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Journal of Mathematical Physics

Operator integrals and phase space observables

PEKKA J. LAHTI, Juha-Pekka Pellonpää, Kari Ylinen

Published Apr 1, 1999• 21 citations
Publisher

Abstract

The operator integral ∫f dE of a complex valued measurable function f with respect to a positive operator measure E is considered. If F is a Neumark dilation of E into a projection measure, then the “projected” operator integral pr(∫ fdF) is a restriction of the operator ∫ f dE. Necessary and sufficient conditions for the equality pr(∫f dF)=∫ f dE are obtained. The results are applied to determine the moment operators of the phase space observables generated by the number states.

DOI
10.1063/1.532858

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The operator integral ∫f dE of a complex valued measurable function f with respect to a positive operator measure E is considered. If F is a Neumark dilation of E into a projection measure, then the “projected” operator integral pr(∫ fdF) is a restriction of the operator ∫ f dE. Necessary and sufficient conditions for the equality pr(∫f dF)=∫ f dE are obtained. The results are applied to determine the moment operators of the phase space observables generated by the number states.
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 Γ(fk), 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 Γ 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.
We consider questions related to the following quantization scheme: a classical variable f: Ω → ℝ on a phase space Ω is associated with a unique semispectral measure E f , such that the kth moment operator of E f is required to coincide with the operator integral L(f k , E) of f k with respect to a certain fixed phase space semispectral measure E. Mainly, we take the phase space Ω to be a locally compact unimodular group. In the concrete case where Ω = ℝ2 and E is a translation covariant semispectral measure, we determine explicitly the relevant operators L(f k , E) for certain variables f. In addition, we consider the question under what conditions a positive operator measure is projection valued.
We consider questions related to a quantization scheme in which a classical variable f:Ω\to R on a phase space Ωis associated with a semispectral measure E^f, such that the moment operators of E^f are required to be of the form Γ(f^k), with Γa suitable mapping from the set of classical variables to the set of (not necessarily bounded) operators in some Hilbert space. In particular, we investigate the situation where the map Γis implemented by the operator integral with respect to some fixed positive operator measure. The phase space Ωis first taken to be an abstract measurable space, then a locally compact unimodular group, and finally R^2, where we determine explicitly the relevant operators Γ(f^k) for certain variables f, in the case where the quantization map Γis implemented by a translation covariant positive operator measure. In addition, we consider the question under what conditions a positive operator measure is projection valued.
The theory of operator integrals is used to determine the moment operators of the Cartesian margins of the phase space observables generated by the mixtures of the number states. The moments of the xxx-margin are polynomials of the position operator and those of the yyy-margin are polynomials of the momentum operator.