On the normal ordering of multi-mode boson operators

Combinatorial aspects of normal ordering for annihilation and creation operators of a multi-mode boson system are discussed. The modes are assumed to be coupled, since otherwise the problem of normal ordering is reduced to the cooresponding problem of the single-mode case. To describe the normal ordering in the multi-mode case, for each mode, a color is introduced, and colored contractions are considered. A depiction for colored contractions via colored linear representations is given. By analogy with the single-mode case, associated colored Stirling numbers are defined as coefficients appearing in the process of normal ordering for powers of the number operators. Several properties of these colored Stirling numbers are discussed.