The classification of decoherence functionals: An analog of Gleason’s theorem

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Abstract
Gell‐Mann and Hartle have proposed a significant generalization of quantum theory with a scheme whose basic ingredients are ‘‘histories’’ and decoherence functionals. Within this scheme it is natural to identify the space U P of propositions about histories with an orthoalgebra or lattice. This raises the important problem of classifying the decoherence functionals in the case where U P is the lattice of projectors P(V) in some Hilbert space V; in effect, one seeks the history analog of Gleason’s famous theorem in standard quantum theory. In the present article the solution to this problem for the case where V is finite dimensional is presented. In particular, it is shown that every decoherence functional d(α,β), α,β∈P(V) can be written in the form d(α,β)=trV⊗V(α⊗βX) for some operator X on the tensor‐product space V⊗V.
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