Are Pilot Wave & Many Worlds The Same Theory? (script) (Patreon)

It’s hard to interpret the strange results of quantum mechanics, though many have tried. Interpretations range from the outlandish—like the multiple universes of Many Worlds, to the almost mundane, like the very mechanical Pilot Wave Theory. But perhaps we’re converging on an answer, because some are arguing that these two interpretations are really the same thing.

[ ~ Space Time jingle ~ ]

If we have perfect knowledge of the universe at any point, we can plug that knowledge into the equations of the laws of physics and determine the exact state of the universe at any other time in the past or future. At least according to classical physics, in which the world is represented as particles and fields that obey the equations of physical law lock-step. Classical physics is what we call deterministic.

And then along came quantum mechanics and messed up everything. In quantum mechanics, we represent the world not as particles and fields, but as wavefunctions. These are abstract mathematical objects that encode the physical state of a system. And like a good physics object wavefunction evolves deterministically—in this case under the Schrodinger equation. But the wavefunction alone doesn’t tell you what you’ll actually see if you try to make a measurement. That action seems to introduce randomness into our deterministic theory.

Take the famous double slit experiment, for example. We send a particle—say a photon— via a pair of slits to land at a single spot on a detector. If we send photon after photon they’ll guide up a series of bands identical to the interference pattern that you’d get by sending not a single particle, but a wave that traverses both slits. The quantum wavefunction behaves exactly as such a wave, and its evolution under the Schrodinger equation accurately predicts the shape of the interference pattern we see. However, it doesn’t seem to predict the first and most obvious result of this experiment – that each photon lands at a single spot on the detector. That is, the quantum wavefunction stays wave-like when it gets to the detector, but we don’t observe this wave. We only observe a localized particle striking the detector at a single location that seems to be picked randomly.

This apparent randomness is a big deal. It suggests that Nature is NOT deterministic after all—that we can’t perfectly predict the future or recover the past simply by knowing the present.

Now, sometimes things that look random aren’t really random—they look that way because we don’t have all the information. For example, flipping a coin only seems random because we’re not smart enough to calculate exactly how the coin will spin before it lands.

But the non-determinism of quantum mechanics is taken to be fundamental—at least by the dominant interpretation of the theory. The Copenhagen interpretation states that reality is fully described by the wavefunction evolving under the Schrodinger equation, and that observation or measurement “collapses the wavefunction” in a way that is truly random. Copenhagen became the default interpretation, or “story behind the math” despite some fundamental problems and despite opposition by the likes of Einstein and even Schrodinger. And some physicists objected enough cared to seek alternative interpretations of quantum mechanics—and some that might save determinism in the process.

The two most famous alternatives to the Copenhagen interpretation are Pilot Wave Theory and the Many Worlds Interpretation. On the surface they seem as different as you can imagine. Pilot Wave theory is almost mundane, while Many Worlds feels more out-there than Copenhagen. And yet there are deep underlying connections between these two ideas that make them much more similar than you’d think. To the point that some proponents of Many-Worlds claim that Pilot Wave Theory is really just Many-Worlds “in a state of chronic denial”.

We’ve talked about both pilot wave theory and many worlds before, and those episodes are worth a look for more detail on the individual interpretations. Today’s episode is about comparing them, and perhaps to see which is the more worthy successor to Copenhagen.

Before we dig into the alternate interpretations, let’s say a bit more about what’s wrong with Copenhagen. For one thing, the Schrodinger equation tells us how the wavefunction evolves, but it doesn’t directly give us the probability of getting a particular result when we make a measurement. That requires something called the Born rule, which just says that you square the wavefunction to get the probability distribution. But Copenhagen doesn’t tell us where that squaring comes from—it’s just accepted because it works. Another problem is that the collapse of the wavefunction has to happen across the entire wavefunction instantaneously. If the photon is detected at this spot on the detector then the chance of it appearing anywhere else becomes instantly zero. This violates something called “locality”, and implies a sort of faster-than-light influence, which on its surface violates the core axiom of Einstein’s relativity.

So, do the alternatives do any better? Let’s start with pilot wave theory. This was first proposed by Louis deBroglie and then fully developed by David Bohm. The idea is that the wavefunction behaves exactly as in standard quantum mechanics, but that it’s not really the wavefunction that is being directly measured. Rather, the wavefunction acts to guide the motion of an actual particle, which we’ll call a corpuscle to avoid confusion. A “entire” particle, like a photon, is a corpuscle riding in its wavefunction, which we’ll start calling the guiding wave when we’re referring to pilot wave theory.

In the double slit experiment, the guiding wave goes through slits and interferes with itself to produce the classic interference pattern. But we don’t measure the guiding wave, only the corpuscle. And the corpuscule only goes through one slit, riding the wave and being deposited on the detector at a single spot, although that location is influenced by the interference pattern.

The guiding wave for consecutive particles might look exactly the same to us, but the corpuscule riding each can take different paths. After many of them they’ll map out the shape of the guiding wave at the detector. But how does each corpuscule know which path to take if it’s not random? In Pilot Wave Theory, the answer to that is the same as the answer to why does a coin land heads versus tails. There’s information that we aren’t aware of that defines the detailed path—information that’s not in the guiding wave alone. That makes Pilot Wave Theory a so-called hidden variable theory.

In Pilot Wave Theory the hidden information is in the form of the exact position of the corpuscule, and that isn’t coded in the wavefunction. We also need to add something called the guiding equation, which helps the corpuscule pick out a particular path. It depends on the wavefunction and on the position of not only the corpuscule in question, but also on every other corpuscule in the universe that this system is entangled with. We’ll come back to that point.

So in the Pilot Wave picture, the “world” is made of these corpuscules being pushed around by these guiding waves under the omniscient overview of the guiding equation. It’s a bit like how the electromagnetic field can push around an electron, but with an enormous difference. When an EM field shoves an electron, the electron shoves back. For pretty much every type of particle and field interaction there’s this sort of back-reaction. Except for pilot waves. The guiding wave is completely unaffected by the corpuscle—and this is necessary if we want the Schrodinger equation to be strictly obeyed. The guiding wave will channel the corpuscule and the guiding wave will interact with itself and with the guiding wave of other particles, but it completely ignores the presence or absence of the corpuscule within it. This is going to be extremely important in a minute.

So if we know the initial positions of all the corpuscles we could use wavefunction to predict the result of any experiment. In pilot wave theory the apparent randomness of quantum mechanics is not fundamental, but instead emerges from our lack of knowledge about the universe.

We’ve now solved the non-determinacy problem of Copenhagen. We’ve also solved the question of what causes the collapse of the wavefunction. It doesn’t need to collapse. In Pilot Wave theory the parts of the guiding without the corpuscle just continue on because, remember, the corpuscle has no effect on it. Presumably we can ignore this empty guiding wave because WE only detect the corpuscle.

Let’s now look at what happens at the detector when the photon wavefunction arrives. For both Copenhagen and pilot wave theory the photon wavefunction reaches the screen and encounters electron wavefunctions in the detector pixels. An electron wavefunction in one of those pixels is excited, nudging electrons further down the circuit pathway so that eventually a photon gets emitted by a computer screen indicating which pixel was excited, and so recording the arrival location of the photon.

But what happens to the photon wavefunction everywhere else on the detector? Well, we know that it's possible for electrons to be in a state of simultaneously excited and not excited—what we call a superposition of states. That would also happen if, for example, we'd placed a single electron inside just one of the slits. If the photon can be in a superposition of passing through both slits, then the electron enters a superposition of being both excited and not excited. That's true for Copenhagen and pilot wave theory. In the latter case, the corpuscle of the photon really does pass through one slit or the other. If it passes through the slit with the electron, then the corpuscle of that electron follows the excited part of the electron wavefunction. If the photon corpuscle passes through the other slit, then the electron wavefunction is excited by an empty segment of the photon wavefunction. The guiding equation then forces the electron corpuscle to remain in the unexcited part of its wavefunction, while the excited part of the electron wavefunction remains empty.

Same at the detector. Electrons in all pixels enter a superposition of excited and not excited as the photon wavefunction reaches all points on the detector. And the part of these electron wavefunctions that become excited will in turn excite neighboring electron wavefunctions. All across the detector, this superposition should propagate through the circuitry--a cascade of interactions that leads to a record of the photon landing at that pixel and at the same time silence from that pixel. So at least for a little bit, the world is in a complex superposition of each pixel being excited and not.

Of course we only observe a signal from one pixel. So, what happens to the cascades from all of the others? Well, Copenhagen would tell us that somewhere between the photon arriving at the detector and us becoming aware of the signal from the screen, all of the extraneous superpositions die off, leaving only one excited pixel. The wavefunction collapses.

However pilot wave theory doesn't collapse the wavefunction. IT would say that an electron wavefunction in all pixels does enter a superposition of excited and not excited. But all of those excited states are empty of corpuscules except for one. Wherever the photon corpuscle arrives, that electron corpuscle follows the excited electron wavefunction. The cascade of excitations from that one pixel all carry corpuscles.

So we end up with this interesting situation where we have many similar cascades of excitation, but only one of them is "real" because it carries corpuscles. The rest are sort of phantoms--empty superpositions. In pilot wave theory, the cascade of superpositions doesn't end at the detector, nor at the computer screen. Instead, we end up with a superposition of our brain in many states--each corresponding to an awareness of the photon landing in a different spot. But only one of these is "real" because it carries corpuscles. The rest are the phantom, corpuscle-empty wavefunctions of our brains as they would have looked if the particle landed in other locations.

This is probably a good time to get to the Many Worlds Interpretation. Fortunately I don’t have to describe how our double-slit experiment looks in Many Worlds because I just described it. Take pilot wave theory and just subtract the corpuscles. The corpuscles served as markers of which of these diverging alternate realities was the real one. But the corpuscles themselves have no influence on the wavefunction, so they really are just labels. Without the corpuscle, all the branches of the wavefunction are empty—all are ghosts, or perhaps none of them are.

The wavefunction in pilot wave theory and in many worlds behaves exactly the same way. And that includes not collapsing, and not being random. The main difference between the two is the existence or not of the corpuscules. In pilot wave theory, corpuscules mark real physical reality,  while many worlds accepts that the wavefunction is the primary and really sole elementary constituent of physical reality.

Another difference is the guiding equation in pilot wave theory. Many worlds doesn’t need it because the evolution of the wavefunction alone only needs the Schrodinger equation. This guiding equation presents its own issues. The main one is that it needs to know about not just the corpuscle in question, but also every other corpuscle connected to it via entanglement. It needs this to ensure that all corpuscles are always funneled into the one “real” world. This means that the Guiding Equation is explicitly non-local, causing instantaneous interactions between corpuscles across arbitrarily large regions of space. In fact, we now know that it’s impossible for any hidden variable theory to preserve locality. Because of this no one has managed to make pilot wave theory play nice with special relativity. However it’s easy enough to make our theory relativistic if we just delete the corpuscles—but then of course we have Many Worlds.

Occam’s razor tells us that the simplest explanation is usually the most correct. Don’t add parameters to your model if they aren’t needed. Some have said that the branching timelines of Many Worlds is an addition that goes against Occam’s razor—but proponents of Many Worlds say that these timelines are a core prediction of raw quantum mechanics. To get rid of them you have to add something: either the collapse of the wavefunction in Copenhagen, or the tagging of the “one true world” with corpuscles in pilot wave theory.

Let’s ask one more question: do any of these interpretations explain the Born rule? We know that Copenhagen just takes it on faith that you need to square the wavefunction to get the probabilities. Do they fall out more naturally in pilot wave theory or many worlds? Well, in pilot wave theory the probability of getting a measurement in a particular location depends on the distribution of corpuscles. It turns out that if you start the universe with the corpuscules distributed according to the Born rule—so their density being proportional to the square of the wavefunction—then at all later times they’ll retain that distribution, and so the Born rule will continue to work. But pilot wave theory doesn’t tell us how the corpuscles got distributed that way in the first place.

In many worlds, the probability of recording a particular measurement value is the probability of you finding yourself in a timeline where that value actually happened. So it’s about the density of timelines—how many alternate realities correspond to your measurement. And the fact is, you CAN recover the born rule with this sort of statistics of alternate worlds. You can calculate the probability of landing in one particular world. And the probability that you’ll learn how to do this close to 100%, as long as your quantum timeline includes a near-future episode of Space Time.