The procedure by which higher-dimensional systems are reduced to lower-dimensional ones is called dimensional reduction. The reason that dimensional reduction is so powerful from the point of view of gauge theory is that it converts coordinate transformations in the full space into gauge transformations in the subspace.

From the recollections of mathematician Herman Goldstine, friend of Hermann Weyl and the great mathematical physicist John von Neumann, and (with von Neumann) one of the developers of the early ENIAC computer:


Hermann Heine Goldstine, 1913-2004

"I always was struck by the difference between Weyl and Johnny von Neumann. There are jokes, one of which Johnny always swore was false. That's the story that, I don't know, Hermann Weyl was going to prove some theorem, a very deep and profound theorem, let's say it was the Riemann-Roch theorem. I don't know if it was the Reimann-Roch theorem, but that was one I always have trouble with, so let's say that was the theorem. And Weyl gave a lecture on why this is a very deep, profound result, and he gave a very complicated proof. And the apocryphal story goes that at the end of the lecture there's this kid who is supposed to have raised his hand at the back of the class and said, 'Professor Weyl, may I show you a proof?' And goes up to the board and goes zip, zip, zip, zip, and in about 15 lines has a brilliant proof of this thing.

"I asked Johnny about it, and he said no, that wasn't true. But it is true, if you talk to Natasha Brunswick, who was in those days Natasha Artin. Natasha says that there was always Johnny with these tight pants on. All of Johnny's life, whatever size suit he bought, he always ate too much, and the suit was always one size smaller than Johnny. Even as a student in Göttingen, his behind was always ready to bulge out of his pants. I guess Natasha and everybody in the class were always charmed.

"But Joachim, who was one of Hermann's children, told me that when Hermann used to work in his house on Mercer Street, in the study in there, you would hear groans coming out of the study. That Weyl worked at things in sort of anguish, that it was hard for him, that he delivered his theorems practically like a woman giving birth to a child. That's so different from Johnny, because when he and I would be working at something, when we'd get stuck, he'd say, 'Okay, that's it, ' and pack it up. It might be that he'd phone at two in the morning to say, 'This is how the proof goes.' But it might be three weeks, a month or so later, or it might even be I who would come in a month or so later and say, 'This is, maybe, how to go.' But he never struggled with something. When he got stuck, he filed it somehow, and it just came out easily. I suspect that Weyl was probably the deeper of the two mathematicians."

[Follow-up to my October 17 post.] While visiting San Francisco recently, I stopped by the city's Main Public Library, which is featuring an exhibit on the American actress Louise Brooks. Brooks, who passed away in 1985, would have turned 100 on November 14.



Astonishingly beautiful, Brooks created the movies' "bobbed" hairstyle look around 1926. I still remember the recollections of my late mother who, as a lovely teenager herself in the late 1920s, begged and begged her parents to get her hair bobbed á la Brooks. But as strict Southern Baptists, such a hairstyle (not to mention even going to the movies!) was absolutely verboten.

Back from SF, I was happy to have finally received Criterion Collection's two-disc DVD set of Brooks' early 1929 psycho-sexual drama Pandora's Box, the actress' signature film (filmed in Berlin by the great German director, Georg Wilhem Pabst). Regarded as one of the top ten greatest silent movies of all time, Criterion's digitally remastered version of the Munich Museum's restored film includes four different musical scores, Lulu in Berlin (a rare filmed interview with the actress, produced in 1984), Looking for Lulu (the one-hour, 1998 Hugh Neeley documentary on Brooks' life), the book Reflections on Pandora's Box, and assorted essays, audio commentaries, interviews and stills. If you're into this actress, this is a must-have DVD.


Brooks at age 64, in a rare 1971 interview with British filmmaker/
film essayist (and Harvard physics graduate!) Richard Leacock

Tragically, parental neglect and childhood sexual abuse (at the age of nine) most likely destroyed Brooks' life, and she went on to become the same kind of woman she portrayed in Pandora's Box and Diary of a Lost Girl (also 1929). An ultra-liberal, chain-smoking, alcoholic, partying sexual abandonee and iconoclastic loner until very late in life, to her enduring credit she renewed her Catholic roots, took up writing and turned herself around. She died of emphysema at the age of 78. May God save her soul.

My candle burns at both ends;
It will not last the night;
But ah, my foes, and oh, my friends--
It gives a lovely light! -- Edna St. Vincent Millay (1920)

Could one really charge the Lord with inconsequence for not seizing the opportunity you have found to harmonize the physical world? I think not. If He had made the world according to you, you see, Weyl II would have come along to address Him reproachfully thus:

"Dear Lord, if it did not suit Thy way to give objective meaning to the congruency of infinitesimal rigid bodies, so that when they are at a distance from one another one cannot know whether or not they were congruent, then why didst Thou, Inscrutable One, not decline to leave this property to the angle or to the similarity? If two infinitely small, initially congruent bodies K, K^' are no longer able to be brought into congruency after K^' has made a round trip through space, why should the similarity between K and K^' remain intact during this round trip? So it does not seem more natural for the transformation of K^' relative to K to be more general than affine."

But because the Lord had already noticed, long before the development of theoretical physics, that He cannot do justice to the opinions of mankind, He simply does as He sees fit.

I am glad that Hans has an intense interest in something. On just what it is directed is less important to me, even if it is engineering, by God!

Ray is not only an extraordinary scientific person, but also an extraordinary human being. Ray treats the janitor in the laboratory with the same friendliness and respect that he does the most senior scientist. And for that, he is loved by his colleagues.

The comedian Steve Martin, who was a philosophy major at California State University at Long Beach (my undergraduate school!), once said that he learned just enough philosophy there to screw him up for the rest of his life. I was luckier than he was -- not only have I never taken a class in the subject, it wouldn't have made any difference anyway, because I just don't get philosophy at all.

Stanford University Professor of Philosophy Thomas Ryckman does get it and, more importantly, one of his specialties is the relationship between the development of general relativity and the state of German philosophy in the early 20th century. He has written a book on the subject, The Reign of Relativity, in which both Einstein and Weyl play prominent roles. Einstein himself was an armchair philosopher, but Weyl was much more active on the subject. He was an early adherent of the great German philosopher Edmund Husserl, and in fact married one of Husserl's students, Helene Joseph.

Both Weyl and his wife were not only good philosophers, they were gifted linguists. In the preface to his seminal book The Classical Groups: Their Invariants and Representations, Weyl tells us

The gods have imposed upon my writing the yoke of a foreign tongue [English] that was not sung at my cradle.

As a mathematician, I must absolutely insist that my geometry is the true, local geometry [reine Nahegeometrie]; the fact that Riemann posited only the special case F_{μ ν} = 0 has no substantive legitimacy other than a merely historical one ... If in the end your views are correct concerning the actual world, then I would regret having to accuse the dear Lord of a mathematical inconsistency.

It is not the purpose of this work to conceal the bedrock on which the house of analysis is founded with a fake wooden structure of formalism -- a structure which can fool the reader and, ultimately, the author himself into believing that [formalism] is the true foundation. Rather, I shall show that this house is, to a great extent, built on sand.

What really interests me is whether God had any choice in the creation of the world. -- Albert Einstein

I just hate it when the media reports carefully vetted scientific data as fact and not as just one of many valid points of view. I'm not asking for them to ignore the opinions of these so-called scientists, but they really should report the fact there's a lot of controversy about whether this kind of evidence is valid. LIke, were you there, huh, Mr. Hotshot Washington Post? As if this ludicrous nonsense - a marble blows up like a baloon [sic] to become the entire universe in a trillionth of a second - is more plausible than Genesis? Give me a break!