants.

Contribution

It develops a novel combinatorial formalism for QFT on graphs, linking Feynman graph sums with spacetime graph mappings and Atiyah-Segal-like gluing techniques.

Findings

Provides a combinatorial interpretation of Feynman graphs on graphs.

Establishes gluing formulas for Green's functions and Laplacian determinants.

Connects first quantization formalism with topological and geometric graph properties.

Abstract

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.