Amazingly great post. But I’m still confused on one point.
Say we want to set up the quantum configuration space for two 1-dimensional particles. So we have a position coordinate for each one, call them x and y. But wait, the two particles aren’t distinguishable, so we really need to look at the quotient space under the equivalence (x,y) ~ (y,x). But this is no longer a smooth manifold is it? At the moment I’m at a loss for a proof that it isn’t, but I certainly can’t find a smooth structure for it. And if it’s not smooth then what the heck do second derivatives of amplitude distributions mean?
Amazingly great post. But I’m still confused on one point.
Say we want to set up the quantum configuration space for two 1-dimensional particles. So we have a position coordinate for each one, call them x and y. But wait, the two particles aren’t distinguishable, so we really need to look at the quotient space under the equivalence (x,y) ~ (y,x). But this is no longer a smooth manifold is it? At the moment I’m at a loss for a proof that it isn’t, but I certainly can’t find a smooth structure for it. And if it’s not smooth then what the heck do second derivatives of amplitude distributions mean?