Deformed Bosons: Combinatorics of Normal Ordering

[A,N ] = A, [A, N ] = −A†, [A,A] = [N + 1]− [N ]. (1) In the above A and A are annihilation and creation operators, respectively, while the number operator N counts particles. It is defined in the Fock basis as N |n〉 = n|n〉 and commutes with AA. Because of that in any representation of (1) N can be written in the form AA = [N ], where [N ] denotes an arbitrary function of N , usually called the ”box” function. For general considerations we do not assume any realization of the number operator N and we treat it as an independent element of the algebra. Moreover, we do not assume any particular form of the ”box” function [N ]. Special cases, like so(3) or so(2, 1) algebras, will provide examples that show how such a general approach simplifies if an algebra and its realization are chosen. In this note we give the solution to the problem of the normal ordering of a monomial (AA) in deformed annihilation and creation operators. A classical result due to J. Katriel [2] is that for canonical bosons a and a, i.e., for the Heisenberg-Weyl algebra, we have