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arXiv:1306.0127

Consistent Histories as Valuations

Yousef Ghazi-Tabatabai

Published Jun 1, 2013• 0 citations
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Abstract

Sorkin's coevent interpretation shifts the focus of quantum logic from the structure of a propositional lattice to the nature of truth valuations thereon. We apply this shift in emphasis to a simple formulation of the consistent histories approach, expressing it in terms of truth valuations which are brought together in a logical framework. We see that these consistent histories valuations are related to Sorkin's multiplicative coevents, and that they can be naturally described by an adaptation of Isham's early topos-theoretic approach.

DOI
10.48550/arxiv.1306.0127

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Sorkin's coevent interpretation shifts the focus of quantum logic from the structure of a propositional lattice to the nature of truth valuations thereon. We apply this shift in emphasis to a simple formulation of the consistent histories approach, expressing it in terms of truth valuations which are brought together in a logical framework. We see that these consistent histories valuations are related to Sorkin's multiplicative coevents, and that they can be naturally described by an adaptation of Isham's early topos-theoretic approach.
Sorkin's coevents can be thought of as the `beables' of a quantum histories theory; in this paper we study the 'logical' implications of taking this claim at face value, constructing a propositional lattice for the space of coevents applicable to a given histories theory and comparing it to the more traditional propositional lattice of events. In particular we focus on multiplicative coevents, and find that the precise nature of their anhomomorphism leads to a particularly simple relationship between the two propositional lattices. Finally, we notice that our constructions contain elements intuitively similar to topos nations of truth values and use this to suggest a means of applying Isham's topos theoretic constructions to multiplicative coevents.
Topos theory has been suggested by Doring and Isham as an alternative mathematical structure with which to formulate physical theories. In particular it has been used to reformulate standard quantum mechanics in such a way that a novel type of logic is used to represent propositions. In this paper we extend this formulation to include temporally-ordered collections of propositions as opposed to single-time propositions. That is to say, we have developed a quantum history formalism in the language of topos theory where truth values can be assigned to temporal propositions. We analyse the extent to which such truth values can be derived from the truth values of the constituent, single-time propositions.
Topos theory has been suggested by Döring and Isham as an alternative mathematical structure with which to formulate physical theories. In particular, it has been used to reformulate standard quantum mechanics in such a way that a novel type of logic is used to represent propositions. In this paper we extend this formulation to include temporally ordered collections of propositions as opposed to single-time propositions. That is to say, we have developed a quantum history formalism in the language of topos theory where truth values can be assigned to temporal propositions. We analyze the extent to which such truth values can be derived from the truth values of the constituent, single-time propositions.
Isham's topos-theoretic perspective on the logic of the consistent-histories theory is extended in two ways. First, the presheaves of consistent sets of history propositions in the topos proposed by Isham are endowed with a Vietoris-type of topology and subsequently they are sheafified with respect to it. The category resulting from this sheafification procedure is the topos of sheaves of sets varying continuously over the Vietoris-topologized base poset category of Boolean subalgebras of the universal orthoalgebra of quantum history propositions. The second extension of Isham's topos consists in endowing the stalks of the aforementioned sheaves, which were originally inhabited by structureless sets, with further algebraic structure, that also enjoys a quantum causal interpretation, so as to arrive at the topos of consistent-histories of quantum causal sets. The resulting quantum causal histories topos is compared with Markopoulou's quantum causal histories, with Mallios and this author's finitary spacetime sheaves of quantum causal sets, as well as with Butterfield and Isham's topos perspective on the Kochen-Specker theorem of quantum logic.
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.