Everettian rationality: defending Deutsch's approach to probability in the Everett interpretation

Abstract

An analysis is made of Deutsch's recent claim to have derived the Born rule from decision-theoretic assumptions. It is argued that Deutsch's proof must be understood in the explicit context of the Everett interpretation, and that in this context, it essentially succeeds. Some comments are made about the criticism of Deutsch's proof by Barnum, Caves, Finkelstein, Fuchs, and Schack; it is argued that the flaw which they point out in the proof does not apply if the Everett interpretation is assumed.A longer version of this paper, entitled Quantum Probability and Decision Theory, Revisted, is available online (Wallace 2002). The present paper will appear in Studies in the History and Philosophy of Modern Physics; confusingly, when it does it will also bear the title Quantum Probability and Decision Theory, Revisited.Keywords: Interpretation of Quantum Mechanics -Everett interpretation; Probability; Decision Theory IntroductionIn recent work on the Everett (Many-Worlds) interpretation of quantum mechanics, it has increasingly been recognized that any version of the interpretation worth defending will be one in which the basic formalism of quantum mechanics is left unchanged. Properties such as the interpretation of the wave-function as describing a multiverse of branching worlds, or the ascription of probabilities to the branching events, must be emergent from the unitary quantum mechanics rather than added explicitly to the mathematics. Only in this way is it possible to save the main virtue of Everett's approach: having an account of quantum mechanics consistent with the last seventy years of physics, not one in which the (December 26, 2002 ) * Magdalen College, Oxford University, Oxford OX1 4AU, U.K. (e-mail: david.wallace@magdalen.ox.ac.uk). 1 edifice of particle physics must be constructed afresh (Saunders 1997, p. 44). 1Of the two main problems generally raised with Everett-type interpretations, the preferred-basis problem looks eminently solvable without changing the formalism. The main technical tool towards achieving this has of course been decoherence theory, which has provided powerful (albeit perhaps not conclusive) evidence that the quantum state has a de facto preferred basis and that this basis allows us to describe the universe in terms of a branching structure of approximately classical, approximately non-interacting worlds. I have argued elsewhere (Wallace 2001a(Wallace , 2001b) that there are no purely conceptual problems with using decoherence to solve the preferred-basis problem, and that the inexactness of the process should give us no cause to reject it as insufficient. In particular, the branching events in such a theory can be understood, literally, as replacement of one classical world with several -so that in the Schrödinger Cat experiment, for instance, after the splitting there is a part of the quantum state which should be understood as describing a world in which the cat is alive, and another which describes a world in which it is dead. This multiplication comes about not as a consequen...

Epistemology Quantized: circumstances in which we should come to believe in the Everett interpretation

“…There is no need for cautious functionalism where the Everett interpretation is concerned. As was originally argued by Deutsch (1999), and is defended in detail in Wallace (2003b) and Wallace (2003c), the principles of decision theory actually entail the fact that weight fits the functional definition. That is: in the Everett interpretation, we can prove that weight=probability.…”

“…But recent work by Deutsch (1999), Saunders (1998), Vaidman (1998), Greaves (2004), myself (Wallace 2003b,Wallace 2003c,Wallace 2005a,Wallace 2005b) and others has made use of considerations from decision theory, personal identity, philosophy of probability, and philosophy of language to provide both conceptual frameworks for thinking about probability in the Everettian context, and -perhaps more surprisingly -concrete mathematical results which purport to be (Everett-interpretation-specific) derivations of the Born rule. Intractability has a certain simplicity.…”

What is Probability?

“…By the same argument µ( P 3 ) = µ( P ′ 4 ) = 0, so by additivity µ( P 1 ) = 1. The case d = 1 is the eigenvector-eigenvalue rule; this result and the method of proof follows closely the mathematical ideas introduced by Deutsch [5] and Wallace [18], [20]. The next two lemmas and Theorem 6 differ in certain respects, however.…”

“…Here we shall follow Wallace [18], who has substantially revised and simplified the argument. A quantum game can be played using any quantum experiment, simply by agreeing on various payoffs (positive or negative) on each possible outcome.…”

“…A principle to this effectt has been called measurement neutrality [18] ; it is the suspicion that measurement neutrality is too strong, or rests on unwarranted or incoherent assumptions as to what it is right to believe in the face of personal division, that prompts one to seek a more direct argument for the general equivalence condition. But better is to seek a direct argument for measurement neutrality.…”