A novel approach to perturbative calculations for a large class of interacting boson theories

摘要

We present a method of calculating the interacting S-matrix to an arbitraryperturbative order for a large class of boson interaction Lagrangians. Themethod takes advantage of a previously unexplored link between the $n$-pointGreen's function and a certain system of linear Diophantine equations. Byfinding all nonnegative solutions of the system, the task of perturbativelyexpanding an interacting $S$-matrix becomes elementary for any number ofinteracting fields, to an arbitrary perturbative order (irrespective of whetherit makes physical sense) and for a large class of scalar boson theories. Themethod does not rely on the position-based Feynman diagrams and promises to beextended to many perturbative models typically studied in quantum field theory.Aside from interaction field calculations we showcase our approach by expandinga pair of Unruh-DeWitt detectors coupled to Minkowski vacuum to an arbitraryperturbative order in the coupling constant. We also link our result to Hafnianas introduced by Caianiello and present a method to list all (2n-1)!! perfectmatchings of a complete graph on 2n vertices.