(PS: I’m given to understand that the Feynman path integral may be more fundamental than the Schrödinger equation: that is, you can derive Schrödinger from Feynman. But as far as I can tell from examining the equations, Feynman is still differentiating the amplitude distribution, and so reality doesn’t yet break down into point amplitude flows between point configurations. Some physicist please correct me if I’m wrong about this, because it is a matter on which I am quite curious.)
Feynman really does give you the amplitude for going from one point distribution to another point distribution. The formula for the path integral doesn’t involve any derivatives of the amplitude distribution. But your fundamental point is still correct. Nature can’t be viewed as classical just by thinking only in terms of point distributions. This is because the point distribution evolves into a non-point distribution. So even if you start out thinking in terms of point distributions you are immediately forced to consider other distributions.
(You might be worried that the point distribution has infinite second derivative, and so can’t be evolved using the Schrodinger equation. But if you turn down your rigour dial you can find the solution:
phi = exp[i x^2 / (4t) ]/sqrt[4 pi i t]
(This is the solution for a free particle in one dimension where I’ve picked the mass hbar/2 for convenience.) One can sort of see how this becomes a point distribution as t tends to zero. The amplitude becomes very oscillatory everywhere except zero, and at zero all those oscillations cancel out. Meanwhile the magnitude increases like 1/sqrt(t) as t tends to zero, so at zero it has the correct value of sqrt(infintiy).)
Feynman really does give you the amplitude for going from one point distribution to another point distribution. The formula for the path integral doesn’t involve any derivatives of the amplitude distribution. But your fundamental point is still correct. Nature can’t be viewed as classical just by thinking only in terms of point distributions. This is because the point distribution evolves into a non-point distribution. So even if you start out thinking in terms of point distributions you are immediately forced to consider other distributions.
(You might be worried that the point distribution has infinite second derivative, and so can’t be evolved using the Schrodinger equation. But if you turn down your rigour dial you can find the solution:
phi = exp[i x^2 / (4t) ]/sqrt[4 pi i t]
(This is the solution for a free particle in one dimension where I’ve picked the mass hbar/2 for convenience.) One can sort of see how this becomes a point distribution as t tends to zero. The amplitude becomes very oscillatory everywhere except zero, and at zero all those oscillations cancel out. Meanwhile the magnitude increases like 1/sqrt(t) as t tends to zero, so at zero it has the correct value of sqrt(infintiy).)