Saturday, May 28, 2022

Chaos: The Real Problem with Quantum Mechanics

[This is a transcript of the video embedded below. Some of the explanations may not make sense without the animations in the video.]

You’ve probably seen a lot of headlines claiming that quantum mechanics is “strange”, “weird” or “spooky”. In the best case it’s “unintuitive” and “no one understands it”. Poor thing. In this video I will try to convince you that the problem with quantum mechanics isn’t that it’s weird. The problem with quantum mechanics is chaos. And that’s what we’ll talk about today.

Saturn has 82 moons. This is one of them, its name is Hyperion. Hyperion has a diameter of about 200 kilometers and its motion is chaotic. It’s not the orbit that’s chaotic, it’s the orientation of the moon on that orbit.

It takes Hyperion about 3 weeks to go around Saturn once, and about 5 days to rotate about its own axis. But the orientation of the axis tumbles around erratically every couple of months. And that tumbling is chaotic in the technical sense. Even if you measure the position and orientation of Hyperion to utmost precision, you won’t be able to predict what the orientation will be a year later.

Hyperion is a big headache for physicists. Not so much for astrophysicists. Hyperion’s motion can be understood, if not predicted, with general relativity or, to good approximation, with Newtonian dynamics and Newtonian gravity. These are all theories which do not have quantum properties. Physicists call such theories without quantum properties “classical”.

But Hyperion is a headache for those who think that quantum mechanics is really the way nature works. Because quantum mechanics predicts that Hyperion’s chaotic motion shouldn’t last longer than about 20 years. But it has lasted much longer. So, quantum mechanics has been falsified.

Wait what? Yes, and it isn’t even news. That quantum mechanics doesn’t correctly reproduce the dynamics of classical, chaotic systems has been known since the 1950s. The particular example with the moon of Saturn comes from the 1990s. (For details see here or here.)

The origin of the problem isn’t all that difficult to see. If you remember, in quantum mechanics we describe everything with a wave-function, usually denoted psi. There aren’t just wave-functions for particles. In quantum mechanics there’s a wave-function for everything: atoms, cats, and also moons.

You calculate the change of the wave-function in time with the Schrödinger equation, which looks like this. The Schrödinger equation is linear, which just means that no products of the wave-function appear in it. You see, there’s only one Psi on each side. Systems with linear equations like this don’t have chaos. To have chaos you need non-linear equations.

But quantum mechanics is supposed to be a theory of all matter. So we should be able to use quantum mechanics to describe large objects, right? If we do that, we should just find that the motion of these large objects agrees with the classical non-quantum behavior. This is called the “correspondence principle”, a name that goes back to Niels Bohr.

But if you look at a classical chaotic system, like this moon of Saturn, the prediction you get from quantum mechanics only agrees with that from classical Newtonian dynamics for a certain period of time, known as the “Ehrenfest time”. Within this time, you can actually use quantum mechanics to study chaos. This is what quantum chaos is all about. But after the Ehrenfest time, quantum mechanics gives you a prediction that just doesn’t agree with what we observe. It would predict that the orientations of Hyperion don’t tumble around but instead blur out until they’re so blurred you wouldn’t notice any tumbling. Basically the chaos gets washed away in quantum uncertainty.

It seems to me that some of you are a little skeptical. It can’t possibly be that physicists have known of this problem for 60 years and just ignored it? Indeed, they haven’t exactly ignored it. The have come up with an explanation which goes like this.

Hyperion may be far away from us and not much is going on there, but it still interacts with dust and with light or, more precisely, with the quanta of light called “photons”. These are each really tiny interactions, but there are a lot of them. And they have to be added to the Schrödinger equation of the moon.

What these tiny interactions do is that they entangle the moon with its environment, with the dust and the light. This means that each time a grain of dust bumps into the moon, this very slightly changes some part of the moon’s wave-function, and afterwards they are both correlated. This correlation is the entanglement. And those little bumps slightly shift the crest and troughs of parts of the wave-function.

This is called “decoherence” and it’s just what the Schrödinger equation predicts. And this equation is still linear, so all those interactions don’t solve the problem that the prediction doesn’t agree with observation. The solution to the problem comes in the 2nd step of the argument. Physicists now say, okay, so we have this wave-function for the moon with this huge number of entangled dust grains and photons. But we don’t know exactly what this dust is or where it is or what the photons do and so on. So we do what we always do if we don’t know the exact details: We make guesses about what the details could plausibly be and then we average over them. And that average agrees with what classical Newtonian dynamics predicts.

So, physicists say, all is good! But there are two problems with this explanation. One is that it forces you to accept that in the absence of dust and light a moon will not follow Newton’s law of motion.

Ok, well, you could say that in this case you can’t see the moon either so for all we can tell that might be correct.

The more serious problem is that taking an average isn’t a physical process. It doesn’t change anything about the state that the moon is in. It’s still in one of those blurry quantum states that are now also entangled with dust and photons, you just don’t know exactly which one.

To see the problem with the argument, let me use an analogy. Take a classical chaotic process like throwing a die. The outcome is an integer from 1 to 6, and if you average over many throws then the average value per throw is 3.5. Just exactly which outcome you get is determined by a lot of tiny details like the positions of air molecules and the surface roughness and the motion of your hand and so on.

Now suppose I write down a model for the die. My model says that the outcome of throwing the die is either 106 or -99 each with probability 1/2. Wait, you say, there’s no way throwing a die will give you minus 99. Look, I say, the average is 3.5, all is good. Would you accept this? Probably not.

Clearly for the model to be correct it shouldn’t just get the average right, but each possible individual outcome should also agree with observations. And throwing a die doesn’t give minus 99 any more than a big blurry rock entangled with a lot of photons agrees with our observations of Hyperion.

Ok but what’s with the collapse of the wave-function? When we make a measurement, then the wave-function changes in a way that the Schrödinger-equation does not predict. Whatever happened to that?

Exactly! In quantum mechanics we use the wave-function to make probabilistic predictions. Say, an electron hits either the left or right side of a screen with 50% probability each. But then when we measure the electron, we know it’s, say, left with 100% probability.

This means after a measurement we have to update the wave-function from 50-50 to 100-0. Importantly, what we call a “measurement” in quantum mechanics doesn’t actually have to be done by a measurement device. I know it’s an awkward nomenclature, but in quantum mechanics a “measurement” can happen just by interaction with a lot of particles. Like grains of dust, or photons.

This means, Hyperion is in some sense constantly being “detected” by all those small particles. And the update of the wave-function is indeed a non-linear process. This neatly resolves the problem: Hyperion correctly tumbles around on its orbit chaotically. Hurray.

But here’s the thing. This only works if the collapse of the wave-function is a physical process. Because you have to actually change something about that blurry quantum state of the moon for it to agree with observations. But the vast majority of physicists today think the collapse of the wave-function isn’t a physical process. Because if it was, then it would have to happen instantaneously everywhere.

Take the example of the electron hitting the screen. When the wave-function arrives on the screen, it is spread out. But when the particle appears on one side of the screen, the wave-function on the other side of the screen must immediately change. Likewise, when a photon hits the moon on one side, then the wave-function of the moon has to change on the other side, immediately.

This is what Einstein called “spooky action at a distance”. It would break the speed of light limit. So, physicists said, the measurement is not a physical process. We’re just accounting for the knowledge we have gained. And there’s nothing propagating faster than light if we just update our knowledge about another place.

But the example with the chaotic motion of Hyperion tells us that we need the measurement collapse to actually be a physical process. Without it, quantum mechanics just doesn’t correctly describe our observations. But then what is this process? No one knows. And that’s the problem with quantum mechanics.

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