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arXiv:1803.08762 · Foundations of Physics

Analysis of Wallace’s Proof of the Born Rule in Everettian Quantum Mechanics II: Concepts and Axioms

André L. G. Mandolesi

Published Nov 21, 2018• 8 citations
arXivHTMLPDF
DOI
10.1007/s10701-018-0226-4

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Everettian quantum mechanics (EQM) results in ‘multiple, emergent, branching quasi-classical realities’ (Wallace [2012]). The possible outcomes of measurement as per ‘orthodox’ quantum mechanics are, in EQM, all instantiated. Given this metaphysics, Everettians face the ‘probability problem’—how to make sense of probabilities and recover the Born rule. To solve the probability problem, Wallace, following Deutsch ([1999]), has derived a quantum representation theorem. I argue that Wallace’s solution to the probability problem is unsuccessful, as follows. First, I examine one of the axioms of rationality used to derive the theorem, ‘branching indifference’ (BI). I argue that Wallace is not successful in showing that BI is rational. While I think it is correct to put the burden of proof on Wallace to motivate BI as an axiom of rationality, it does not follow from his failing to do so that BI is not rational. Thus, second, I show that there is an alternative strategy for setting one’s credences in the face of branching which is rational and which violates BI. This is ‘branch counting’ (BC). Wallace is aware of BC and has proffered various arguments against it. However, third, I argue that Wallace’s arguments against BC are unpersuasive. I conclude that the probability problem in EQM persists. 1 Introduction 2 Branching Indifference 2.1 The positive argument for branching indifference 2.2 The negative argument for branching indifference 2.3 Branching indifference section summary 3 Branch Counting 3.1 Branch counting is rational under the subjective-uncertainty viewpoint 3.2 Branch counting is rational under the objective-determinism viewpoint 3.3 Branch counting section summary 4 Number of Branches 4.1 Veracity of the framework 4.2 No such thing as the number of branches 4.2.1 The number of branches is indeterminate 4.2.2 The number of branches is indeterminable 4.3 Rationality and weight 5 Conclusion

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One World Versus Many: The Inadequacy of Everettian Accounts of Evolution, Probability, and Scientific Confirmation

Adrian Kent

Abstract There is a compelling intellectual case for exploring whether purely unitary quantum theory defines a sensible and scientifically adequate theory, as Everett originally proposed. Many different and incompatible attempts to define a coherent Everettian quantum theory have been made over the past fifty years. However, no known version of the theory (unadorned by extra ad hoc postulates) can account for the appearance of probabilities and explain why the theory it was meant to replace — Copenhagen quantum theory — appears to be confirmed, or more generally why our evolutionary history appears to be Born-rule typical. This chapter reviews some ingenious and interesting recent attempts in this direction by Wallace, Greaves–Myrvold, and others, and explains why they don't work. An account of one-world randomness, which appears scientifically satisfactory, and has no many-worlds analogue, is proposed. A fundamental obstacle to confirming many-worlds theories is illustrated by considering some toy many-worlds models. These models show that branch weights can exist without having any role in either rational decision-making or theory confirmation, and also that the latter two roles are logically separate. Wallace's proposed decision theoretic axioms for rational agents in a multiverse and claimed derivation of the Born rule are examined. It is argued that Wallace's strategy of axiomatizing a mathematically precise decision theory within a fuzzy Everettian quasiclassical ontology is incoherent. Moreover, Wallace's axioms are not constitutive of rationality either in Everettian quantum theory or in theories in which branchings and branch weights are precisely defined. In both cases, there exist coherent rational strategies that violate some of the axioms.

arXiv:0905.0624•Jun 24, 2010•116 citations
Everettian quantum mechanics (EQM) results in ‘multiple, emergent, branching quasi-classical realities’ (Wallace [2012]). The possible outcomes of measurement as per ‘orthodox’ quantum mechanics are, in EQM, all instantiated. Given this metaphysics, Everettians face the ‘probability problem’—how to make sense of probabilities and recover the Born rule. To solve the probability problem, Wallace, following Deutsch ([1999]), has derived a quantum representation theorem. I argue that Wallace’s solution to the probability problem is unsuccessful, as follows. First, I examine one of the axioms of rationality used to derive the theorem, ‘branching indifference’ (BI). I argue that Wallace is not successful in showing that BI is rational. While I think it is correct to put the burden of proof on Wallace to motivate BI as an axiom of rationality, it does not follow from his failing to do so that BI is not rational. Thus, second, I show that there is an alternative strategy for setting one’s credences in the face of branching which is rational and which violates BI. This is ‘branch counting’ (BC). Wallace is aware of BC and has proffered various arguments against it. However, third, I argue that Wallace’s arguments against BC are unpersuasive. I conclude that the probability problem in EQM persists. 1 Introduction 2 Branching Indifference 2.1 The positive argument for branching indifference 2.2 The negative argument for branching indifference 2.3 Branching indifference section summary 3 Branch Counting 3.1 Branch counting is rational under the subjective-uncertainty viewpoint 3.2 Branch counting is rational under the objective-determinism viewpoint 3.3 Branch counting section summary 4 Number of Branches 4.1 Veracity of the framework 4.2 No such thing as the number of branches 4.2.1 The number of branches is indeterminate 4.2.2 The number of branches is indeterminable 4.3 Rationality and weight 5 Conclusion
Abstract There is a compelling intellectual case for exploring whether purely unitary quantum theory defines a sensible and scientifically adequate theory, as Everett originally proposed. Many different and incompatible attempts to define a coherent Everettian quantum theory have been made over the past fifty years. However, no known version of the theory (unadorned by extra ad hoc postulates) can account for the appearance of probabilities and explain why the theory it was meant to replace — Copenhagen quantum theory — appears to be confirmed, or more generally why our evolutionary history appears to be Born-rule typical. This chapter reviews some ingenious and interesting recent attempts in this direction by Wallace, Greaves–Myrvold, and others, and explains why they don't work. An account of one-world randomness, which appears scientifically satisfactory, and has no many-worlds analogue, is proposed. A fundamental obstacle to confirming many-worlds theories is illustrated by considering some toy many-worlds models. These models show that branch weights can exist without having any role in either rational decision-making or theory confirmation, and also that the latter two roles are logically separate. Wallace's proposed decision theoretic axioms for rational agents in a multiverse and claimed derivation of the Born rule are examined. It is argued that Wallace's strategy of axiomatizing a mathematically precise decision theory within a fuzzy Everettian quasiclassical ontology is incoherent. Moreover, Wallace's axioms are not constitutive of rationality either in Everettian quantum theory or in theories in which branchings and branch weights are precisely defined. In both cases, there exist coherent rational strategies that violate some of the axioms.