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Hype about a formula for eigenstates

OK, Charles wrote e-mails about this topic to me and I think that there's something to say. Quanta Magazine's Natalie Wolchover has announced that

Neutrinos Lead to Unexpected Discovery in Basic Math
It's a story revolving around the July paper
Eigenvalues: the Rosetta Stone for Neutrino Oscillations in Matter
by Denton, Parke, Zhang (who are smiling on a photograph). They have played with the neutrino oscillations whose relevant mathematics revolves around eigenstates and eigenvalues of \(3\times 3\) matrices. Just to avoid misunderstandings, these are tables with \(3\times 3=9\) numbers and the eigenvalue \(\lambda\) of the matrix \(M\) and the eigenstate \(\vec v\) are objects that obey \(M\cdot \vec v = \lambda \vec v\) where the dot is the matrix multiplication (entries of the result are the sums of products of elements of a row of the left matrix and entries of a column of the right matrix; you combine all columns with all rows and which ones you choose decides where you write the resulting sum of products in the resulting matrix) – that was my crash course of linear algebra LOL.

And they have found an explicit formula for \(|U_{\alpha i}|^2\) squared matrix elements of a mixing matrix – which is basically equivalent to a matrix whose rows or columns are some eigenstates. Their formula is a product of differences divided by another product of differences. In the denominator, the differences are differences between the eigenvalues of the actual neutrino Hamiltonian matrix. In the numerator, some of these eigenvalues in the differences are replaced with eigenvalues of some submatrices.

This verbal description arguably conveys more information than the formula\[

|\hat U_{\alpha i}|^2 = \frac{ (\lambda_i - \xi_\alpha) (\lambda_i - \chi_\alpha) }{ (\lambda_i-\lambda_j)(\lambda_i-\lambda_k) }

\] where \(j,k=1,2,3\) are chosen so that \(i\neq j \neq k \neq i\). Great. With this formula, one may also write some mixing angles explicitly. It's a formula for a solution (well, only the absolute value of entries of a solution...) that I haven't ever seen or at least I don't realize it. And I don't really "know" the formula even now when I see it. I sincerely hope that it's a correct formula. At least heuristically, it's a matter of seconds to become certain that the formula works (if it works). Pick a random enough matrix, calculate the eigenstates numerically, and check whether it agrees with the claim above. When 200 digits agree, it won't be an accident.

The real question is whether the discovery is important or interesting. And whether it is surprising.

Well, I tend to say "Not really". Just to be sure, this statement isn't terribly unusual. If you look at Google Scholar, you will see that after 4 months, the paper has 1 citation and it is a self-citation by the same authors. That followup, a September 2019 paper, doesn't have any citation so far.

So the bibliometric analysis seems to disagree with the ambitious assertions that it's a revolution in basic mathematics inspired by neutrinos etc.

You know, it's a formula for the squared absolute values of the eigenstates' coordinates; and given some knowledge of rather unusual data. There are lots of "inverse problems" of similar kinds that you may invent for \(3\times 3\) matrices or for matrices of other sizes and lots of variations of these problems. It's obvious that almost no physicist remembers the solutions for all of them. Physicists remember the definitions of objects, basic equations of the physical laws, and identities that are used all the time (like the solution to the quadratic equation LOL). But when they need something more special and cumbersome, they generally derive it or find it somewhere.

The general question for each such problem is whether the solution may be written explicitly or analytically – or not. In this case, it can be written explicitly. Good for them. Given the answer is "Yes", we may discuss the form of the solution. I find it natural that it is a ratio of rather similar things. And I find it natural that the denominator includes the product of differences of eigenvalues – because when the eigenvalues collide, the eigenstates must become ill-defined due to the degeneracy.

There are lots of formulae involving products of differences between the eigenvalues or matrix entries, starting with the Vandermonde determinant and various formulae for inverse matrices (subdeterminant over determinant). So the general Ansatz of the formula they find looks rather natural and compatible with the instincts – at least mine. Amplitudes in perturbative string theory or the amplituhedron integrands (and some resulting amplitudes) are ratios, too.

OK, why does Wolchover claim that it is interesting or surprising? Because they gave that formula to Terence Tao, a Fields Medal winner, for him to say something and he didn't know the formula and even expressed doubts. So it must be very surprising or important. OK, please, give me a break. Terence Tao is far from being omniscient and it's utterly normal for him – and any other randomly chosen mathematician or theoretical physicist – not to know such identities or explicit formulae for the solutions to rather contrived problems. If such formulae were known to physicists and mathematicians who are equally far from them as Tao, Denton, Parke, and Zhang, then Denton, Parke, and Zhang wouldn't have to "work" on them, either.

The conclusion that the formula is interesting, important, and surprising is based purely on totally irrational sociological memes – mainly on some cult of personality of Terence Tao as an omniscient being. Wolchover's article simultaneously assumes this bizarre Tao Cult; and it helps to build it and expand it, too. Sorry, this has nothing to do with science. Most mathematicians and physicists just won't know a technical result that researchers who are just working on some special, different questions discovered now. There is nothing special about physicists' and mathematicians' not knowing something that others have just found; and there is nothing special about Tao's not knowing, either.

On top of that, I don't really think that Tao is among the best people in the world who may give reasonable opinions on such questions about the formulae for eigenvalues – although he was able to produce two proofs within hours, good for him.

So another part of Wolchover's discussion is whether this exact formula has appeared somewhere. They found very similar formulae but not quite the same. Who really cares about the modest differences if any? The point is that the similar identities weren't super-hyped in the media in the past – so there is no good reason to hype this one, either. There was no real demand for a similar identity – which is why one should expect that when there is suddenly supply, it's not too important! (If there had been a demand, people would simply construct it.)

At the end, after Wolchover's article was published, they found a Piet Van Mieghem's paper from 2014 which contains the exact same formula.

To emphasize this resolution: This super-duper mysterious, surprising, revolutionary formula that has impressed even the semigod Terence Tao was already written down more than five years ago in a preprint by a nearly unknown author. Back in 2014, no one hyped that preprint or its author. Piet Van Mieghem wasn't photographed as a smiling priest whose result had the honor of employing semigod Terence Tao for hours. Nothing. We don't know what Piet Van Mieghen looks like and his name still appears only once in a footnote under Wolchover's article. Strictly speaking, I don't even know whether Piet is a he. ;-)

And that is the case despite the fact that by now, Mieghem's paper actually has nontrivial 20 citations according to Google Scholar. That's many more than "zero" of Zhang, Denton, Parke (although their paper is newer).

My point is that something is totally corrupt about the way hot the popular science media work these days. Why was this effectively zero-citation paper singled out for this amazing hype? Just think about the events and their causal relationships. When you do it right, the answer is: Because someone randomly sent a formula from a particle physics paper to Terence Tao!

So that's how it works. If you want to promote a new chocolate candybar, send a formula – or the nutritional values from the package – to someone like Terence Tao, to a darling of the media. The popular science writers will do the rest. No, this is really sick. And it is bad that I must point it out – that it wasn't done by Tao himself. The degree of excitement and hype (and faith in the validity) that the popular science writers spread should at least approximate the degree of excitement (and faith in the validity) according to the actual researchers and the importance that they assign to various results. These two things – the hype according to the actual scientists and the excitement conveyed by the journalists – have basically completely decoupled from each other.

And that's very pathological because this decoupling also implies the decoupling between the readers of the media (including newspapers and the Quanta Magazine) and the actual scientists. The public is led to believe that science is something completely different than it actually is. It is led to be incorporated into cults and "faith in omniscience" that the actual scientists don't believe. Examples of this fact are omnipresent (claims about omniscience are more omnipresent than the omniscience itself) but I wanted to show that even when it comes to some totally straightforward results, the media approach the scientific work totally incorrectly, irrationally, and dishonestly.

Bonus: textbook stuff?

Incidentally, Tao said:
Something this short and simple — it should have been in textbooks already. So my first thought was, no, this can’t be true.
A rational reader who has seen this story would be assured that it's wrong to assume that Terence Tao is capable of correctly describing the current status of a random eigenvalue identity. But the rational thinking is marketed as a politically incorrect thing by the contemporary journalists. Wolchover wants the readers to believe that there was nothing wrong about her wrong approach and I am almost sure that it wasn't the last time when the omniscient Tao was asked a similar question and when his answer will be considered very important.

Tao's textbook comment is silly, too. I have co-written a linear algebra textbook and I can make you sure that there is no reason for this non-fundamental identity to be in basic textbooks of linear algebra. It could only belong to a book with many exercises; or some very non-basic textbooks. But textbooks of what? When you write more advanced textbooks about topics based on linear algebra, these textbooks are usually not about the classification of increasingly complex problems that you may invent (calculate this and that in an eigenvalue problem given this and that).

Unless such a problem becomes omnipresent in research, identities such as this one simply don't belong to any textbooks (and they don't belong to physicists' and mathematicians' memory) because they're utterly non-elementary. Solutions to such random cumbersome problems are a part of the life but they are not a part of the universal learning or teaching because the universal learning should only cover the foundations that will be used again and again; the learning shouldn't cover every possible event or problem that you will encounter in your life. In physics, there's even nothing wrong about determining similar parameters numerically (especially because we know that matrix entries of the real-world neutrino mass matrix with some precision now and there is only one matrix that is "right").

A physicist doesn't even need to know whether an analytic formula of this form exists. A mathematician may like to solve similar problems but this direction of extending his expertise won't really make him a leading expert in any important sense – the habit of solving many such problems is really recreational mathematics, not deep professional mathematics.

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