Eliezer Yudkowsky wonders aloud just what the Born probabilities in quantum mechanics are. It is, I think, an object lesson that nobody should try to understand quantum mechanics: it simply cannot be done.
We hope he recovers someday:
Overcoming Bias: The Born Probabilities: One serious mystery... is where the Born probabilities come from, or even what they are probabilities of. What does the integral over the squared modulus of the amplitude density have to do with anything?... A professor teaching undergraduates might say: "The probability of finding a particle in a particular position is given by the squared modulus of the amplitude at that position."
This is oversimplified in several ways. First, for continuous variables like position, amplitude is a density, not a point mass. You integrate over it. The integral over a single point is zero. (Historical note: If "observing a particle's position" invoked a mysterious event that squeezed the amplitude distribution down to a delta point, or flattened it in one subspace, this would give us a different future amplitude distribution from what decoherence [theory] would predict. All interpretations of QM that involve quantum systems jumping into a point/flat state, which are both testable and have been tested, have been falsified. The universe does not have a "classical mode" to jump into; it's all amplitudes, all the time.)
Second, a single observed particle doesn't have an amplitude distribution. Rather the system containing yourself, plus the particle, plus the rest of the universe, may approximately factor into the multiplicative product of (1) a sub-distribution over the particle position and (2) a sub-distribution over the rest of the universe. Or rather, the particular blob of amplitude that you happen to be in, can factor that way. So what could it mean, to associate a "subjective probability" with a component of one factor of a combined amplitude distribution that happens to factorize?...
If a whole gigantic human experimenter made up of quintillions of particles interacts with one teensy little atom whose amplitude factor has a big bulge on the left and a small bulge on the right, then the resulting amplitude distribution, in the joint configuration space, has a big amplitude blob for "human sees atom on the left", and a small amplitude blob of "human sees atom on the right.... [T]he Born probabilities seem to be about finding yourself in a particular blob, not the particle being in a particular place. But what does the integral over squared moduli have to do with anything? On a straight reading of the data, you would always find yourself in both blobs, every time. How can you find yourself in one blob with greater probability? What are the Born probabilities probabilities of? Here's the map - where's the territory?
I don't know. It's an open problem. Try not to go funny in the head about it. This problem is even worse than it looks because the squared-modulus business is the only non-linear rule in all of quantum mechanics...