A Primer on the Interpretations of Quantum Mechanics, Part 4

I was going to wrap things up with this article, but I decided that the Everett-tradition deserves its own post, so I’ll wait until next week for a final post taking a broader view of these issues and tackling what it all means.

Here are the previous parts in this series if you missed them:

Part 1 (Introduction to QM)

Part 2 (Collapse Interpretations)

Part 3 (De Broglie – Bohm Interpretation)

In 1957, Hugh Everett III submitted, for his PhD thesis, a dissertation arguing that Quantum Mechanics can be understood without invoking wave function collapse and without adding anything to the theory over and above the wave function. His proposal thus stands in contrast to both the orthodox, collapse-based interpretations and hidden variable interpretations like Bohm’s. He called this the “relative state formulation” of Quantum Mechanics.

The trouble is that no one can quite agree on how to interpret Everett’s interpretation1. Indeed, you’ll often see people claim that Everett proposed the Many Worlds Interpretation; but in fact, his paper says nothing whatsoever about “worlds”. As with Niels Bohr and the Copenhagen Interpretation, it’s an interesting exercise to read Everett’s thesis and try to discern exactly what his view actually was. But for the purpose of this article, that’s beside the point. Instead, let’s look at the various interpretations that have been proposed as ways of understanding Everett.

The basic idea behind all of these interpretations is that the superpositions never go away, and that, strange as it might seem, measurements don’t have determinate outcomes. Everett’s central claim is that QM nevertheless provides a perfectly good description of the universe, and in particular that beings with brains will go around acting as if experiments have determinate outcomes. In Everett’s terminology, there is a subjective impression of definite outcomes while the objective physical state of the universe remains in a superposition.

The Bare Theory

The boldest reading of Everett is what’s called “the Bare Theory”. To introduce it, I want to think again about the kind of state that the linear dynamics of QM predicts will occur after a measurement of the x-spin of an electron (and let’s name the experimenter Alice):

ψ = a|x-up>|pointer-up>|Alice-sees-up> + b|x-down>|pointer-down>|Alice-sees-down>

Up until now, we’ve arrived at this state and immediately said, “This makes no sense. Alice has to either see the pointer pointing to ‘up’ or pointing to ‘down’. It can’t be two things.”

But let’s think a little bit about what this state would actually be like. In particular, let’s imagine that we have another experimenter, Barbara. Suppose that Barbara does not look at the device, but asks Alice about what she saw when she looked at the measuring device. We can assume that Alice is a competent observer and reporter of her own mental states, and is going to honestly answer Barbara’s questions.

Let’s suppose Barbara asks Alice what result she saw for her measurement. In essence, what Barbara is doing by asking Alice a question is performing a “measurement” of Alice’s brain. If Alice is a competent “measuring device”, then she will answer “up” if she’s in the state
|Alice-sees-up>, and that will put the combined Alice/Barbara system in the state
|Alice-sees-up>|Barbara-hears-up>. Similarly, if she’s in the state |Alice-sees-down>, then she’ll answer “down”, and the Alice/Barbara system will end up in the state

So, if we start with the superposed state that we get when Alice performs her measurement, then from the linearity of the Schrödinger Equation, we know that after Barbara asks her question, we will end up in the state:

ψ = a|x-up>|pointer-up>|Alice-sees-up>|Barbara-hears-up>
+ b|x-down>|pointer-down>|Alice-sees-down>|Barbara-hears-down>

That is to say, once Alice tells Barbara what result she saw, Barbara becomes entangled with Alice (and the pointer, and the electron). This doesn’t solve anything; we’ve just extended the strange superposition further.

But let’s suppose that instead of asking what result she saw, Barbara says to Alice “Don’t tell me what result the experiment had. Just tell me, did the measurement have a definite result?” What will Alice’s answer be? Well, if Alice is in the state |Alice-sees-up>, she will of course say, “Yes, the experiment did have a definite result, and I saw that result and am thinking of it right now.” Let’s call Barbara’s state after hearing that answer
|Barbara-hears-yes>. Similarly, if Alice is in the state |Alice-sees-down>, she will say exactly the same thing. But, again, because the dynamics of QM are linear, we can conclude that if Alice starts off in our superposed state and Barbara asks that question, we will end up in the state:

ψ = a|x-up>|pointer-up>|Alice-sees-up>|Barbara-hears-yes>
+ b|x-down>|pointer-down>|Alice-sees-down>|Barbara-hears-yes>

This is a state where Alice is in a superposition of seeing different results, but nevertheless Barbara got a definite answer to her question, and that answer was “Yes, the measurement had a definite result.” The experiment did not have a result, but Alice (who, remember, we are assuming to be completely honest) has, unambiguously, just told Barbara that it did. And this is a really neat little trick. Alice, in other words, believes2 that the measurement had a definite result. So we have just shown that, even though measurements in this version of QM never have results, experimenters will believe that they do.

Let’s imagine now that Barbara does ask Alice what particular result she saw, and then Barbara looks at the measurement device herself to see if it agrees with what Alice told her. Then a third researcher, Chelsea, asks Barbara if she agrees with Alice about what result the measurement had. It’s pretty easy to see that the state in which the pointer is pointing to “up”, and Alice tells Barbara she saw the pointer pointing to “up”, is one that will lead to Barbara also seeing that the pointer is pointing “up”. Barbara will therefore agree with Alice about the result of the experiment, and will say as much to Chelsea. The same applies to the “down” state. And so, by a similar argument to that above, we can also conclude that in the superposed state, Barbara will unambiguously report that, after checking the result herself, she can confirm that Alice was correct about the measurement.

So even though these superpositions never go away, everyone ends up acting as if the measurement had a definite result, even to the point of agreeing with each other about what that result was. And in a sense, this looks like it might be a perfectly good solution to the measurement problem. The idea is that we’ve shown that there never was a measurement problem, because the impression that measurements have definite outcomes is an illusion. This proposal is the “Bare Theory” interpretation.

The problem with this is that it is hard to figure out how to make sense of probabilities in this interpretation. Our whole reason for believing in Quantum Mechanics, after all, is that the results of our measurements follow, statistically, the probabilities given by the square amplitudes of the wave function. Now, it can be shown that, in the Bare Theory, in the limit where an experimenter repeats an experiment a very large number of times, her belief about the statistical distribution of the results will approach that predicted by standard Quantum Mechanics. What that means is this: suppose Barbara keeps giving Alice electrons that are in a state like a|x-up> + b|x-down>, which she keeps feeding into her measuring device to measure the x-spin. And suppose Barbara ask her not about the result of any particular measurement, but rather “What fraction of the measurements have given you a result of x-up?” There won’t be a determinate answer for that; different ‘branches’ of her wave function will give Barbara various answers (and thus entangle Barbara with her state). But as the number of measurements increases, the amplitude for the answer “a2” (the probability predicted by QM) will increase, and if she could perform an infinite number of such measurements, then her answer would be a definite “a2“.

That’s as strong a statement about probabilities as the Bare Theory can make, and it’s not a very strong one at all. And indeed the problem is deeper than that; in a sense, the Bare Theory would seem to get things backward. A physical theory is supposed3 to describe and predict the empirical experiences of observers. Those experiences are the given – the data – and are supposed to be the one sort of thing that is not in dispute. The Bare Theory, on the other hand, would seem to say that our very experiences are not what we think they are, and that seems to me to be a contradiction in terms. And in particular, if we say, as Bare Theory does, that our impression of empirical observations is an illusion, then we have undermined the only reason for believing in Quantum Mechanics (including Bare Theory) in the first place.

Many Worlds

Other variations of Everett’s basic idea start from the same premise – that the wave functions constitute a complete physical description of the universe, and that they always obey the linear dynamics of the Schrödinger Equation – but reinterpret, to one degree or another, what a superposition means. And the most famous of these is the proposal that we should interpret a state like this:

ψ = a|x-up>|pointer-up>|Alice-sees-up> + b|x-down>|pointer-down>|Alice-sees-down>

. . . as saying that there are two universes, one in which the electron has turned out to have x-spin up and another in which it’s turned out to have x-spin down. And because of the way the linear dynamics work, if this superposed state continues to evolve according to the Schrödinger Equation, then in each universe, it will seem to each version of Alice as if the original superposition has collapsed, just as in standard Quantum Mechanics.

This is the Many Worlds Interpretation, put forward by Bryce DeWitt in 1970, and called by him the “EWG interpretation” after Everett, John Wheeler (Everett’s graduate adviser), and R. Neill Graham (DeWitt’s graduate student), and it soon became the standard way of understanding Everett’s original proposal4. Let’s repeat its core claim, because it’s so outlandish: it claims that reality as a whole is described by the wave function and the Schrödinger Equation, but that reality consists of many equally real worlds, one corresponding to each term in the universal wave function.

These worlds are, in principle, not causally isolated, because it’s always possible for interference phenomena to arise between terms in the universal wave function. A simple example would be a photon in the two-slit experiment; the world splits into two, one in which the photon passes through the left slit and one in which it passes through the right slit5, but these two worlds interact with one another to create the interferenc in the wave function that leads (after many repetitions of the process) to the pattern of light and dark bands. In practice, however, it will be incredibly unlikely for interference between macroscopically different worlds to occur6, so in effect, these different worlds are mutually unobservable.

Now, there’s a technical problem here, in that there are different, mathematically equivalent, ways of writing down a wave function, and these will generally contain a different number of superposed terms. That’s fine under most interpretations, but in many-worlds, different ways of writing down the universal wave function would seem to imply that there were different numbers of universes. For example, imagine that reality contained just one electron in a state of definite x-spin, ψ = |x-up>. That would seem to be one universe, when written that way. But if we write that same wave function in terms of the y-spin of the electron, it would be ψ = 1/√ 2 |y-up> + 1/√ 2 |y-down>, which would seem to suggest two universes. But the number of universes that exist seems like it ought to be a definite number. Similarly, a particle in a state of definite position would seem to imply one universe, but we know (from the Uncertainty Principle) that if it’s in a state of definite position, it’s in a maximally indeterminate state with respect to momentum, which would seem to imply infinite universes. The only way to solve this problem is to assume that there is a “preferred basis” – a special set of definite states, so that we can say that each part of the universal wave function corresponding to one of those states constitutes its own universe. Position would seem to be a natural choice for this, and indeed we’ve seen that position plays a special role in both GRW and Bohmian mechanics as well.

A bigger problem is, as with the Bare Theory, how to make sense of probabilities in the Many Worlds Interpretation. If all possible results of a measurement do, deterministically, happen, then it what sense can we associate a probability with each branch? And since it is the (apparent) fact that the results of measurements do, statistically, conform to the probabilities of QM, the Many Worlds Interpretation would seem to undermine itself in the same way as the Bare Theory.

On top of these technical issues, of course there’s the fact that, if understood in anything like a literal way, the Many Worlds Interpretation is ostentatiously un-parsimonious. It entails, after all, the metaphysical baggage of an infinite (or at least mind-bogglingly huge) number of equally real universes. Various writers have tried to soften this by construing things in a less literal way, leading to the so-called “Many Histories” or “Many Threads” interpretations, the idea of which is, more or less, that although there is only one universe, there are many mutually exclusive, but equally true, stories that can be told about that universe. And sure, this makes things sound slightly less ridiculous, but it’s hard to see what such a claim might actually mean – and, of course, this does nothing to ameliorate the problem of probabilities or of a preferred basis.

Many Minds (and Single Mind)

As a way of reducing the extravagant metaphysics of an Everett-type interpretation, we might posit that, while there is only one universe, the parts of the wave function containing |Alice-sees-up> and |Alice-sees-down> represent, essentially, different Alices co-existing in the same universe. It turns out that (as was first pointed out by David Albert and Barry Loewer in 1988) something along these lines can also be used to make sense of the probabilities in an Everett interpretation.

Suspend your sense of absurdity for a moment and imagine that each physical brain in the universe corresponds to an infinite number of minds. When a brain enters a superposition of physical states that imply different mental states, each of those infinite minds probabilistically ends up in one of those possible mental states, following the expected probabilities from the wave function. If the number of minds associated with each brain is infinite, then we are guaranteed that the distrubition of mental states across those minds will match the square amplitudes of the wave function. This is somewhat like a version of Many Worlds where, instead of taking something like position as the “preferred basis”, the preferred basis is that of mental states.

The key here to making sense of the probabilities is that instead of a single world branching into multiple worlds, we start with an infinite number of minds. If it’s a single world branching, then there’s nothing you can sensibly ask the probability of. But in the Many Minds picture, you can point to some one of those infinite minds and ask, “What is the probability of this particular mind ending up in that particular mental state?” I think it’s worth noting, though (and I haven’t seen this pointed out elsewhere), that it’s not the fact that we’ve gone from Worlds to Minds that makes the probabilities understandable. One could just as easily play the same trick within Many Worlds, and posit that, rather than a universe that branches when wave functions enter superpositions, we start with an infinite number of universes, and each one probabilistically ends up in a state corresponding to one term in the wave function.

That aside, Many Minds would seem to be almost as metaphysically extravagant as Many Worlds. A more parsimonious version of it can easily be constructed, though – instead of an infinite number of minds, imagine that there’s just a single mind associated with each brain, and that it probabilistically ends up in one of the possible mental states corresponding to the wave function of its brain. (For that matter, a solipsistic version could posit only a single mind associated with one brain). It might be objected that, even in the Single Mind version, this picture suffers from a bad case of dualism. But I would point out that, first, the problem of how mental states supervene on brain states is something that will always be an issue anyway (quite apart from Quantum Mechanics) and, second, that if one takes a truly positivist view (that scientific theories are ultimately just descriptions/predictions of empirical experience) then this cannot be said to be a problem.

So it seems that some version of an Everett interpretation (which, depending on what particular metaphysical baggage you want to associate with it, you might call Many Worlds, Many Histories, Many Minds, or Single Mind) could do the trick and provide a way of understanding Quantum Mechanics without collapse and with only – or perhaps, almost only – the wave function.

That’s all I’m going to say about that now, but next week I’ll have one last part in the series, where first I’ll talk about a few odds and ends that I didn’t mention elsewhere and then I’ll briefly talk about where all this leaves us.