On extending the Quantum Measure

TL;DR

This paper investigates the conditions under which quantum measures and decoherence functionals can be extended from algebraic structures to larger sigma algebras, highlighting cases where extension fails and discussing physical implications.

Contribution

It establishes a link between the extendability of decoherence functionals and associated vector measures, providing criteria and examples of non-extendable quantum systems.

Findings

Decoherence functional extension is equivalent to vector measure extension.

Examples show certain quantum systems' decoherence functionals do not extend.

Extension conditions may be unphysically restrictive.

Abstract

We point out that a quantum system with a strongly positive quantum measure or decoherence functional gives rise to a vector valued measure whose domain is the algebra of events or physical questions. This gives an immediate handle on the question of the extension of the decoherence functional to the sigma algebra generated by this algebra of events. It is on the latter that the physical transition amplitudes directly give the decoherence functional. Since the full sigma algebra contains physically interesting questions, like the return question, extending the decoherence functional to these more general questions is important. We show that the decoherence functional, and hence the quantum measure, extends if and only if the associated vector measure does. We give two examples of quantum systems whose decoherence functionals do not extend: one is a unitary system with finitely many states, and the other is a quantum sequential growth model for causal sets. These examples fail to extend in the formal mathematical sense and we speculate on whether the conditions for extension are unphysically strong.