I Intuitive/classical picture of electron spin g-factor of 2?

[AndreasC]

TL;DR Summary

Pretty much what the title says.

It's been troubling me for a while, is there some kind of intuitive heuristic picture of why the electron spin g-factor is 2? I remembered this question because of the thread about the nature of spin. One of the early models of spin that were proposed was that it represented the electrons spinning around their axis. Under that model, how does one justify the 2 showing up?

 

[vanhees71]

There is no intuitive heuristic picture except the mathematics of electrodynamics as a gauge theory and using the minimal-coupling principle.

First you need the idea of spin-1/2 particles. For spin we don't have an intuition either. You can derive it from the Lie algebra of the rotation group and the fact that in quantum theory we deal with unitary ray representations rather with unitary representations. The ray representations can be lifted to unitary representations of the covering group of SO(3), which is SU(2), and the fundamental representation of SU(2) realizes spin 1/2.

If you stick to non-relativistic quantum mechanics, the right heuristics is to write the free-particle Hamiltonian in the form
$$\hat{H}=\frac{1}{2m} (\vec{\sigma} \cdot \hat{\vec{p}})^2,$$
where $\vec{\sigma}$ are the three Pauli matrices and $\hat{\vec{p}}=-\mathrm{i} \hbar \vec{\nabla}$.

Now you do minimal substitution by gauging the symmetry of the wave function under multiplication with phase factors, i.e., you make this symmetry local, which forces you to introduce the gauge potentials $\phi$ and $\vec{A}$ and define covariant derivatives,
$$\mathrm{D}_t=\partial_t + \mathrm{i} q \phi/\hbar, \quad \mathrm{D}_j=\partial_j-\mathrm{i} q A_i/\hbar.$$
Then you get a covariant "Schrödinger equation",
$$\mathrm{i} \hbar \mathrm{D}_t \psi(t,\vec{x})=-\frac{\hbar^2}{2m} (\vec{\sigma} \cdot \vec{\mathrm{D}})^2.$$
If you work that out you get the Pauli equation including the correct gyrofactor of 2.

The analogous heuristic "derivation" works famously for the relativistic case too, where you just gauge the phase invariance of the free-particle Dirac equation finding again the correct gyrofactor of 2.

In the relativistic case, of course, you have to use quantum field theory for a consistent picture (or reinterpret the classical Dirac equation via Dirac's hole theory to a many-body theory, but that's very inconvenient compared to use modern QFT methods right away), from which you get corrections to the gyrofactor of 2, the famous "anomalous magnetic moment" of the electron (or the muon).

 

[AndreasC]

There is no intuitive heuristic picture except the mathematics of electrodynamics as a gauge theory and using the minimal-coupling principle.

What about Thomas precession though? I'm not SURE what it is about exactly but I have heard it referenced as having explained the factor within the confines of the (inaccurate) intuitive picture of electron spin as the electron spinning around itself. But I don't really understand how that factors in. I am possibly confusing it with something different.

 

[vanhees71]

Sure, if you argue semi-classically about the fine structure you have to take into account the Wigner rotation in the composition of two Lorentz non-collinear boosts when transforming from the instantaneous inertial rest frame of the electron back to the lab frame at different times, which leads to the Thomas precession. For me that's too tedious to really call it intuitive though.

To get intuition for QT you have to train it by learning to think mathematically in terms of Lie-group and Lie-algebra theory in connection with symmetries (Noether's theorems). That's a pretty abstract level of intuition but it's one that works!

 

[BvU]

I know we shouldn't trust wikipedia, but isn't what I encountered here applicable ?:

The spin g-factor gs = 2 comes from the Dirac equation, a fundamental equation connecting the electron's spin with its electromagnetic properties.

I find it back in the very last chapter of Merzbacher QM (needless to say I didn't get that far as an experimentalist, but ...curiosity )

$\$

 

[vanhees71]

Of course, that's the standard story being told that $g=2$ follows from the Dirac equation. It's interesting to see that it also comes out within non-relativistic quantum mechanics by just "gauging" the symmmetry under multipolation of the wave function with a local phase factor for a Pauli spinor in an analogous way. In the case of the Dirac equation it's, however, much more convincing, because there you get it without rewriting the free-particle equation first using the trick with the Pauli matrices and there's no other choice to include the em. interaction using the minimal-coupling principle in the Dirac case, while for the Pauli equation it's not unique.