We address some usually overlooked issues concerning the use of $*$-algebras in quantum theory and their physical interpretation. If $\mathfrak{A}$ is a $*$-algebra describing a quantum system and $\omega\colon\mathfrak{A}\to\mathbb{C}$ a state, we focus in particular on the interpretation of $\omega(a)$ as expectation value for an algebraic observable $a=a^*\in\mathfrak{A}$, studying the problem of finding a probability measure reproducing the moments $\{\omega(a^n)\}_{n\in\mathbb{N}}$. This problem enjoys a close relation with the self-adjointeness of the (in general only symmetric) operator $\pi_\omega(a)$ in the GNS representation of $\omega$ and thus it has important consequences for the interpretation of $a$ as an observable. We provide physical examples (also from QFT) where the moment problem for $\{\omega(a^n)\}_{n\in\mathbb{N}}$ does not admit a unique solution. To reduce this ambiguity, we consider the moment problem for the sequences $\{\omega_b(a^n)\}_{n\in\mathbb{N}}$, being $b\in\mathfrak{A}$ and $\omega_b(\cdot):=\omega(b^*\cdot b)$. Letting $\mu_{\omega_b}^{(a)}$ be a solution of the moment problem for the sequence $\{\omega_b(a^n)\}_{n\in\mathbb{N}}$, we introduce a consistency relation on the family $\{\mu_{\omega_{b}}^{(a)}\}_{b\in\mathfrak{A}}$. We prove a 1-1 correspondence between consistent families $\{\mu_{\omega_{b}}^{(a)}\}_{b\in\mathfrak{A}}$ and positive operator-valued measures (POVM) associated with the symmetric operator $\pi_\omega(a)$. In particular there exists a unique consistent family of $\{\mu_{\omega_{b}}^{(a)}\}_{b\in\mathfrak{A}}$ if and only if $\pi_\omega(a)$ is maximally symmetric. This result suggests that a better physical understanding of the notion of observable for general $*$-algebras should be based on POVMs rather than projection-valued measure (PVM).

中文翻译：

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可观察的概念和 $$*$$*-代数的矩问题及其 GNS 表示

我们解决了一些通常被忽视的问题，这些问题涉及在量子理论中使用 $*$-代数及其物理解释。如果 $\mathfrak{A}$ 是描述量子系统的 $*$-代数，而 $\omega\colon\mathfrak{A}\to\mathbb{C}$ 是一个状态，我们特别关注 $ 的解释\omega(a)$ 作为代数可观测值 $a=a^*\in\mathfrak{A}$ 的期望值，研究寻找再现矩的概率度量的问题 $\{\omega(a^n)\ }_{n\in\mathbb{N}}$。这个问题与 $\omega$ 的 GNS 表示中的（通常只是对称的）算子 $\pi_\omega(a)$ 的自邻接关系密切，因此它对 $a 的解释具有重要意义$ 作为可观察的。我们提供了物理示例（也来自 QFT），其中 $\{\omega(a^n)\}_{n\in\mathbb{N}}$ 的矩问题不承认唯一解决方案。为了减少这种歧义，我们考虑序列 $\{\omega_b(a^n)\}_{n\in\mathbb{N}}$ 的矩问题，即 $b\in\mathfrak{A}$ 和$\omega_b(\cdot):=\omega(b^*\cdot b)$。令 $\mu_{\omega_b}^{(a)}$ 是序列 $\{\omega_b(a^n)\}_{n\in\mathbb{N}}$ 的矩问题的解，我们在 $\{\mu_{\omega_{b}}^{(a)}\}_{b\in\mathfrak{A}}$ 族上引入了一致性关系。我们证明了一致族 $\{\mu_{\omega_{b}}^{(a)}\}_{b\in\mathfrak{A}}$ 和正运算符值度量 (POVM) 之间的 1-1 对应关系) 与对称运算符 $\pi_\omega(a)$ 相关联。特别地，存在一个唯一一致的 $\{\mu_{\omega_{b}}^{(a)}\}_{b\in\mathfrak{A}}$ 当且仅当 $\pi_\omega (a)$ 是最大对称的。该结果表明，对一般 $*$-代数的可观察概念的更好物理理解应该基于 POVM，而不是投影值测度 (PVM)。