## The nature of spin, magnetism, and monopoles

Spin is a rather interesting parameter (in case you didn’t already know that).  It is generally believed to be an inherent property much like mass and charge.  Nevertheless, it does appear to affect certain systems in much the same way as a more traditional angular momentum would.  The reason, of course, that we don’t think of it in these terms – i.e. that it actually is the angular momentum of a “spinning” particle – is that the electron would need to be spinning faster than the speed of light if spin really were associated with a rotation (this was first noticed by Pauli, criticizing Ralph Kronig’s interpretation of Pauli’s “two degrees of freedom” for the electron, though neither Kronig’s suggestion nor Pauli’s response was published.  Independently, Sam Goudsmit and George Uhlenbeck made the same observation).  Ultimately this led to the proposition that the electron had no size, i.e. it was a point particle (which was the only way to have it obey relativity and still have an angular momentum).

But here’s a curious thing.  Physicists were presented with a conundrum: if spin is the angular momentum of a spinning, non-point particle then the electron would have to be spinning so fast it violated relativity.  So, should they abandon the notion of spin as an angular momentum or should they abandon the non-point particle interpretation?  In fact, over the course of time, they did both.

Part of the reason for this is surely due to the fact that we now know that whole atoms and even molecules can have a quantized spin and this simply makes no sense if spin is just plain, old angular momentum.  So we’ll agree, then, that spin isn’t just angular momentum.  On the other hand, it is clearly related since we know, for instance, that the total angular momentum of an electron in an atom can be given by the sum of the orbital angular momentum and the spin.

So where do we draw the line on the comparison between spin and angular momentum?  I say that we shouldn’t be too hasty to disassociate the two since treating it more like an angular momentum neatly explains two interesting ideas: the origin of magnetism and the non-existence of magnetic monopoles (I acknowledge the inspiration of Tom Moore regarding these ideas).

So, first, let’s assume that it does represent some kind of angular momentum behavior despite its obvious deviations from true classical angular momentum behavior.  Let’s also assume that, in all instances in which we observe spin, it is associated with something (an atom, a point-particle, etc.) that, whether or not it has any true size, possesses a spherical symmetry.  In this way we draw an analogy to a black hole which is technically a single point but that possesses a spherical symmetry in the extent of its field.

Relativity tells us that massive, rotating bodies actually “drag” spacetime around themselves in a process known as “frame-dragging” (whether spacetime truly rotates or not is debated but, given the recent results of the Gravity Probe B experiment, we take it as such since it seems the only logical way to explain why photons appear to behave differently in such a situation depending on which direction they are going).  If we take spacetime to be the gravitational field (i.e. the old “dimples on a bedsheet” interpretation which is admittedly debated) then frame dragging can be viewed as the motion of the actual field itself in a kind of twisting sense (or, perhaps more accurately, like a vortex).

Given that, but irrespective of the relationship between the electromagnetic field and gravitational field, the “rotation” of an electron ought to include rotation of its electric field.  This rotation means that there is always relative motion between an electron’s electric field and any observer and it is well-known that changing electric fields produce magnetic fields (and vice-versa).  As such, at the most fundamental level, magnetism is a purely relativistic effect.  A colleague of mine originally disputed this saying that one could simply imagine an isolated Dirac monopole doing the same thing, thus making electricity relativistic.  But Dirac monopoles have never been observed.  Furthermore, we know that stationary charge always has an electric field but not necessarily a magnetic field.  It is not clear that a similar intrinsic property exists for magnetism (though some have conjectured spin fits the bill, but it depends on how we interpret spin!).

Now, recalling that we are assuming a spherical symmetry (or ellipsoidal if we use the frame-dragging analogy), the moment any sphere is rotated on an axis, one immediately ends up with two possible directions it can spin on that single axis.  This is a simple fact of rotational symmetry and, when combined with spherical symmetry, one sees that any rotating sphere rather naturally has two poles!  Thus, if magnetism is truly nothing but a relativistic process as proposed above, then magnetic monopoles ought not exist.  Since magnetic (Dirac) monopoles have ever been found, the evidence seems to suggest that magnetism is purely relativistic and that spin is more like angular momentum than we like to think it is.

### 2 Responses to “The nature of spin, magnetism, and monopoles”

1. BlackGriffen Says:

For any electron in a pure state, as I understand any single electron must be, we can associate it’s spin state as corresponding to spin up in some direction. As such, an electron imagined as a point particle should always produce a pure dipole magnetic field. You’ll note that sticking a point-like magnetic dipole in the same place as a point-like electric monopole implies a circulating Poynting vector.

My questions. I’ve heard that QFT radiative corrections can screw up the “spin up in some direction” interpretation. This comes from an off-handed comment at a colloquium by Leggett. Would you happen to have a brief explanation handy for this? If not, no big deal.

The other question is do the radiative corrections address the fact that the angular momentum stored in the electromagnetic field would seem to exceed $\hbar/2$? I tend to assume that renormalization does so, to the extent that renormalization can be considered to have solved the problem of divergences by stuffing them all away into other parameters. I have assumed that this is so without investigating because it seemed unlikely that this wasn’t addressed. Looking at the expression for the classical level angular momentum:

$\bar{L} = \frac{g e^{2} \hbar c^{2}}{48 \varepsilon_{0} m_{e}}\hat{z} \int_{0}^{\infty} \mathrm{d}r \frac{1}{r^2}$

My guess would be that the renormalization of $e$ and $m_e$ balance out the divergent integral, somehow. If we assume that all of the angular momentum associated with the electron’s spin is, in fact, carried by the electromagnetic field around it then we get:
$\frac{m_e}{e^2} = \frac{g c^2}{24 \varepsilon_0 \hbar} \Lambda_{\mathrm{cutoff}}$

the interesting part of this is that the constant of proportionality between the momentum cutoff, charge, and mass is inconsistent with what you’d get with the derivation of the classical radius of the electron (ie setting the energy stored in the elctromagnetic field produced by the electron equal to $m_e c^2$), although the inclusion of the magnetic dipole term doesn’t really effect the answer because the contribution from the magnetic field vanishes as the ratio $\Lambda_{\mathrm{max}} / m_e \rightarrow 0$, as it must to be consistent with $e \rightarrow \infty$. Specifically, if you use the classical radius of the electron as the inner radius cutoff in the formula for the angular momentum in the field you get that the angular momentum in the field is: $\vec{L} = \frac{g \pi}{6}\hbar$. Naturally this doesn’t prove anything since renormalization should alter the functional form of the electric and magnetic fields at small radii. Still, it seems odd that the answer is approximately 1.

BlackGriffen

2. quantummoxie Says:

Hmmm. Never heard that before. Maybe Matt, if he’s still lurking about, has an explanation for that.

As for your other question, I think the answer is ‘yes,’ but I’m not positive. I think it stems from the fact that the charge gets renormalized, if I’m correct.