Field With One Element
June 12th, 2008 by WaltLieven LeBruyn has a series of posts about the “field with one element”, here, here, and here. The field with one element does not exist, of course, but Tits pointed out a long time ago that you can think of the symmetric groups as Lie groups over a field with one element. I thought that there was all there was to the idea, but apparently there’s much more you can do with the idea.
June 14th, 2008 at 12:15 am
I wrote a bunch about this idea in week186 of This Week’s Finds. For one thing, the symmetric groups are just the An case of a game that works for every Dynkin diagram.
June 15th, 2008 at 1:26 am
A couple of links away from this post David Corfield has a quote from Hilbert:
Similarly…
But Feyman never gave up on this project, and produced a relatively elementary explanation almost 20 years later in his Dirac memorial lecture, Elementary Particles and the Laws of Physics, in 1986.
How long will we have to wait before someone performs the same trick with F1, and explains it without “decategorified flag varieties?”
It may also be worth wondering what the most advanced “complete theory” in mathematics may be, according to Hilbert’s criterion of making it “so clear that you can explain it to the first man whom you meet on the street.” A persnickety observer might even take it as a reproach to the entire mathematical community that in comparison to the rapidly expanding horizon of mathematical research, the circle of mathematical objects conforming to the criterion of “Hilbert completeness” has contracted almost to a point.
June 15th, 2008 at 5:55 am
What examples of “complete theories” did Hilbert have in mind, then? And did Hilbert ever actually try the “man-in-the-street” experiment? Or did he intend this as idealistic sentiment?
I think there’s something basically wrong with the idea that a complete theory can be transplanted wholesale from one mind to another, unless the other is pretty well prepared to receive it in the first place. This truism applies not just to those who would learn mathematics, but to Feynman’s (highly motivated Cal Tech) freshmen as well. Theories are comprehended or reconstructed in the mind of another piece by piece, and yes, some jargon (as shorthand) may be necessary on occasion, whether it’s “fermions” or “decategorified flag varieties”.
June 15th, 2008 at 1:15 pm
Todd: Feyman didn’t keep trying to clarify the connection between fractional spin and Fermi-Dirac statistics for 20 years just because he couldn’t think of an excuse why it couldn’t be clarified, and along the way to the beautiful exposition in his Dirac lecture he also arrived at a deeper understanding of another mystery about mysterious half-spinning little entities like electrons: Why do we have to rotate them 720 degrees to get back to where we began, when ordinary blobs like you and me recover our original orientation after only half as much rotating?
June 15th, 2008 at 2:53 pm
Yes, I agree Jacob: Feynman was a master of exposition, and we should all emulate his efforts to prepare “freshman lectures”. And in suggesting that that statement of Hilbert was “idealistic”, I don’t exclude the fact it’s an ideal to aim for, however elusive it may be!
I’d go on to say that among mathematicians, John Baez tries harder than most to realize that ideal. I did wonder, reading your first comment, how literally you were interpreting Hilbert, and also whether you thought John’s efforts to discuss geometry over F_1 were particularly deserving of reproach along those lines. And whether you could do better.
June 15th, 2008 at 2:54 pm
Actually, the question is why we don’t have to be rotated 720 degrees when ordinary stuff like electrons takes twice as long to return to its original orientation.
June 15th, 2008 at 4:45 pm
Feynman thought long and hard about what an electron actually was. That’s why he was so perturbed at the exitence of the muon. “Who ordered THAT?” he asked. The muon was one of the top 10 or 20 puzzles that he shuffled and reshuffled in his mind, always looking for a new trick or method or experiment that would resolve his puzzlement. He was sure than any explanation for the muon would also explain the tau lepton, which was detected in a series of experiments between 1974 and 1977 by Martin Lewis Perl with his colleagues at the SLAC-LBL group. That means he had a decade before he died to wonder about the tau.
I get in trouble on the blogosphere by often writing some snippet of private conversation between myself and Feynman. I feel that I’m injecting into print an anecdote of a great mind that may not have been previously published. Blogmasters usually see this as gratuitous namedropping, ignoring how much time I’d spent alone with Feynman in conversation, and I usually apologize on the blog.
Jack Safartti cites otherwise unpublished interviews with Feynman in Jagdish Mehra’s book, The Beat of a Different Drum on the life and work of Richard Feynman, Oxford, 1994, as including such thoughts as:
“The parameters of mass and charge associated with the electron in the formalism of electrodynamics are not the quantities measured under ordinary conditions. A free electron is accompanied by an electromagnetic field which effectively alters the inertia of the system, and an electromagnetic field is accompanied by a current of electron-positron pairs which effectively alters the strength of the field and of all charges. Hence a process of renormalization must be carried out, in which the initial parameters are eliminated in favor of those with immediate physical significance.” Simple subtraction does not work relativistically because:
17. “the difference of two individually divergent terms is generally ambiguous. It was necessary to subject the conventional Hamiltonian electrodynamics to a transformation designed to introduce a proper description of single electron and photon states, so that the interactions among these particles would be characterized from the beginning by experimental parameters. … to the first significant order of approximation in the electromagnetic coupling, the electron acquired new electrodynamic properties, which were completely finite. These inclued an energy displacement in an external magnetic field corresponding to an additiona spin magnetic moment, and a displacement of energy levels in a Coulomb field. Both predictions were in good accord with experiment …”
18. “However, the Coulomb calculation disclosed a serious flaw; the additional spin interaction that appeared in an electrostatic field was not that from the relativistic transformation … of the supplementary spin magnetic moment …”
Renormalization must be consistent with both special relativity and gauge invariance. The divergent terms introduce ambiguities so that use of a particular choice of gauge could violate special relativity (i.e., violate “covariant features”) in doing perturbation theory.
And so forth.
Feynman did indeed insist that something was not understood until it could be explained to “the man on the street” — not meaning San Pasqual through Caltech.
Richard Feynman said that he worried about what he was supposed to say to reporters to explain why he was nominated for the Nobel Prize in physics. On the way to the meeting, Feynman explained the situation to his taxi cab driver and got the sage advise, “Don’t explain it. Tell ‘em if you could explain it you wouldn’t got it in the foist place.” Which is what Feynman did.
June 15th, 2008 at 7:37 pm
JVP: far be it from me to question, but wasn’t it Isidor Rabi who asked about the muon when dividing up the particle physics check?
June 15th, 2008 at 9:03 pm
Here, with the “field with one element”, we really don’t understand what’s going on, for any definition of “understand”.
I ordered the muon. I was embarrassed to admit it at the time, but it’s time for me to come clean, and make a break with my checkered past. Sorry, guys.
June 16th, 2008 at 12:32 am
Todd: I just mentioned “decategorified flag varieties” because it sounds funny, and I didn’t intend to disparage the weirdly intelligent John Baez or his curious reduction of the symmetry groups of projective geometry into a row of “Dynkin dots.”
Even Feynman’s freshman-level explanations were only successful in principle, and the stupefied freshmen in his audience were quickly replaced by faculty connoisseurs, as if the lecture hall had fallen into some sort of accelerated time warp, with beards and bifocals sprouting out of faces that were fresh and smooth only yesterday.
June 16th, 2008 at 12:15 pm
I agree, “decategorified flag varieties” sounds funny, as does most math jargon. I vividly remember having lunch in a crowded English pub with John Baez; we were having a fairly intense discussion about n-categories, A_\infty spaces, the Eckmann-Hilton lemma, and so on. A family of four sat down across from us (no other place to sit), and had no choice but to listen in on this extremely strange conversation about… Lord knows what (I wonder if they could tell it was mathematics!). For a while they found us entertaining — the daughters had a little trouble controlling their laughter — but at length they seemed to grow a little uncomfortable and weirded out by the scene. [I could feel their eyes on us all the while, but avoided returning even a glance -- that would have ruined everything!]
Regarding the Feynman lectures on physics (that eventually became the famous three volumes), David Goodstein and Gerry Neugebauer wrote, “Through the distant veil of memory, many of the students and faculty attending the lectures have said that having two years of physics with Feynman was the experience of a lifetime. But that’s not how it seemed at the time. Many of the students dreaded the class, and as the course wore on, attendance by the registered students started dropping alarmingly. But at the same time, more and more faculty and graduate students started attending. The room stayed full, and Feynman may never have known he was losing some of his intended audience. But even in Feynman’s view, his pedagogical endeavor did not succeed. He wrote in the 1963 preface to the Lectures: ‘I don’t think I did very well by the students.’ Rereading the books, one sometimes seems to catch Feynman looking over his shoulder, not at his young audience, but directly at his colleagues, saying ‘Look at that! Look how I finessed that point! Wasn’t that clever?’ But even when he thought he was explaining things lucidly to freshmen or sophomores, it was not really they who were able to benefit most from what he was doing. It was his peers — scientists, physicists, and professors — who would be the main beneficiaries of his magnificent achievement, which was nothing less than to see physics through the fresh and dynamic perspective of Richard Feynman.”
June 17th, 2008 at 8:42 am
Jacob Freeze is mostly correct about the effects of and audiences for Feynman’s lectures (and the books built from their transcripts).
WHILE he was speaking, most of the audience felt that it understood most of the contents. He flattered our attention, and was supremely effective in boosting our intuition with simple language, clear narration, vivid images, clever mathematical tricks.
AFTER he’d spoken, we found ourselves unable to replicate what he’d said when non-attendees asked, and it was notoriously hard to solve homework problems after the lectures, without lecture notes were printed with conventional problem-solution examples.
Part of the glory of Feynman as one of the great teachers of the 20th century was that he reached freshmen and tenured faculty simultaneously, at several levels, and the success of his Lecture Notes in Physics verifies that — popular as undergrad textbooks, yet deep enough to be of constant interest to experts.
Gian-Carlo Rota’s “Ten lessons I wish I had been taught explains Feynman, “tricks”, and genius.
The whole paper is worth reading. But this summary therein by the author of what Feynman told me in somewhat more colloquial terms is as follows:
“Richard Feynman was fond of giving the following advice on how to be a genius. You have to keep a dozen of your favorite problems constantly present in your mind, although by and large they will lay in a dormant state. Every time you hear a new trick or a new result, test it against each of your twelve problems to see whether it helps. Every once in a while there will be a hit and people will say, ‘How did he do it? He must be a genius!’”
June 17th, 2008 at 2:47 pm
In the lecture Jonathan links in his comment, Gian Carlo Rota describes an exchange between Dirk Struik and Eugene Calabi, after one of Calabi’s lectures:
“Give us something we can take home!” said Struik. Calabi obliged, and in the next five minutes he explained in beautifully simple terms the gist of his lecture.
It’s just a reminder that freshmen and “the man in the street” aren’t the only audience that gets lost in mathematics, and even a very good geometer like Dirk Struik is happier going home with one little diamond in his pocket, instead of a bag of coal on his back.