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 ( a n ) } n ∈ N . This problem enjoys a close relation with the selfadjointness 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 ∈ N does not admit a unique solution. To reduce this ambiguity, we consider the moment problem for the sequences { ω b ( a n ) } n ∈ N , being b ∈ A and ω b ( · ) : = ω ( b ∗ · b ) . Letting μ ω b ( a ) be a solution of the moment problem for the sequence { ω b ( a n ) } n ∈ N , we introduce a consistency relation on the family { μ ω b ( a ) } b ∈ A . We prove a 1-1 correspondence between consistent families { μ ω b ( a ) } b ∈ 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 ∈ 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.