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

Mar 18, 2019
44 pages
Published in:
  • Lett.Math.Phys. 110 (2020) 7, 1711-1758
  • Published: Feb 26, 2020
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Abstract: (Springer)
We address some usually overlooked issues concerning the use of ∗*-algebras in quantum theory and their physical interpretation. If A{\mathfrak {A}} is a ∗*-algebra describing a quantum system and ω:A→C\omega : {\mathfrak {A}}\rightarrow {\mathbb {C}} a state, we focus, in particular, on the interpretation of ω(a)\omega (a) as expectation value for an algebraic observable a=a∗∈Aa=a^*\in {\mathfrak {A}}, studying the problem of finding a probability measure reproducing the moments {ω(an)}n∈N\{\omega (a^n)\}_{n\in {\mathbb {N}}}. This problem enjoys a close relation with the selfadjointness of the (in general only symmetric) operator πω(a)\pi _\omega (a) in the GNS representation of ω\omega 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\{\omega (a^n)\}_{n\in {\mathbb {N}}} does not admit a unique solution. To reduce this ambiguity, we consider the moment problem for the sequences {ωb(an)}n∈N\{\omega _b(a^n)\}_{n\in {\mathbb {N}}}, being b∈Ab\in {\mathfrak {A}} and ωb(⋅):=ω(b∗⋅b)\omega _b(\cdot ):=\omega (b^*\cdot b). Letting μωb(a)\mu _{\omega _b}^{(a)} be a solution of the moment problem for the sequence {ωb(an)}n∈N\{\omega _b(a^n)\}_{n\in {\mathbb {N}}}, we introduce a consistency relation on the family {μωb(a)}b∈A\{\mu _{\omega _{b}}^{(a)}\}_{b\in {\mathfrak {A}}}. We prove a 1-1 correspondence between consistent families {μωb(a)}b∈A\{\mu _{\omega _{b}}^{(a)}\}_{b\in {\mathfrak {A}}} and positive operator-valued measures (POVM) associated with the symmetric operator πω(a)\pi _\omega (a). In particular, there exists a unique consistent family of {μωb(a)}b∈A\{\mu _{\omega _{b}}^{(a)}\}_{b\in {\mathfrak {A}}} if and only if πω(a)\pi _\omega (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.
Note:
  • 44 pages, no figures, accepted for publication in Letters in Mathematical Physics
  • Algebraic quantum field theory
  • Moment problem
  • Star-algebras
  • GNS construction
  • Selfadjointness
  • moment
  • perturbation theory
  • star algebra
  • algebra: C*
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