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.

中文翻译：

---

在多项式代数上定义的线性算子的位置动量分解

我们首先介绍一组换向器关系，涉及由有限的随机变量族生成的，具有所有阶的有限矩的联合量子，半量子和数字算子，并展示如何使用这些换向器来恢复联合量子算子来自半量子运算符。我们证明，在多项式或多项式随机变量的代数上定义的任何线性算子，由具有所有阶的有限矩的有限随机变量族生成，可以唯一地表示为所生成的乘法算子的无穷和通过这些随机变量和偏导数运算符。以该总和的形式，每个乘法运算符放置在每个偏导数运算符的左侧。