TY - JOUR

T1 - Phase space quantization and the operator moment problem

AU - Kiukas, J.

AU - Lahti, P.

AU - Ylinen, K.

N1 - Copyright:
Copyright 2006 Elsevier B.V., All rights reserved.

PY - 2006/7/12

Y1 - 2006/7/12

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=33746781659&partnerID=8YFLogxK

U2 - 10.1063/1.2211931

DO - 10.1063/1.2211931

M3 - Article

AN - SCOPUS:33746781659

SN - 0022-2488

VL - 47

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

IS - 7

M1 - 072104

ER -