A product formula and combinatorial field theory

TL;DR

This paper introduces a novel product formula for boson operators that simplifies the normally ordering problem, leading to a graphical combinatorial approach with applications to physical Hamiltonians.

Contribution

It develops a new product formula for formal power series involving boson operators, extending Taylor expansion techniques to combinatorial field theory.

Findings

Derived a double exponential formula for boson operator ordering

Established a graphical method for generating functions of combinatorial sequences

Applied the approach to specific physical Hamiltonians

Abstract

We treat the problem of normally ordering expressions involving the standard boson operators a, a* where [a,a*]=1. We show that a simple product formula for formal power series - essentially an extension of the Taylor expansion - leads to a double exponential formula which enables a powerful graphical description of the generating functions of the combinatorial sequences associated with such functions - in essence, a combinatorial field theory. We apply these techniques to some examples related to specific physical Hamiltonians.