Position–momentum decomposition of linear operators defined on algebras of polynomials

Abstract

We present first a set of commutator relationships involving the joint quantum, semi-quantum, and number operators generated by a finite family of random variables, having finite moments of all orders, and show how these commutators can be used to recover the joint quantum operators from the semi-quantum operators. We show that any linear operator defined on an algebra of polynomials or the polynomial random variables, generated by a finite family of random variables, having finite moments of all orders, can be written uniquely as an infinite sum of compositions of the multiplication operators, generated by these random variables, and the partial derivative operators. In the terms of this sum, each multiplication operator is placed to the left side of each partial derivative operator. We provide many examples concerning the decomposition of some classic operators.