Abstract This introductory chapter summarizes the principal positive theses of the book (Section 1), namely, that the unitary evolution of the quantum state, for sufficiently large systems of particles in thermal environments, yields a multiplicity of branching structures (worlds) obeying approximately classical equations, a result that follows from decoherence theory with no special assumptions or additions to the quantum formalism; and that further, the branching of worlds has a natural interpretation in terms of probability, as quantified by the Born rule in terms of ratios of squared norms of branch amplitudes. In particular, an agent who knows the branching structures of the quantum state that will result from his actions is rationally compelled to order his preferences among actions by the expected utilities of those branching structures as defined by the Born rule. He is, moreover, entitled to view a branching process as involving uncertainty as to which branch he will find himself. But independent of the truth of these claims, the theory is still testable: an agent who does not know the branching structure of the state, and who is indeed undecided between a many-worlds and a one-world theory, is rationally bound to update her credences in such theories by conditionalisation on observed statistical evidence, just as she is bound to do in the case of conventional quantum mechanics. Section 2 summarizes some of the arguments and counter-arguments for these claims as developed in subsequent chapters. Section 3 provides some more mathematical and historical background to the Everett interpretation, including a brief introduction to the decoherent histories formalism.
Abstract This introductory chapter summarizes the principal positive theses of the book (Section 1), namely, that the unitary evolution of the quantum state, for sufficiently large systems of particles in thermal environments, yields a multiplicity of branching structures (worlds) obeying approximately classical equations, a result that follows from decoherence theory with no special assumptions or additions to the quantum formalism; and that further, the branching of worlds has a natural interpretation in terms of probability, as quantified by the Born rule in terms of ratios of squared norms of branch amplitudes. In particular, an agent who knows the branching structures of the quantum state that will result from his actions is rationally compelled to order his preferences among actions by the expected utilities of those branching structures as defined by the Born rule. He is, moreover, entitled to view a branching process as involving uncertainty as to which branch he will find himself. But independent of the truth of these claims, the theory is still testable: an agent who does not know the branching structure of the state, and who is indeed undecided between a many-worlds and a one-world theory, is rationally bound to update her credences in such theories by conditionalisation on observed statistical evidence, just as she is bound to do in the case of conventional quantum mechanics. Section 2 summarizes some of the arguments and counter-arguments for these claims as developed in subsequent chapters. Section 3 provides some more mathematical and historical background to the Everett interpretation, including a brief introduction to the decoherent histories formalism.
Few, if any, issues in physics have engendered as much discussion as the measurement problem in quantum mechanics. It is generally agreed that the `normal' dynamical evolution of the state vector in quantum mechanics is given by a unitary map. The linearity of this map implies that the state vector will, in general, be found in a superposition of eigenstates of a given observable (or, similarly, that the density matrix describing a subsystem will not correspond to a definite value of this observable). However, when we make a measurement of an observable, we always obtain a define value—although it is impossible to predict with certainty which value will be obtained. The traditional response to this issue is to postulate that when a measurement is made, the wavefunction `collapses' to an eigenstate of the observable being measured, perhaps due to the inherent classicality of the measuring apparatus (Bohr), or to the consciousness of the observer (Wigner), or possibly to some modification of quantum dynamics that occurs even when observations are not being made. The main motivation for the Everett (`many worlds') interpretation is to avoid introducing any such collapse postulate. This volume commemorates the 50th anniversary of the publication of Everett's paper in 1957 and contains 20 original articles as well as the transcripts of several discussions that took place at meetings devoted to the Everett interpretation at Oxford University and the Perimeter Institute. The attractiveness of the Everett interpretation is very succinctly summarized by a sentence from Vaidman's contribution (p 582): `The collapse, with its randomness, non-locality and the lack of a well-defined moment of occurrence, is such an ugly scar on quantum theory, that I, along with many others, am ready to follow Everett and deny its existence.' But the main drawback of the interpretation is then equally succinctly stated in the next sentence: `The price is the many worlds interpretation, i.e., the existence of numerous parallel worlds.' In other words, if the state vector evolves to have a non-zero amplitude for two different values of an observable that I `measure' (i.e., interact with in an appropriate way), each of these alternatives remains equally `real', i.e., there is a `branch' of the wavefunction in which the observable takes the first value and I believe that it has this first value and a branch of the wavefunction where it takes the second value and I believe that it has this second value. Each of these `me's' sees a definite value for the observable, thus nicely accounting for what I believe I experience without invoking collapse. But then, it seems that I have to believe that there are many other `me's' around—even if they are very difficult for this particular me to access—and no way I can predict which `me' I will be in the future. Indeed, Saunders writes on p 192, `If Alice is a person (of course she is) then we must say, even prior to branching at tj that there are many Alices present, atom for atom duplicates...Alice should be uncertain after all— each Alice should be uncertain—for each as of tj does not (and as a matter of principle, cannot) know which of these branching persons she is.' Thus, if taken seriously, the Everett viewpoint leads directly to questions like `What determines which ``me'' I am now and/or will feel that I am in the future?' The response `You are all of them' somehow doesn't seem satisfying, and further pursuit of these issues would seem to lead to questions as fruitless as questions like `Why am I ``me'' and not ``you''?' I believe that this accounts for why the Everett interpretation does not have universal appeal. But enough of `me' for now (more later); let's discuss the book. Unlike many edited volumes and conference proceedings in physics—which often consist of hastily slapped-together re-hashes of previous papers—this volume meets very high standards in the choice of topics and authors and in the quality of the writing. Although each of the articles is closely tied in with the Everett interpretation, there are many articles of considerable value and interest in their own right, such as the discussion of quasiclassical realms by Hartle and and the discussion of the de Broglie–Bohm interpretation by Valentini. There is also an excellent introduction by Saunders to the quantum measurement problem and the main issues that arise in the Everett interpretation. A central issue in the Everett interpretation is the status of the `Born rule', which asserts that, for state ψ, the probability of obtaining a particular outcome of a measurement is ||Pψ||2, where P is the projection operator onto the eigensubspace associated with the measurement outcome. In traditional interpretations, the Born rule is simply postulated as part of the collapse hypothesis. In the Everett interpretation, it is far from obvious that the Born rule even has any meaning—if all outcomes occur, how can one talk about the probability of a particular outcome? Given the importance of this issue, it is highly appropriate that four chapters of the book (by Saunders, Papineau, Wallace, and Greaves and Myrvold) are devoted to addressing probability and the Born rule from the Everett viewpoint, and three chapters (by Kent, Albert, and Price) are devoted to criticising these views. As indicated by the above quote from Saunders' article, uncertainty and probability can enter the Everett interpretation via each Alice being uncertain as to which Alice she is or will be. However, in view of the difficulties to which this kind of argumentation will lead, it is perhaps not surprising that in the 123 pages of combined discussion by Saunders, Papineau, Wallace, and Greaves and Myrvold, there are only fleeting references to the use of probability along these lines. Rather, the approach taken is reminiscent of that of adept politicians: if you can't answer a question, then answer a different question, and answer it with conviction and detail. Instead of considering probabilities of outcomes—a highly problematical notion when all of the possible outcomes actually occur—consider only decision theory issues. In particular, Wallace gives a precise and detailed mathematical proof that, under certain hypotheses, the only rational strategy that a quantum observer can follow in making decisions is to weight the possible outcomes by Born rule probabilities. To illustrate this more graphically, suppose tomorrow's weather were determined by some quantum mechanical, Stern–Gerlach-like process. I would like to know whether it is likely to rain tomorrow. However, when I ask an Everett follower (who knows the state vector and has done the Born rule calculation), all she will tell me in direct response to my question is that `it will rain and it won't rain.' Nevertheless, she is willing tell me whether or not it is rational for me to take an umbrella. I suppose that if she tells me that it would be highly irrational for me to take an umbrella, I can take the hint and deduce that it is very unlikely to rain. However, there does seem something wrong with the fact that she is not allowed to say this directly. In any case, if the conclusion of a mathematically correct argument is that rational decision strategies require the Born rule, then there must be quite a bit lying in the assumptions. The articles by Kent, Albert, and Price do a good job of fleshing out these assumptions and pointing out the weaknesses and flaws in the probability and decision theory discussions within the Everett framework. I hope it is clear from what I have written above that this is an outstanding volume, which should be of great interest to anyone interested in the foundations of quantum theory. Indeed, I feel quite fortunate to be the particular `me' that happens to exist in the same branch of the wavefunction of the universe in which this book was published.
Abstract What does realism about the quantum state imply? What follows when quantum theory is applied without restriction, if need be, to the whole universe? These are the questions which are debated in this book. All the chapters start from the point of realism, and based on the need, or the aspiration, for a theory that unites micro- and macroworlds, at least in principle. But the further claim argued by some is that if you allow the Schrödinger equation unrestricted application, supposing the quantum state to be something physically real, then this universe is one of countlessly many others, constantly branching in time, all of which are real. The result is the many worlds theory, also known as the Everett interpretation of quantum mechanics. The contrary claim sees this picture of many worlds as in no sense inherent in quantum mechanics, even when the latter is allowed unrestricted scope and even given that the quantum state itself is something physically real. For this picture of branching worlds fails to make physical sense, let alone common sense, even on its own terms. The status of these worlds, what they are made of, is never adequately explained. Ordinary ideas about time and identity over time become hopelessly compromised. The concept of probability itself is brought into question. This picture of many branching worlds is inchoate, it is a vision, an error. There are realist alternatives to many worlds, some even that preserve the Schrödinger equation unchanged. Twenty chapters, accompanied by commentaries and discussions, examine these claims and counterclaims in depth. They focus first on the question of ontology, the existence of worlds, second on the interpretation of probability, and third on alternatives or additions to many worlds. The introduction offers a helpful guide to the arguments for the Everett interpretation, particularly as they have been formulated in the last two decades.
A defence is offered of a version of the branch-counting rule in the Everett interpretation (otherwise known as many worlds interpretation) of quantum mechanics that both depends on the state and is continuous in the norm topology on Hilbert space. The well-known branch-counting rule, for realistic models of measurements, in which branches are defined by decoherence theory, fails this test. The new rule hinges on the use of decoherence theory in defining branching structure, and specifically decoherent histories theory. On this basis ratios of branch numbers are defined, free of any convention. They agree with the Born rule and deliver a notion of objective probability similar to naive frequentism, save that the frequencies of outcomes are not confined to a single world at different times, but spread over worlds at a single time. Nor is it ad hoc : it is recognizably akin to the combinatorial approach to thermodynamic probability, as introduced by Boltzmann in 1879. It is identical to the procedure followed by Planck, Bose, Einstein and Dirac in defining the equilibrium distribution of the Bose–Einstein gas. It also connects in a simple way with the decision-theory approach to quantum probability.
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
Abstract This chapter argues for the view that the Everett (‘many worlds’) interpretation is nothing more or less than a literal reading of the formalism of unitary quantum mechanics. The central component of this story is decoherence: the process by which the evolution of the quantum state acquires a branching structure. The chapter gives a conceptual introduction to decoherence and its conceptual role, emphasizing the significance of its effects on the dynamics of degrees of freedom which are undergoing decoherence. It then argues that, although the branching structure generated by decoherence is ‘for all practical purposes’ in the sense that it is not part of the microscopic formalism of quantum theory, nonetheless it is a perfectly real, observer-independent feature of reality — it is emergent from that microscopic formalism in a way very familiar from many other parts of science.
Abstract There is a compelling intellectual case for exploring whether purely unitary quantum theory defines a sensible and scientifically adequate theory, as Everett originally proposed. Many different and incompatible attempts to define a coherent Everettian quantum theory have been made over the past fifty years. However, no known version of the theory (unadorned by extra ad hoc postulates) can account for the appearance of probabilities and explain why the theory it was meant to replace — Copenhagen quantum theory — appears to be confirmed, or more generally why our evolutionary history appears to be Born-rule typical. This chapter reviews some ingenious and interesting recent attempts in this direction by Wallace, Greaves–Myrvold, and others, and explains why they don't work. An account of one-world randomness, which appears scientifically satisfactory, and has no many-worlds analogue, is proposed. A fundamental obstacle to confirming many-worlds theories is illustrated by considering some toy many-worlds models. These models show that branch weights can exist without having any role in either rational decision-making or theory confirmation, and also that the latter two roles are logically separate. Wallace's proposed decision theoretic axioms for rational agents in a multiverse and claimed derivation of the Born rule are examined. It is argued that Wallace's strategy of axiomatizing a mathematically precise decision theory within a fuzzy Everettian quasiclassical ontology is incoherent. Moreover, Wallace's axioms are not constitutive of rationality either in Everettian quantum theory or in theories in which branchings and branch weights are precisely defined. In both cases, there exist coherent rational strategies that violate some of the axioms.