On extending the Quantum Measure

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.

23 pages, 2 figures

References in corpus (9)

• Quantum Mechanics as Quantum Measure Theory
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• An exercise in "anhomomorphic logic"
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Cited by in corpus (13)

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• Schwinger's Picture of Quantum Mechanics III: The statistical interpretation
• A Dynamics for Discrete Quantum Gravity
• A manifestly covariant framework for causal set dynamics
• Two-Site Quantum Random Walk
• Lorentzian Spectral Geometry with Causal Sets
• An Argument for Strong Positivity of the Decoherence Functional
• Ontological models for quantum theory as functors
• Observables for cyclic causal set cosmologies
• In-in correlators and scattering amplitudes on a causal set
• The coevent formulation of quantum theory
• Models for Discrete Quantum Gravity
• Timelike boundary and corner terms in the causal set action