A novel approach to perturbative calculations for a large class of interacting boson theories. (English) Zbl 1380.81192
Summary: We present a method of calculating the interacting \(S\)-matrix to an arbitrary perturbative order for a large class of boson interaction Lagrangians. The method takes advantage of a previously unexplored link between the \(n\)-point Green’s function and a certain system of linear Diophantine equations. By finding all nonnegative solutions of the system, the task of perturbatively expanding an interacting \(S\)-matrix becomes elementary for any number of interacting fields, to an arbitrary perturbative order (irrespective of whether it makes physical sense) and for a large class of scalar boson theories. The method does not rely on the position-based Feynman diagrams and promises to be extended to many perturbative models typically studied in quantum field theory. Aside from interaction field calculations we showcase our approach by expanding a pair of Unruh-DeWitt detectors coupled to Minkowski vacuum to an arbitrary perturbative order in the coupling constant. We also link our result to Hafnian as introduced by Caianiello and present a method to list all \((2 n - 1)!!\) perfect matchings of a complete graph on \(2n\) vertices.
MSC:
| 81T10 | Model quantum field theories |
| 81U20 | \(S\)-matrix theory, etc. in quantum theory |
| 81T15 | Perturbative methods of renormalization applied to problems in quantum field theory |
| 81T18 | Feynman diagrams |
| 35J08 | Green’s functions for elliptic equations |
| 11D04 | Linear Diophantine equations |
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