Decoherence Functionals for von Neumann Quantum Histories: Boundedness and Countable Additivity

Abstract:

Gell–Mann and Hartle have proposed a significant generalisation of quantum theory in which decoherence functionals perform a key role. Verifying a conjecture of Isham–Linden–Schreckenberg, the author analysed the structure of bounded, finitely additive, decoherence functionals for a general von Neumann algebra A (where A has no Type I₂ direct summand). Isham et al. had already given a penetrating analysis for the situation where A is finite dimensional. The assumption of countable additivity for a decoherence functional may seem more plausible, physically, than that of boundedness. The results of this note are obtained much more generally but, when specialised to L(H), the algebra of all bounded linear operators on a separable Hilbert space H, give:

Let d be a countably additive decoherence functional defined on all pairs of projections in L(H). If H is infinite dimensional then d must be bounded. By contrast, when H is finite dimensional, unbounded (countably additive) decoherence functionals always exist for L(A).