Chris Frisella said:
Ok. How do you define "unphysical" here?
A tentative definition would be: An object that contains ambiguous information.
However, I don't really need a definition, since I introduced this word only for the purpose of avoiding math. In the case of quantum mechanics, there is a precise meaning to it: The "wave function" can be multiplied by an phase factor without changing the physics of the system. This is similar to the case of the electric potential, which is ambiguous up to an arbitrary number (and even more). However, from these ambigous quantities, you can compute new quantities that don't have this ambiguity anymore. In the case of the wave function, this would be the density matrix and in the case of the electric potential, this would be the electric field. In quantum mechanics, only the density matrix needs to obey reasonable laws such as being invariant under 360° rotations. The wave function needn't satisfy these requirements, since you are allowed to do arbitrary nonsense to the unphysical phase factor. The reason for why the wave function is not invariant under 360° degree rotations is that the phase factor isn't invariant, roughly speaking. However, if you strip off this unphysical factor, you end up with an object (the density matrix), which behaves like it should.
stevendaryl said:
However, it seems to me that the minus sign would come into play in interference effects. I don't actually know how one would accomplish this, but conceptually, you can imagine an experiment similar to the two-slit experiment, where an electron make take one of two paths to get to a destination. Along one of the paths, the electron is rotated 360o by its journey, while along the other path, no rotation takes place. Then (assuming an electron is equally likely to take either path), there would be total destructive interference at the destination, so the electron would have no probability of arriving there.
I'm not sure what would reliably rotate an electron, though. Maybe one path has a magnetic field, and the other doesn't?
I hoped nobody would bring this up.

What you do when you "rotate" one of the fermions is not to apply a rotation (##U(R)##) to it, but rather apply a time evolution (##e^{-\frac{\mathrm i}{\hbar}t H}##) to the whole system consisting of both fermions, which happens to change their relative phase. The point of Wigner is that physics should mathematically be invariant under rotations of the coordinate system (and their active counterparts), since the choice of a coordinate system is a mathematical ambiguity introduced by the physicist. However you can't change the coordinate systems of the individual particles. Spin ##\frac{1}{2}## particles behave differently than spin ##1## particles, since they have more degrees of freedom. I wouldn't consider sending a fermion through a Stern-Gerlach apparatus a rotation. It's just an interaction with the magnetic field, which also influences the spin degrees of freedom. It doesn't rotate the particle.
strangerep said:
You seem to be saying that fermions are unphysical, which is clearly not correct since one can observe effects in, e.g., neutron interferometers (by using a magnetic field to rotate the neutrons on one side of the interferometer and then recombining).
Perhaps you meant something else?
No, that's not what I meant.

Of course fermions are physical. What is unphysical is the wave function, since it contains ambiguous information. The physical information sits in the density matrix or the ray defined by the wave function. It's a kind of gauge freedom.
(And let's not forget ye olde
Plate trick, aka "Belt trick" or "Dirac Scissors", which shows that there are macroscopic situations where rotation by 360deg is not equivalent to rotation by 720deg.)
The Plate trick demonstrates this nicely: It's not really just a rotation. It is also an up/down movement that takes place at the same time. The plate itself returns to its original state after a 360° rotation. It's just that the additional up/down degree of freedrom returns only after two periods. The relation to quantum mechanics is: The physical information returns to its initial state after a 360° rotation, but the additional unphysical phase degree of freedom needs two periods. If you strip off the unphysical degree of freedrom (by computing the density matrix), you get an object that only needs a 360° degree rotation to get back to its initial state. Of course in case of the plate trick, the up/down movement isn't unphysical, but then again the whole movement is not just a rotation. In QM, ##SU(2)## is of course not the rotation group, but if you apply the covering map to any ##SU(2)## matrix, you extract the part of the ##SU(2)## transformation that can be considered a rotation.