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How to Prove the Born Rule

David Wallace

Published Jun 24, 2010• 68 citations
Publisher

Abstract

Abstract This chapter develops the decision-theoretic approach to quantum probability, originally proposed by David Deutsch, into a mathematically rigorous proof of the Born rule in (Everett interpreted) quantum mechanics. It sketches the argument informally, then proves it formally, and lastly considers a number of proposed ‘counter-examples’ to show exactly which premises of the argument they violate.

DOI
10.1093/acprof:oso/9780199560561.003.0010

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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.

arXiv:quant-ph/0604191•Apr 26, 2006•93 citations
Abstract This chapter develops the decision-theoretic approach to quantum probability, originally proposed by David Deutsch, into a mathematically rigorous proof of the Born rule in (Everett interpreted) quantum mechanics. It sketches the argument informally, then proves it formally, and lastly considers a number of proposed ‘counter-examples’ to show exactly which premises of the argument they violate.
Building on work originally carried out by David Deutsch, the chapter develops a mathematically rigorous framework for discussing the quantum probability rule in terms of the preferences between various courses of action of rational agents. Within this framework, the chapter proves formally that any rational agent who believes in the truth of Everettian quantum mechanics and who knows (or thinks they know) the current quantum state of their branch will act as if future branching was a probabilistic process with the probabilities given by the standard quantum-mechanical probability rule (the ‘Born rule’). The chapter concludes by analysing other proposed rules for action in a quantum universe and showing where they violate the axioms for quantum decision making which it proposes and defends.
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 major problem facing no-collapse interpretations of quantum mechanics in the tradition of Everett is how to understand the probabilistic axiom of quantum mechanics (the Born rule) in the context of a deterministic theory in which every outcome of a measurement occurs. Deutsch claims to derive a decision-theoretic analogue of the Born rule from the non-probabilistic part of quantum mechanics and some non-probabilistic axioms of classical decision theory, and hence concludes that no probabilistic axiom is needed. I argue that Deutsch’s derivation begs the question.
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.