Homogeneous decoherence functionals in standard and history quantum mechanics

TL;DR

This paper analyzes the mathematical properties of decoherence functionals in history quantum mechanics, focusing on standard and homogeneous cases, and establishes their representations and limitations within Hilbert space frameworks.

Contribution

It provides a detailed analysis of the standard and homogeneous decoherence functionals, including their representations and the non-existence of certain extensions, advancing the mathematical understanding of history quantum theories.

Findings

Proves the non-existence of finitely valued extensions for the standard decoherence functional.

Derives a representation of the standard decoherence functional as an unbounded quadratic form.

Establishes an ILS-type representation for the standard decoherence functional.

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.