Question

We know that Spin is described with SU(2) and that SU(2) is a double cover of the rotation group SO(3). This suggests a simple thought experiment, to be described below. The question then is in three parts:

1. Is this thought experiment theoretically sound?

2. Can it be conducted experimentally?

3. If so what has been the result?

The experiment is to take a slab of material in which there are spin objects e.g. electrons all (or most) with spin ?. Then rotate that object 360 degrees (around an axis perpendicular to the spin direction), so that macroscopically we are back to where we started. Measure the electron spins. Do they point ??

Answer 1

I think that you are confused. When you rotate something by 360 degrees, you won't change the direction in space of anything. You will only change the wave function to minus itself - if there is an odd number of fermions in the object (which is usually hard to count for large objects).

If you have electrons with spins pointing up and you rotate them around the vertical axis by any angle, whether it's 360 degrees or anything else, you will still get electrons with spin pointing up. This is about common sense - many spins with spin up give you a totally normal, "classical" angular momentum that can be seen and measured in many ways.

The flip of the sign of the wave function can't be observed by itself because it is a change of phase and all observable probabilities only depend on the density matrix in which the phase (or minus sign) cancels. The phase - or minus sign - has nothing to do with directions in space. It is just a number. In particular, it is incorrect to imagine that complex numbers are "vectors", especially if it leads you to think that they're related to directions in spacetime. They're not.

You would have to prepare an interference experiment of an object that hasn't rotated with the "same" object rotated by 360 degrees - and it's hard for macroscopic objects because the "same" object quickly decoheres and you must know whether it has rotated or not, so no superpositions can be produced. ;-)

However, all detailed measurements of the spin with respect to any axis indirectly prove that the fermions transform as the fundamental representation of . In particular, if you create a spin-up electron and measure whether it's spin is up with respect to another axis tilted by angle , the probability will be . The only sensible way to obtain it from the amplitude is that the amplitude goes like and indeed, this function equals for equal to 360 degrees.