How can electrons spin if they have no volume?

Question

My understanding of electrons is that they have no volume, eg they are point particles. If this is true, how can a point spin?

Answer 1

The electron spin corresponds to a classical angular momentum, but it is not related to any "rotation" of the electron. The spin of the electron is a quantum mechanical phenomenon and can only have the value of + or - $\frac {\hbar}{2}$ in a given direction. The electron is a point-like particle, it has no finite spatial dimensions that could rotate.

Answer 2

If you ignore the location/velocity of an electron, each electron carries one bit (or, more precisely, q-bit) of information: whether it is spin up or spin down with respect to a chosen axis. This information is encoded in a vector with two components and complex components. That is, the "spin state" of the electron can be written as

$$(\alpha, \beta) \in \mathbb{C}^2$$

where $\alpha$ and $\beta$ are complex numbers, and $|\alpha|^2 + |\beta|^2 = 1$. There is nothing "spinning," nothing is "dynamical." If you rotate the electron (or even just rotate your reference frame) the spin state of the electron will have to "rotate" accordingly as well. (How do you rotate an electron? You can manipulate it with a magnetic field, or you can just give up and mathematically pretend you are rotating it.)

Any appropriate "rotation" of a vector in $\mathbb{C}^2$ will be an $SU(2)$ matrix. The 3 dimensional rotation group, $SO(3)$, must somehow be mapped into $SU(2)$. (The map can only be accomplished up to a physically unmeasurable sign ambiguity, but whatever. This is called the "spin 1/2" (projective) representation of $SO(3)$.)

If that went over your head, consider this: the spin state is, in an extremely crude sense, pointing in some direction, and this little arrow, pointing in space, must be rotated somehow. In classical mechanics, aside from velocity, the only arrow you can really associate with an object is its angular momentum, which will will only be non-zero if the object is "spinning." Unlike a classical object, which can be made to stop spinning, a spin 1/2 particle like an electron, always has this little arrow pointing somewhere that we call spin.