The Classification of Decoherence Functionals: An Analogue of Gleason’s Theorem

Gell-Mann and Hartle have proposed a significant generalisation of quantum theory with a scheme whose basic ingredients are ‘histories’ and decoherence functionals. Within this scheme it is natural to identify the space UP of propositions about histories with an orthoalgebra or lattice. This raises the important problem of classifying the decoherence functionals in the case where UP is the lattice of projectors P(V) in some Hilbert space V; in effect we seek the history analogue of Gleason’s famous theorem in standard quantum theory. In the present paper we present the solution to this problem for the case where V is finite-dimensional. In particular, we show 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.