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arXiv:quant-ph/0212147 · Journal of Mathematical Physics

Positive operator valued measures covariant with respect to an Abelian group

Gianni Cassinelli, Ernesto De Vito, Alessandro Toigo

Published Dec 19, 2003• 16 citations
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Abstract

Given a unitary representation U of an Abelian group G and a subgroup H, we characterize the positive operator valued measures based on the quotient group G/H and covariant with respect to U.
DOI
10.1063/1.1631081

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Positive-operator-valued observables in a complex Hilbert space H based on a topological homogeneous space G/H and G-covariant with respect to a strongly continuous unitary representation on H of a locally compact group G are investigated. The goal is to determine whether they are the weak integrals of some positive-operator-valued densities relative to a G-quasi-invariant measure μ on G/H. To this end, the notion of a kernel Hilbert space is generalized to spaces of equivalence classes with respect to μ of mappings of G/H into a complex Hilbert space K.