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arXiv:quant-ph/9506034 · Physical Review A

Approximate and exact consistency of histories

Jim McElwaine

Published Apr 1, 1996• 12 citations
arXivHTMLPDF

Abstract

The consistent histories formalism is discussed using path-projected states. These are used to analyze various criteria for approximate consistency. The connection between the Dowker-Halliwell criterion and sphere packing problems is shown and used to prove several bounds on the violation of probability sum rules. The quantum Zeno effect is also analyzed within the consistent histories formalism and used to demonstrate some of the difficulties involved in discussing approximate consistency. The complications associated with null histories and infinite sets are briefly discussed. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.53.2021

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The consistent histories formalism is discussed using path-projected states. These are used to analyze various criteria for approximate consistency. The connection between the Dowker-Halliwell criterion and sphere packing problems is shown and used to prove several bounds on the violation of probability sum rules. The quantum Zeno effect is also analyzed within the consistent histories formalism and used to demonstrate some of the difficulties involved in discussing approximate consistency. The complications associated with null histories and infinite sets are briefly discussed. \textcopyright{} 1996 The American Physical Society.
This dissertation investigates questions arising in the consistent histories formulation of the quantum mechanics of closed systems. Various criteria for approximate consistency are analysed. The connection between the Dowker-Halliwell criterion and sphere packing problems is shown and used to prove several new bounds on the violation of probability sum rules. The quantum Zeno effect is also analysed within the consistent histories formalism and used to demonstrate some of the difficulties involved in discussing approximate consistency. The complications associated with null histories and infinite sets are briefly discussed. The possibility of using the properties of the Schmidt decomposition to define an algorithm which selects a single, physically natural, consistent set for pure initial density matrices is investigated. The problems that arise are explained, and different possible algorithms discussed. Their properties are analysed with the aid of simple models. A set of computer programs is described which apply the algorithms to more complicated examples. Another algorithm is proposed that selects the consistent set (formed using Schmidt projections) with the highest Shannon information. This is applied to a simple model and shown to produce physically sensible histories. The theory is capable of unconditional probabilistic prediction for closed quantum systems, and is strong enough to be falsifiable. Ideas on applying the theory to more complicated examples are discussed.
The Consistent Histories (CH) formalism attempts to construct a quantum framework which can be used without the need to introduce observers external to the studied system. The prime motivation in mind is the application of the formalism to the universe as a whole. In order to achieve this, CH maintains that a formulation of quantum mechanics should allow for the assignment of probabilities to alternative histories of a system. Therefore, it provides an observer-independent criterion to decide which sets of histories can be given probabilities and states rules to determine them. The framework establishes that each realm, that is, each set of histories to which probabilities can be assigned, provides a valid quantum-mechanical account of a system. Furthermore, the version of CH first presented in [1, 2] proposes an "evolutionary" explanation of our existence in the universe and of our preference for quasiclassical descriptions of nature. The present work critically evaluates claims to the effect that the formalism offered in [1, 2] solves many interpretational problems in quantum mechanics. In particular, it is pointed out that the interpretation of the proposed framework leaves vague two crucial points, namely, whether realms are fixed or chosen and the link between measurements and histories. The claim of this work is that by doing so, CH overlooks the main interpretational problems of quantum mechanics. Furthermore, we challenge the evolutionary explanation offered and we critically examine the proposed notion of a realm-dependent reality.
In a recent paper Griffiths claims that the consistent histories interpretation of quantum mechanics gives rise to results that contradict those obtained from the Bohm interpretation. This is in spite of the fact that both claim to provide a realist interpretation of the formalism without the need to add any new mathematical content and both always produce exactly the same probability predictions of the outcome of experiments. In constrasting the differences Griffiths argues that the consistent histories interpretation provides a more physically reasonable account of quantum phenomena. We examine this claim and show that the consistent histories approach is not without its difficulties.
A system of quantum reasoning for a closed system is developed by treating nonrelativistic quantum mechanics as a stochastic theory. The sample space corresponds to a decomposition, as a sum of orthogonal projectors, of the identity operator on a Hilbert space of histories. Provided a consistency condition is satisfied, the corresponding Boolean algebra of histories, called a framework, can be assigned probabilities in the usual way, and within a single framework quantum reasoning is identical to ordinary probabilistic reasoning. A refinement rule, which allows a probability distribution to be extended from one framework to a larger (refined) framework, incorporates the dynamical laws of quantum theory. Two or more frameworks which are incompatible because they possess no common refinement cannot be simultaneously employed to describe a single physical system. Logical reasoning is a special case of probabilistic reasoning in which (conditional) probabilities are 1 (true) or 0 (false). As probabilities are only meaningful relative to some framework, the same is true of the truth or falsity of a quantum description. The formalism is illustrated using simple examples, and the physical considerations which determine the choice of a framework are discussed. \textcopyright{} 1996 The American Physical Society.
The consistent histories approach was initially proposed by Robert Griffiths in 1984. Roland Omnes then developed a more mathematically rigorous version based on the rules of classical probability and logic. In the consistent histories terminology, each projective decomposition defines a framework, which is, in essence, a context for discussing the system. As Griffith describes it: In quantum mechanics, each Boolean event algebra constitutes what is in effect a “language” out of which one can construct a quantum description of some physical system. A crucial and distinctive feature of the consistent histories approach is the single framework rule , which provides guidance for quantum reasoning. From the perspective of consistent histories, the emergence of classical physics from the quantum level is linked to the suppression of interference, so that the normal rules of probability apply.
We present a decomposition of the general quantum-mechanical evolution operator that corresponds to the path decomposition expansion, and interpret its constituents in terms of the quantum Zeno effect (QZE). This decomposition is applied to a finite-dimensional example and to the case of a free particle in the real line, where the possibility of boundary conditions more general than those hitherto considered in the literature is shown. We reinterpret the assignment of consistent probabilities to different regions of space-time in terms of the QZE. The comparison of the approach of consistent histories to the problem of time of arrival with the solution provided by the probability distribution of Kijowski shows the strength of the latter point of view.