Am I correct in assuming that this is independent of (observations, “wave function collapses”, or whatever it is when we say that we find a particle at a certain point)?
Wavefunction collapses are unpredictable. The claim in The Quantum Arena, if your summary is right, is that subsequent amplitude distributions are predictable if you know the entire current amplitude distribution. The amplitude distribution is the wavefunction. Since wavefunction collapses are unpredictable but the wavefunction’s progression is claimed to be predictable, wavefunction collapses are logically barred from existing if the claim is true. From this follows the no-collapse “interpretation” of quantum mechanics, a.k.a. the many-worlds interpretation. Eliezer’s claim, then, is expressing the many-worlds interpretation of QM. The seeming collapse of the wavefunction is only apparent. The wavefunction has not, in fact, collapsed. In particular, when you find a particle at a certain point, then the objective reality is not that the particle is at that point and not at any other point. The you which sees the particle at that one point is only seeing a small part of the bigger objective picture.
If I observe the particle on the opposite side of the moon at some point (i.e. where the amplitude is non-zero, but still tiny), does the particle still have the same probability as before of “jumping” back onto the line from x to y?
Yes and no. Objectively, it still has the same probability of being on the line from x to y. But the you who observes the particle on the opposite side of the moon will from that point forward only see a small part of the bigger objective picture, and what that you will see will (almost certainly) not be the particle jumping back onto the line from x to y. So the subjective probability relative to that you is not the same as the objective probability.
Now, let me correct my language. Neither of these probabilities is objective. What I called “objective” was in fact the subjective probability relative to the you who witnessed the particle starting out at x but did not (yet) witness the particle on the other side of the moon.
Am I correct in assuming that this is independent of (observations, “wave function collapses”, or whatever it is when we say that we find a particle at a certain point)?
Wavefunction collapses are unpredictable. The claim in The Quantum Arena, if your summary is right, is that subsequent amplitude distributions are predictable if you know the entire current amplitude distribution. The amplitude distribution is the wavefunction. Since wavefunction collapses are unpredictable but the wavefunction’s progression is claimed to be predictable, wavefunction collapses are logically barred from existing if the claim is true. From this follows the no-collapse “interpretation” of quantum mechanics, a.k.a. the many-worlds interpretation. Eliezer’s claim, then, is expressing the many-worlds interpretation of QM. The seeming collapse of the wavefunction is only apparent. The wavefunction has not, in fact, collapsed. In particular, when you find a particle at a certain point, then the objective reality is not that the particle is at that point and not at any other point. The you which sees the particle at that one point is only seeing a small part of the bigger objective picture.
If I observe the particle on the opposite side of the moon at some point (i.e. where the amplitude is non-zero, but still tiny), does the particle still have the same probability as before of “jumping” back onto the line from x to y?
Yes and no. Objectively, it still has the same probability of being on the line from x to y. But the you who observes the particle on the opposite side of the moon will from that point forward only see a small part of the bigger objective picture, and what that you will see will (almost certainly) not be the particle jumping back onto the line from x to y. So the subjective probability relative to that you is not the same as the objective probability.
Now, let me correct my language. Neither of these probabilities is objective. What I called “objective” was in fact the subjective probability relative to the you who witnessed the particle starting out at x but did not (yet) witness the particle on the other side of the moon.