GETTING TO BASECAMP WITH EVERETTIAN QUANTUM MECHANICS

According to the standard way of thinking about quantum mechanics, certain properties (so-called observables) of material particles—such as position, momentum, and spin—don’t assume any determinate values until an external observer attempts to measure the value of one of these properties. For example, there may be no fact of the matter about whether a particle is located here or there until I perform some action that constitutes a ‘measurement’ of its position. This is reflected in the fact that quantum mechanics describes the state of a particle in terms of a mathematical object called its wavefunction, which maps possible particle positions to (complex) numbers called amplitudes. Before a position measurement is performed, this wavefunction can be ‘smeared out’ in space, meaning that it assigns a non-zero number to many different points in space at which the particle could be located. A particle described by such a wavefunction is said to be in a superposition of all of these different possible locations. All that we can say about the position of the particle in such a state (allegedly!) is that there are certain probabilities that the particle will be found here or there should a measurement be performed. Until then, the particle has no determinate location in space whatsoever. (And similar stories are told about the value of other observables, such as momentum, spin, energy, and so on.)

Besides being an affront to our common intuitions, this story faces a significant conceptual challenge. The standard picture tells us that sometimes the wavefunction of a particle evolves in strict adherence to a deterministic, linear differential equation called the Schrödinger equation. But at other times, the wavefunction evolves stochastically, randomly ‘collapsing’ to one or another determinate state. These quantum–mechanical probabilities are given by the Born rule, which says that the probability of observing a quantum system in a particular state is equal to the square of the magnitude of wavefunction amplitude corresponding to that outcome. If the quantum–mechanical wavefunction does indeed stochastically collapse, then the Schrödinger equation can’t apply all of the time: whenever you perform a measurement, the linearity and determinism of the Schrödinger equation breaks down. In other words, once the probability rule ‘kicks into gear’, the deterministic rule that tells you how this mysterious wavefunction changes over time must, as a mathematical fact, stop applying.

The problem with the standard picture is that it has nothing precise to say about the conditions under which the Schrödinger equation breaks down. All it says is that the collapse rule applies when measurements are performed; but what are measurements? Can measurements only be carried out by conscious observers, or do they occur whenever our particle interacts with some ‘macroscopic’ device? If the former, what counts as a conscious observer? Dogs? Cats? Worms? If the latter, what counts as macroscopic? It turns out to be exceedingly difficult to answer these questions in a precise and non-arbitrary way. But if this issue can’t be resolved, then the standard picture will suffer a kind of vagueness unbefitting of a fundamental physical theory of the world.

In light of the apparent failure of the standard story to offer a complete, consistent, and precise theory, the central task in the foundations of quantum mechanics is to offer a complete physical theory that has a precise and consistent set of dynamical laws, and which recovers the quantum–mechanical probabilities. This puzzle has earned the moniker ‘the measurement problem’.

My article is aimed at assessing one of the most popular candidate solutions to the measurement problem, namely, the Everettian formulation of quantum theory, also known as the many-worlds interpretation. The central innovation of Everettian quantum mechanics is that the theory postulates nothing in addition to a wavefunction for the entire universe evolving in perfect accordance with the Schrödinger equation. What happens, then, when I perform a measurement of some quantum observable is not that the system collapses to one or another state upon being observed, but rather that I branch into a superposition of seeing the system in one state and seeing it in another, and so a number of ‘copies’ of me are created on these different ‘branches’ of the wavefunction, each of whom sees a different outcome! More generally: when the state of a system becomes sufficiently correlated with the state of its environment (which happens, for example, through observation by an observer), the entire universe branches into two (or more) distinct ‘worlds’: one ‘world’ in which the original system is in state A (and the observers of that world see the system in state A, believe the system to be in state A, and so on), and another in which the original system is in state B, and so forth. Wavefunctions, then, don’t collapse at all. The reason they appear to collapse—that is, the reason we only ever see systems in one determinate state or another—is because the universal wavefunction branches into approximately distinct, non-interacting states, in which macroscopic objects like brains, measuring devices, and so on are well localized. Everett’s theory was a stroke of genius. It recovers the Schrödinger equation in its fullest generality: absolutely everything always obeys the Schrödinger equation and at the fundamental level, no fuzzy distinctions between ‘microscopic’ and ‘macroscopic’, ‘system’ and ‘environment’, and so on are necessary.

But Everettian quantum mechanics faces its deepest challenge in making sense of the quantum–mechanical probabilities found in the Born rule. These probabilities don’t follow from the dynamics the way they do in collapse theories, since Everettian theory does away with the probabilistic collapse postulate altogether. If the Born rule probabilities can’t be recovered in some way, then the theory will be empirically inadequate: our empirical reasons for believing in quantum mechanics in the first place come in part from the statistical distributions of experimental outcomes that we in fact observe and which are predicted, to a high degree of accuracy, by the Born rule. If Everettians can’t explain why these experimental results are expectable, given their theory, then they will fail to sufficiently connect the theory with our experience. And to show that these experimental results are to be expected is just to show that our expectations ought rationally, in an Everettian universe, to conform to the Born rule. In other words, the task for Everettians is to show why our credences, or degrees of belief, are required, given plausible assumptions about rationality, to match the Born rule probabilities.

The aim of my article is to propose and defend a rule for setting credences that in fact diverges from the Born rule. The basic idea behind this rule—which I call indexed branch-counting—is that when we observe the results of some quantum–mechanical experiment, we should assign equal credence to the proposition that we are located on any one branch of the wavefunction as to any another. For example, if we believe that our evidence is consistent with our being on any one of three branches, two of which host outcome A and one of which hosts outcome B, then our degree of belief that we will see outcome A should be two-thirds—and we should be indifferent between taking and not taking a bet at 2:1 odds in favour of A. I offer two independent defences of indexed branch-counting as a rationally required (or, at least, rationally permissible) strategy for setting credences, each of which starts with a different assumption about rationality.

Without pursuing details here, the basic assumption in the first justification of indexed branch-counting is that we should seek to maximize our utility across branches, whereby we weigh the utility of each of our successors on each branch equally. In other words, we should not care any more about the pain, pleasure, epistemic success, longevity, or whatever else of our low-amplitude successors than our high-amplitude successors, just by virtue of their having lower quantum amplitude. The basic assumption of the second justification of indexed branch-counting is that our credences should be ‘exchangeable’, meaning (in the present context) that they assign equal probability to any two outcomes that agree on the frequency of observers who see this or that outcome, but who disagree about how those observers are ‘ordered’ across branches. After laying out each justification and defending their assumptions, I respond to the common worry that branch-counting rules are somehow incoherent in light of the fact that there is no well-defined ‘number’ of branches, and hence it is impossible for agents to ‘count’ them in setting their credences. Finally, I address and critique an innovative proposal from Simon Saunders that attempts to make branch-counting consistent with the Born rule.

What is the upshot of my defence of indexed branch-counting? Well, it certainly is not that I believe Everettian quantum theory to be true, and that we in fact ought rationally to set our credences in a way that contradicts the Born rule. On the contrary: we must use the Born rule, because we have overwhelming empirical evidence that the Born rule works. (So, if you start betting against the Born rule as a result of reading my article and end up losing money, please don’t blame me!) Instead, what I am considering is what credences I would adopt if Everettian quantum theory were true, or if I believed that it were true. If my arguments succeed, they would provide some reason to believe that if Everettian quantum mechanics were true, then rational credences would deviate from those that we already know to be supported by our empirical evidence. Consequently, they would give us reason not to believe in Everettian quantum theory in the first place.

One can, of course, deny my assumptions about rationality and opt to promote and defend different assumptions that end up being more favourable to Everettian quantum theory. But I think that the arguments of this article put some pressure on the idea that obeying the Born rule would be the only rational thing to do in an Everettian universe. And since this idea is crucial to reconciling Everettian quantum theory with our experience, the arguments of this article deepen the probability problem for Everettian quantum theory, and push us to further examine alternative solutions to the measurement problem.