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Probability and Uncertainty in the Everett Interpretation

How to Prove the Born Rule (June 2009)

In Simon Saunders, Jon Barrett, Adrian Kent and David Wallace (ed.), Many Worlds? Everett, Quantum Theory, and Reality (OUP, 2010)

I develop 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. I sketch the argument informally, then prove it formally, and lastly consider a number of proposed "counter-examples" to show exactly which premises of the argument they violate.

The preprint version linked to below was published under the less catchy, but more informative, title A Formal Proof of the Quantum Probability Rule from Decision-Theoretic Assumptions.

Saunders and Wallace Reply (2008)

(By Simon Saunders and DW)

British Journal for the Philosophy of Science 59 (2008), pp. 315-317

A reply to comments on Saunders and Wallace (below) by Paul Tappenden (this paper)

Branching and Uncertainty (June 2007)

(By Simon Saunders and DW)

British Journal for the Philosophy of Science 59 (2008), pp. 293-305

Following Lewis, it is widely held that branching worlds differ in important ways from diverging worlds. There is, however, a simple and natural semantics under which ordinary sentences uttered in branching worlds have much the same truth values as they conventionally have in diverging worlds. Under this semantics, whether branching or diverging, speakers cannot say in advance which branch or world is theirs. They are uncertain as to the outcome. This same semantics ensures the truth of utterances typically made about quantum mechanical contingencies, including statements of uncertainty, if the Everett interpretation of quantum mechanics is true. The 'incoherence problem' of the Everett interpretation, that it can give no meaning to the notion of uncertainty, is thereby solved.

Epistemology Quantised: circumstances in which we should come to believe in the Everett interpretation (July 2005)

British Journal for the Philosophy of Science 57 (2006), pp. 655-689

I consider exactly what is involved in a solution to the probability problem of the Everett interpretation, in the light of recent work on applying considerations from decision theory to that problem. I suggest an overall framework for understanding probability in a physical theory, and conclude that this framework, when applied to the Everett interpretation, yields the result that that interpretation satisfactorily solves the measurement problem.

Language Use in a Branching Universe (July 2005)

Online only (cite as http://philsci-archive.pitt.edu/2554/).

I investigate the consequences for semantics, and in particular for the semantics of tense, if time is assumed to have a branching structure not out of metaphysical necessity (to solve some philosophical problem) but just as a contingent physical fact, as is suggested by a currently-popular approach to the interpretation of quantum mechanics.

(Note: For various reasons I never got around to publishing this; it's now largely superseded by chapter 7 of The Emergent Multiverse.)

Three Kinds of Branching Universe (July 2005)

Unpublished (cite via this page).

In the light of recent work suggesting that the quantum probability rule can be derived in the Everett interpretation via decision theory, I consider what physical features of quantum mechanics make this possible. I analyse the status of the probabiliy rule in three different models of branching universes, each somewhat more complicated than the last, and conclude that only in the last model --- in which the branching structure, as in quantum mechanics, emerges in a somewhat imprecise way from the underlying physical reality --- is it possible to derive a probability rule, or indeed to behave in any rational way at all.

(Note: I wrote this in 2005 intending to submit it fairly shortly afterwards, but for various reasons I still haven't got around to it. My views have evolved some way since then and I'm now unlikely to publish it, but it's occasionally been cited and discussed so it seems sensible to leave it available here.)

Quantum Probability from Subjective Likelihood: improving on Deutsch's proof of the probability rule (originally October 2003; significantly revised June 2005)

Studies in the History and Philosophy of Modern Physics 38 (2007), pp. 311--332.

I present a proof of the quantum probability rule from decision-theoretic assumptions, in the context of the Everett interpretation. The basic ideas behind the proof are those presented in Deutsch's recent proof of the probability rule, but the proof is simpler and proceeds from weaker decision-theoretic assumptions. This makes it easier to discuss the conceptual ideas involved in the proof, and to show that they are defensible.

Everettian Rationality: defending Deutsch's approach to probability in the Everett interpretation (December 2002)

Studies in the History and Philosophy of Modern Physics 34 (2003), pp. 415-438.

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.

Quantum Probability and Decision Theory, Revisited (October 2002)

Online only (cite as arxiv:quant-ph/0211104)

An extended analysis is given of the program, originally suggested by Deutsch, of solving the probability problem in the Everett interpretation by means of decision theory. Deutsch's own proof is discussed, and alternatives are presented which are based upon different decision theories and upon Gleason's Theorem. It is argued that decision theory gives Everettians most or all of what they need from `probability'. Some consequences of (Everettian) quantum mechanics for decision theory itself are also discussed.

Note: this long (70 pages) and occasionally rambling paper has been almost entirely superseded by material in the above papers; if something is not included in them it usually means that I have had second thoughts. I include it for completeness only.