Epistemic status: very curious non-physicist.
Here's what I find weird about the Born rule.
Eliezer very successfully thought about intelligence by asking "how would you program a computer to be intelligent?". I would frame the Born rule using the analogous question for physics: "if you had an enormous amount of compute, how would you simulate a universe?".
Here is how I would go about it:
- Simulate an Alternate Earth, using quantum mechanics. The simulation has discrete time. At each step in time, the state of the simulation is a wavefunction: a set of
(amplitude, world) pairs. If you would have two pairs with the same
world in the same time step, you combine them into one pair by adding their
amplitudes together. Standard QM, except for making time discrete, which is just there to make this easier to think about and run on a computer.
- Seed the Alternate Earth with humans, and run it for 100 years.
- Select a world at random, from some distribution. (!)
- Scan that world for a physicist on Alternate Earth who speaks English, and interview them.
The distribution used in step (3) determines what the physicist will tell you. For example, you could use the Born rule: pick at random from the distribution on worlds given by P(amplitude)=|amplitude2|. If you do, the interview will go something like this:
Simulator: Hi, it's God.
Physicist: Oh wow.
Simulator: I just have a quick question. In quantum mechanics, what's the rule for the probability that an observer finds themselves in a particular world?
Physicist: The probability is proportional to the square of the magnitude of the amplitude. Why is that, anyways?
Simulator: Awkwardly, that's what I'm trying to find out.
Physicist: ...God, why did you make a universe with so much suffering in it? My child died of bone cancer.
Simulator: Uh, gotta go.
Remember that you (the simulator) were picking at random from an astronomically large set of possible worlds. For example, in one of those worlds, photons in double slit experiments happened to always go left, and the physicists were very confused. However, by the law of large numbers, the world you pick almost certainly looks from the inside like it obeyed the Born rule.
However, the Born rule isn't the only distribution you could pick from in step 3. You could also pick from the distribution given by P(α)=|α| (with normalization). And frankly that's more natural. In this case, you would (almost certainly, by the law of large numbers) pick a world in which the physicists thought that the Born rule said P(α)=|α|. By Everett's argument, in this world probability does not look additive between orthogonal states. I think that means that its physicists would have discovered QM a lot earlier: the non-linear effects would be a lot more obvious! But is there anything wrong with this world, that would make you as the simulator go "oops I should have picked from a different distribution"?
There's also a third reasonable distribution: ignore the amplitudes, and pick uniformly at random from among the (distinct) worlds. I don't know what this world looks like from the inside.
I think b) is what I always assumed was meant by the Born rule being called mysterious?