Neutrinos Lead to Unexpected Discovery in Basic Math
November 15, 2019 8:55 AM   Subscribe

In a way, it’s not surprising that a new insight into centuries-old mathematical objects came from physicists. Nature has inspired mathematical thinking ever since humans started counting on 10 fingers. “For math to thrive, it has to connect to nature,” Vu said. “There is no other way.” Natalie Wolchover writes 2000 words for Quanta Magazine.
posted by cgc373 (13 comments total) 35 users marked this as a favorite
“Centuries old” is a stretch - eigenvalues and eigenvectors come out of Hilbert’s work in the early 20th century, when he named them that. Obviously he built on work that built on work that stretches back to the dawn of time, but that is how everything is.
posted by w0mbat at 9:24 AM on November 15, 2019 [1 favorite]

*for some definitions of basic
posted by Dr. Twist at 9:25 AM on November 15, 2019 [2 favorites]

That is more basic than I was expecting! Though I did take linear algebra.
posted by little onion at 9:34 AM on November 15, 2019

Strange that there’s no mention of the reddit r/math threads where the physicists originally started asking for help: Part I, Part II, Part III. The third part contains the lovely quote:
At this point I was two proofs, a corollary, and some other new things behind. I hacked my way through the new information and was about to send a v2 of the draft the next day when he sends another proof ... now I'm three proofs behind... . At some point during this story, a colleague of mine who straddles physics and math said, “He (Tao)’s famously like a cheery firehose of mathematics, Guess he’s power-washing you today.” I felt clean.
posted by pharm at 10:24 AM on November 15, 2019 [31 favorites]

Frequently I want to make the argument that everything is terrible, and getting worse all the time, but have to grudgingly add a footnote that, ok, fair point, Quanta Magazine is pretty good, but I mean everything else.
posted by Wolfdog at 10:52 AM on November 15, 2019 [11 favorites]

Physicists famously use mathematically sketchy shortcuts (renormalization, etc.) Get Terence Tao on that! There's a goldmine there.
posted by sjswitzer at 1:42 PM on November 15, 2019 [2 favorites]

If you want some background on the concepts here, you could do worse than consulting 3blue1brown. I hope he treats this new result eventually.
posted by sjswitzer at 1:48 PM on November 15, 2019 [1 favorite]

/r/math seems to have somewhat mixed feelings about the significance level as well as Quanta and, well, non-(pure)math folks.

(gosh they are so cute serious, lol : QED)
posted by sammyo at 3:10 PM on November 15, 2019

Physicists famously use mathematically sketchy shortcuts

“If it’s on top, make it zero. If it’s on the bottom, it’s a 1.“
posted by schadenfrau at 3:15 PM on November 15, 2019 [9 favorites]

I've been wondering for a while what applications to physics Cantor's theory of transfinite numbers might ever find, but this article got me thinking along similar lines about the Riemann Hypothesis, and it turns out Natalie Wolchover also wrote an article a couple of years ago about an attempt to use quantum physics to prove the Riemann Hypothesis:
As mathematicians have attacked the hypothesis from every angle, the problem has also migrated to physics. Since the 1940s, intriguing hints have arisen of a connection between the zeros of the zeta function and quantum mechanics. For instance, researchers found that the spacing of the zeros exhibits the same statistical pattern as the spectra of atomic energy levels. In 1999, the mathematical physicists Michael Berry and Jonathan Keating, building on an earlier conjecture of David Hilbert and George Pólya, conjectured that there exists a quantum system (that is, a system with a position and a momentum that are related by Heisenberg’s uncertainty principle) whose energy levels exactly correspond to the nontrivial zeros of the Riemann zeta function. Each of these energy levels, En, corresponds to a zero of the form Zn = ½ + iEn, which has a real part equal to ½ and an imaginary part formed by multiplying En by the imaginary number i.

If such a quantum system existed, this would automatically imply the Riemann hypothesis. The reason is that energy levels of quantum systems are always real numbers (as opposed to imaginary), since energy is a physically measurable quantity. And since the En’s are purely real, they become purely imaginary when multiplied by i in the formula for the corresponding Zn’s. There is never a case where an imaginary part of En is multiplied by i, canceling out its imaginary property and rendering it real, so that it then contributes to the real component of Zn and changes it from ½ to something else. Since energy levels are always real, the real parts of the zeros of the zeta function would always be ½, and the Riemann hypothesis would therefore be true.
The argument for the existence of such a system was also framed in terms of matrices and eigenvalues, and associated symmetries, and is apparently incomplete, and it would be amazing, though I presume very unlikely, if the discovery of the identity of the FPP led to progress toward proving the Riemann Hypothesis.
posted by jamjam at 4:16 PM on November 15, 2019 [2 favorites]

I've been wondering for a while what applications to physics Cantor's theory of transfinite numbers might ever find

OMG, jamjam, we must be on the same wavelength or something! I was just going to post a comment that's virtually word-for-word, erm, if you subscribe to an appropriate analogue of the "zero on top, one on the bottom" policy re: mathematics that is attributed to some physicists...
posted by spacewrench at 8:45 PM on November 15, 2019 [2 favorites]

Terry Tao's blog post on the subject (and subsequent comments) are well worth reading. After the Quanta article, Manjari Naryan pointed out that these formulas had already been published many times but never become widely known. Besides Van Mieghem's nice 2014 paper applying the result to graph metrics, the oldest mentioned by Tao is Thompson-McEnteggert (1968), which already contains the most illuminating proof by adjugates, and relates the identity to Cauchy interlacing inequalities.
posted by metaplectic at 2:09 AM on November 16, 2019 [3 favorites]

Mathematics really is its own language, and I wish I were more fluent in it.

This is pretty cool anyway.
posted by snuffleupagus at 5:16 PM on November 16, 2019 [1 favorite]

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