Author: Kowalski

Partly motivated as a follow-up to the Oppenheimer biography I recently read, and partly to get a clearer idea of Quantum Mechanics than I had (for my course on Spectral Theory in Hilbert Spaces, which just ended; though the posted lecture notes are a bit lagging, I will post a complete version soon…), I have just finished reading “The conceptual development of quantum mechanics“, by M. Jammer (a book which seems unfortunately to have almost no online presence; I got the reference from another book and borrowed it from the ETH library).

This was extremely interesting; the author assumed rather more knowledge of classical physics than I could claim to remember from my studies, but since I had been reading other accounts of (modern) Quantum Mechanics for mathematicians, I was not entirely lost, and I was quite fascinated. I won’t say anything about the physical aspects, but one quite amazing thing that emerges is how quickly the formalism emerged from 1925 to 1929. Not only did matrix mechanics (Heisenberg, 1925), Dirac’s formalism 1925–26), Schrödinger’s wave mechanics (1926) all come out in barely more than a year, but also Nordheim, Hilbert and von Neumann had the time to do a first mathematical reformulation in 1927 before von Neumann gave the formalization in terms of the spectral theorem for unbounded self-adjoint operators in 1929. (I note in passing that von Neumann was quite footnote-happy, at least in that paper).

Another thing I didn’t know is that the Diract δ function was first discussed by Kirchhoff in 1882 (in a paper on optics), who already explained its origin as a limit of Gaussians with variance going to 0.

During the discussion about the correct way to pronounce friable (in English) to avoid creating the impression that integers without large prime factors are to be considered as (presumably unealthy) food, it seems of some interest to indicate what type of outrageous French accent would be suitable. For this purpose, I suggest watching the very funny Bugs Bunny cartoon entitled French Rarebit (especially the part with Monsieur François and Monsieur Louie fighting over the rabbit). I can’t help picturing two irascible French mathematicians fighting over priority: “Non, non, non, Monsieur François, ze Theorem, it is mine!”, “Au contraire, Monsieur Louie, le Théorème, it is mine, not yours!”

(When I was growing up, Bugs Bunny cartoons appeared only sparingly on French television, and were a particular treat, as were the ones of Tex Avery; nowadays, I guess all of them can be found just as simply on the internet…)

The last two papers I’ve published (found here and here) turned out to both appear in journals published by Oxford University Press. This publisher’s footnote policy is apparently quite rigid: they will not countenance them. This is quite unfortunate, since I have a tendency to indulge in them, and I have to admit that I can’t really imagine the reasoning behind this prohibition. Are footnotes frivolous? Distracting? Do they encourage our baser instincts? Is the path to damnation liberally strewn with pedantic footnotes? It is hard for me to believe that this can be true when, to take but one example, one can learn more from the footnotes in some papers of Barry Mazur (such as this one or this other one) than from the complete works of many a lesser mathematician.

Oxford’s editorial policy with respect to footnote is therefore to take this carefully selected material and move it to a parenthetical aside in the main body of the text; this, it must be said, is often a very decent solution — suggesting maybe that many of my footnotes (even in the joint paper which is the second of the two I mentioned, most of them were introduced by me) could be dispensed with –, but it may lead to bizarre, convoluted, or downright incomprehensible sentences if one is not careful. (In fact, it is only by reading one of these at the proof stage that I became conscious of the anti-footnote principle).

I’ve just read a recentish biography of Robert Oppenheimer, about whom I didn’t know much before (the authors are K. Bird and M.J. Sherwin). It was quite interesting, but not entirely what I expected: the emphasis was fairly strongly on the “political” sides of the story, in particular the problems Oppenheimer had with the government after the war and his rather unpleasant security-clearance hearings in 1953–54 (which I knew essentially nothing about).

Reading from a scientific point of view, I was surprised how I couldn’t quite get a good mental picture of Oppenheimer’s true level as a physicist in his youth. The book mentions that, of course, he did not get a Nobel prize (and relates that this was apparently thought to be a problem when it came to put him in charge of the Manhattan project, since a number of Nobel winners would maybe resent having to take orders from him…), and seems to suggest that his strongest work might have been a fairly obscure paper which was a precursor of the study of Black Holes. I’m curious to learn more about him (and his contemporaries) as a scientist, so if some kind readers can suggest further books, please do so…

Of course, any time one reads a biography involving complicated political issues and possibly hidden agendas, a basic question is how far one should trust the authors. They are clearly rather sympathetic towards Oppenheimer (though they do not avoid discussing less flattering information). I would typically have expected to use as an indicator of reliability the quality of the more scientific information, but since there was not much of that, this was not really doable. One seed of doubt lies in their discussion of Oppenheimer’s period as director of the Institute for Advanced Study, from 1947 to 1966. Apparently, he didn’t get along at all with the mathematics faculty. Now the discussion of this (page 385 of the paperback edition of the book) left me a bit dubious: the authors claim that the problem (having to do with academic affairs and appointments outside mathematics in particular) was due to the fact that the mathematicians were mostly past their creative peaks and thus didn’t have much better to do than devoting themselves to “other affairs” (whereas historians and social scientists, at the same middle age, had “little interest or time for such academic intrigues”). Thinking that this period encompasses Selberg’s work on the trace formula, the birth of the Langlands program in which Weil participated, and quite a few other mathematical achievements in which IAS mathematicians were actively involved, this doesn’t seem quite correct.

A token mathematician is given the occasion (through a quote from another book) to give his side of the story: it is Weil, who is identified as a “great French mathematician”, but then is said to be “typical of the bloated egos Oppenheimer encountered at the Institute”, and to have been “arrogant, acerbic and demanding”. (Of course, Weil was actually appointed in 1958, in the middle of Oppenheimer’s directorship; and it is quite amusing to think that Weil was certainly one of the few members of the Institute with whom Oppenheimer could have spoken Sanskrit and discussed the Bhagavad Gita, which they had both studied…).

There was another bizarre story about Oppenheimer that I didn’t know, and which echoes disturbingly Turing’s death by poisoned apple: at some point in the 1920’s, during an unhappy postdoc in Cambridge, Oppenheimer apparently left a poisoned apple (how deadly is not clear…) on his mentor’s desk…

Yesterday, while walking a bit around Zurich with A. Saha (one of the new postdocs, who will arrive at ETH in September) I found a slide-rule in an antique shop; since I had never actually seen one before, and it was very cheap, I bought it immediately. It is in very good shape, and seems to be fairly sophisticated, and I hope to learn a bit how to use it for the fun of it.

I was interested to see that π is indicated on some of the scales, but then we also noticed another mark indicating a value between 1.74 and 1.75, highlighted by a symbol which I had never seen before (click for larger picture):

Trial-and-error led us to the conclusion that this number is simply

$latex \frac{100\pi}{180}=1.7453292519943295769236907684886127134\ldots$

and so is to be used to convert between degrees and radians. Further searches revealed a number of pictures of (essentially) identical slide-rules with a small tick to indicate this constant. However, I haven’t found a picture yet with the strange symbol. Is it really standard? Has anyone seen it before? Is it somewhere in Unicode?