Classical geometry tells us that when a rigid sphere rotates in three dimensional

space, it is impossible for all portions of the surface to be moving with the same

speed at the same time; indeed, to be moving at all, because there always must be

an axis. The axis can lay any direction, but an axis there must be. It is related

to the principle that the divergence of the curl of a vector field is inherently

zero. Trying to get around this principle by allowing the sphere to be non-rigid

does not help a whole lot, because there still must be at least one pair of

entrance and exit whorls of opposing handedness, again related to the same vector

principle.

But given the uncertainties in quantum mechanics, it is worth asking whether in the

quantum world, what is classically impossible might be possible. That is, having a

structure which spins but in such a way that there is no axis; all directions are

equivalent. That would be analogous to a sphere all portions of whose surface are

moving about the center at the same speed at the same time.

In the case of particles of spin 1/2, it appears that this is not possible, but

that any (stationary) spin state of such a particle must have one and only one

definite axis about which the spin projection component is fully unambiguous.

States can be found which will place the axis in any direction we choose, but must

always exist and is always the same direction for the specific chosen state.

In the case of particles of spin 1, things appear more interesting. About a given

axis, a state can be chosen such that the component of spin about it is

unambiguously zero. This state will have the characteristic that the spin about any

other axis is always ambiguous. Therefore the only axis of unambiguity is an axis

of non-rotation, from whence it follows that the rotation is axisless. However, it

is not quite fully symmetrical, in that the rotation does not appear identical for

all axes. For axes at right angles to that of non-rotation, there is a 50%--50%

chance that a measurement will show spin "up" or spin "down". For axes more closely

aligned with the axis of non-rotation, the probability of "up" or "down"

diminishes, equally for both, while the probability of "no rotation" increases,

approaching 100% as the axis approaches that of total irrotation. And although it

is clear that this achieves axisless rotation, there does not seem to be a way to

generalize the state sufficiently to achieve total symmetry in all directions. The

best we can do seems to be some kind of ellipsoid of ambiguity having 3 mutually

perpendicular axes along which the probability profile is somehow special.