Abstract: This paper develops and advocates a rule for assigning self-locating credences in quantum branching scenarios, called Indexed Branch-Counting. It is argued that Indexed Branch-Counting can be justified on both accuracy-theoretic grounds and on the grounds that it satisfies a requirement of exchangeability for probability assignments. Since Indexed Branch-Counting diverges from the Born Rule, this poses trouble for Everettian approaches to probability. The paper also addresses a common argument against branch-counting, namely that the rule is incoherent in light of putative vagueness in the number of branches. Finally, the paper addresses a recent proposal from Simon Saunders that aims to reconcile branch-counting with the Born Rule, arguing that the proposal faces challenges.
Abstract: This paper develops and advocates a rule for assigning self-locating credences in quantum branching scenarios, called Indexed Branch-Counting. It is argued that Indexed Branch-Counting can be justified on both accuracy-theoretic grounds and on the grounds that it satisfies a requirement of exchangeability for probability assignments. Since Indexed Branch-Counting diverges from the Born Rule, this poses trouble for Everettian approaches to probability. The paper also addresses a common argument against branch-counting, namely that the rule is incoherent in light of putative vagueness in the number of branches. Finally, the paper addresses a recent proposal from Simon Saunders that aims to reconcile branch-counting with the Born Rule, arguing that the proposal faces challenges.
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
The Everett interpretation of quantum mechanics faces a well‑known challenge: how can probabilities emerge in a deterministic theory where all measurement outcomes occur? A prominent response appeals to self‑locating uncertainty (SLU): after branching, copies of an observer are uncertain about which branch they inhabit during the post-measurement, pre-observation period, and this uncertainty is then quantified to yield the Born rule. This paper examines whether the notion of SLU can be coherently instantiated in the Everettian framework. It argues that the two conditions required for SLU—having a branch‑relative identity (Recording Condition) and being uncertain about that identity (Uncertainty Condition)—are difficult to satisfy simultaneously in any physical system. Any attempt to realize SLU appears to force a decomposition of the observer into a memory module that records the outcome (and thus has branch‑relative identity) and a decision module that is uncertain (and thus lacks branch‑relative identity). Neither module individually possesses SLU, and it is not obvious that they can be combined into a unified epistemic agent that does. The analysis complements and deepens existing critiques and suggests that SLU‑based derivations of the Born rule may rest on a problematic foundation.
Everettian quantum mechanics tells us that the fundamental dynamics of the universe are deterministic. So what are the `probabilities' that the Born rule describes? One popular answer has been to treat these probabilities as rational credences. A recent alternative, Isaac Wilhelm's centered Everett Interpretation (CEI), takes the Born probabilities to be centered chances: the objective chances that some centered propositions are true. Thus, the CEI challenges the `orthodox assumption’ that fundamental physical laws concern only uncentered facts. I provide three arguments against the centered Everett Interpretation. First, I argue that the CEI is in apparent tension with a significant motivation for adopting Everettian quantum mechanics: rejecting the attribution of special significance to observers or agents in fundamental physics. I suggest the CEI can avoid this tension, but only at the cost of sacrificing its central claim that there are objective chances in an Everettian multiverse. My second argument concerns the CEI’s claim that the centered Born rule is a fundamental physical law. I provide two plausible notions of fundamentality for physical laws, and I argue that the centered Born rule satisfies neither. My final argument is that the CEI’s branch-relative laws cannot explain or constrain an agent’s rational credences in the way that the CEI claims.
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.
Abstract In Everettian quantum mechanics, justifications for the Born rule appeal to self-locating uncertainty or decision theory. Such justifications have focused exclusively on a pure-state Everettian multiverse, represented by a wavefunction. Recent works in quantum foundations suggest that it is viable to consider a mixed-state Everettian multiverse, represented by a (mixed-state) density matrix. Here, we discuss the conceptual foundations for decoherence and branching in a mixed-state multiverse, and extend arguments for the Born rule to this setting. This extended framework provides a unification of ‘classical’ and ‘quantum’ probabilities, and additional theoretical benefits, for the Everettian picture.
Everettian quantum mechanics (EQM) results in ‘multiple, emergent, branching quasi-classical realities’ (Wallace [2012]). The possible outcomes of measurement as per ‘orthodox’ quantum mechanics are, in EQM, all instantiated. Given this metaphysics, Everettians face the ‘probability problem’—how to make sense of probabilities and recover the Born rule. To solve the probability problem, Wallace, following Deutsch ([1999]), has derived a quantum representation theorem. I argue that Wallace’s solution to the probability problem is unsuccessful, as follows. First, I examine one of the axioms of rationality used to derive the theorem, ‘branching indifference’ (BI). I argue that Wallace is not successful in showing that BI is rational. While I think it is correct to put the burden of proof on Wallace to motivate BI as an axiom of rationality, it does not follow from his failing to do so that BI is not rational. Thus, second, I show that there is an alternative strategy for setting one’s credences in the face of branching which is rational and which violates BI. This is ‘branch counting’ (BC). Wallace is aware of BC and has proffered various arguments against it. However, third, I argue that Wallace’s arguments against BC are unpersuasive. I conclude that the probability problem in EQM persists. 1 Introduction 2 Branching Indifference 2.1 The positive argument for branching indifference 2.2 The negative argument for branching indifference 2.3 Branching indifference section summary 3 Branch Counting 3.1 Branch counting is rational under the subjective-uncertainty viewpoint 3.2 Branch counting is rational under the objective-determinism viewpoint 3.3 Branch counting section summary 4 Number of Branches 4.1 Veracity of the framework 4.2 No such thing as the number of branches 4.2.1 The number of branches is indeterminate 4.2.2 The number of branches is indeterminable 4.3 Rationality and weight 5 Conclusion