Homogeneous decoherence functionals in standard and history quantum mechanics
arXiv:math-ph/9807024 · doi:10.1007/s002200050645
Abstract
General history quantum theories are quantum theories without a globally defined notion of time. Decoherence functionals represent the states in the history approach and are defined as certain bivariate complex-valued functionals on the space of all histories. However, in practical situations -- for instance in the history formulation of standard quantum mechanics -- there often is a global time direction and the homogeneous decoherence functionals are specified by their values on the subspace of homogeneous histories. In this work we study the analytic properties of (i) the standard decoherence functional in the history version of standard quantum mechanics and (ii) homogeneous decoherence functionals in general history theories. We restrict ourselves to the situation where the space of histories is given by the lattice of projections on some Hilbert space H. Among other things we prove the non-existence of a finitely valued extension for the standard decoherence functional to the space of all histories, derive a representation for the standard decoherence functional as an unbounded quadratic form with a natural representation on a Hilbert space and prove the existence of an Isham-Linden-Schreckenberg (ILS) type representation for the standard decoherence functional.
19 pages, LaTeX2e, uses times, amssymb, amsmath, babel, mathptm, and latexsym packages
References in corpus (9)
- Classical Equations for Quantum Systems
- Quantum temporal logic and decoherence functionals in the histories approach to generalised quantum theory
- Continuous Histories and the History Group in Generalised Quantum Theory
- Topos Theory and Consistent Histories: The Internal Logic of the Set of all Consistent Sets
- The Classification of Decoherence Functionals: An Analogue of Gleason's Theorem
- Alternative Decohering Histories in Quantum Mechanics
- Symmetries of Decoherence Functionals
- On the Consistent Effect Histories Approach to Quantum Mechanics
- Consistent Histories and Operational Quantum Physics