Noncrossing Normal Ordering for Functions of Bosons

Normally ordered forms of functions of boson operators are important in many contexts mainly concerning quantum field theory and quantum optics. Beginning with the seminal work of Katriel [Lett. Nuovo Cimento 10(13):565–567, 1974], in the last few years, normally ordered forms have been shown to have a rich combinatorial structure [see P. B lasiak, quant-ph/0507206]. In this paper, we apply the linear representation of noncrossing partitions to contractions of normally ordered forms. In this way, we define the notion of noncrossing normal ordering. This appear to be a natural mathematical concept and it results indeed to be linked with well-known combinatorial objects. We explicitely give the noncrossing normally ordered form of the functions (ar(a†)s)n) and (a + (a†)s)n, plus various special cases. We establish bijections between noncrossing contractions of these functions, k-ary trees and sets of lattice paths. Physical applications of the noncrossing normal ordering are desiderata. PACS numbers: 02.10.Ox