SELF-LOCATING BELIEFS WON’T MOVE EVERETT’S MOUNTAIN

The problem of accounting for quantum probabilities is widely regarded as the most serious challenge against the Everett ‘many worlds’ interpretation of quantum mechanics. A recent joint article by a famous physicist and a philosopher of physics claims to have solved this problem by appealing to ‘self-locating uncertainty’, as familiar from the literature on cosmological multiverse theories. We show that their argument does not work: the principle on which their derivation rests is implausible in the light of the very assumption that they use to motivate it.

Everettian quantum mechanics makes the stunning claim that all possible outcomes of a quantum measurement actually happen: at the time of the measurement, the wave function that represents the system does not collapse on one outcome but splits into ‘branches’, each of which corresponds to one of the physically allowed outcomes. Our perception that just one outcome materializes is due to the fact that each outcome is realized on a branch that includes a full set of ‘copies’ of experimenters and observers who only observe this one outcome. The idea, presented by Everett in the 1950s, was neglected for a long time but has attained a very influential position in the debate on the foundations of quantum mechanics in recent decades. Today, the Everett interpretation of quantum mechanics is very popular, mainly because it combines two attractive features. First, it allows us to understand quantum mechanics as describing objective reality in a straightforward sense. Second, it does not need more formal ingredients than the standard quantum formalism, besides those required to apply quantum mechanics in practice.

But the Everett interpretation also faces two fundamental challenges. The first challenge is to individuate the different branches that supposedly form the Everettian ‘multiverse’: what counts as a branch in the quantum formalism, how are branches formed, and how can we reconcile our experience of a single world with the story of a multi-branch reality?

A remarkable characteristic of quantum mechanics called decoherence is widely agreed to be the key in any promising Everettian response to this challenge. Decoherence means that so-called quantum entanglement dissipates quickly into the environment in complex interacting quantum systems. The state of a quantum system that is coupled to a complex environment such as a measurement apparatus will, after a very brief period of time, in some sense appear classical in that environment. Applied to Everettian quantum mechanics, this means that branches can be seen as terms of the universal quantum state between which quantum entanglement quickly vanishes due to interactions such as those that occur in measurement. If it weren’t for decoherence, we would be able to measure strong effects of other branches of an Everettian multiverse, something we do not see in our actual quantum measurements. Decoherence thus is essential for having an empirically viable Everettian theory of quantum mechanics.

The second fundamental challenge to the Everett interpretation is to understand how the probabilities of measurement outcomes that quantum mechanics allows us to compute arise in the Everettian framework. Quantum mechanics is based on the understanding that the probabilistic distributions of measurement outcomes correspond to the absolute squares of certain coefficients appearing in quantum states. This connection between quantum states and probabilities, called the Born rule, is essential for making contact between the theory of quantum mechanics and the statistical characteristics of quantum measurement outcomes. In other words, this connection encodes the physical content of quantum states. A time-honoured view of quantum mechanics assumes that measurements, or the dynamics of quantum states, are associated with a ‘collapse’ of the quantum state when represented as a wave function. In collapse accounts, the Born rule is introduced as a basic feature of the dynamics of the wave function. The probability distributions of our measurements follow from the probabilistic characteristics of the collapse mechanism. (Collapse models have problems of their own that we won’t address here.)

The question now is: how can Everettian quantum mechanics explain the Born rule? Everettian quantum mechanics does not allow for the explanation of the Born rule along the lines suggested by collapse models. Unlike these models, Everettian quantum mechanics asserts that each outcome of a quantum measurement corresponds to a different branch of the wave function with a different instantiation of the agent doing the experiment. Statistics plays no role whatsoever in the Everettian representation of the dynamics of the wave function. So how can the probabilistic nature of the Born rule be extracted from a theory that has a fully deterministic dynamics? (Note that decoherence on its own does not provide an answer to this question since the way that the suppression of effects of other branches can be related to decoherence already relies on a probabilistic interpretation of the wave function.)

A number of attempts have been made in recent decades to extract the Born rule from the basic principles of Everettian quantum mechanics. A recent, original proposal has been presented in a joint article by the philosopher Chip Sebens and the physicist Sean Carroll ([2018]). Following an idea by Lev Vaidman ([1998]), they understand quantum probabilities as self-locating probabilities on the branches of the wave function. Based on this understanding, they identify the Born rule as a consequence of a more general principle of ‘self-locating belief’. Such a principle helps one to assign probabilities in a wide range of situations where one is uncertain of who one is among a number of independently specified agents. This, at the face of it, looks like a plausible suggestion. It seems correct that guiding principles of self-location, such as the principle of indifference (‘assigning equal probability to all possible outcomes’), have a peculiar status. They are not analytical principles of reasoning but nevertheless seem more general than what is implied by an individual scientific theory. Sebens and Carroll suggest that the Born rule should be derived from principles of the same kind. Such a derivation would then have the same status as the derivation of probabilistic statements from principles of self-location in the cosmic multiverse.

In order to succeed, however, Sebens and Carroll must overcome the problem that the principle of indifference applied in the cosmic multiverse context presupposes that we can count Everettian branches. But these branches are either ill defined or the recipe leads to conflict with the empirical evidence. Sebens and Carroll therefore propose a new principle, which they call the QM epistemic separability principle (ESP–QM), from which they derive the Born rule. ESP–QM, intuitively, states that a quantum agent’s self-locating degrees of belief should only depend on the quantum state (more precisely: reduced density matrix) of the part of the Everettian multiverse with respect to which the agent has self-locating uncertainty. According to them, ESP–QM is plausible because it is a natural implementation, in the Everettian quantum context, of a more general and independently plausible epistemic separability principle (ESP). This general ESP states, roughly speaking, that an agent’s rational self-locating credences should not be affected by features of the environment to which the agent has no access. This seems plausible: we cannot hold an agent responsible for failing to take into account features of the world about which she cannot possibly know anything.

In our article, we criticize this argument in two steps. First, we argue that ESP–QM is not an implementation of ESP in the Everettian context. Contrary to what Sebens and Carroll claim, no cogent line of reasoning leads from ESP to the ESP–QM, and the apparent connection between the two relies on a superficial analogy. In fact, we argue that ESP–QM is implausible in the light of ESP: if we cannot hold an agent responsible for taking into account features of the world to which she has no epistemic access, we cannot hold her responsible for taking into account the quantities associated with Everettian branches (called the ‘Born weights’) that determine self-locating probabilities on branches. We clearly have no epistemic access to those Born weights if we do not assume the Born rule from the start. Far from being a special case of ESP, ESP–QM, which is needed in the derivation of the Born rule, is in tension with it.

Second, absent its ESP-based motivation, the plausibility of ESP–QM depends entirely on the empirical success of quantum mechanics and therefore, indirectly, on the Born rule. There is no basis for viewing ESP–QM as an independently attractive principle of rational reasoning. It rather appears as a mere restatement, within the Everettian framework, of the Born rule in terms of the freshly introduced concept of ‘self-location probability’. As a consequence, establishing that the Born rule can be derived from ESP–QM does not solve the probability problem for Everettian quantum theory. And in view of this problem, the Everett interpretation in its current form falls short of providing a satisfactory understanding of quantum mechanics.