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 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 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) = tr V⊗V (α ⊗ βX) for some operator X on the tensor product space V ⊗ V.

We discuss the use of histories labelled by a continuous time in the approach to consistenthistories quantum theory in which propositions about the history of the system are represented by projection operators on a Hilbert space. This extends earlier work by two of us [1] where we showed how a continuous time parameter leads to a history algebra that is isomorphic to the canonical algebra of a quantum field theory. We describe how the appropriate representation of the history algebra may be chosen by requiring the existence of projection operators that represent propositions about time average of the energy. We also show that the history description of quantum mechanics contains an operator corresponding to velocity that is quite distinct from the momentum operator. Finally, the discussion is extended to give a preliminary account of quantum field theory in this approach to the consistent histories formalism. *

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The basic ingredients of the 'consistent histories' approach to quantum theory are a space U P of 'history propositions' and a space D of 'decoherence functionals'. In this article we consider such history quantum theories in the case where U P is given by the set of projectors P(V) on some Hilbert space V. We define the notion of a 'physical symmetry of a history quantum theory' (PSHQT) and specify such objects exhaustively with the aid of an analogue of Wigner's theorem. In order to prove this theorem we investigate the structure of D, define the notion of an 'elementary decoherence functional' and show that each decoherence functional can be expanded as a certain combination of these functionals. We call two history quantum theories that are related by a PSHQT 'physically equivalent' and show explicitly, in the case of history quantum mechanics, how this notion is compatible with one that has appeared previously.

The basic ingredients of the 'consistent histories' approach to a generalized quantum theory are 'histories' and decoherence functionals. The main aim of this program is to find and to study the behaviour of consistent sets associated with a particular decoherence functional d. In its recent formulation by Isham [4] it is natural to identify the space U P of propositions about histories with an orthoalgebra or lattice. When U P is given by the lattice of projectors P(V) in some Hilbert space V, consistent sets correspond to certain partitions of the unit operator in V into mutually orthogonal projectors {α 1 , α 2 , . . .}, such that the function d(α, α) is a probability distribution on the boolean algebra generated by {α 1 , α 2 , . . .}. Using the classification theorem for decoherence functionals proven in [6] we show that in the case where V is some separable Hilbert space there exists for each partition of the unit operator into a set of mutually orthogonal projectors, and for any probability distribution p(α) on the corresponding boolean algebra, decoherence functionals d with respect to which this set is consistent and which are such that for the probability functions d(α, α) = p(α) holds.