[MooMooMilkParty]

I'm a little bit curious here. My understanding is that the wave function has complex values and the "state" is described by the wave function times its complex conjugate. Couldn't they have meant the factor of -1 to be on the imaginary portion and then things still work out to have the same probability distribution?

[amluto]

Suppose you have a very simple system with two indistinguishable particles in one dimension and no spin or any other complications. The wavefunction is just a complex valued function of two variables:

f(x0, x1)

where x0 and x1 are coordinates of the two particles.

The fact that the particles are indistinguishable means that no possible experiment can tell them apart. After a bit of math, this means that f(x0, x1) = k * f(x1, x0) where k is a constant. All wavefunctions are normalized, so |k|^2=1, which constrains k to be on the unit circle.

But you can go farther: it's possible, at least in principle, to come up with an experiment that measures k. So you start with the particles in their initial state, do your experiment, and learn k. (Maybe you need to repeat it a bunch of times -- no big deal.) Now you repeat the same experiment but start with the two particles having their roles swapped. Since the particles are indistinguishable, you had better measure the same value for k. This means:

f(x0, x1) = k * f(x1, x0)

and also

f(x1, x0) = k * f(x0, x1) = k^2 * f(x1, x0)

So k^2 = 1, and that rules out everything except k=1 or k=-1.

[MiroF]

Why does the factor of -1 just have to be on the imaginary proportion? seems like this would be true if you multiplied the whole thing by -1.