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 A is a ∗-algebra describing a quantum system and ω:A→C a state, we focus, in particular, on the interpretation of ω(a) as expectation value for an algebraic observable a=a∗∈A, studying the problem of finding a probability measure reproducing the moments {ω(an)}n∈N. This problem enjoys a close relation with the selfadjointness of the (in general only symmetric) operator πω(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(an)}n∈N does not admit a unique solution. To reduce this ambiguity, we consider the moment problem for the sequences {ωb(an)}n∈N, being b∈A and ωb(⋅):=ω(b∗⋅b). Letting μ(a)ωb be a solution of the moment problem for the sequence {ωb(an)}n∈N, we introduce a consistency relation on the family {μ(a)ωb}b∈A. We prove a 1-1 correspondence between consistent families {μ(a)ωb}b∈A and positive operator-valued measures (POVM) associated with the symmetric operator πω(a). In particular, there exists a unique consistent family of {μ(a)ωb}b∈A if and only if πω(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.