Playing games in many worlds - Part II
In the previous article we looked at an intriguing result, developed by David Deutsch and David Wallace, which claims to make sense of the Born rule in the many-worlds interpretation of quantum mechanics. But does it?
Proponents of the many-worlds view argue that the Deutsch/Wallace result makes sense of the Born rule in the many-worlds context. The numbers it attaches to each outcome of a measurement, or more generally to each branch of the universe, are no longer redundant or mystifying. They are the numbers that any rational person must use in their decision making.
The idea that human behaviour can explain quantities that have popped out of theoretical physics might seem weird at first. In the traditional interpretation of quantum mechanics the numbers given by the Born rule are interpreted as probabilities. Probability is something solid and objective, so we are happy to let that stand. But human behaviour? Solid and objective?
Rationality and probability
Decision theory can be used to understand the calculus of probability.
There is a two-part answer to this qualm. In their arguments Deutsch and Wallace assume that people's preferences stick to certain basic principles of rationality. For example, when faced with a variety of actions, a person should be able to order them according to preference. The result says that as long as these rationality principles are adhered to, there is only one possible course of action when you are faced with a set of decisions in a branching universe. This course of action involves the numbers given by the Born rule. Viewed in this way, the Born rule begins to acquire the flavour of inevitability we expect from a physical law.
The second answer is that even in an ordinary, non-branching, world probability, as a concept, is far from solid and objective. We say that the probability of heads coming up when you toss a perfectly symmetric coin is 1/2. But how can you justify this statement? You can toss that coin many, many times and note that it comes up heads in roughly half of the tosses. But it's hardly ever going to be exactly half of the tosses. So how can we insist that probability is something fixed and absolute? "Probability seems to be something perfectly objective, it's not just a matter of opinion," says Wallace. "But equally it's not clear what that something objective is."
People have looked to decision theory as a way of making sense of probability long before Deutsch and Wallace adapted the idea for branching universes. "One way of seeing what somebody's own assessments of the probabilities are is to ask them what bet they would take in a given circumstance," says Wallace. "Even if you don't understand probability, you can still understand action, preferences, choices and decisions. This kind of strategy as a way of understanding why probability calculus has the form it does has been very influential."
If you are happy to think of probability in terms of the role it plays in rational decision making, then there is not much difference between the interpretation of the Born rule in the many-worlds view and its interpretation in the one-world view. "The question of whether [the Born rule numbers] are really probabilities, or just [pretend] probabilities, starts to collapse into a question of language," says Wallace. And, crucially, there is no reason to reject this interpretation in the many-worlds view if you accept it in the one-world view.
And there is something else. Traditional decision theory, applied in a one-world situation where a coin comes up heads or tails, but not both, tells you how to make rational decisions, like accepting a bet on a coin toss, based on what you believe the probabilities are. If you happen to falsely believe that the probability of heads is 0.99, then that's your problem. It doesn't contravene any of the rationality axioms.
But the Deutsch/Wallace result goes further. Not only does it tell you how to make optimal decisions using some probabilities (if we call them that), it tells you that those probabilities must be the ones given by the Born rule. So what started out as a weakness of the many-worlds view, not knowing what the Born rule meant, lead to a result that is stronger than its counter-part in ordinary decision theory.
Not everybody is convinced however. The Born rule is something we observe experimentally. If it says that the probabilities of observing spin-up and spin-down are both 1/2, and you repeat the experiment of measuring spin many times, then roughly half of the times you will measure spin-up and the other half spin-down. Any decent scientific theory should explain these experimental observations directly. "[The Deutsch/Wallace result] doesn't tell you why you see experimental outcomes that follow [the Born rule]," argues Adrian Kent, a quantum physicist at the University of Cambridge who opposes the many-worlds view. "What we need is some story about probabilities, or some other concept that replaces probabilities, that has direct scientific use. The whole thing about decision theory is answering the wrong question."
Probability is a slippery concept. Image: Diacritica.
Another sticking point are the principles of rationality that underlie the Deutsch/Wallace result. "[What if] I don't believe the [principles] of rationality?" asks Kent. "Why are they right? In the end no-one can prove that they are. I think there are perfectly well-defined strategies for which you can make a rational case and which don't satisfy those axioms."
As an example, one of those principles says that there should be no conflict of interest between yourself and one of your future selves. If one of your future selves prefers being rich to being poor, then your present self should also prefer that future self to be rich rather than poor.
But suppose you can ensure that your future selves are either filthy rich or poor but decent. You may prefer being filthy rich, but for moral reasons you may still decide to make sure that half of your future selves are poor. This violates the principle: your present self wants some of your future selves to be poor but the poor future selves might themselves want to be rich. But can this strategy really be deemed irrational?
Kent also points out that mathematically it is not entirely clear exactly how the world splits into branches and which numbers the Born rule attaches to each branch. How can a person possibly make decisions about whether or not to accept a bet without that information?
The debates surrounding these issues are subtle and technical. For proponents of the many-worlds view, the Deutsch/Wallace result is a triumph. By suggesting a meaning for a major component of the mathematics of quantum mechanics it has removed a major obstacle to taking the many-worlds view seriously. And there is no doubt that the result is intriguing at the very least.
Other evidence for many worlds?
Ultimately though hardline opponents of the many-words view will only be swayed by more direct experimental evidence. Is there any on the horizon? The many-worlds theory hinges on the idea that superposition exists not only in very small systems, involving small particles or molecules, but all the way up to the scale of the universe. So whenever an instance of superposition in a larger system is confirmed in the lab, that can be counted as evidence for the many-worlds view.
But the tell tale signs of superposition are very delicate: you can only observe them in systems that are extremely well isolated from their surroundings (see this article for more). For larger systems this level of isolation is currently beyond reach. But experimentalists are hard at work, not least because we need such methods to build superfast quantum computers, which exploit superposition. It's a long way away, but perhaps one day the shadowy signatures of our other selves will be revealed.
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