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arXiv · 1011.1694

Hilbert Space Representations of Decoherence Functionals and Quantum Measures

Stan Gudder
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Abstract

We show that any decoherence functional $D$ can be represented by a spanning vector-valued measure on a complex Hilbert space. Moreover, this representation is unique up to an isomorphism when the system is finite. We consider the natural map $U$ from the history Hilbert space $K$ to the standard Hilbert space $H$ of the usual quantum formulation. We show that $U$ is an isomorphism from $K$ onto a closed subspace of $H$ and that $U$ is an isomorphism from $K$ onto $H$ if and only if the representation is spanning. We then apply this work to show that a quantum measure has a Hilbert space representation if and only if it is strongly positive. We also discuss classical decoherence functionals, operator-valued measures and quantum operator measures.

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Stan Gudder. 2010-11-08. Hilbert Space Representations of Decoherence Functionals and Quantum Measures. https://doi.org/10.2478/s12175-012-0074-1

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