Exploring Orbital Angular Momentum & Magnetic Moments

[MadRocketSci2]

So, I've been looking into orbital angular momentum and magnetic moments, (which, at least in normal space with a normal angular topology seems limited to integer values of spin). (My model so far has been a parabolic potential harmonic oscillator in 3d, and the sort of spinning modes you can construct from a basis of degenerate modes at each energy level).

I've sort of been stubbornly clinging to the idea that there is a connection between the orbital angular momentum of scalar particles in a potential well of some sort (one where there must be some large energy difference between the ground state and any other state, otherwise you would expect to see series of particles at different spin energies), and the sort of angular momentum exhibited by "intrinsic particle spin." (Points don't have moments, dangit! Except as some sort of limit around a ball going to zero size.)

My understanding is that for spin-1/2 particles, the sort of "internal azimuthal wavefunction" for the scalar sub-particle-in-some-sort-of-well has to be distributed over a space with a weird topology. (In the language that I've read so far: The Spin-Group has the same set of generators as the normal rotation group, but a different topology such that it takes a rotation of 4pi along any combination of the unit-infintessimal generators to return to what is regarded as an equivalent point.) (I have to wonder if in reality, then, we actually have more rotational scope than is apparent, hidden from us by some symmetry)

Okay, so we can generate a "double-covering" space, where two distinct (opposed) points on the sphere correspond to the same points on the normal 3-sphere. (For points away from the origin, there are circles and tracks of "equivalent points in 3-space" that are not equivalent in the double covering space). Do I have this so far?

Why just a double covering space? It seems to me, if I am free to chop the topology of the space within which this wavefunction exists any way I want, there would be nothing preventing me from positing "quadruple-covering-spaces" and spin 1/4 particles, "8x covering spaces" and spin 1/8 particles and so on. Why do we only end up with spin 1/2?

(Again, pursuing my idea that there is something down there actually spinning around, given that it must be happening at something approaching the speed of light, I wonder if a general relativistic effect can explain the weird topology of this space.)

 

[Bill_K]

(Points don't have moments, dangit! Except as some sort of limit around a ball going to zero size.)

It's not just a point. An elementary particle with spin is described by a multicomponent quantity (spinor, vector, etc) at that point, and under a rotation the components go into linear combinations of each other. In QM, if an object changes under rotations, that is exactly what we mean by angular momentum.

The Spin-Group has the same set of generators as the normal rotation group, but a different topology such that it takes a rotation of 4pi along any combination of the unit-infintessimal generators to return to what is regarded as an equivalent point.)
Okay, so we can generate a "double-covering" space, where two distinct (opposed) points on the sphere correspond to the same points on the normal 3-sphere. Why just a double covering space? It seems to me, if I am free to chop the topology of the space within which this wavefunction exists any way I want, there would be nothing preventing me from positing "quadruple-covering-spaces" and spin 1/4 particles, "8x covering spaces" and spin 1/8 particles and so on. Why do we only end up with spin 1/2?

An element in the 3-D rotation group SO(3) can be represented by a vector pointing along the axis of rotation, with length equal to the angle of rotation. The angle is at most π. Such vectors fill up a sphere of radius π, and diametrically opposite points on the surface of the sphere are considered the same point.

Consider closed paths (loops) in this group space, the interior of the sphere. These represent a continuous series of group operations. Some can be shrunk all the way down to a single point. Any series of group operations that can be deformed to a point is the identity. Some cannot be reduced to the identity in this way. For example a diameter is a closed loop, since its two points on the surface are identified, and cannot be shrunk. But a path that goes through the sphere twice actually can be deformed and shrunk to a point. In fact, a path that goes through any even number of times can be shrunk, while a path that goes through 3 times, or any odd number of times, cannot be. This means the rotation group is doubly connected.

So there exists another group, SU(2), the universal covering group of SO(3), which covers it twice and itself is simply connected. All loops in SU(2) can be deformed to a point. An object which forms a representation of SU(2) may require a double loop in SO(3) (a 4π rotation) to go into itself, and is consequently a half-integer spin object.

But this doubling can't be repeated, because SU(2) is simply connected. ALL loops in SU(2) can be shrunk to a point, and consequently all objects must go into themselves under a single loop. So there can't be any such thing as a fractional spin object, except for half-integer ones.

(Again, pursuing my idea that there is something down there actually spinning around, given that it must be happening at something approaching the speed of light, I wonder if a general relativistic effect can explain the weird topology of this space.)

Idle nonsense.