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Studies in History and Philosophy of Science Part B Studies in History and Philosophy of Modern Physics

Uncertainty and probability for branching selves

Peter J. Lewis

Published Jun 20, 2006• 73 citations
Publisher
DOI
10.1016/j.shpsb.2006.02.001

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Abstract David Lewis cautions that although a no-collapse interpretation of quantum mechanics entails immortality for trans-world selves, the nature of the branching leaves us crippled, lonely, deathly ill (although never dead), and mentally infirm, meaning that immortal life, on such terms, amounts to an existence in eternal torment. This paper argues that the problem Lewis points to is in fact one of individuation and that a synthesis of Lewis’ own notion of perdurance and Robert Nozick’s closest continuer theory, when cast in the mould of a deterministic multiverse (as conceived by the Oxford quantum physicist David Deutsch), individuates trans-world selves in such a way as to allow to prune the infinitesimal expectation of a miserable eternal existence from the histories of most trans-world agents. Thus, contrary to Lewis’ warning that if personal identity is a trans-world notion, then we should all shake in our shoes, this paper argues that even if we are trans-world selves, we almost certainly have nothing to worry about.

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Vaidman's self-locating uncertainty (SLU) represents an innovative and influential approach to deriving Born rule probabilities in the Many-Worlds Interpretation (MWI) of quantum mechanics, offering a promising epistemic foundation for understanding probabilities in a deterministic multiverse. This paper examines a potential inconsistency in the SLU framework: while the concept relies on observer branching to generate multiple copies and thus enable uncertainty about branch location, such branching results in pure, deterministic local states for each copy, making the mixed reduced density matrix (RDM)—a key element in deriving amplitude-squared probabilities—unavailable. Conversely, if the mixed RDM is retained, the observer remains unentangled and spans all branches, eliminating the multiplicity essential to SLU. The derivations by McQueen and Vaidman appear to draw on both SLU (which presupposes branching) and the mixed RDM (which presupposes no branching), without fully addressing this tension. Our analysis suggests that the inconsistency poses a significant challenge to grounding the Born rule purely in SLU, independent of specific branching models. The results highlight unresolved questions in the foundations of Everettian probability and point toward the need for further refinement or supplementation of the SLU approach.

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Self-locating Uncertainty and the Origin of Probability in Everettian Quantum Mechanics

Charles T. Sebens, Sean M. Carroll

A longstanding issue in attempts to understand the Everett (many-worlds) approach to quantum mechanics is the origin of the Born rule: why is the probability given by the square of the amplitude? Following Vaidman, we note that observers are in a position of self-locating uncertainty during the period between the branches of the wave function splitting via decoherence and the observer registering the outcome of the measurement. In this period, it is tempting to regard each branch as equiprobable, but we argue that the temptation should be resisted. Applying lessons from this analysis, we demonstrate (using methods similar to those of Zurek’s envariance-based derivation) that the Born rule is the uniquely rational way of apportioning credence in Everettian quantum mechanics. In doing so, we rely on a single key principle: changes to the environment alone do not affect the probabilities one ought to assign to measurement outcomes in a local subsystem. We arrive at a method for assigning probabilities in cases that involve both classical and quantum self-locating uncertainty. This method provides unique answers to quantum Sleeping Beauty problems, as well as a well-defined procedure for calculating probabilities in quantum cosmological multiverses with multiple similar observers. 1. Introduction2. Preliminaries: Many-Worlds, Self-locating Uncertainty, and Branch-counting 2.1. The many-worlds interpretation2.2. Self-locating uncertainty and the Everettian multiverse2.3. Indifference and the quantitative probability problem2.4. Against branch-counting3. The Epistemic Irrelevance of the Environment 3.1. The epistemic separability principle3.2. Deriving the Born rule4. Varieties of Uncertainty 4.1. Epistemic separability principle and indifference4.2. Mixed uncertainties4.3. Large universe cosmology and the quantum multiverse5. Probability in Practice 5.1. Betting and branching5.2. Theory confirmation6. Comparison to Other Approaches 6.1. Zurek’s envariance-based derivation of the Born rule6.2. The decision-theoretic programme7. ConclusionAppendix

arXiv:1405.7577•The British Journal for the Philosophy of Science•Jul 5, 2016•104 citations
Abstract David Lewis cautions that although a no-collapse interpretation of quantum mechanics entails immortality for trans-world selves, the nature of the branching leaves us crippled, lonely, deathly ill (although never dead), and mentally infirm, meaning that immortal life, on such terms, amounts to an existence in eternal torment. This paper argues that the problem Lewis points to is in fact one of individuation and that a synthesis of Lewis’ own notion of perdurance and Robert Nozick’s closest continuer theory, when cast in the mould of a deterministic multiverse (as conceived by the Oxford quantum physicist David Deutsch), individuates trans-world selves in such a way as to allow to prune the infinitesimal expectation of a miserable eternal existence from the histories of most trans-world agents. Thus, contrary to Lewis’ warning that if personal identity is a trans-world notion, then we should all shake in our shoes, this paper argues that even if we are trans-world selves, we almost certainly have nothing to worry about.
Vaidman's self-locating uncertainty (SLU) represents an innovative and influential approach to deriving Born rule probabilities in the Many-Worlds Interpretation (MWI) of quantum mechanics, offering a promising epistemic foundation for understanding probabilities in a deterministic multiverse. This paper examines a potential inconsistency in the SLU framework: while the concept relies on observer branching to generate multiple copies and thus enable uncertainty about branch location, such branching results in pure, deterministic local states for each copy, making the mixed reduced density matrix (RDM)—a key element in deriving amplitude-squared probabilities—unavailable. Conversely, if the mixed RDM is retained, the observer remains unentangled and spans all branches, eliminating the multiplicity essential to SLU. The derivations by McQueen and Vaidman appear to draw on both SLU (which presupposes branching) and the mixed RDM (which presupposes no branching), without fully addressing this tension. Our analysis suggests that the inconsistency poses a significant challenge to grounding the Born rule purely in SLU, independent of specific branching models. The results highlight unresolved questions in the foundations of Everettian probability and point toward the need for further refinement or supplementation of the SLU approach.
A longstanding issue in attempts to understand the Everett (many-worlds) approach to quantum mechanics is the origin of the Born rule: why is the probability given by the square of the amplitude? Following Vaidman, we note that observers are in a position of self-locating uncertainty during the period between the branches of the wave function splitting via decoherence and the observer registering the outcome of the measurement. In this period, it is tempting to regard each branch as equiprobable, but we argue that the temptation should be resisted. Applying lessons from this analysis, we demonstrate (using methods similar to those of Zurek’s envariance-based derivation) that the Born rule is the uniquely rational way of apportioning credence in Everettian quantum mechanics. In doing so, we rely on a single key principle: changes to the environment alone do not affect the probabilities one ought to assign to measurement outcomes in a local subsystem. We arrive at a method for assigning probabilities in cases that involve both classical and quantum self-locating uncertainty. This method provides unique answers to quantum Sleeping Beauty problems, as well as a well-defined procedure for calculating probabilities in quantum cosmological multiverses with multiple similar observers. 1. Introduction2. Preliminaries: Many-Worlds, Self-locating Uncertainty, and Branch-counting 2.1. The many-worlds interpretation2.2. Self-locating uncertainty and the Everettian multiverse2.3. Indifference and the quantitative probability problem2.4. Against branch-counting3. The Epistemic Irrelevance of the Environment 3.1. The epistemic separability principle3.2. Deriving the Born rule4. Varieties of Uncertainty 4.1. Epistemic separability principle and indifference4.2. Mixed uncertainties4.3. Large universe cosmology and the quantum multiverse5. Probability in Practice 5.1. Betting and branching5.2. Theory confirmation6. Comparison to Other Approaches 6.1. Zurek’s envariance-based derivation of the Born rule6.2. The decision-theoretic programme7. ConclusionAppendix