Undergrad Why can't an electron be spinning?

[Vampyr]

I'm trying to understand electron spin. I understand that if you try to explain the intrinsic angular momentum of an electron, you end up with the surface spinning several times the speed of light.

However, calculations always seem to be done classically. Shouldn't it be done relatavistically? Is there a way to model a spinning ball consistent with relativity? Presumably the speed of the spin would be capped at c, but the mass of the electron would have to increase from rest mass.

 

[vanhees71]

As any elementary particle an electron cannot be understood as a bullet-like miniature classical extended body. In fact in classical physics all "point particles" are idealized simplified desriptions of extended bodies. The simplification is in the fact that for many motions like that of the planets around the Sun of our solar systems, we can with good approximation describe the motion of the bodies center of masses around the Sun (also described as a "point particle" by only considering its center of mass) without taking into account the finite extension of the planet and the Sun.

This is wrong for an electron. As far as we know today from scattering experiments at the highest available energies in particle accelerators an electron is really an elementary particle and thus under the circumstances where you consider only a few particles and scattering experiments with them (usually you have 2 particles in the initial state and then look at a "spray" of particles created in an ultra-high-energy collisions) you have to describe the by quantum theory.

For small energies, we can look at the electron in the non-relativistic approximation. Then a single electron can be described by a wave function, which is a socalled SU(2) spinor field. A convenient complete set of compatible observablesin this non-relativistic approximation are the three Cartesian components of the momentum of the electron, and one component of the spin, which is an angular momentum realized by the fundamental 2D representation of the group SU(2).

Physicswise the spin manifests itself in the fact that an electron besides carrying one negative elementary charge $-e$ it also carries a magnetic moment of approximately one Bohr magneton. The fact that it is one Bohr magneton, i.e.,
$$\\vec{\mu}=\frac{-e g_s}{2m_e} \vec{s}$$
with a gyro-factor $g_s \simeq 2$ and not a gyro-factor of 1, as you'd expect from a classical "ring-current model" a la Ampere (indeed the gyro-factor for the magnetic moment of electrons within an atom associated with the orbital angular momentum is 1, and that's the quantum description of Ampere's old idea), shows that the electron as an elementary particle cannot be understood in any classical terms.

 

[hilbert2]

Or more simply put, there's no way to measure an electron's spatial orientation, like you can do with macroscopic bodies and define the angular velocity as a time derivative of an angle describing that.