2019/03/18 by Nicolò Drago, Valter Moretti · 1 voice
Computer Science · Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Quantum Information and Cryptography #Quantum Mechanics and Applications #hep-th #math-ph #math.OA
paper · pdf · doi:10.1007/s11005-020-01277-x
arxiv published 2019/03/18 · openalex created_date 2019/03/22 · arxiv updated 2020/02/17 · openalex publication_date 2020/02/26 · openalex updated_date 2026/08/01
We address some usually overlooked issues concerning the use of *-algebras in quantum theory and their physical interpretation. If \mathfrakA is a *-algebra describing a quantum system and ω\colon\mathfrakA→ℂ a state, we focus in particular on the interpretation of ω(a) as expectation value for an algebraic observable a=a^*∈\mathfrakA, studying the problem of finding a probability measure reproducing the moments \ω(an)\n∈ℕ. This problem enjoys a close relation with the self-adjointeness 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∈\mathfrakA and ωb(⋅):=ω(b^*⋅ b). Letting μωb(a) be a solution of the moment problem for the sequence \ωb(an)\n∈ℕ, we introduce a consistency relation on the family \μ_ωb(a)\_b∈\mathfrakA. We prove a 1-1 correspondence between consistent families \μ_ωb(a)\_b∈\mathfrakA and positive operator-valued measures (POVM) associated with the symmetric operator πω(a). In particular there exists a unique consistent family of \μ_ωb(a)\_b∈\mathfrakA 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 (PVM).