- ... ball.1
- If you want to be really concrete, imagine a spinning gyroscope
fitting snugly in a box. Rotate the box.
- ... (frames).2
- In other words, a element of
determines a mapping of to . As it happens,
the action is faithful, i.e., the mapping determines the element of
- ... moment.3
- A quick review: write for the equivalence
class of . We will associate either a complex number
or else with each class . If , then
, where . All pairs of the form belong to the
class . So we associate with if , and
with . Mapping the complex plane plus to the
Riemann sphere via the usual stereographic projection completes the trick.
Some sample points to bear in mind: the north pole is ; the south
pole is ; points on the equator have the form
(For the purist, the special treatment of rankles. It is not
singled out on either the complex projective line or on the Riemann sphere.
Later I will show how to set up the correspondence without this blemish.)
- ... simple4
the infinite dimensional case, you have to use the spectral decomposition
of instead of an orthonormal basis of eigenvectors; another reason why
spin is simpler than position.
- Why wouldn't the electron simply snap into
alignment with the magnetic field? Answer: the spinning electron would act
like a gyroscope, and precess in response to the torque exerted by the
field. Thus it would maintain its angle of inclination to the field.