The existence of a real linear-space structure on the set of observables of a quantum system -- i.e., the requirement that the linear combination of two generally non-commuting observables $A,B$ is an observable as well -- is a fundamental postulate of the quantum theory yet before introducing any structure of algebra. However, it is by no means clear how to choose the measuring instrument of the composed observable $aA+bB$ ($a,b\in \mathbb{R}$) if such measuring instruments are given for the addends observables $A$ and $B$ when they are incompatible observables. A mathematical version of this dilemma is how to construct the spectral measure of $f(aA+bB)$ out of the spectral measures of $A$ and $B$. We present such a construction with a formula which is valid for generally unbounded selfadjoint operators $A$ and $B$, whose spectral measures may not commute, and a wide class of functions $f: \mathbb{R} \to \mathbb{C}$. We prove that, in the bounded case the Jordan product of $A$ and $B$ can be constructed with the same procedure out of the spectral measures of $A$ and $B$. The formula turns out to have an interesting operational interpretation and, in particular cases, a nice interplay with the theory of Feynman path integration and the Feynman-Kac formula.

中文翻译：

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量子理论和相关构造中两个非交换可观测量之和的运算构造

在量子系统的可观测量集上存在实线性空间结构——即，要求两个通常不可交换的可观测量 $A,B$ 的线性组合也是一个可观测量——是一个基本原理在引入任何代数结构之前，量子理论的假设。但是，如果给加数可观察量$A$($a,b\in \mathbb{R}$) 的测量工具，那么如何选择组合可观察量$aA+bB$($a,b\in\mathbb{R}$)和 $B$ 当它们是不兼容的 observable 时。这个困境的数学版本是如何从 $A$ 和 $B$ 的频谱度量中构建 $f(aA+bB)$ 的频谱度量。我们用一个公式提出这样的构造，该公式对一般无界自伴随算子 $A$ 和 $B$ 有效，其谱测量可能不会交换，以及一大类函数 $f：\mathbb{R} \to \mathbb{C}$。我们证明，在有界情况下，$A$ 和 $B$ 的 Jordan 乘积可以用相同的程序从 $A$ 和 $B$ 的谱测度中构造出来。结果证明，该公式有一个有趣的操作解释，在特定情况下，它与费曼路径积分理论和费曼-卡克公式有很好的相互作用。