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arXiv:1704.01173

Everettian Decoherent Histories and Causal Histories

André L. G. Mandolesi

Published Apr 4, 2017• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.1704.01173

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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.
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.
The purpose of this thesis is to examine different conceptions of decoherence and their significance within interpretations of quantum mechanics. I set out three different conceptions of decoherence found in the literature and examine the relations between them. I argue that only the weakest of these conceptions is empirically well supported, and that the other two rely on claims about the structure of the histories (robust patterns within the wavefunction) which we occupy which require justification. \n \nI also examine the ways in which conceptions of decoherence are used to solve aspects of the quantum measurement problem and support modern interpretations of quantum mechanics. I focus particularly on Wallace's Everettian interpretation of quantum mechanics. I argue that while decoherence is generally successful in supporting this interpretation in a variety of ways, the very strong conception of decoherence on which he relies is itself difficult to justify. \n \nI consider a variety of possible approaches to justifying the use of this strong conception of decoherence and argue that many of them are either unconvincing, or rely on controversial cosmological claims. Finally, I suggest that the best way to justify the use of this strong conception of decoherence is by appealing directly to its indispensability to an otherwise very attractive interpretation of quantum mechanics.
Abstract In Everettian quantum mechanics, justifications for the Born rule appeal to self-locating uncertainty or decision theory. Such justifications have focused exclusively on a pure-state Everettian multiverse, represented by a wavefunction. Recent works in quantum foundations suggest that it is viable to consider a mixed-state Everettian multiverse, represented by a (mixed-state) density matrix. Here, we discuss the conceptual foundations for decoherence and branching in a mixed-state multiverse, and extend arguments for the Born rule to this setting. This extended framework provides a unification of ‘classical’ and ‘quantum’ probabilities, and additional theoretical benefits, for the Everettian picture.
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.
A defence is offered of a version of the branch-counting rule in the Everett interpretation (otherwise known as many worlds interpretation) of quantum mechanics that both depends on the state and is continuous in the norm topology on Hilbert space. The well-known branch-counting rule, for realistic models of measurements, in which branches are defined by decoherence theory, fails this test. The new rule hinges on the use of decoherence theory in defining branching structure, and specifically decoherent histories theory. On this basis ratios of branch numbers are defined, free of any convention. They agree with the Born rule and deliver a notion of objective probability similar to naive frequentism, save that the frequencies of outcomes are not confined to a single world at different times, but spread over worlds at a single time. Nor is it ad hoc : it is recognizably akin to the combinatorial approach to thermodynamic probability, as introduced by Boltzmann in 1879. It is identical to the procedure followed by Planck, Bose, Einstein and Dirac in defining the equilibrium distribution of the Bose–Einstein gas. It also connects in a simple way with the decision-theory approach to quantum probability.