Paper Review: The measurement theory of Everett and de Broglie’s pilot wave

I’ve written before on this blog about quantum mechanics. We’ve looked at a questionable interpretation of a recent experiment, the way in which quantum mechanics is radically nonlocal, and certain theoretical constructs needed for the thought experiments used in physics.

One of the main reasons why quantum physics fascinates people is because the phenomenon themselves violate our intuitions. Indeed, when first confronted with the empirical results, it is challenging to imagine a theory that can handle them. Of course, such theories have been developed. However, we have yet to converge on a single theory as the correct theory of quantum mechanics. Instead, we find ourselves faced with a constellation of competing theories. The field of quantum mechanics is not a monolith.

One of the people who thought deeply about these issues was John S. Bell, a physicist. Actually one of the three paper reviews I linked to above was about one of his papers, and another about one of the fundamental results he proved in quantum physics. In the paper I’ll review in this post, Bell is interested in the connection between two prominent and competing theories of quantum mechanics: Everett’s and de Broglie’s.

***The original paper can be found here.***

The central claim of Bell’s paper is that “the elimination of arbitrary and inessential elements from Everett’s theory leads back to, and throws new light on, the concepts of de Broglie.” (p. 93).

This, then, is what Bell thinks the relationship between these two theories is. If we take away some bits of Everett (some of which we will see are actually absences of structure as opposed to structure), we recover de Broglie. It is also clear where Bell’s sympathies lie through his choice of words. Furthmore, anyone familiar with Bell’s work will know he is a big fan of the pilot wave theory.

Bell starts off by given a brief summary of the history and reasoning that led to Everett’s theory. In the standard theory of quantum mechanics there is a strong separation between an observer and the rest of the world. The dynamical laws of quantum mechanics are framed in this way. Bell writes, “this usual interpretation refers only to the statistics of measurement results for an observer intervening from outside the quantum system” (p. 93).

Everett, however, was concerned with a quantum theory of cosmology. This applies pressure to the standard theory for, when we are doing cosmology, we are trying to describe the whole universe. But if “that system is the whole world, there is nothing outside: (p. 93). Thus there is no observer outside the universe, and it is unclear how the standard theory can deal with this.

This led Everett to develop a theory that treated the world and the observer in exactly the same way. Thus, it could describe the universe as a whole, because the same dynamical laws were applied to any and all parts of the universe. There was no need for an observer-external world reality.

I’m trying to stay fairly high level here, but I do want to give a quick sketch of the theory so that we can better understand Bell’s argument for how it connects to the pilot wave theory. The sketch will be sketchy, and all too brief.

In the standard theory physical systems can be prepared in certain states called super positions. For example, a particle could be in a superposition of being both inside and outside a certain box. We encounter nothing like this in our ordinary experience of the world. To us, it seems like properties like position are always well defined for an object. However, in the standard theory this is not true. A particle can fail to be in the box and fail to be outside of the box, instead being in a kind of combination in both of these states.

However, whenever we go to look for the particle, we either find it inside the box or outside. But we just said it was neither. So what happened here? In the standard theory, whenever we measure some property — like whether or not the particle is in the box or outside of it — we always end up with a determinate result, even if the particle was in a superposition, because our measurement itself causes a collapse of the superposition to one of the possible measurement outcomes. I will call this the collapse dynamics. Furthermore, if a measurement does not occur, then the system evolves in a different, collapse-free way. I will call this the linear dynamics.

Everett, in contrast, throws out the collapse dynamics — he sticks with just the linear dynamics. This removes the strong metaphysical separation between observer and external world. All situations can be treated by the same law.

This leads to another puzzle though. If the particle is in a superposition, what happens when we look in the box? Bell writes

Everett disposes of this vaguely defines suspension of the [linear dynamics] with the following bold proposal: it is just an illusion that the physical world makes a particular choice among the many macroscopic possibilities contained in the expansion; they are all realized, and no reduction of the wave function occurs. He seems to envisage the world as a multiplicity of ‘branch’ worlds…

p. 95

This “reduction” Bell writes about is the collapse dynamics. So in throwing out the collapse dynamics, instead of finding either the particle in the box or out, I find it in both places. Or, rather, there is one branch world in which I find it in the box, and one in which I find it out of the box. Of course I would not notice this. For when I carry out the measurement, my experience in a sense splits into the two different experiences. One Daniel finds it in the box, one out. Up until then it was the same stream of experience; after the branch it is two, stemming from a common source.

Of course, we can describe this talk of “experience” physically in terms of memory of a physical system (stored on a magnetic tape, for example), and this is precisely what Everett does. This, then, is a very quick and dirty introduction to Everett’s theory, sometimes called “many worlds”.

Bell is unsatisfied with this account for two reasons. The first is a problem with what he calls “expansions”. This is a tricky one to explain without resorting to the mathematics of the theory, but the following is an attempt. I’ll rely on the explanation I gave here — scroll down to the pictures of the clocks for the relevant section.

The issue is that when we are decomposing the vector describing the quantum state — that is, writing it out as a sum of some other vectors, each of which corresponds to a certain observable feature — there are an infinite number of ways to do this.

This is abstract, so here is what I hope is a helpful analogy to get the point. We have some physical system, like a metal rod. It might be a certain length, say 1 meter. My choice of using meters doesn’t fix the length. I could describe it in feet, or furlongs, or football fields. The method of description doesn’t affect the actual physics.

This should be the same for our choice of how to decompose the quantum state into a sum of other states. In Everett’s theory, it is the summands of this sum that are the equally real worlds. However, on one reading of Everett’s theory, this implies that the physical interpretation does depend on the way in which the state is decomposed, or expanded. Since there are an infinite number of ways, Bell claims that Everett is implicitly smuggling in a preferred basis — a preferred set of summands — into his theory.

This is the first point in which Bell draws a connection between Everett’s many worlds theory of quantum mechanics and de Broglie’s pilot wave theory of quantum mechanics. In the pilot wave theory a particular observable — position — is made determinant. That is, contra my example earlier about the particle not being in the box or out of the box (as in standard quantum mechanics), in de Broglie the particle actually does have a determinate position at every point in time.

We can see this connection clearly when Bell writes

This preference for a particular set of operators is not dictated by the mathematical structure of the wave function ψ. It is just added (only tacitly by Everett, and only if I have not misunderstood) to make the model reflect human experience. The existence of such a preferred set of variables is one of the elements in the close correspondence between Everett’s theory and de Broglie’s — where the positions of particles have a particular.

p. 96

Bell’s second concern is that

if instrument readings are to be given such a fundamental role should we not be told more exactly what an instrument reading is, or indeed, an instrument, or a storage unit in a memory, or whatever? In diving the world into pieces A and B Everett is indeed following an old convention of abstract quantum measurement theory, that the world does fall neatly into such pieces — instruments and systems.

p. 96

Bell thinks that Everett’s theory relies again on a kind of dichotomy between the measurement system and the environment. Furthermore, he thinks that de Broglie has given a formulation that does not suffer from the duality.

I agree with Bell on the second point, but I don’t agree on the first. Of course I might be missing something here that Bell has seen, but my best understanding of Everett’s theory is that the choice to treat one system as measurement device and the other as the thing being measured is going to depend on what specific thing we are trying to calculate, and is not a fundamental part of his theory. Indeed, by representing the quantum states of the two systems as one state in the join Hilbert space allows one to apply the linear dynamics to the joint system, thus removing this duality. However, given how sophisticated a reasoner Bell is, I definitely want to think about this more to see if there is anything I am missing.

Now Bell turns his attention toward de Broglie’s theory. This theory is definitely challenging to explain concisely, but here is a shot at giving an intution. Before with Everett we had that the state of the universe at any point in time was described by a wave function. All elements of this function (speaking loosely) were equally real. If there was a superposition of the particle being in the box and out of the box, both of these were equally real. In de Broglie’s theory, on the other hand, each particle has a specific location at each time. The evolution over time of the particle’s position is governed by the wave function very similar to the linear dynamics; a helpful analogy is that it moves like a massless particle being pushed around by an infinitely compressible fluid.

In this picture there are also no collapses, and certainly no deep division between observer and observed. We can also think of this theory (a little informally) as picking out a particular term of the wave function to be the “real” one (when it comes to position), in contrast to Everett’s assertion that all are equally real. Bell writes

This model is like Everett’s in employing a world wave function and an exact Schrödinger equation, and in superposing on this wave function an additional structure involving a preferred set of variables.

p. 97

So we see with Bell that there is a deep similarity here, with the addition of extra structure in the de Broglie pilot wave theory that picks out a unique position to be the actual one.

Bell then notes three main differences between Everett’s theory and de Broglie’s:

  1. Everett’s theory is “vaguely anthropocentric” (p. 97), whereas de Broglie’s is not, since we can average over the positions of all particles composing a composite object to get a course grain description of the macroscopic world from the microscopic. The comment about Everett being anthropocentric or dualist in some way was the one with which I disagreed earlier.
  2. Everett’s theory has that all elements of the superposition are realized “each in the appropriate branch universe” (p. 97), whereas de Broglie’s theory picks out a unique state to be the real one by specifying an actual position for each particle. This is what I mentioned just before this list.
  3. Everett doesn’t really have a solid account of the continuous trajectories of objects, whereas, since in de Broglie’s theory each particle has a determinate position, de Broglie can provide an account with continuous deterministic trajectories.

These, then, are the main relationships between the two theories. Bell ends the paper by remarking that one could do something like de Broglie’s theory without trajectories if one wanted, but writes

But I do not like it. Emotionally, I would like to take more seriously the past of the world (and of myself) than this theory would permit. More professionally, I am uneasy about the possibility of incorporating relativity in a profound way.

p. 98

We see here then a crucial lesson for science and the philosophy thereof: whenever evaluating a physical theory, we have to ask ourselves “what do we want from it?” Bell is laying his cards on the table, stating what he finds attractive and unattractive about the different possibilities. This is an essential step when formulating theories about the world.

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