Electron's electric quadrupole moment

If you think something is badly wrong with the title, we're on the same frequency

I've answered over 1,000 questions at Physics Stack Exchange, many of them are interesting, most of them show some elementary misconceptions among the laymen, but only some of them show that there exist insanely wrong papers in the literature. The latter category usually covers topics that have been discussed on this blog many times.

A new category of such questions was opened by Grant Teply who asked the following question today:

I understand an electric quadrupole moment is forbidden in the standard electron theory. In this paper considering general relativistic corrections (Kerr-Newman metric around the electron), however, there is a claim that it could be on the order of \(Q=−124e\cdot{\rm b}\). That seems crazy large to me, but I can't find any published upper limits to refute it. Surely someone has tested this? Maybe it's hidden in some dipole moment data? If not, is anyone planning to measure it soon?

The huge quadrupole moment is obviously wrong so I instantly started to write an answer saying how many things would be different – including energy levels of the Hydrogen atom – if the quadrupole moment were this high. In a few minutes, however, I regained my common sense and realized that the quadrupole moment has to be zero, of course.

First, I will repost my answer and then I will discuss the 2004 paper and the wrong culture in which it was written.




First, my answer:

I think that the paper is completely wrong and the conclusions are preposterous. The paper argues that when one models the vicinity of the electron as a rotating black hole, he will get new effects.

However, the black hole corresponding to the electron mass – which is much lighter than the Planck mass – would have a much smaller radius than the Planck length. It really means that the Einstein-Hilbert action can't be trusted and all the quantum corrections are important. It also implies that the typical distance scale in any hypothetical electric quadrupole moment of the electron would be much shorter than the Planck scale – surely not a femtometer. Also, the black holes with masses, charges, and spins similar to those of electrons would heavily violate the extremality bound – something that would be a problem for astrophysical black holes but it isn't a problem in particle physics because the classical general theory of relativity can't be trusted for such small systems.

The facts in the previous paragraphs are just different perspectives on the universal facts that gravity may be neglected in any observable particle physics, a fact that the author of the paper tries to deny.

Proof of the vanishing of the quadrupole moment

More seriously, one may prove from quantum mechanics that the quadrupole moment for an electron, a spin-1/2 particle, has to vanish because of the rotational symmetry. The quadrupole moment is a traceless symmetric tensor and because the electron's spin is the only quantum number of the particle that breaks the rotational symmetry, one would have to express the quadrupole moment as a function of the spin, i.e. as\[

Q_{ij}=\gamma\cdot(3S_i S_j+3S_j S_i−2S^2 \delta_{ij})

\]However, in the rest frame, \(S_i\) simply act as multiples of Pauli matrices (with respect to the up/down basis vectors of the electron's spin) and the anticommutator \(\{S_i,S_j\}\) above – needed for the symmetry of the tensor – is nothing else than the multiple of the Kronecker delta symbol, so it cancels against the last term. \(Q_{ij}=0\) for all spin-1/2 objects (and similarly, of course, for all spin-0 objects). Only particles (nuclei) with the spin at least equal to \(j=1\) (the case of deuteron) may have a nonzero electric quadrupole moment; the spin matrices \(S_i\) no longer anticommute with each other for \(j\geq 1\). This simple group-theoretical selection rule is the reason why you won't find any experiments trying to measure the electron's (or proton's or neutron's or other spin-1/2 particles') electric quadrupole moment. Such experiments would be as nonsensical as the paper quoted by the OP.

Note that unlike the case of the electron's dipole moment, one doesn't have to rely on any C, P, or CP-symmetry (which are broken) to show that the quadrupole vanishes. To deny the vanishing, one would have to reject the rotational symmetry.

Let me wrap by saying that the quadrupole moment may always be interpreted as some "squeezed" or "elliptical shape" of the object or particle. This ellipsoid would be stretched along some axes and shrunk along other axes. However, the electron's spin-up and spin-down state really pick the same preferred axis in space – the sign doesn't matter for the quadrupole – so they can't have different values of the quadrupole moment. In other words, the quadrupole moment doesn't depend on the spin, and because the spin is the only rotational-symmetry-breaking quantum number that the electron has, the quadrupole moment has to be zero. (A Pauli-matrix-free proof.)

The author of the 2004 paper was Kjell Rosquist who is actually a physics professor in Stockholm. As far as I can say, he or she shouldn't have passed qualifying exams as a PhD student because the paper reveals his or her complete misunderstanding of many key topics in graduate physics – such as how far certain theories can be trusted; and angular momentum in quantum mechanics.

But those things "don't matter" in a certain culture. Kjell Rosquist is a "relativist" and it's still considered "kosher" for "relativists" to say many clearly preposterous things about particle physics or the range of validity of classical general relativity, and related things. After all, Albert Einstein – the father of relativity – has said many similar things as well so why shouldn't his followers enjoy the same right?

Well, that's a good question and it has an even better answer. They shouldn't because more than half a century of progress has shifted the physics research after the death of Albert Einstein. Stupid propositions about the electron that were "OK" during Einstein's life simply aren't OK today. Not to mention that Einstein wasn't massively humiliated for some of the statements because of his amazing achievements in many parts of theoretical physics.



During Einstein's life, they didn't even have this impressive human-powered Rube Goldberg machine that turns on a TV. Via Jorge P.

So I believe it's just totally wrong to allow whole groups of people to justify their junk physics by claims that they belong to a "different culture". They shouldn't have the right to belong to a "different culture of science" where junk physics is tolerated simply because junk physics isn't a legitimate branch of science. And of course, "pure relativists" often produce lots of junk physics. This example of an "electron as a rotating charged black hole" with a huge "electric quadrupole moment" was perhaps "politically neutral". But loop quantum gravities, spin foams, and various other would-be "unifying theories" belong to the very same category of junk physics born in the heads of "relativists" who sometimes make excursions into topics they have no idea about.

So, Mr or Ms Kjell Rosquist, spin-1/2 particles can't have a nonzero electric (or magnetic, or any other) quadrupole moment. And elementary particles can't be described as classical black holes – classical solutions to the equations of the general theory of relativity – because they would be hugely "superextremal" and at the distance scale that corresponds to the tiny elementary particles' masses (the corresponding Schwarzschild radii), Einstein's equations are simply not applicable. (Heavily excited string modes are "marginally" describable as black holes.) This statement shouldn't be culture-dependent. Inequalities that determine what the parameters must satisfy for certain approximate theories to be applicable are calculable and demonstrable facts of physics – and all physicists should learn them and agree about them just like they agree about other insights of physics.

Meanwhile, it really looks like Mr or Ms Kjell Rosquist hasn't been told for 8 years that spin-1/2 particles can't have a nonzero quadrupole moment. I think it's an elementary selection rule that every graduate student of physics and good undergraduate students of physics should know. However, in certain "different cultures", it is either a blasphemy, a politically inconvenient truth, or something that is left to personal preferences.

This ignorance and self-confidence of the ignorant people is so overwhelming that even David Zaslavsky, a very sensible and educated moderator of the discussion forum, got immediately carried away and was immediately assuming that there have to be lots of papers that measure the "electron's electric quadrupole moment". The misconceptions are so audacious that even otherwise sensible people don't even dare to doubt them.

This guy also insists that he wants to measure the effect, anyway. How can one even try to measure something that is mathematically impossible?