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Non-Probabilistic Decoherent Histories and Causal Histories in Everettian Quantum Mechanics

André L. G. Mandolesi

Published Apr 4, 2017• 0 citations
OpenAlex

Abstract

D. Wallace has combined the Decoherent Histories formalism and Everettian Quantum Mechanics, in an attempt to solve the preferred basis problem. This in turn serves as a foundation for his proof of the Born rule. But approximations used in decoherence depend on the probabilistic interpretation of Born weights, resulting in a circular reasoning. We contest his arguments that the approximations are valid even without probabilities, and show that, without them, the combination of those theories leads to unacceptable results. There is a proliferation of ill behaved histories, compromising his proof of the Born rule. We propose the development of a new causal histories formalism, which might improve the situation. Histories of small amplitude are discarded, not on probability grounds, but for suffering too much interference, which blurs causal relations. The remaining histories might still suffer small amounts of interference, leaving open the possibility of experimental verification.

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Everettian Decoherent Histories and Causal Histories

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D. Wallace has tried to use decoherence to solve the preferred basis problem of Everettian Quantum Mechanics, and this solution lays the foundation for his proof of the Born rule. But this is a circular argument, as approximations used in decoherence usually rely on the probabilistic interpretation of the Hilbert space norm. He claims the norm can measure approximations even without probabilities, but this assumption has not been properly justified. Without it, the combination of the Everettian and decoherent histories formalisms leads to strange consequences, such as a proliferation of small amplitude histories with lots of macroscopic quantum jumps. Still, this erratic behavior may provide a way to justify the approximations, in a new histories formalism, in which macroscopic causal relations play a central role. Small histories, suffering too much interference, may lose causality, being thus discarded as invalid. The remaining branches can present some small interference, opening the possibility of experimental verification.

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D. Wallace has combined the Decoherent Histories formalism and Everettian Quantum Mechanics, in an attempt to solve the preferred basis problem. This in turn serves as a foundation for his proof of the Born rule. But approximations used in decoherence depend on the probabilistic interpretation of Born weights, resulting in a circular reasoning. We contest his arguments that the approximations are valid even without probabilities, and show that, without them, the combination of those theories leads to unacceptable results. There is a proliferation of ill behaved histories, compromising his proof of the Born rule. We propose the development of a new causal histories formalism, which might improve the situation. Histories of small amplitude are discarded, not on probability grounds, but for suffering too much interference, which blurs causal relations. The remaining histories might still suffer small amounts of interference, leaving open the possibility of experimental verification.
D. Wallace has tried to use decoherence to solve the preferred basis problem of Everettian Quantum Mechanics, and this solution lays the foundation for his proof of the Born rule. But this is a circular argument, as approximations used in decoherence usually rely on the probabilistic interpretation of the Hilbert space norm. He claims the norm can measure approximations even without probabilities, but this assumption has not been properly justified. Without it, the combination of the Everettian and decoherent histories formalisms leads to strange consequences, such as a proliferation of small amplitude histories with lots of macroscopic quantum jumps. Still, this erratic behavior may provide a way to justify the approximations, in a new histories formalism, in which macroscopic causal relations play a central role. Small histories, suffering too much interference, may lose causality, being thus discarded as invalid. The remaining branches can present some small interference, opening the possibility of experimental verification.
It is often objected that the Everett interpretation of QM cannot make sense of quantum probabilities, in one or both of two ways: either it can't make sense of probability at all, or it can't explain why probability should be governed by the Born rule. David Deutsch has attempted to meet these objections. He argues not only that rational decision under uncertainty makes sense in the Everett interpretation, but also that under reasonable assumptions, the credences of a rational agent in an Everett world should be constrained by the Born rule. David Wallace has developed and defended Deutsch's proposal, and greatly clarified its conceptual basis. In particular, he has stressed its reliance on the distinguishing symmetry of the Everett view, viz., that all possible outcomes of a quantum measurement are treated as equally real. The argument thus tries to make a virtue of what has usually been seen as the main obstacle to making sense of probability in the Everett world. In this note I outline some objections to the Deutsch-Wallace argument, and to related proposals by Hilary Greaves about the epistemology of Everettian QM. (In the latter case, my arguments include an appeal to an Everettian analogue of the Sleeping Beauty problem.) The common thread to these objections is that the symmetry in question remains a very significant obstacle to making sense of probability in the Everett interpretation.
It is often objected that the Everett interpretation of QM cannot make adequate sense of quantum probabilities, in one or both of two senses: either it cannot make sense of probability at all, or cannot explain why probability should be governed by the Born rule. David Deutsch has attempted to meet these objections. He argues not only that rational decision under uncertainty makes sense in the Everett interpretation, and that under reasonable assumptions, the credences of a rational agent in an Everett world should be constrained by the Born rule. David Wallace has recently developed and defended Deutsch's proposal, and greatly clarified its conceptual basis. In this note I outline some concerns about the Deutsch argument, as presented by Wallace, and about related proposals by Hilary Greaves. In particular, I argue that the argument is circular, at a crucial point.
A longstanding issue in the Everettian (Many-Worlds) interpretation is to justify and make sense of the Born rule that underlies the statistical predictions of standard quantum mechanics. The paper offers a reappraisal of Everett’s original account in light of the recent literature on the concept of typicality. It argues that Everett’s derivation of the Born rule is sound and, in a certain sense, even an optimal result, and defends it against the charge of circularity. The conclusion is that Everett’s typicality argument can successfully ground post-factum explanations of Born statistics, while questions remain about the predictive power of the Many-Worlds interpretation.
The emergence of intrinsic probability has long been one of the most important and puzzling problems in quantum mechanics, and the law most directly related to this problem is the Born rule. For a century, there have been many attempts to derive the Born rule as a theorem rather than postulating it. However, existing derivations of the Born rule are each based on different frameworks and have attracted different criticisms. The assumptions from which they start are also highly divergent, and the connections between them have not been sufficiently studied. These possible connections are very likely to be the key to answering questions about the origin of probability in quantum mechanics. This paper focuses on proving the necessity and indispensability of the additivity assumption in the derivation of the Born rule. This supports the view that the Born rule cannot be derived solely from other non-probabilistic quantum or additional postulates. We first prove that additivity cannot be derived from two other commonly used non-probabilistic additional assumptions, non-contextuality and normalization. Then we analyze the crucial role of the additivity assumption in five important existing derivations of the Born rule. These include Gleason's Theorem, Busch's extension of Gleason's Theorem, the Deutsch-Wallace Theorem, Zurek's envariance proof, and the Finkelstein-Hartle Theorem. We show that these derivations either depend heavily on the additivity assumption or lead to obvious loopholes due to the lack of additivity. We also point out some problems arised from the lack of a non-contextuality assumption. Our results provide a novel insight into the important role of additivity assumption in quantum measurement, as well as into the origin of probability in quantum mechanics.