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

David John Baker

Lina Jansson

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

Adrien Barton

Alexander Ma, Gabriel Wong

The measurement problem remains one of the most profound puzzles in the foundation of quantum mechanics. It contains two distinct aspects: the collapse of the wavefunction and the origin of probabilities. While Everettian Quantum Mechanics (EQM) and the Many Worlds Interpretation (MWI) work around the collapse aspect, by positing that the universe branches through pure unitary evolution, it leaves unresolved the question of why measurement outcomes occur with frequencies given by the Born Rule. This paper explores a recent derivation of the Born Rule by Carroll and Sebens based on principles of Self-Locating Uncertainty (SLU) and the Epistemic Separability Principle (ESP). By examining this framework in detail, beginning with an equal amplitude case and extending to an unequal amplitude case, we show how these principles account for the probabilities given by the Born Rule, hence showing that SLU and the ESP, within the scope of EQM, offer a solution to the measurement problem. We further show the significance and implications of deriving the Born Rule and highlight the relevance of the measurement problem to more recent developments like quantum gravity.

Nadia Blackshaw