Consistent histories and operational quantum theory

In this work a generalization of the consistent histories approach to quantum mechanics is presented. We first critically review the consistent histories approach to nonrelativistic quantum mechanics in a mathematically rigorous way and give some general comments about it. We investigate to what extent the consistent histories scheme is compatible with the results of the operational formulation of quantum mechanics. According to the operational approach, nonrelativistic quantum mechanics is most generally formulated in terms of effects, states, and operations. We formulate a generalized consistent histories theory using the concepts and the terminology which have proven useful in the operational formulation of quantum mechanics. The logical rule of the logical interpretation of quantum mechanics is generalized to the present context. The algebraic structure of the generalized theory is studied in detail.     

Timeless Configuration Space and the Emergence of Classical Behavior

The inherent difficulty in talking about quantum decoherence in the context of quantum cosmology is that decoherence requires subsystems, and cosmology is the study of the whole Universe. Consistent histories gave a possible answer to this conundrum, by phrasing decoherence as loss of interference between alternative histories of closed systems. When one can apply Boolean logic to a set of histories, it is deemed ‘consistent’. However, the vast majority of the sets of histories that are merely consistent are blatantly nonclassical in other respects, and further constraints than just consistency need to be invoked. In this paper, I attempt to give an alternative answer to the issues faced by consistent histories, by exploring a timeless interpretation of quantum mechanics of closed systems. This is done solely in terms of path integrals in non-relativistic, timeless, configuration space. What prompts a fresh look at such foundational problems in this context is the advent of multiple gravitational models in which Lorentz symmetry is not fundamental, but only emergent. And what allows this approach to overcome previous barriers to a timeless, conditional probabilities interpretation of quantum mechanics is the new notion of records—made possible by an inherent asymmetry of configuration space. I outline and explore consequences of this approach for foundational issues of quantum mechanics, such as the natural emergence of the Born rule, conservation of probabilities, and the Sleeping Beauty paradox.     

A Bayesian account of quantum histories

We investigate whether quantum history theories can be consistent with Bayesian reasoning and whether such an analysis helps clarify the interpretation of such theories. First, we summarise and extend recent work categorising two different approaches to formalising multi-time measurements in quantum theory. The standard approach consists of describing an ordered series of measurements in terms of history propositions with non-additive ‘probabilities.’ The non-standard approach consists of defining multi-time measurements to consist of sets of exclusive and exhaustive history propositions and recovering the single-time exclusivity of results when discussing single-time history propositions. We analyse whether such history propositions can be consistent with Bayes’ rule. We show that certain class of histories are given a natural Bayesian interpretation, namely, the linearly positive histories originally introduced by Goldstein and Page. Thus, we argue that this gives a certain amount of interpretational clarity to the non-standard approach. We also attempt a justification of our analysis using Cox’s axioms of probability theory.     

The Three-Box “Paradox” and Other Reasons to Reject the Counterfactual Usage of the ABL Rule

An apparent paradox proposed by Aharonov and Vaidman in which a single particle can be found with certainty in two (or more) boxes is analyzed by way of a simple thought experiment. It is found that the apparent paradox arises from an invalid counterfactual usage of the Aharonov-Bergmann-Lebowitz (ABL) rule and effectively attributes conflicting properties not to the same particle but no different particles. A connection is made between the present analysis and the consistent histories formulation of Griffiths. Finally, a critique is given of some recent counterarguments by Vaidman against the rejection of the counterfactual usage of the ABL rule.     

On the consistent histories approach to quantum mechanics

We review the consistent histories formulations of quantum mechanics developed by Griffiths, Omnès, and Gell-Mann and Hartle, and describe the classification of consistent sets. We illustrate some general features of consistent sets by a few simple lemmas and examples. We consider various interpretations of the formalism, and examine the new problems which arise in reconstructing the past and predicting the future. It is shown that Omnès' characterization of true statements—statements which can be deduced unconditionally in his interpretation—is incorrect. We examine critically Gell-Mann and Hartle's interpretation of the formalism, and in particular their discussions of communication, prediction, and retrodiction, and conclude that their explanation of the apparent persistence of quasiclassicality relies on assumptions about an as-yetunknown theory of experience. Our overall conclusion is that the consistent histories approach illustrates the need to supplement quantum mechanics by some selection principle in order to produce a fundamental theory capable of unconditional predictions.     

Contrary Inferences in Consistent Histories and a Set Selection Criterion

The best developed formulation of closed system quantum theory that handles multiple-time statements, is the consistent (or decoherent) histories approach. The most important weaknesses of the approach is that it gives rise to many different consistent sets, and it has been argued that a complete interpretation should be accompanied with a natural mechanism leading to a (possibly) unique preferred consistent set. The existence of multiple consistent sets becomes more problematic because it allows the existence of contrary inferences [1]. We analyse the conceptual difficulties that arise from the existence of multiple consistent sets and provide a suggestion for a natural set selection criterion. This criterion does not lead to a unique physical consistent set, however it evades the existence of consistent sets with contrary inferences. The criterion is based on the concept of preclusion and the requirement that probability one propositions and their inferences should be non-contextual. The allowed consistent sets turn-out to be compatible with coevents which are the ontology of an alternative, histories based, formulation [2–4].     

Incompatible and Contradictory Retrodictions in the History Approach to Quantum Mechanics

We illustrate two simple spin examples which show that in the consistent histories approach to quantum mechanics one can retrodict with certainty incompatible or contradictory propositions corresponding to non-orthogonal or, respectively, orthogonal projections.     

Primordial fluctuations from inflation: a consistent histories approach

We show how the quantum-to-classical transition of the cosmological fluctuations produced during an inflationary stage can be described using the consistent histories approach. We identify the corresponding histories in the limit of infinite squeezing. To take the decaying mode into account, we propose an extension to coarse-grained histories.     

Spontaneous Collapse Models on a Lattice

We present spontaneous collapse models of field theories on a 1+1 null lattice, in which the causal structure of the lattice plays a central role. Issues such as “locality,” “nonlocality,” and superluminal signaling are addressed in the context of the models which have the virtue of extreme simplicity. The formalism of the models is related to that of the consistent histories approach to quantum mechanics.     

Glafka 2004: Spacetime Topology from the Tomographic Histories Approach I: Non-Relativistic Case

The tomographic histories approach is presented. As an inverse problem, we recover in an operational way the effective topology of the extended configuration space of a system. This means that from a series of experiments we get a set of points corresponding to events. The difference between effective and actual topology is drawn. We deduce the topology of the extended configuration space of a non-relativistic system, using certain concepts from the consistent histories approach to Quantum Mechanics, such as the notion of a record. A few remarks about the case of a relativistic system, preparing the ground for a forthcoming paper sequel to this, are made in the end.     

Glafka 2004: Spacetime Topology from the Tomographic Histories Approach

The tomographic histories approach is presented. As an inverse problem, we recover in an operational way the effective topology of the extended configuration space of a system. This means that from a series of experiments we get a set of points corresponding to events. The difference between effective and actual topology is drawn. We deduce the topology of the extended configuration space of a non-relativistic system, using certain concepts from the consistent histories approach to Quantum Mechanics, such as the notion of a record. A few remarks about the case of a relativistic system, preparing the ground for a forthcoming paper sequel to this, are made in the end.     

Preferred Consistent History Sets

The theme of this paper is the multiplicity ofthe consistent sets of histories emerging in theconsistent-histories approach to quantum mechanics. Wepropose one criterion for choosing preferred families among them: that the physically realizablequasiclassical domain ought to be one corresponding toclassical histories. We examine the way classicalmechanics arises as a particular window, and theimportant role played by the canonical group and theHamiltonian. We finally discuss a possible implicationof our selection criterion: that only a class of Hilbertspace operators corresponds to physical quantities, and hence the full nondistributivity of thelattice of quantum propositions is redundant.     

Decoherent Histories and Realism

We reconsider the decoherent histories approach to quantum mechanics and analyze some problems related to its interpretation which we believe have not been adequately clarified by its proponents. We put forward some assumptions which, in our opinion, are necessary for a realistic interpretation of the probabilities that the formalism attaches to decoherent histories. We prove that such assumptions, unless one limits the set of the decoherent families which can be taken into account, lead to a logical contradiction. The line of reasoning we follow is conceptually different from other arguments which have been presented and which have been rejected by the supporters of the decoherent histories approach. The conclusion is that the decoherent histories approach, to be considered as an interesting realistic alternative to the orthodox interpretation of quantum mechanics, requires the identification of a mathematically precise criterion to characterize an appropriate set of decoherent families which does not give rise to any problem.     

Topos theory and consistent histories: The internal logic of the set of all consistent sets

A major problem in the consistent-histories approach to quantum theory is contending with the potentially large number of consistent sets of history propositions. One possibility is to find a scheme in which a unique set is selected in some way. However, in this paper the alternative approach is considered in which all consistent sets are kept, leading to a type of ‘many-world-views’ picture of the quantum theory. It is shown that a natural way of handling this situation is to employ the theory of varying sets (presheafs) on the spaceB of all nontrivial Boolean subalgebras of the orthoalgebraUP of history propositions. This approach automatically includes the feature whereby probabilistic predictions are meaningful only in the context of a consistent set of history propositions. More strikingly, it leads to a picture in which the ‘truth values’ or ‘semantic values’ of such contextual predictions are not just two-valued (i.e., true and false) but instead lie in a larger logical algebra—a Heyting algebra—whose structure is determined by the spaceB of Boolean subalgebras ofUP. This topos-theoretic structure thereby gives a coherent mathematical framework in which to understand the internal logic of the many-world-views picture that arises naturally in the approach to quantum theory based on the ideas of consistent histories.     

Extended Application of the Velocity Superselection Rule

We have applied the velocity superselection rule proposed by Georgi, bgur and Wise to some transition matrix element calculations in an extended way. We have found that by the approach f_B, is about 0.6-0.8f_K which is consistent with the potential model calculations.     

Relativistic invariant approach to the parity violation in np→dγ capture

It is shown that the consistent relativistic invariant approach to the parity violation in np→dγ capture of thermal neutrons leads to results which differ essentially from those of the nonrelativistic approach. In particular the isospin selection rule established earlier for the photon circular polarization is charged due to the contribution of “relativistic” parity violating components of the deuteron and np-scattering wave function.