On Jun 24, 2014, at 9:45 AM, James Propp <jamespropp@gmail.com> wrote:
I feel like I'm lacking context for the operation of symmetrization/antisymmetrization, but I'm not sure if I can explain what I mean by context. I'm guessing Veit means something like "If you symmetrize it, you'll get a wave-function that's just as descriptive of reality; the universe won't notice the difference." Or "If you symmetrize it, you'll still get a solution to the relevant equation, but if it isn't unitary then it won't correspond to anything we see in the universe."
Veit, can you clarify? One of the “rules”, when constructing a quantum state for a system of identical particles, is that it transform as one of the 1-dimensional representations of the symmetric group. If you are dealing with fermions, then the only valid multi-particle wave function has the property it changes sign when the particle labels are given an odd permutation. This is Nature telling you that the particles do not actually have labels!
If you are given a multi-particle wave function without this group representation property, say psi(x1,x2), then you can “symmetrize” or “antisymmetrize” it so it does: psi_B(x1,x2) = psi(x1,x2) + psi(x2,x1) psi_F(x1,x2) = psi(x1,x2) - psi(x2,x1) (These are not normalized, and you may end up with zero — no wave function.) In math I believe this is called “Fourier analysis with respect to the characters of a finite group”. -Veit