In the quantum measure approach, quantum theory is viewed as a generalisation of classical stochastic theory rather than of classical deterministic theory. An observable or "be-able" is thus a quantum measurable set in an appropriate event algebra. Thus, the observables in quantum cosmology are covariant quantum measurable sets. In a quantum system with a strongly positive quantum measure or decoherence functional one can construct a vector valued measure whose domain is the algebra of events or physical questions. Using results in the literature, this allows us to specify when a vector valued measure on finite time events can be extended to a measure over infinite time, or equivalently, covariant events. For the class of complex percolation dynamics we show that there is a generic obstruction to this process and that covariant observables can be constructed only for a very restricted class of dynamics.
In the quantum measure approach, quantum theory is viewed as a generalisation of classical stochastic theory rather than of classical deterministic theory. An observable or "be-able" is thus a quantum measurable set in an appropriate event algebra. Thus, the observables in quantum cosmology are covariant quantum measurable sets. In a quantum system with a strongly positive quantum measure or decoherence functional one can construct a vector valued measure whose domain is the algebra of events or physical questions. Using results in the literature, this allows us to specify when a vector valued measure on finite time events can be extended to a measure over infinite time, or equivalently, covariant events. For the class of complex percolation dynamics we show that there is a generic obstruction to this process and that covariant observables can be constructed only for a very restricted class of dynamics.
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.
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.
Abstract The classical sequential growth model for causal sets provides a template for the dynamics in the deep quantum regime. This growth dynamics is intrinsically temporal and causal, with each new element being added to the existing causal set without disturbing its past. In the quantum version, the probability measure on the event algebra is replaced by a quantum measure, which is Hilbert space valued. Because of the temporality of the growth process, in this approach, covariant events (or observables) are measurable only if the quantum measure extends to the associated sigma algebra of events. This is not always guaranteed. In this work we find a criterion for extension (and thence covariance) in complex sequential growth models for causal sets. We find a large family of models in which the measure extends, so that all covariant events/observables are measurable.
In this thesis we will explore the development of a realist quantum theory based on the decoherence functional using the co-event interpretation of Quantum Measure Theory. The Sum-Over-Histories theory of quantum mechanics will provide the bedding for a Hilbert-space-free stochastic-like theory that can accommodate spacetime-like objects, and can therefore be applied to quantum gravity and cosmology, as well as give an alternative perspective on quantum phenomena. The primitive objects of the theory are histories, which give different accounts of a system’s evolution, and the decoherence functional, which sums the quantum interference between these histories. Quantum Measure Theory and Generalised Quantum Mechanics (a theory close to Decoherent Histories) then give alternative interpretations of the decoherence functional’s relation to reality. In these theories, the decoherence functional is mathematically constrained in analogue to probability measures. However, one of the conditions, called weak positivity, can be lost under composition of isolated systems. We will extend this composition argument to take the case for a stronger condition of strong positivity for decoherence functionals. The bulk of the report will then focus on the co-event interpretation of Quantum Measure Theory, where co-events give full accounts of which events do or do not occur for a given system. The quantum nature of reality is expressed through the breaking of classical logic within these co-event descriptions. We will focus on evolving co-event schemes, which dynamically construct co-events to describe the reality of an evolving system in tandem with its progression. The evolving co-event schemes will be shown to reproduce classical logic when they are applied to classical systems. Moreover, similar to classical stochastic theory, these schemes will be shown to be invariant under the inclusion or exclusion of non-interfering histories. We will also explore a number of outstanding problems for these schemes, and will propose some potential modifications.
We sketch a quantum-mechanical framework for the universe as a whole. \nWithin that framework we propose a program for describing the ultimate \norigin in quantum cosmology of the "quasiclassical domain" of familiar experience and for characterizing the process of measurement. Predictions \nin quantum mechanics are made from probabilities for sets of alternative \nhistories. Probabilities (approximately obeying the rules of probability theory) can be assigned only to sets of histories that approximately decohere. \nDecoherence is defined and the mechanism of decoherence is reviewed. Decoherence requires a sufficiently coarse-grained description of alternative \nhistories of the universe. A quasiclassical domain consists of a branching set \nof alternative decohering histories, described by a coarse graining that is, in \nan appropriate sense, maximally refined consistent with decoherence, with \nindividual branches that exhibit a high level of classical correlation in time. \nWe pose the problem of making these notions precise and quantitative. A \nquasiclassical domain is emergent in the universe as a consequence of the \ninitial condition and the action function of the elementary particles. It is \nan important question whether all the quasiclassical domains are roughly \nequivalent or whether there are various essentially inequivalent ones. A \nmeasurement is a correlation with variables in a quasiclassical domain. An "observer" (or information gathering and utilizing system) is a complex \nadaptive system that has evolved to exploit the relative predictability of a \nquasiclassical domain, or rather a set of such domains among which it cannot discriminate because of its own very coarse graining. We suggest that \nresolution of many of the problems of interpretation presented by quantum \nmechanics is to be accomplished, not by further scrutiny of the subject as it \napplies to reproducible laboratory situations, but rather by an examination \nof alternative histories of the universe, stemming from its initial condition, \nand a study of the problem of quasiclassical domains.
The physical--observation dual-axis (PODA) structure treats physical and observation configurations as distinct arguments of joint dynamics. We study the analytic conditions under which this structure admits rigorous path integrals and finite complex measures for sequential quantum histories. On compact smooth joint geometry, a specified covariant Hamiltonian has a Feynman--Kac representation. A uniform complex-time product estimate permits simultaneous removal of Abel damping and time slicing at arbitrary relative speeds. For fixed finite bounded-readout protocols, class operators converge strongly and the trace-class branch-pair operators converge in trace norm. Finite instruments with complete Kraus families also yield trace-norm convergence of their record states. Energy-form regularity gives quantitative error bounds. A completed cylinder-test space carries a consistent, strongly positive history functional. The stronger finite-measure question is controlled by covariant transport: calibrated rank-one readouts on a finite-dimensional fibre satisfy the sharp estimate ∥μγ∥TV≤e2cγ, where cγ is the off-diagonal covariant variation. In the specified common-action class, conserved energy and smooth, globally labelled rank-one readouts imply this regularity when the kinetic geometry is complete and the effective potential is bounded below, or when the chosen energy confines the trajectory to a compact set. In infinite dimension, finitely many readout sectors allow unbounded internal energies and bounded transitions. The construction extends to countable outcomes when the operator of block norms is bounded on ℓ2; nonatomic readouts admit a double-Dyson construction when an absolute integral kernel is bounded on L2. In these Dyson constructions, pure preparations with rank-one discrete readouts or scalar continuous fibres have exact variation ∥eTBv∥22. Whole-sector and spatial-domain cutoffs converge in variation, whereas nonatomic readout quantization converges weakly and remains separated in variation norm. Mixed curvature survives in the covariant constructions, and invariant quantum embeddings recover physical-sector histories. Unrestricted basis switching, a bounded-Hamiltonian countable example and free-particle position histories establish the limits of finite-measure existence; exact physical recovery transfers the last obstruction to the joint theory. Analytic two-axis models exhibit the resulting dynamics and verify the distinct convergence statements. A proposed two-mode trapped-ion realization links the magnetic oscillator's transport phase to independent spectral and driven-response tests. Exact finite-time Gaussian evolution shows that nearly complete wave-packet return can coexist with a substantial phase correction. The protocol therefore infers the coupling from finite-time interference and predicts the other observables without refitting, with calibration and uncertainty requirements explicit. These are model predictions, not measured data. Each construction is tied to its stated geometric, dynamical and readout hypotheses.