Pilot Wave Theory

There’s one interpretation of the meaning of quantum mechanics that somehow manages to skip a lot of the wildly extravagant, or near mystical ideas of the mainstream interpretations: it’s DeBroglie-Bohm Pilot-Wave theory. Despite it’s alluring intuitive nature, for some reason it remains a fringe theory.

Misinterpretation of the ideas of Quantum Mechanics has spawned some of the worst quackery, Pseudo-Science, hoo-ha, and unfounded mystical story telling of any scientific theory. It’s easy to see why, there are far out there explanations for the processes at work behind the incredible successful mathematics of quantum mechanics.

These explanations claim stuff like: things about waves and particles at the same time, the act of observation defines reality, cats supposed to be alive and dead, and even that the universe is constantly splitting into infinite alternate realities. The weird results of quantum experiments seem to demand weird explanations of the nature of reality.

There is one interpretation of quantum mechanics that remains comfortably, almost stodgily, physical, that’s DeBroglie-Bohm Pilot-Wave theory. Pilot-Wave Theory, also known as Bohmian Mechanics stands in striking contrast to the much more main stream ideas, as for example the Copenhagen, and Many-World interpretations.

Copenhagen Interpretation of Quantum Mechanics  ↓Many-Worlds Interpretation of Quantum Mechanics  ↑

Pilot-Wave Theory is perhaps the most solidly physical, even mundane, of the complete and self-consistent interpretation of quantum mechanics. But at the same time it’s considered one of the least orthodox. Why so? Because orthodoxy equals radicalism plus time. The founding fathers of the Copenhagen interpretation of quantum mechanics, Werner Heisenberg and Niels Bohr were radicals. When quantum theory was coming together in the twenties, the were fervent about the need to reject all classical thinking in interpreting the strange results of early quantum experiments.

(left) – Werner Heisenberg and Niels Bohr – (right)

One aspect of that radical thinking was that the wave function is not a wave in anything physical, but an abstract distribution of probabilities. Bohr and Heisenberg insisted that in the absents of measurement the unobserved universe is only a suite of possibilities of the various states it could take, were a measurement to be made. Then upon measurement, fundamental randomness determines the properties of say, the particle that would emerge from it’s wave function.

This required an almost mystical duality between the wave and the particle like nature of matter. Not everyone was so sure, Einstein famously hated the idea of fundamental randomness, but to counter Bohr and Heisenberg there needed to be a full theory that described how a quantum object could show both wave and particle behavior at the same time without being fundamentally probabilistic.

That theory came from Louis DeBroglie, the guy who originally proposed the idea that matter could be described as waves right at the beginning of the quantum revolution. DeBroglie theory reasons that, there was no need for quantum objects to transition in a mystical way between non-real waves and real particles, why not just have real waves just push around real particles.

This is Pilot-Wave Theory. In it, the wave function describes a real wave of some stuff, this wave guides the motion of a real point-like particle that has a define location at all times. Importantly, the wave function in Pilot-Wave Theory evolves exactly according to the Schrödinger equation. That’s the equation at the heart of all quantum mechanics that tells the wave function how to change across space and time.

Schrödinger Equation: Describes How a Physical System Will Change Over Time

This means that Pilot-Wave Theory makes the same basic predictions as any other breed of quantum mechanics. For example, it’s guiding wave has all the usual wavy stuff, like forming interference patterns when it passes through a pair of slits. Because particles follow the paths etched out by the wave, it will end up landing according to that pattern. The wave defines a set of possible trajectories and the particle takes one of those trajectories. But the choice of paths isn’t random, if you know the exact particle position and velocity at any point you can figure out it’s entire future trajectory.

Apparent randomness arises because we can’t ever have a perfect measurement of initial position, velocity or other properties. This hypothetical predictability means that a Pilot-Wave universe is completely deterministic. When DeBroglie presented his still incomplete theory at the famous Solway conference of 1927 it didn’t go down so well. Technical objections were raised and Niels Bohr doubled down on the probabilistic interpretation. DeBroglie was so convinced, and he dropped Pilot-Waves all together. The idea was forgotten for decades and Copenhagen became the orthodoxy.

Solvay conference (1927): Schrödinger, Bohr, Eisenberg, De Broglie, Dirac, Lorentz, Einstein..

It took until 1952 for another physicist, David Bohm to feel very uncomfortable with some of the wackiness of Copenhagen and to re-discover DeBroglie’s old idea. Bohm took off where DeBroglie left off and completed the theory. The result was Bohmian Mechanics, also known as DeBroglie-Bohm Pilot-Wave theory. These days, more and more serious physicist are favoring Bohm’s ideas. However, it’s far from being broadly accepted. DeBroglie himself remained firmly in the Copenhagen camp even after Bohm’s efforts.

Although Pilot-Wave theory makes all the usual predictions of Quantum Mechanics, it has some really fundamental differences. Those differences are in a sort of “special thinking” you need do, in order to accept Pilot-Waves over other interpretations. In fact most of the arguments fore or against it are about this “special thinking”. Are you more or less comfortable with the oddness of Pilot-Waves versus the oddness say, of Copenhagen, or Many Worlds. So what uncomfortable thinking does Pilot-Wave theory require. For one thong, it needs a teensy bit of extra math that mainstream interpretations don’t. As well as the Schrödinger equation that tells the wave function how to change, it also has a Guiding (Velocity) Equation that tells the particle how to move within that wave function.

Schrödinger Equation and Guiding (Velocity) Equation

That “extra math” is considered un-parsimonious to some, a needles added complexity. However, the Guiding Equation is derived directly from the wave function, so some would argue that it was there all along. A more troubling requirement of Bohmian Mechanics is that it does contain real complexity that’s not uncoded in the wave function. That’s something that Niels Bohr was so fervently against.

Bohmian Mechanics, has so called “hidden variables” details about the state of the particles that are not described by the wave function. According to Pilot-Wave Theory the wave function just describes the possible distribution of those variables given our lack of perfect knowledge. But hidden variables have a bad wrap in quantum mechanics.

Distribution of Variables Inside a Wave Function

Pretty soon after DeBroglie first proposed Pilot-Waves, the revered mathematician John Von Neumann published a proof, showing that hidden variable explanation for the wave function, just couldn’t work. That proclamation contributed to the long shelving of Pilot-Wave Theory. In fact Von Neumann didn’t get the full answer, it turns out that the restriction against hidden variables, only applies to local hidden variables. So, there can’t be extra hidden information about specific region of the wave function that the rest of the wave function doesn’t know.

This was figured out pretty soon after Von Neumann paper, by German mathematician Grete Hermann. Although her repudiation wasn’t noticed until it was re-derived by John Bell in the 1960’s. This helped the resuscitation of Pilot-Wave Theory, because Bohmian Mechanics doesn’t use local hidden variables, it’s hidden variables are global. The entire wave function knows the location, velocity and spin of each particle. This non-locality is another weird thing you have to believe, in order to accept Pilot-Waves.

Grete Hermann                                                                         John Bell

Not only does the entire wave function knows the properties of the particle, but the entire wave function could be affected instantaneously. So a measurement at one point in the wave function will effect it’s shape elsewhere. This can therefore effect the trajectories and properties of the particles carried by that wave, potentially very far away. Quantum entanglement experiment show this sort of “Spooky Action” at a distance, is a very real phenomena.

Again, I’ve gone into the non-locality of entangle particles in detail before. Also worth a look. It’s a though idea to swallow, but experiments indicate that some type of non locality is real, wether or not we accept Pilot-Waves. It would be remit of me to talk about Pilot-Waves without mentioning the amazing analogy that was discovered in bouncing droplets on a vibration pool of oil.

Quantum Phenomena on the Macroscopic Level

This is pretty amazing, we see many of the familiar quantum phenomena appear in this macroscopic system with suspended oil droplets following it’s own Pilot-Wave. Now we shouldn’t take a macroscopic analogy as proof of microscopic reality. But it does demonstrate that this sort of thing does happen in this universe, at least on some scales. I should also add that DeBroglie-Bohm Pilot-Wave theory is certainly wrong, and I don’t think anyone could deny that, because it doesn’t account for relativity, either special or general.

That means that at best it’s incomplete, while regular mechanics has Quantum Field theory and it’s relativistic version, Pilot-Wave theory hasn’t quite got there yet. Quantum Field theory pretty explicitly requires that all possible particle trajectories be considered equally real. Pilot-Wave theory postulates that the particle really takes a single actual trajectory, the Bohm trajectory. This is not consistent with quantum field theory and so there isn’t a complete relativistic formulation of Bohmian Mechanics, yet. But there is good effort in that direction.

Now let’s not even start talking about gravity, as no version of Quantum Mechanics has that sorted out. Also, we can’t ignore the fact that the initial motivation behind Pilot-Wave theory was to preserve the idea of real particles. We need to be dubious about that sort of classical bias. All that said, Pilot-Wave theory does do something remarkable. It shows as that it’s possible to have a consistent interpretation of Quantum Mechanics, that is both physical and deterministic, no hoo-ha needed. Maybe something like Pilot-Waves really do drive the microscopic mechanics of space-time.

Particle(s) Taking Bohm Trajectory While Surfing Pilot Waves

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