D.70 More awkwardness about spin

How about that? A note on a note.

The previous note brought up the question: why can you only change the spin states you find in a given direction by a factor $\vphantom0\raisebox{1.5pt}{$-$}$1 by rotating your point of view? Why not by ${\rm i}$, say?

With a bit of knowledge of linear algebra and some thought, you can see that this question is really: how can you change the spin states if you perform an arbitrary number of coordinate system rotations that end up in the same orientation as they started?

One way to answer this is to show that the effect of any two rotations of the coordinate system can be achieved by a single rotation over a suitably chosen net angle around a suitably chosen net axis. (Mathematicians call this showing the “group” nature of the rotations.) Applied repeatedly, any set of rotations of the starting axis system back to where it was becomes a single rotation around a single axis, and then it is easy to check that at most a change of sign is possible.

(To show that any two rotations are equivalent to one, just crunch out the multiplication of two rotations, which shows that it takes the algebraic form of a single rotation, though with a unit vector ${\vec n}$ not immediately evident to be of length one. By noting that the determinant of the rotation matrix must be one, it follows that the length is in fact one.)