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

Abstract

We address some usually overlooked issues concerning the use of $\ast$-algebras in quantum theory and their physical interpretation. If $\mathfrak{A}$ is a $\ast$-algebra describing a quantum system and $\ddot{I}‰:\mathfrak{A}\to \mathbb{C}$ a state, we focus in particular on the interpretation of $\ddot{I}‰(a)$ as expectation value for an algebraic observable $a=a^{\ast }\in \mathfrak{A}$, studying the problem of finding a probability measure reproducing the moments $\{\ddot{I}‰(a^n){\}}_{n\in \mathbb{N}}$. This problem enjoys a close relation with the self-adjointeness of the (in general only symmetric) operator $\ddot{I}{€}_{\ddot{I}}‰(a)$ in the GNS representation of $\ddot{I}‰$ 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 $\{\ddot{I}‰(a^n){\}}_{n\in \mathbb{N}}$ does not admit a unique solution. To reduce this ambiguity, we consider the moment problem for the sequences $\{\ddot{I}{‰}_b(a^n){\}}_{n\in \mathbb{N}}$, being $b\in \mathfrak{A}$ and $\ddot{I}{‰}_b(\cdot ):=\ddot{I}‰(b^{\ast }\cdot b)$. Letting $\hat{I}{¼}_{\ddot{I}{‰}_b}^{(a)}$ be a solution of the moment problem for the sequence $\{\ddot{I}{‰}_b(a^n){\}}_{n\in \mathbb{N}}$, we introduce a consistency relation on the family $\{\hat{I}{¼}_{\ddot{I}{‰}_b}^{(a)}{\}}_{b\in \mathfrak{A}}$. We prove a 1-1 correspondence between consistent families $\{\hat{I}{¼}_{\ddot{I}{‰}_b}^{(a)}{\}}_{b\in \mathfrak{A}}$ and positive operator-valued measures (POVM) associated with the symmetric operator $\ddot{I}{€}_{\ddot{I}}‰(a)$. In particular there exists a unique consistent family of $\{\hat{I}{¼}_{\ddot{I}{‰}_b}^{(a)}{\}}_{b\in \mathfrak{A}}$ if and only if $\ddot{I}{€}_{\ddot{I}}‰(a)$ is maximally symmetric. This result suggests that a better physical understanding of the notion of observable for general $\ast$-algebras should be based on POVMs rather than projection-valued measure (PVM).

44 pages, no figures, accepted for publication in Letters in Mathematical Physics