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Positive Operator Valued Measures Covariant With Respect to an Irreducible Representation

Gianni Cassinelli, Ernesto De Vito, Alessandro Toigo

Published Jan 1, 2003• 28 citations
OpenAlex

Abstract

Abstract. Given a unitary representation U of an Abelian group G and a subgroup H, we characterise the positive operator valued measures based on the quotient group G/H and covariant with respect to U. 1.

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Given a topological group GGG and a unitary representation UUU of GGG, we consider the problem of classifying the positive operator measures which are based on a GGG-homogeneous space XXX and covariant with respect to the representation UUU. We completely solve the problem in two cases: 1) when GGG is abelian; 2) when GGG is not abelian, XXX has compact stbiliser in GGG and the representation UUU is irreducible. We then give an application of the above results to the characterisation of covariant position and momentum observables in quantum mechanics, and to the determination of those covariant position and momentum observables that are jointly measurable.
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.