Okay, I've been looking at my String Theory text when I've been a bad boy and not looking up ideas for the next Math Challenge. I'm only skimming it for the moment but it has brought up a number of questions in my mind and I've noticed I can't even tell you what branch of Mathematics this is all coming from. (Actually that kind of excites me. again.)
Anyway I've reduced some of this to something I can actually ask questions for.
1) When we solve the Schrodinger equation (or any of the relativistic wave equations as well) what kind of space do the kets live in? I've never considered that one. Is it a (possibly infinite) Hilbert space? And if it is a finite space can we sit it inside a manifold? (I really don't know why you would, but there it is.)
2) How would you solve the wave equation in different spaces? (And I don't even know what kind of space it should be. The text keeps referencing a 26 dimensional conformal space and an "heterotic" space. (I'm getting a headache. Too many symmetries and too many Noether charges.)
Okay, back to the Challenge Problems!