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Journal of Mathematical Physics

The structure of decoherence functionals for von Neumann quantum histories

J. D. Maitland Wright

Published Oct 1, 1995• 28 citations
Publisher

Abstract

Gell-Mann and Hartle have proposed a significant generalization of quantum theory in which decoherence functionals perform a key role. Isham–Linden–Schreckenberg have given a penetrating analysis of decoherence functionals for L(H), where H is finite dimensional (dimension greater than 2). In this note their conjecture on the significance of boundedness is verified. In particular, it is shown that when d is a bounded decoherence functional associated with a von Neumann algebra A, then, provided A has no direct summand of type I2, d can be represented as the difference between semi-innerproducts on A.

DOI
10.1063/1.531268

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Gell-Mann and Hartle have proposed a significant generalization of quantum theory in which decoherence functionals perform a key role. Isham–Linden–Schreckenberg have given a penetrating analysis of decoherence functionals for L(H), where H is finite dimensional (dimension greater than 2). In this note their conjecture on the significance of boundedness is verified. In particular, it is shown that when d is a bounded decoherence functional associated with a von Neumann algebra A, then, provided A has no direct summand of type I2, d can be represented as the difference between semi-innerproducts on A.
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 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, 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.
A general quantum history theory can be characterized by the space of histories and by the space of decoherence functionals. In this note we consider the situation where the space of histories is given by the lattice of projection operators on an infinite dimensional Hilbert space H. We study operator representations for decoherence functionals on this space of histories. We first give necessary and sufficient conditions for a decoherence functional being representable by a trace class operator on H⊗H, an infinite dimensional analogue of the Isham–Linden–Schreckenberg representation for finite dimensions. Since this excludes many decoherence functionals of physical interest, we then identify the large and physically important class of decoherence functionals which can be represented, canonically, by bounded operators on H⊗H.
The recent suggestion that a temporal form of quantum logic provides the natural mathematical framework within which to discuss the proposal by Gell-Mann and Hartle for a generalized form of quantum theory based on the ideas of histories and decoherence functionals is analyzed and developed herein. Particular stress is placed on properties of the space of decoherence functionals, including one way in which certain global and topological properties of a classical system are reflected in a quantum history theory.