Phase space quantization and the operator moment problem

J. Kiukas*, P. Lahti, K. Ylinen

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

15 Citations (Scopus)

Abstract

We consider questions related to a quantization scheme in which a classical variable f:Ω→ ℝ on a phase space Ω is associated with a (preferably unique) semispectral measure Ef, such that the moment operators of Ef are required to be of the form Λ(f k), with Γ a suitable mapping from the set of classical variables to the set of (not necessarily bounded) operators in the Hilbert space of the quantum system. In particular, we investigate the situation where the map Γ is implemented by the operator integral with respect to some fixed positive operator measure. The phase space Ω is first taken to be an abstract measurable space, then a locally compact unimodular group, and finally ℝ2, where we determine explicitly the relevant operators Γ(fk) for certain variables f, in the case where the quantization map F is implemented by a translation covariant positive operator measure. In addition, we consider the question under what conditions a positive operator measure is projection valued.

Original languageEnglish
Article number072104
JournalJournal of Mathematical Physics
Volume47
Issue number7
DOIs
Publication statusPublished - 12 Jul 2006
Externally publishedYes

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