Quantum Statistical Field Theory and Combinatorics

We investigate elements of combinatorics from the point of view of the algebra for the canonical commutation relations. We find novel interpretations for Stirling numbers of both type within this context. This allows us to develop compact formulas for vacuum averages and normal orderings of products of Bose fields. Two applications are mentioned: an uniform estimate for Dyson series expansions occurring in quantum open systems; multiple (quantum) stochastic integral decompositions based on a new Möbius inversion formula (theorem 2.7) for sums over partitions.