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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?

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snail feedback (18) :

reader Dilaton said...

Nice answer +1 :-)

Reading the abstract of the paper just confused me, I had no clue what he was talking about ... Now I see that I dont have to have a clue :-P

reader Prathyush Manchala said...

Mr. Motl Please can you look into this question and tell me your views, I dont know any other way to contact you.

reader Luboš Motl said...

Hi, I posted an answer over there...

reader Mitchell said...

In English, especially British English, "I understand that X is true" can just mean "I hear that X is true; people tell me that X is true".
See definition 2: "to believe, based on information".

So Grant Teply was just saying that he already knew that the electron isn't supposed to have a quadrupole moment; he wasn't claiming that he already understood *why*.

reader Peter F. said...

Kjell is a male Swede - one of all those who ought to feel ashamed of themselves. ;->

reader Mitchell Porter said...

I checked Rosquist's paper, and he does in fact acknowledge that "a spin 1/2 particle cannot have an electric quadrupole moment" (page 7). So he knows the textbook QM. What he's doing is calculating the quadrupole moment that should exist, if space around the electron is curved at the scale of the Compton wavelength. That's a more interesting discussion because it would focus on the difference between this "Kerr-Newman" model of the electron's gravitational field, and a properly quantum-gravitational description.

reader anna v said...

With all due respect Lubos, most of failed physics theories, before declared obsolete by new data, mathematically declared certain things as impossible . Al That is how a later experiment falsified them. So your last statement is not true. All theories modelling physics are mathematically self consistent and closed. It is the experiments that validate them for physics, not the mathematical QED.

In this particular case I agree that the mathematics of the validated theory is such that even to envisage a violation forces a change in the attribution either of spin or of what an electron is, something that experimental data drastically excludes.

reader Luboš Motl said...

Except that it's a self-evident lie. He hadn't known that the quadrupole moment had to be zero when he originally asked the question at SE.

reader Luboš Motl said...

Dear Mitchell, you clearly misunderstand the character of the QM derivation, too. The derivation proving that Q_{ij} = 0 for j=1/2 doesn't make any assumptions about the internal structure of the object, its gravitational field, or anything else. It is completely universal - it is group-theoretical, and as such, it must be 100% true in every consistent rotationally symmetric theory whether it's special relativistic, QFT, string theory, or anything else you may find!

reader Luboš Motl said...

Dear Anna, your comment is just one from the crackpot category "everything goes". The failed theory declared obsolete here is *classical physics* (i.e. childish models from the Swedish paper), not quantum mechanics, something that you and many other stubbornly misled people refuse to even try to understand.

reader Raimond said...

A lot of garbage out there, indeed.
Here is the biggest garbage I've ever come across, he's a physics professor !?
enjoy !

reader Mitchell Porter said...

How do you use the group-theoretical argument in curved space? It just seems to me that the situation becomes more complicated once you include gravity. There could be an "anomalous electric quadrupole moment". Rosquist doesn't consider issues of QFT in curved space, he just substitutes the spin into classical formulas for the Kerr-Newman solution. Probably that's wrong because he gets the large quadrupole. But could there be a small anomalous quadrupole?

reader Vlad said...

There's NO electron's electric quadrupole moment at all.
It's elementary particle without structure.

reader Luboš Motl said...

Well, an important point but only approximately true. Electron has no structure but it still has a dipole moment - a tiny one - due to CP-violation. Symmetries dictate what is zero and what is nonzero.

reader Luboš Motl said...

Dear Mitchell, yours is a meaningless question. The electric quadrupole moment is *defined* from the asymptotic expansion of the electric field at infinity - and one has to assume that the space is asymptotically (i.e. at infinity) flat. On the other hand, what happens in the "bulk" of the space isn't constrained at all.

In particular, the group-theoretical argument isn't compromised in any way by the fact that the particle itself induces a gravitational field around it. It doesn't matter at all. The "full electron", including the gravitational field around it, is still a j=1/2 doublet and the operators of the angular momentum - the generators of the symmetry - act by the Pauli matrices *exactly* as indicated. There is no approximation here. So no, gravity doesn't induce any quadrupole of the spin-1/2 particle, not even a tiny one.

The analysis he made is wrong because of dozens of reasons. In particular, the black holes he considers are super-extremal and they're not allowed even by general relativity. As I said, this is no inconsistency because there's no reason why elementary particles should be describable as classical solutions to GR. They are not and if you want to fully describe the fields around them etc., you need a theory of quantum gravity in which the corrections are more important than the Einstein-Hilbert action and which easily circumvents the extremality bound by totally modifying it for such light particles (which are not really black holes). Just look what's happening in string theory.

One may study classical or quantum field theory in curved spaces and it brings complications but they have really nothing to do with the issue here which is that the intrinsic quadrupole moment of a j=1/2 particle exactly vanishes.

reader Rehbock said...

On reading your remarks yesterday contrasting the electron dipole moment, I googled. I too had always assumed like Vlad.. I noticed that this still small beyond experiment moment may soon have its experimental moment in quantifying the percent C and P violations. Paywalls and constraints on time limit me but seems interesting.

reader Sage Basil said...

so I hear that SO(3) has representations of any odd dimension n, which are described as 2^(½(n-1))-pole (monopole, dipole, quadrupole, octopole). Monopoles are one scalar; the claim appears to have been made that a dipole is a 3-vector and a quadrupole is a symmetric rank 2 tensor over a 3-dimensional vector space. Here's such a tensor: xi⊗i + yj⊗j + zk⊗k + a(i⊗j+j⊗i) + b(i⊗k+k⊗i) + c(j⊗k+k⊗j). I'm counting 6 components.

If, on the other hand, you were claiming that a dipole is a 4-vector with one dimension projected out to normalize it, that would have the necessary 3 components to be acted on by a third degree representation (and, being a projective object, could never deviate from its unit norm). We can do the same thing to an alternating 2-tensor, a(h⊗i-i⊗h) + b(h⊗j-j⊗h) + c(h⊗k-k⊗h) + d(i⊗j-j⊗i) + e(i⊗k-k⊗i) + f(j⊗k-k⊗j), and get a 5-dimensional vector space.

reader Diego Floor said...

Oh boy...
I didn't read the paper but I was going to try to defend it a little by saying it might be just a fun mathematical/thought experiment of "what if the universe was different?" But trying to measure something kind of defeats my good intentions.