The Quantum Mechanics Canonically Associated to Free Probability II: The Normal and Inverse Normal Order Problem

In the first part of the present paper, we have proved that, in order to find an explicit expression for the action, on the [Formula: see text]-orthogonal polynomials, of the 1-parameter unitary groups [Formula: see text] and [Formula: see text], the solution of the inverse normal order problem on the quantum algebra canonically associated to the classical semi-circle random variable is required. In this paper we solve this problem. The solution is obtained in two steps. First we determine the explicit form of normal order in the more general framework of [Formula: see text]-mode-type interacting Fock spaces. Then this result is applied to solve the inverse normal order problem in the semi-circle case.