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The notion of observable and the moment problem for *-algebras and their GNS representations

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 $ω\colon\mathfrak{A}\to\mathbb{C}$ a state, we focus in particular on the interpretation of $ω(a)$ as expectation value for an algebraic observable $a=a^*\in\mathfrak{A}$, studying the problem of finding a probability measure reproducing the moments $\{ω(a^n)\}_{n\in\mathbb{N}}$. This problem enjoys a close relation with the self-adjointeness of the (in general only symmetric) operator $π_ω(a)$ in the GNS representation of $ω$ 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 $\{ω(a^n)\}_{n\in\mathbb{N}}$ does not admit a unique solution. To reduce this ambiguity, we consider the moment problem for the sequences $\{ω_b(a^n)\}_{n\in\mathbb{N}}$, being $b\in\mathfrak{A}$ and $ω_b(\cdot):=ω(b^*\cdot b)$. Letting $μ_{ω_b}^{(a)}$ be a solution of the moment problem for the sequence $\{ω_b(a^n)\}_{n\in\mathbb{N}}$, we introduce a consistency relation on the family $\{μ_{ω_{b}}^{(a)}\}_{b\in\mathfrak{A}}$. We prove a 1-1 correspondence between consistent families $\{μ_{ω_{b}}^{(a)}\}_{b\in\mathfrak{A}}$ and positive operator-valued measures (POVM) associated with the symmetric operator $π_ω(a)$. In particular there exists a unique consistent family of $\{μ_{ω_{b}}^{(a)}\}_{b\in\mathfrak{A}}$ if and only if $π_ω(a)$ 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).