Last week I introduced the idea of classical correlations: that when two particles collide, if we know their position and momentum before the collision then because of conservation of energy and momentum all we need to know is the momentum or energy of one of the particles to also and instantaneously know the momentum or energy of the other particle, no matter how long ago the collision took place or how far away the other particle is. So long as neither particle undergoes any subsequent interaction they will remain perfectly correlated to the end of time and the far side of the universe.
This is no problem because there is a fact of the matter about what happens after a collision: the two pucks bounce off each other and each travel along a uniquely defined trajectory. This obeys the Law of Non-contradiction: a thing cannot both be and not be the same thing at the same time and in the same respect.
The problem is reality doesn't work like this.
Recall that conservation of energy and momentum were enough to fix the outcome of the classical pucks. Those two conservation laws told us that after the collision the pucks had to have velocities given by:
v′ = v(m − M)/(m + M)
V = 2mv/(m+M)
where v is the velocity of the incoming puck with mass m that hits a stationary puck of mass M.
There is a big assumption that goes into this however. If we consider the collision from the point of view of the zero-momentum frame of reference, where both pucks are moving toward each other with velocities in the ratio m/M, so the positive momentum of one cancels out the negative momentum of the other, there are obviously two ways to conserve momentum and energy after the collision: one is for the pucks to bounce off each other, so the signs of their momenta are flipped but they still sum to zero. The other is for the pucks to pass through each other and sail on as they were before.
If we plot both puck positions along the line of approach we get a graph that looks like this, where the x-axis is the first puck's position and the y-axis is the second puck's position. The pucks are moving on a line, so both of these positions are along the same direction in real space—call it X_real—which is more than a little confusing, but this is the way reality wants us to think. Each particle has its own spacial coordinate:
Puck 2 sits still at X_real = 0 until the collision occurs and then it moves in the negative X_real direction while puck 1, which has approached along the positive X_real axis, bounces back along it. Puck 1 is at position 1.5 when the collision happens because I’ve given the pucks finite size in the simulation that generated the graph. They aren’t points, but have diameters of 2 (for puck 2) and 1 (for puck 1), giving them a closest-approach distance of 1.5.
That’s how the world of experience works. The world we can’t experience behaves differently.
When two electrons collide, they don't just bounce off each other, they also pass through each other, which gives us a graph that looks like this:
Both of these post-collision states conserve energy and momentum. For light, simple, objects like electrons, quantum theory is based on the idea that they have a wave-like nature, and this wave-like aspect allows both the case where the particles bounce off each other and the case where the particles "pass through each other" to happen. This is what we observe in reality.
We don't see this kind of thing with big pucks because... reasons.
What those reasons are is not clear. This is generally the case for the use of simple, conceptual arguments as to why things happen. For example, it is very difficult to give a simple and correct explanation of why aircraft fly, although it is false to say "no one can explain why planes stay in the air", which is the kind of egregious lie that only an editor at Scientific American would write.
An explanation is a causal account that starts from sufficiently basic observations that we're willing to take them for granted. In the case of heavy objects not passing through each other, the traditional argument is to say that the object as a whole has a very short wavelength compared to its size, and objects can only pass through a distance of a few wavelengths successfully.
Unfortunately we have no warrant for treating what we know to be a composite object as a single mass. This is a common problem with "explanations" that try to reduce something either difficult (fluid mechanics) or impossible (quantum mechanics) to understand to something that will fit within the tiny confines of human intuition.
Our brains are essentially Procrustean in nature: we mangle reality to make it fit our comprehension, unless we are willing and able to go the human-scale reductionist route and focus on tiny bits of reality that fit within our comprehension, and build up a bigger picture from those minuscule brush-strokes. When we attempt an anti-human "holistic" grasp of a problem we always end up seeing nothing but our own distorted priorities because we can’t fit undistorted reality into our brains. It just doesn’t fit.
That said, there are other ways of looking at why we don't see air-hockey pucks pass through each other. If we consider the composite nature of the pucks, it is still the case that the wavelengths of the electrons and atoms that make them up are very small compared to the size of the puck, and even if the outer-most atoms do partially pass through each other they get bounced off the next atoms in, and so on, until the amount of transmission drops to practically zero before we've even got past the surface roughness.
I'm still not convinced by this explanation--it leaves bits of matter waves intertwined in ways that may create problems I'm not smart enough to see--but it's better than the "wavelength of a whole puck" argument.
But let's stay focused on electrons and the like, which we know exhibit this behaviour of both bouncing off and passing through each other. Here is the result of a simulation of two particles, one with the mass of an electron, one with twice that mass, with equal and opposite momenta running into each other. They interact with a force that I made up for the purposes of demonstration that is fairly short-range and relatively easy to handle numerically. Schrodinger's equation, which describes the wave-like aspect of matter, is notoriously hard to solve numerically, so finding simplifications that make it easier while still capturing the essential physics is very helpful. The result is quite pretty:
This is a view from almost directly above the x1/x2 plane, where x1 is the position of particle 1 and x2 is the position of particle 2. As the two particles approach each other there is a single bump in their joint wavefunction. They interact when both of them get to a point where x1 = x2 = 5 (the length units are "hartrees", which are about half an angstrom or 0.05 nano-metres.) If you look carefully at the point of interaction you'll see the wavefunction kind of smushes up into a peak as the particles run into each other.
This is not a real plane, remember: we are are looking at two particles on a line, but plotting their positions as if they were on separate axes. This is one of the strangest aspects of reality: every particle has its own set of spacial co-ordinates. But there is only one time. If I understood this I'd try to explain it, but I don't. Neither does anyone else. I suspect it's at the heart of the mystery.
If we look at the X_real axis, the only X-axis we can actually see, and project both particle’s wavefunctions onto it, then before the collision we see the two particles approaching each other:
It's what happens after the collision that makes the mystery manifest: the single bump that defines the two particles in the X1/X2 plane splits. There is one part that represents the case where the particles bounce off each other, and one that represents the part where the particles pass through each other. Projected onto X_real they look like this:
The projection hides the fact of the correlation. The two bumps labeled “Bounced Off” are correlated, so we if we measure one particle as having bounced off we know the other one has as well, and the two bumps labelled “Passed Through” are correlated, so likewise if we see one particle has passed through the other particle must have done so as well.
But both the “Bounced Off” and “Passed Through” cases "happen".
There is no simple, single, classical fact of the matter about the outcome, no non-contradictory classical result.
And yet we can never see both of them. For reasons.
What we can know—thanks to Bell’s work—is that both sets of correlated outcomes are "there" in some diaphanous unknowable way. Not either/or, but both, unknowable. As soon as we bring one into the known by an act of “measurement”—whose outcome seems to pick randomly between the two—the other falls outside of even the feeble light of Bell-like statistical tests, and vanishes from our ken forever.
But remember: there are two particles here, which we can measure independently and at an arbitrarily large distance from each other, at any time after they have interacted. Look at the lump that moves away to the right after the collision on the colourful X1/X2 plot, which represents the case where the particles bounce off each other. Particle 2 is headed in the negative-x direction. Particle 1--the heavier of the pair--is moving more slowly, but toward the positive.
The lump that represents the transmitted case, where the particles pass through each other, moves to the left: particle 2 going in the positive direction on the X2 axis, particle 1 going in the negative direction on the X1 axis... both of which, to us, are simply "the x axis". We don't have access to this mysterious space where every particle has its own dimension, which is not a true spacial dimension but... I'm tempted to write in a Rod Sterling voice and say, "...a dimension of the imagination..." Except it isn't. It's beyond imagining.
Both these cases "exist"--in some unbounded sense of "existence" that we don't have language for--but when we "measure" one particle--whatever "measure" means--we immediately know what the other particle is doing, no matter how far away it is or how long after their interaction we make the "measurement". And we know which correllated case we are dealing with.
If we measure a particle that bounced off, we know the other particle must have bounced off as well. If we measure a particle that passed through, we know the other particle must have passed through as well. This is a consequence of the conservation of momentum.
When we do that, when we measure one particle and find it has either bounced off or passed through, we not only know that the other particle did the same, we instantaneously lose the possibility of knowing anything at all about the other case... which up until that moment remained a possible outcome, as real—or not—as the one we measured.
And we don't know why.
The process by which this happens does not respect the law of causality because it is instantaneous over an arbitrarily large distance. It is sometimes reified into something called "wavefunction collapse", which sounds better than "magic" but is really no more meaningful. From a different point of view, according to the Many Worlds interpretation, the case we don't measure is always still there but becomes undetectable, which implies the universe is constantly dividing into finer and finer parts. Many people--including me--find this implausible, though not for any very good reason.
And for some reason, the frequency with which we see each case--bouncing off or passing through--depends on the square of the magnitude of the lump that describes it. This association of wavefunction magnitude with probability is called "Born's Rule" after Max Born, the German physicist who discovered it.
But neither Born nor anyone else can explain why this rule should be the case.
These are some of the questions that lie at the heart of the mystery: Why do we see one outcome and not both? What happens to the other one and when? Why Born’s Rule? What is "measurement", anyway?
And why is there a even a classical world for us to experience?
We've known about these questions for very nearly a century now, and other than Bell's theoretical work in the early '60s and its experimental realization in the '80s and '90s, we have elected to mostly not think about them too much, because they are hard problems with little prospect of tenure.
Me, I think about them a lot, and have done so for decades. I've said to a number of people in the course of writing this piece that trying to make this topic even a little bit comprehensible to laypeople was making me regret my life choices in that regard, but really, I wouldn't have it any other way.
Someone has to think about the hardest problems, and why would you do something easy and financially rewarding if you could do something far more difficult, at great personal cost, and with no chance of success? That, at least, is a question that practically answers itself!
But when it comes to the nature of reality, even the act of formulating what questions we can most usefully ask is a hard problem, which is the topic I'll turn to next week.
🤯🫣