We propose a realistic, spacetime interpretation of quantum theory in which reality constitutes a *single* history obeying a "law of motion" that makes definite, but incomplete, predictions about its behavior. We associate a "quantum measure" |S| to the set S of histories, and point out that |S| fulfills a sum rule generalizing that of classical probability theory. We interpret |S| as a "propensity", making this precise by stating a criterion for |S|=0 to imply "preclusion" (meaning that the true history will not lie in S). The criterion involves triads of correlated events, and in application to electron-electron scattering, for example, it yields definite predictions about the electron trajectories themselves, independently of any measuring devices which might or might not be present. (So we can give an objective account of measurements.) Two unfinished aspects of the interpretation involve *conditonal* preclusion (which apparently requires a notion of coarse-graining for its formulation) and the need to "locate spacetime regions in advance" without the aid of a fixed background metric (which can be achieved in the context of conditional preclusion via a construction which makes sense both in continuum gravity and in the discrete setting of causal set theory).
We propose a realistic, spacetime interpretation of quantum theory in which reality constitutes a single history obeying a "law of motion" that makes definite, but incomplete, predictions about its behavior. We associate a "quantum measure" |S| to the set S of histories, and point out that |S| fulfills a sum rule generalizing that of classical probability theory. We interpret |S| as a "propensity", making this precise by stating a criterion for |S|=0 to imply "preclusion" (meaning that the true history will not lie in S). The criterion involves triads of correlated events, and in application to electron-electron scattering, for example, it yields definite predictions about the electron trajectories themselves, independently of any measuring devices which might or might not be present. (So we can give an objective account of measurements.) Two unfinished aspects of the interpretation involve conditonal preclusion (which apparently requires a notion of coarse-graining for its formulation) and the need to "locate spacetime regions in advance" without the aid of a fixed background metric (which can be achieved in the context of conditional preclusion via a construction which makes sense both in continuum gravity and in the discrete setting of causal set theory).
These are the author's lectures at the 1992 Les Houches Summer School, "Gravitation and Quantizations". They develop a generalized sum-over-histories quantum mechanics for quantum cosmology that does not require either a preferred notion of time or a definition of measurement. The "post-Everett" quantum mechanics of closed systems is reviewed. Generalized quantum theories are defined by three elements (1) the set of fine-grained histories of the closed system which are its most refined possible description, (2) the allowed coarse grainings which are partitions of the fine-grained histories into classes, and (3) a decoherence functional which measures interference between coarse grained histories. Probabilities are assigned to sets of alternative coarse-grained histories that decohere as a consequence of the closed system's dynamics and initial condition. Generalized sum-over histories quantum theories are constructed for non-relativistic quantum mechanics, abelian gauge theories, a single relativistic world line, and for general relativity. For relativity the fine-grained histories are four-metrics and matter fields. Coarse grainings are four-dimensional diffeomorphism invariant partitions of these. The decoherence function is expressed in sum-over-histories form. The quantum mechanics of spacetime is thus expressed in fully spacetime form. The coarse-grainings are most general notion of alternative for quantum theory expressible in spacetime terms. Hamiltonian quantum mechanics of matter fields with its notion of unitarily evolving state on a spacelike surface is recovered as an approximation to this generalized quantum mechanics appropriate for those initial conditions and coarse-grainings such that spacetime geometry
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.
I will propose that the reality to which the quantum formalism implicitly refers is a kind of generalized history, the word history having here the same meaning as in the phrase sum-over-histories. This proposal confers a certain independence on the concept of event, and it modifies the rules of inference concerning events in order to resolve a contradiction between the idea of reality as a single history and the principle that events of zero measure cannot happen (the Kochen-Specker paradox being a classic expression of this contradiction). The so-called measurement problem is then solved if macroscopic events satisfy classical rules of inference, and this can in principle be decided by a calculation. The resulting conception of reality involves neither multiple worlds nor external observers. It is therefore suitable for quantum gravity in general and causal sets in particular. Quantum gravity and quantal reality Why, in our attempts to unify our theories of gravity and the quantum, has progress been so slow? One reason, no doubt, is that it's simply a very hard problem. Another is that we lack clear guidance from experiments or astronomical observations. But I believe that a third thing holding us back is that we haven't learned how to think clearly about the quantum world in itself, without reference to “observers” and other external agents. Because of this we don't really know how to think about the Planckian regime where quantum gravity is expected to be most relevant. We don’t know how to think about the vacuum on small scales, or about the inside of a black hole, or about the early universe. Nor do we have a way to pose questions about relativistic causality in such situations.
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.
Quantum measure theory can be introduced as a histories based reformulation (and generalisation) of Copenhagen quantum mechanics in the image of classical stochastic theories. These classical models lend themselves to a simple interpretation in which a single history (a single element of the sample space) is deemed to be 'real'; we require only that this real history should not be ruled out by the dynamics, the axioms of which ensure that not all histories are precluded. However, applying this interpretation naively to quantum measure theory we can find experimentally realisable systems (notably the Peres-Kochen-Specker system) in which every history is ruled out by the dynamics, challenging us to formulate a deeper realist framework. Our first response is to hold on to our existing interpretative framework and attempt a revision of the dynamics that would reduce quantum measure theory to a classical dynamics. We explore this approach by examining the histories formulation of a stochastic-collapse model on a simple (discrete) null-lattice, concluding that the drawbacks of this approach outweigh the benefits. Our second response is to abandon our classically inspired interpretation in favour of Sorkin's 'co-events', a more general ontology that still allows for strict realism. In this case the 'potentially real' objects of the theory (the 'beables' in Bell's language) are not individual histories but truth valuation maps, or co-events. We develop & evaluate various co-event schemes that have been suggested to date, finally adopting the multiplicative scheme; the current working model of co-event theory and a promising interpretation of quantum measure theory, though still a work in progress. We conclude by exploring the expression of the dynamics & predictions in this new framework.
Ultimately, any explanation of quantum measurement must be extendable to relativistic quantum mechanics (RQM), since many precisely confirmed experimental results follow from quantum field theory (QFT), which is based on RQM. Certainly, the traditional "collapse" postulate for quantum measurement is problematic in a relativistic context, at the very least because, as usually formulated, it violates the relativity of simultaneity. The present paper addresses this with a relativistic model of measurement in which the state of the universe is decomposed into decoherent histories of measurements recorded within it. The approach is essentially Everettian, in the sense that it uses the unmodified, unitary quantum formalism of RQM. But it addresses the difficulty with typical "many worlds" interpretations on how to even define probabilities over different possible ``worlds''. To do this, Zurek's concept of envariance is generalized to the context of relativistic spacetime, giving an objective definition of the probability of any one of the quantum histories, consistent with Born's rule. It is then shown that the statistics of any repeated experiment within the universe also tend to follow the Born rule as the number of repetitions increases. The wave functions that we actually use for such experiments are local reductions of very coarse-grained superpositions of universal eigenstates, and their "collapse" can be re-interpreted as simply an update based on additional incremental knowledge gained from a measurement about the "real" eigenstate of our universe.