Completeness of Decoherence Functionals

Abstract

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 it is natural to identify the space $\UP$ of propositions about histories with an orthoalgebra or lattice. When $\UP$ is given by the lattice of projectors $\PV$ in some Hilbert space $\V$, consistent sets correspond to certain partitions of the unit operator in $\V$ into mutually orthogonal projectors $\{\a_1,\a_2,\ldots\}$, such that the function $d(\a,\a)$ is a probability distribution on the boolean algebra generated by $\{\a_1,\a_2,\ldots\}$. Using the classification theorem for decoherence functionals, proven previously, 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 $p(\a)$ 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 $d(\a,\a)=p(\a)$ holds.

11 pages, Latex; some comments and a reference added. version to appear in JMP 1995

References in corpus (3)

• Quantum Logic and the Histories Approach to Quantum Theory
• Quantum temporal logic and decoherence functionals in the histories approach to generalised quantum theory
• The Classification of Decoherence Functionals: An Analogue of Gleason's Theorem

Cited by in corpus (5)

• Symmetries of Decoherence Functionals
• Symmetry and History Quantum Theory: An analogue of Wigner's Theorem
• On Tracial Operator Representations of Quantum Decoherence Functionals
• Temporal Quantum Theory
• Symmetries and conservation laws in histories-based theories