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Journal of Mathematical Physics

Densities of covariant observables

U. Cattaneo

Published Apr 1, 1982• 15 citations
Publisher

Abstract

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.

DOI
10.1063/1.525413

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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.
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.
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.
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.
The space POVM H(X) of positive operator-valued probability measures on the Borel sets of a compact (or even locally compact) Hausdorff space X with values in B(H), the algebra of linear operators acting on a d-dimensional Hilbert space H, is studied from the perspectives of classical and nonclassical convexity through a transform Γ that associates any positive operator-valued measure ν with a certain completely positive linear map Γ(ν) of the homogeneous C*-algebra C(X)⊗B(H) into B(H). This association is achieved by using an operator-valued integral in which nonclassical random variables (that is, operator-valued functions) are integrated with respect to positive operator-valued measures and which has the feature that the integral of a random quantum effect is itself a quantum effect. A left inverse Ω for Γ yields an integral representation, along the lines of the classical Riesz representation theorem for linear functionals on C(X), of certain (but not all) unital completely positive linear maps φ:C(X)⊗B(H)→B(H). The extremal and C*-extremal points of POVM H(X) are determined.
We represent quantum observables as normalized positive operator valued measures and consider convex sets of observables which are covariant with respect to a unitary representation of a locally compact Abelian symmetry group G. The value space of such observables is a transitive G-space. We characterize the extreme points of covariant observables and also determine the covariant extreme points of the larger set of all quantum observables. The results are applied to position, position difference, and time observables.
We study a projective unitary representation of the product G=G˜×G, where G is a locally compact Abelian group and G^ is its dual consisting of characters on G. It is proven that the representation is irreducible, which allows us to define a covariant positive operator-valued measure (covariant POVM) generated by orbits of projective unitary representations of G. The quantum tomography associated with the representation is discussed. It is shown that the integration over such a covariant POVM defines a family of contractions which are multiples of unitary operators from the representation. Using this fact, it is proven that the measure is informationally complete. The obtained results are illustrated by optical tomography on groups and by a measure with a density that has a value in the set of coherent states.