First, minor editing thingie: The bit at the top, that’s supposed to be a link to “classical configuration spaces” is just text, not a link.
Second, I think you’re wrong about “If you had a point mass of amplitude, an infinitely sharp spike in the quantum arena, the amplitude distribution would not be twice differentiable and the future evolution of the system would be undefined. The known laws of physics would crumple up like tinfoil. Individual configurations don’t have quantum dynamics; amplitude distributions do.”
Specifically, I do seem to remember learning how to deal with the time evolution of a dirac delta (constructed as the appropriate limit of a gaussian amplitude distribution). Basically (at least in non relativistic QM) it instantly flattens.
Also, I’m not entirely sure you’re completely right about amplitudes over positions in space being the actual physical reality underneath the math.
Specifically, wouldn’t relativity come in and say “eh? say what? positions in space? What do you mean?”
No prefered reference frame and all that. You could have them over configurations of stuff in spacetime, sure, but space itself? What one observer considers a single point at different times, a different observer will consider different points. (This obviously holds even in gallilean relativity) so I’d be very hesitant to claim that the amplitudes over configurations of positions in space has a deeper correspondence to the underlying reality than the various transforms that put the amplitudes over momentum space or whatever.
This bit was very helpful though: “This kind of independence-structure is one of several keys to recovering the illusion of individual particles from quantum amplitude distributions. If the amplitude distribution roughly factorizes, has subsystems A and B with Amplitude(X,Y) ~ Amplitude(X) * Amplitude(Y), then X and Y will seem to evolve roughly independently of each other.”
ie, a question that has kinda been bugging me is “well… then why does it look like amplitudes over configurations of ‘billiard balls’? why configurations of ‘billiard balls’ at all?” and that one bit helped be understand a little bit better where the billiard ball illusion is really coming from.
First, minor editing thingie: The bit at the top, that’s supposed to be a link to “classical configuration spaces” is just text, not a link.
Second, I think you’re wrong about “If you had a point mass of amplitude, an infinitely sharp spike in the quantum arena, the amplitude distribution would not be twice differentiable and the future evolution of the system would be undefined. The known laws of physics would crumple up like tinfoil. Individual configurations don’t have quantum dynamics; amplitude distributions do.”
Specifically, I do seem to remember learning how to deal with the time evolution of a dirac delta (constructed as the appropriate limit of a gaussian amplitude distribution). Basically (at least in non relativistic QM) it instantly flattens.
Also, I’m not entirely sure you’re completely right about amplitudes over positions in space being the actual physical reality underneath the math.
Specifically, wouldn’t relativity come in and say “eh? say what? positions in space? What do you mean?”
No prefered reference frame and all that. You could have them over configurations of stuff in spacetime, sure, but space itself? What one observer considers a single point at different times, a different observer will consider different points. (This obviously holds even in gallilean relativity) so I’d be very hesitant to claim that the amplitudes over configurations of positions in space has a deeper correspondence to the underlying reality than the various transforms that put the amplitudes over momentum space or whatever.
This bit was very helpful though: “This kind of independence-structure is one of several keys to recovering the illusion of individual particles from quantum amplitude distributions. If the amplitude distribution roughly factorizes, has subsystems A and B with Amplitude(X,Y) ~ Amplitude(X) * Amplitude(Y), then X and Y will seem to evolve roughly independently of each other.”
ie, a question that has kinda been bugging me is “well… then why does it look like amplitudes over configurations of ‘billiard balls’? why configurations of ‘billiard balls’ at all?” and that one bit helped be understand a little bit better where the billiard ball illusion is really coming from.