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.)