2005-2007 Archives

 Weyl and Emmy -- Posted by wostraub on Wednesday, December 21 2005 Here's a photo taken in the early 1930s of Weyl and his wife Hella, their son Joachim, Emmy Noether and several friends, colleagues and students. Reproduced from the 1981 book Emmy Noether: A Tribute to Her Life and Work, James Brewer and Martha Smith (eds.).

 Weyl Left -- Posted by wostraub on Tuesday, December 20 2005 Hermann Weyl was a patriotic German citizen, but when Hitler came to power in 1933 Weyl saw the writing on the wall. As a respected mathematical physicist and law-abiding Christian, he had nothing to fear himself, but his wife Helene had a Jewish background which placed her in jeopardy. They gave up their bank accounts and all their possessions, packed their bags, and left for Princeton. Albert Einstein and Emmy Noether weren't far behind them. Now that Bush is turning America into Nazified Amerika, where would Weyl go? My guess is back to Germany or Switzerland. He wouldn't have anything to do with this Bush regime. It's still not so bad, they say. You can still speak out against the Bush regime without worrying about being taken away in the middle of the night. Or can you? According to Bush, you're either with him or you're with the terrorists. The Democratic and Independent parties are not with Bush, so they must be for the terrorists. Bush's latest crime is to spy on Americans without a court order. My guess is that he will now authorize his goons at the NSA to spy on these parties to keep them from gaining power in 2006 and 2008. The Republican Party, in the guise of a Frist or Hastert or DeLay or Sessions or Hunter or Inhofe, will then become Dictator for Life. George Orwell may have been off by only 24 years. If Bush is successful, and I see no reason to believe that he won't be unless he is stopped, then you can say goodbye to the America you once knew and loved. Say goodbye also to the Constitution, which Bush recently referred to as "just a goddamned piece of paper." Say goodbye also to the middle class, which will be taxed out of existence to pay off Bush's monstrous deficits. You can also kiss off human rights, the environment and legitimate science, because these niceties have no place in BushWorld. As for me, I'm going to fight like hell in 2006 to keep these nightmares from becoming reality, and I hope your New Year's resolutions are along the same lines. If we fail, we won't recognize the place we're living in. God help us all.

 The Spin Connection in Weyl Space, Again -- Posted by wostraub on Saturday, December 10 2005 I've completely rewritten my article on Weyl and the spin connection from the point of view of non-metric-compatible geometry. In this article, I express my doubts not only about the validity of Weyl's original theory but that of non-metric-compatible theories as well. Connection.pdf

 Connections in a Weyl Space -- Posted by wostraub on Friday, December 2 2005 While updating my previous write-up on Weyl's spin connection, I started looking seriously at the concept of a generalized Weyl space and its relationship to variable vector magnitude under parallel transfer. It does not look encouraging, and I'm beginning to suspect that vector magnitude is a fixed quantity after all. In his 1918 theory, Weyl argued that vector length under physical transplantation varies in an electromagnetic field. If the length of some arbitrary vector Vμ is given by L2 = gμνVμVν, then Weyl's theory basically says that under parallel transport this goes over to 2LdL = gμν αVμVνdxα or dL = AμdxμL, where gμν α is the covariant derivative of the metric tensor and Aμ is the electromagnetic 4-potential. However, I have not been able to find a symmetric connection term Γαμν (Weyl or otherwise) that allows for a non-zero dL and a vanishing Kronecker delta tensor under covariant differentiation. It goes without saying that dL = 0 kills Weyl's theory before it even gets started. This is not deep stuff, and I'm surprised that I've seen no real attempt in the literature to address what appears to be an obvious discrepancy of Weyl space. At the same time, I've read Weyl for years and never given this issue a second thought! Of course, everyone knows that Weyl's 1918 was wrong anyway, but the argument that killed it (due to Einstein) was based on physical, not mathematical, considerations. Einstein himself got wrapped up years later in the same old game when he tried to find a non-symmetric connection for parallel transport in spacetime. Indeed, the last sheet of paper he ever wrote on (while in the hospital where he died) is covered with non-symmetric connections, which were integral to his final (and failed) unified field theory. I like to think that when Einstein stood before God, the Almighty asked him "With the mind I gave you, why on Earth did you waste the last 30 years of your life on this nonsense?!" A colossal waste of time, but fun stuff. "The use of general connections means asking for trouble." -- Abraham Pais, Subtle is the Lord PS: Very big game tomorrow for my old school, USC. I love my kids (UCLA grads), but -- Go Trojans!!

 Newton Routs Einstein -- Posted by wostraub on Friday, November 25 2005 Yesterday, the Royal Society announced the results of a "popularity contest" between Sir Isaac Newton and Albert Einstein. When asked which scientist made the most contributions to science, 86.2% of the Royal Society's voting scientists opted for Newton. When the same question was posed to the general public, Newton again beat out Einstein, with 61.8% voting for Newton. Intestingly, when asked which scientist made the most contributions to humanity, only 60.9% of the 345 Royal Society voting scientists voted for Newton, while the public vote was virtually tied. Newton was elected to the Royal Society in 1672, whereas Einstein came in as a foreign member in 1921. Although this is the 100th anniversary of Einstein's annus mirabilis, or miracle year of 1905 (he wrote five fundamental papers that year, including the ones on special relativity and the photoelectric effect), Newton's achievements were deemed more remarkable overall.

 Black Hole in the Milky Way -- Posted by wostraub on Thursday, November 3 2005 Chinese researchers using a bank of ten radio telescopes spread across the United States have found further evidence that a supermassive black hole inhabits the center of our galaxy, in the constellation Sagittarius. Most scientists now believe that galactic cores host such objects, whose sizes may range from hundreds of thousands to many millions of solar masses. The object at the center of our Milky Way Galaxy was estimated to be about 4 million times the mass of the sun. Using the formula for the radius of a black hole, R = 2GM/c^2, the black hole's event horizon would fit neatly between the earth and the sun. This is great stuff, but in order to get the general public excited about it, newspapers and magazines have to write stupid things like "black holes are cosmic vacuum cleaners that gobble up stars and everything else in their vicinity." But black holes do not suck! They are collapsed stars whose gravity is so great that the star literally shrinks down to ZERO VOLUME and INFINITE DENSITY. Outside the black hole, however, these point-like objects behave like ordinary stars, except they don't shine because they're essentially dead stars (and any light couldn't escape their gravity, anyway). In fact, if our sun were to suddenly become a black hole, the earth and other planets would continue in their orbits as usual, although the sky would be darker than we've ever seen it. Also, these articles never talk about the true nature of a black hole, which is one of the most bizarre physical objects of God's creation the human mind has ever encountered. The mathematics that describes them, Einstein's theory of general relativity, is of course rarely mentioned to the public. Event horizons, ergospheres, Hawking radiation, time travel? No -- give us talk about cosmic vacuum cleaners!

 Theory of Matter in a Weyl Manifold -- Posted by wostraub on Sunday, October 30 2005 While cleaning out some boxes today, I came across a reprint of a paper I received years ago entitled Theory of Matter in Weyl Spacetime by David Hochberg and Gunter Plunien of Vanderbilt University [Phys. Rev. D 43 3358 (1991)]. It's neat to see Weyl's original spacetime gauge theory pop up from time to time in research papers, and this is one of the better ones. The authors demonstrate how a Lagrangian that is linear (not quadratic) in Weyl's version of the Ricci scalar R can be coupled with a scalar field $\phi(x)$ to derive Einstein's gravitational field equations. But the authors then go on to develop a Lagrangian in spinor form that couples the Weyl gauge vector to fixed-chirality spinors that are identified with neutrinos. I think Weyl would have found that really interesting, since his massless form of the Dirac equation anticipated the existence and eventual discovery of these guys! Hochberg and Plunien conclude from their investigation that spacetime is actually Weylian (and only approximately Riemannian) and that the Weyl field is a form of dark matter. Neat stuff! I have the article in pdf format and will post the thing if I can get permission from the American Physical Society. It's a relatively easy paper to follow and I think the effort is worth it (and it might just take your mind off the Bush cabal for a while).

 Atiyah on Weyl -- Posted by wostraub on Monday, October 24 2005 In 2002 the noted mathematician Sir Michael Atiyah wrote a biographical sketch of Hermann Weyl that included reflections on Weyl's interests in philosophy and writing. Here is the article in pdf format: Hermann Weyl

 Weyl Relativity -- Posted by wostraub on Monday, October 10 2005 This morning I was contacted by the great-granddaughter of Hermann Weyl, Elizabeth T. Weyl of Mount Holyoke College in Massachusetts. She informed me that she is aware of only several direct descendants of the great mathematician now living in this country. Why so few? Weyl was married twice. His first wife, Helene (nickname Hella) Joseph, was a philosophy student at the University of Gottingen in Germany under Edmund Husserl, who held the philosophy chair at the school. Weyl's early love of philosophy appears to have sprung at least in part because of the influence of his wife, whom he married in 1913. Weyl and Helene subsequently had two sons, but I have not been able to learn anything about their lives. (Elizabeth Weyl wrote that she is the daughter of the son of one of Weyl's and Helene's two boys.) Helene passed away in 1948, and in 1950 Weyl remarried, this time to Ellen Lohnstein (or Lowenstein) Bar of Zurich (she was a sculptor). At the time, Weyl was 64 and not yet retired from his position at the Institute for Advanced Study in Princeton, New Jersey. He did retire in 1952, and the couple traveled between Zurich and Princeton until Weyl's untimely death in 1955 while in Zurich. Even at the age of seventy, God took him too soon! I think I mentioned some time ago that during Weyl's days at the ETH in Zurich (where he held the chair in mathematics), the German-born Weyl was drafted by Germany to serve in the First World War. Fortunately, the Swiss government secured an exemption for Weyl, and he was allowed to stay in Zurich to continue his research. Also during these days, Weyl and the 1933 Nobel Laureate Erwin Schrodinger became best friends. I've read unsubstantiated (but probably true) claims that Weyl was the source of mathematical inspiration for Schrodinger's wave equation. Unlike many scientists, Schrodinger was a good-looking, well-dressed bon vivant and a Don Juan of sorts, and I've even seen some reports that Weyl's first wife, Helene, fell under his spell, while, at the same time, Schrodinger's (probably long-suffering) wife Anny was enamoured of Weyl!

 Weyl and Antimatter -- Posted by wostraub on Tuesday, October 4 2005 In his famous paper Eleckron und Gravitation (Zeit.f. Physik 56), Weyl wrote   It is reasonable to expect that in the two-component pairs of the Dirac field, one pair should correspond to the electron and the other to the proton. Furthermore, there should appear two electrical conservation laws, which (after quantization) should state the separate conservation of the number of electrons and protons. These would have to correspond to a two-fold gauge invariance involving two arbitrary functions. I find it remarkable that only one year after the appearance of Dirac's relativistic electron theory, Weyl had the temerity to infer that the four-component Dirac spinor referred to the electron and the only other positively-charged particle then known, the proton. Of course, Dirac had also considered this possibility, but I am not aware of any rash statements he made to that effect so early in the game. Neither scientist at that time knew the correct explanation intimately involved the existence of the positively-charged antielectron or positron, the first antimatter particle to be discovered (which was found by Anderson in 1932). Nevertheless, Weyl's gutsy if incorrect 1929 prediction shows how bold an erstwhile pure mathematician could be in a field not originally his own. Courageous, too, because Weyl's equally-erroneous 1918 metric gauge theory had seemingly predisposed him to mockery when he resurrected the idea (although as quantum phase invariance) in his 1929 paper.

 Birthday Quiz -- Posted by wostraub on Thursday, September 29 2005 Here's a photo of Einstein and some friends taken at the Institute for Advanced Study at Princeton on the occasion of Einstein's 70th birthday (March 14, 1949). Weyl is the gentleman in the back, third from the left. Can you identify the others? The answer is below. Yes, it's the film director Visconti, 5 points. Oops, that's from an old Monty Python routine. From the left, they are: H.P. Robertson, Eugene Wigner, Hermann Weyl, Kurt Gödel, Isador Rabi, Einstein, R. Ladenburg, J.R. Oppenheimer, and G.M. Clemence. Note how these gentlemen range in appearance from dapper to advanced geek. Particularly geeky is the mathematician Gödel (pronounced girdle), whose famous 1931 incompleteness theorems proved that in principle not all math problems are solvable. Einstein looks not only nerdy here but ancient as well; maybe it's just his hair. He got the Nobel Prize in Physics in 1921. Rabi won the prize in 1944, I believe, while Wigner got it in 1963. Some pretty smart folks.

 30K for Katrina Relief -- Posted by wostraub on Wednesday, September 28 2005 My son Kristofer's Internet site BlankLabel raised almost $30,000 for Hurricane Katrina disaster relief. The money went directly to the American Red Cross. May God bless the efforts of you and your colleagues, Kris!  Warped Universes, Warped Lives -- Posted by wostraub on Monday, September 26 2005 I've been in Dana Point for several weeks sailing and just goofing off, but during this time I had the opportunity to read Lisa Randall's fascinating new book Warped Passages: Unraveling the Mysteries of the Universe's Hidden Dimensions. Randall is Professor of Theoretical Physics at Harvard University. A Harvard PhD at 25, she's exceptionally intelligent (as well as young and beautiful) and has some neat ideas to share, which is why she wrote the book. I'm thinking of writing and then posting a "book report" on this, but we'll see about that. For now, I'll share an observation I've had for some time about women scientists (hopefully you've already read my comments about Emmy Noether). When I was in physics graduate school, there was a fellow student I got to know who was simply light years beyond everyone else. Angelyn was only 20 at the time (and also beautiful), but she knew ten times as much then as I do now about quantum mechanics. She seemed to always know the answers, and they came off the top of her head seemingly without any effort. She finished her PhD in physics at UC Riverside, and is now a senior scientist at JPL. Later, a female civil engineer worked for me who likewise stood head and shoulders above all the others in the office (she was also beautiful). Julie had the highest GRE score of anyone I'd ever seen, and when she decided to go to graduate school she was immediately picked up at Stanford, where she received her doctorate a few years later. (I could also add that my daughter Sheryl, a California attorney-at-law, is also smart and beautiful, but I'm too biased to say it.) It boggles my mind to think that Randall is almost certainly several orders of magnitude beyond these gals. How can some women be so smart (and beautiful)? I think the answer lies in the fact that they're really no different than men, at least intelligence-wise. I also think all this talk about male mathematical/science superiority is a lot of nonsense. Women can do anything men can do, and often better. They also seem much less prone than men to start wars. [Note: I am not suggesting that Laura "Stepford" Bush run for president, though she'd probably be an infinitely better pick than her s**t-for-brains husband.] In the introduction to Randall's book, she briefly describes how she became hooked on science and her lifelong fascination with math and physics. I think that's all it takes -- a few brains, an unquenchable curiosity of the world we live in, and a burning desire to understand it from first principles (this is almost a direct quote from Einstein). It's a shame that great women scientists like Noether, Lise Meitner, Rosalind Franklin and scores of others were denied Nobel Prizes and other honors simply because of their sex. In closing, I can't help but make an additional (though negative) comment pertaining to female achievement, as I feel it's very appropriate. Dana Point, California is a beautiful place, but it's marred by legions of idle "Orange County women" whose goals in life seem to revolve around shopping, beauty parlors, constant cell phone use, and the acquisition of expensive cars and homes -- all on a middle-class income. In Orange County, they justify these excesses by calling them "family values." Enough said, I'm in trouble now!  Weyl's Theory and Early Quantum Theory -- Posted by wostraub on Wednesday, September 21 2005 Weyl's 1918 gauge theory essentially stated that the magnitude of a vector quantity was not absolute but variable from point to point in a 4-dimensional manifold, and that the electromagnetic four-vector was responsible for this variability. Einstein at first lauded Weyl's idea, but then realized that time, not just length, would also be variable. Einstein noted that time would then depend upon a particle's history, and that atomic spectral lines (which are fixed) would vary from atom to atom depending upon their individual histories. Correspondence between Weyl and Einstein on this point has been preserved, and it shows how desperate Weyl was to reclaim his theory despite the fact that Einstein was obviously correct. Out of his desperation, Weyl suggested that particle time and position were in some sense unobservable, and he briefly postulated that his gauge theory was correct after all and that certain gauge-affected observables (like time) required a more general definition. Of course, it was all nonsense. Or was it? Weyl's basic idea was that Nature employs a gauge symmetry in which a rescaled metric tensor does not affect any essential physics:$g_{\mu \mu} --> \lambda(x) g_{\mu \nu}$where$\lambda(x)$is an arbitrary function of spacetime. Of course, the components of the metric tensor$g_{\mu \nu}$are real and observable. As is well-known, Weyl's theory was reinvented as the phase invariance concept of quantum mechanics, perhaps the most profound symmetry known in modern physics. Weyl's gauge theory works in QM precisely because the wave function is unobservable and can involve an arbitrary phase function. My contention is that Weyl's original gauge idea didn't work only because the metric tensor is a real, observable quantity, and that Weyl actually anticipated the existence of the wave function eight years prior to Schrodinger's celebrated wave equation. After all, it was only one year after the 1926 wave equation that physicists (including Weyl, London, and even Schrodinger himself) began to realize that Weyl's gauge concept was workable in QM and that it was in fact required in order to incorporate electrodynamics into the then-developing quantum theory.  Amalie's Ashes -- Posted by wostraub on Tuesday, September 6 2005 This year marks the 70th anniversary of the death of Amalie (Emmy) Noether, colleague of Weyl, Einstein and countless other great 20th-century scientists, and generally regarded as the greatest female mathematician who ever lived. I just finished reading a chapter on Noether in Nobel Prize Women in Science: Their Lives, Struggles, and Momentous Discoveries, Second Edition, J.H. Henry Press (1993). I realize now that I did not give her adequate credit in my little write-up (see Weyl & Higgs), and I now stand in awe of the woman, both in terms of her gifts as a mathematician and as a human being. In spite of the harsh, ongoing prejudice she experienced firsthand even as one of Germany's top mathematicians in the teens and 1920s, Noether doggedly pursued her field with little or no regard for her own well-being. In recognition of her greatness as a mathematician, she was invited by Hilbert and Weyl to teach at the University of Gottingen. But for many years she was an unpaid, untenured, unpensioned nichtbeamteter ausserordenticher Professor, which roughly translates to "unofficial, unprivileged third-class instructor" (not unlike adjunct faculty!) Out of a total faculty of 237, Noether was one of only two female professors at the school (the other was a physicist). As I mentioned in my earlier write-up, Noether was a pacifist, left-wing Jewish female, and these traits did not endear her to the Nazis. When Hitler was appointed Chancellor of Germany in January 1933, Noether was one of the first professors to be fired. She and numerous other colleagues at the University of Gottingen tried to hang on, but brownshirted Nazi students successfully boycotted her and other Jewish professors -- “Aryan students want Aryan mathematics, not Jewish mathematics!.” Denied of a livelihood, Noether (with the assistance of Weyl) formed the German Mathematicians’ Relief Fund, and for a while taught secretly from her apartment. Even Weyl (a Christian) was forced to leave, as his wife was a Jew. Moving to the Institute for Advanced Study in Princeton in 1933, Weyl mourned the resulting Nazification of science and mathematics and witnessed the destruction of German preeminence in science, philosophy, psychology and mathematics with a broken heart. In 1933, Noether too fled, to Bryn Mawr College in Pennsylvania, where she was given a limited professorship at three-quarters pay. She died there in 1935 following the surgical removal of a large ovarian cyst. Although the college neglected to preserve her papers, it did manage to preserve her ashes. In 1982, on the centennial anniversary of her birth, the school buried her ashes under a brick walkway near the library’s cloisters. I see a terrible parallel to the madness Noether faced in Germany with events in this country today: anti-intellectual, fundamentalist fervor is demonizing stem-cell research and evolution (even geology) in favor of mystical, irrational, evangelical creationist theories, including “intelligent design.” Like the anti-intellectual, anti-feminist Nazis, narrow-minded idealogues like Pat Robertson, Jerry Falwell and Bill Frist are beating the drums for the destruction of modern science and rational thought in America. In their foaming hatred of feminism, I hear clear echoes of the words of Nazi Propaganda Reich Minister Josef Goebbels: “The mission of women is to be beautiful and to bring children into the world.” In a recent issue of Physics Today, physicist Lawrence Krauss addressed the lack of any contemporary Einsteins. Sadly, no one of the moral and intellectual stature of Noether, Einstein or Weyl exists today. No doubt, if these great people were alive now, they would be quickly ostracized by the Bushies and their media whores as intellectual peaceniks. They would also be ignored by the American public, which largely prefers reality TV to reality. The Republican War on Science  Dirac's Burial Plaque -- Posted by wostraub on Sunday, August 28 2005 Just thought I'd show this, which is located in Westminster Abbey, not far from where Newton rests. This and Boltzmann's headstone are the only markers I know of that celebrate great scientists with famous equations! The grave of Boltzmann (who committed suicide in 1906) is honored with his entropy equation S = k log W, while the above photo expresses Dirac's relativistic electron equation, which is arguably the most beautiful equation in physics. The "OM" stands for Order of Merit, an honor that Dirac was particularly proud of. He was also elected a member of the Royal Society in 1930 at the age of 28. One of the utter shames of this world is that the average person has never heard of Paul Dirac, whose name should be as well-known as Newton's and Einstein's. For more information on Dirac and his equation, see my write-up on Weyl spinors.  Weyl and Overdetermination -- Posted by wostraub on Saturday, August 27 2005 In one of my write-ups I glossed over the fact that Weyl's theory of the combined gravitational-electrodynamic field relies upon the square of the Ricci scalar,$R^2$. In terms of the metric tensor, this quantity is of the fourth order in$g_{\mu \nu}$and its first and second derivatives. Einstein and many others objected to Weyl's theory for this reason, since solutions of the Weyl action tend to be overdetermined (i.e., non-physical "ghost" fields can appear). I've looked all over for a detailed response from Weyl on this issue. Clearly, he understood its relevance yet he didn't seem to be overly concerned about it. However, if you calculate the equations of motion from the free-field Weyl action principle, you find that you can divide out an R term (assuming it is a non-zero constant), which leaves second-order equations of motion! I don't know if Weyl was aware of this or whether he dismissed the overdetermination issue out of preference for the essential beauty of his theory. Nature seems to prefer second-order equations, whether one is dealing with classical physics or quantum mechanics. There are exceptions, however. The one that comes immediately to my mind (which any structural engineer will instantly relate to) concerns the equations governing the elastic bending of beams. Indeed, loaded beams are described by a fourth-order differential equation. Fourth-order equations also result from perturbative expansions in quantum mechanics, but these don't qualify! Ghost fields in quantum mechanics are generally frowned upon. I've always looked upon the scalar Higgs field as a kind of ghost field, but it results from symmetry breaking rather than any inherent defect in the associated action quantity.  The Snapping of String Theory? -- Posted by wostraub on Friday, August 5 2005 This month's Discover magazine has an article by Michio Kaku on the future of string theory. Kaku addresses the fact that string theory has now been around for over 35 years without a shred of experimental evidence to back up the theory's many predictions. He also recognizes the fact that most of the world's top physicists seem to be gravitating toward string theory, thus depriving other fields (notably particle physics) of upcoming talent. Many notable physicists, including Lawrence Krauss and Sheldon Glashow, feel that string theory is a mathematically beautiful but ultimately empty concept that should be either verified once and for all or abandoned. Kaku describes a few experiments that might provide some support for string theory (involving dark matter, gravitational waves, and the Large Hadron Collider), but for now the theory's only support seems to be its beautiful mathematics. I for one disagree, because I feel that the math is just too confounding (but I'm a mediocre hack, so who am I to judge?) String theory verification may ultimately require energies that are simply beyond what mankind will ever muster. We can currently probe spacetime down to a distance of around 10(-18) meter, but strings typically involve distances a billion billion times smaller than that. What good is a theory if it predicts structures and hidden dimensions that are on the order of the Planck scale? We'll never get own that far! It's too bad that Kaku's article wasn't handed to Scientific American, which always goes into things much deeper than popular science magazines like Discover. Popularized accounts of the quantum theory and gravitation are rarely interesting nowadays, mostly because the mathematics can be understood by undergraduates. But string theory is so damned confounding that only experts can work in the field, and even they have confessed that they don't know what the hell they're doing. Consequently, popularized accounts of strings are so dumbed-down that they're essentially useless. Kaku (himself one of the experts) is one of the better expositors, but his article in Discover really doesn't tell me anything.  Weyl and Higgs -- Posted by wostraub on Sunday, July 24 2005 Here's a very simple derivation of the Lagrangian for quantum electrodynamics along with a description of the Higgs mechanism (and why Weyl should get a lot of the credit for both of them). Weyl/Higgs  Net Energy -- Posted by wostraub on Tuesday, July 19 2005 The July 17 Los Angeles Times Magazine ran a great article on the likely future of hybrid cars, focusing primarily on rapidly-developing technologies will allow these cars to be plugged in overnight to charge batteries, rather than have the cars' own gas engines do the charging. AeroVironment, a Monrovia, California company (www.AeroVironment.com) has received a$170,000 grant to retrofit Toyota Prius hybrids with an additional 180-lb battery pack that can be charged separately. Additional tinkering with the car's electronic controls allows the car to run on battery power only for the first 30 miles or so (I have a new Prius, and I think this is a fantastic idea). Overall, the company's prototype Prius is getting slightly over 100 MPG using the new system. While messing with the hybrid energy drive voids the car's warranty, Toyota appears smitten with the idea and has indicated a willingness to work with the company regarding the warranty issue. The article goes on to state that jazzed-up hybrid vehicles might soon achieve up to 500 MPG and beyond. Great news, when gasoline is running around $2.67 a gallon (at least here in Pasadena, CA). However, that 500 MPG figure does not take into account the gasoline energy equivalent to charge a hybrid's batteries off the grid. A more recent article, put out by the Environmental News Network, demonstrates that the net energy output of a system needs to take such things into account. This is especially true when considering the production of ethanol from corn, which has lately been widely touted as a cost effective new gasoline additive. The article states that researchers at Cornell University and the University of California at Berkeley have concluded that it takes 29 percent more fossil energy to turn corn into ethanol than the amount of fuel the process produces. Similarly, it requires 27 percent more energy to turn soybeans into biodiesel fuel, while more than double that to do the same to sunflower plants, the study found. "Ethanol production in the United States does not benefit the nation's energy security, its agriculture, the economy, or the environment," according to the study by Cornell's David Pimentel and Berkeley's Tad Patzek. The universities concluded that the country would be better off investing in solar, wind and hydrogen energy. The researchers included such factors as the energy used in producing the crop, costs that were not used in other studies that supported ethanol production, and they also took into account some$3 billion in omitted state and federal government subsidies that go toward ethanol production in the United States each year. Believe me, I'd love to see America producing cars that get 100 MPG, and I sincerely think it's technologically possible. But like all things, let's consider the whole picture before we get too optimistic. Article

 More Fizzicks Fun -- Posted by wostraub on Monday, July 18 2005 You gotta just love the new Hewlett-Packard Pavilion notebook computer commercial. The setting is a university lecture hall. A physics professor is droning on monotonously (a la Ben Stein as the teacher in "The Wonder Years") on atomic physics. The cute young thing in the front row is busy with her new HP Pavilion computer, but she's not taking notes -- she's watching DVD videos, including tattooed rock singers who magically jump out and writhe suggestively on her desk, obliterating the boring physics lecture. Remember Malibu Stacy's response to a Simpson's math question? "Don't ask me -- I'm only a girl (tee hee)!" Ms. Stacy must be HP's target demographic. No wonder America's students are going down the drain in math and science. If a student of mine had acted like this, I'd have kicked her out of the class forthwith (and probably gotten myself fired in the process). Earlier I gave a bad review of Tom Friedman's new book The World is Flat, but one of the book's many good points is that it accurately assesses the awful state of math & science education in the United States and how we are being rapidly being taken over academically by other countries, notably China. Hewlett-Packard is a high-tech US firm. What in hell are they doing putting out ads like this?! Update 19 Jul 2005 HP announced this morning that it would lay off 14,500 workers and freeze employee pensions. Guess the commercial's not working.

 Goenner Again -- Posted by wostraub on Thursday, July 14 2005 If you have any real interest in physics, particularly its evolution from Einstein's geometrical approach to quantum theory, you simply must read On the History of Unified Field Theories by Hubert Goenner of the University of Gottingen. Pretty much all of this involves the progress of theoretical physics from 1920 to 1929, concluding with Weyl's historic 1929 paper on gauge symmetry. Of particular interest are the efforts to to incorporate Dirac's relativistic electron theory, which appeared in 1928, into Einstein's ideas of spacetime geometry. Even the five-dimensional theory of Kaluza-Klein was given the Einstein treatment, to no avail. By the end of 1929, it was all too clear that Einstein's general relativity just did not mesh with quantum mechanics. I mentioned Goenner's paper earlier on this site. I finally finished reading the whole thing, and I have to admit that he's got a lot more in his one paper on Weyl than I have on my whole stupid website (at least he doesn't seem to be adversely distracted by the Bush Reich, like I am). Early Field Theories I cannot get over the sheer amount of intellectual effort that went into the various attempts to reconcile gravitation with quantum theory, or the optimism that reigned regardless of the fact that nobody really seemed to know what was going on. I think it can be traced to the fact that there were only two forces known at the time: gravitation, which was elucidated by Einstein, and electrodynamics, which Weyl had seemingly unified with gravity in 1918. In the end, neither could be reconciled with quantum theory, at least in terms of what was known by the time the 1920s ended. Part 2 of Goenner's excellent overview of unified theory is yet to come; I welcome it enthusiastically.

 World Oil Production -- Posted by wostraub on Sunday, July 10 2005 The attached link summarizes world oil production data for the period 1860-2003, as obtained from the US Department of Energy (Energy Information Agency). I assume it's reliable, although the numbers are a tad higher than those given by Deffeyes. My statistical analysis for the Gaussian regression is included; the graphic shows the data (open circles) along with the superimposed regressed normal distribution curve (grey line), which fits rather nicely. However, note that, according to this analysis, Peak Oil occurred in 1998! Production Analysis for 1860-2003 [Note: I used a non-linear multivariate regression program called NLREG to do the analysis.] Obviously my preliminary analysis is not very realistic, but it's intended only to get you thinking about the Peak Oil issue, anyway. The actual situation is more complicated because it involves oil reserves and discoveries (that may or may not be included in the EIA data) and not just produce-and-use data. At any rate, this will give you some idea of how the data are being viewed by a number of researchers (and many of them are alarmed at what they're seeing). One thing that is not in question is that once the oil production curve starts to fall over from its exponential rise, the Peak Oil phenomenon will be inevitable. This will then signal the end of cheap oil, the commodity that runs the modern world. What will replace it? I haven't a clue. God gave us something like 2 trillion barrels of oil, and we've gone through about half of that. God's gift should have been used to develop a more sustainable energy source (such as solar), but instead it went to Hummers and their kin. Now it looks like oil wars are inevitable. My advice is that you get up on the issue and decide for yourself.

 Peak Oil -- Posted by wostraub on Wednesday, July 6 2005 Recently, I compiled a table of world oil production data for the period 1860 to 2004 and did a regression analysis on the data assuming a normal distribution (Gaussian) model. I think I know why the Peak Oil doomsayers do not use this model. Using a nonlinear regression program called NLREG for the Gaussian model, my results show a decent data fit (the r^2 statistic is about 0.98) with a standard deviation of approximately 25 years and a total production of about 1.7 trillion barrels (this is the amount of oil contained in all the planet's reservoirs). However, the peak year comes out to be 1998, seven years ago! [This really isn't as embarrassing as it may seem, because it's doubtful that any simple model will be within 10 years of the actual peak, anyway.] Of course, oil production hasn't peaked yet, as far as we know. For my pathetic little model, post-1998 oil production data overshoot the model, but this doesn't necessarily mean it's wrong. Most of the models I've seen use the logistic function, which is often used in population projections. I have absolutely no idea why it should be preferred over the Gaussian function for making oil projections. The logistic models typically give the peak year at around 2005 to 2015, with a total production of about 2 trillion barrels. Since these peaks are in the future, maybe that's the reason. I've found a way to express the data as a rate plot, a device that Deffeyes explains in his excellent book Hubbert's Peak. It's basically just an x-y plot using specially transformed data, which gives a straight line. The x-intercept provides another method of obtaining the total production. It too gives 1.7 trillion barrels. What's not in doubt is the amount of oil we've burned since the famous Titusville, Pennsylvania oil well started producing in 1860 (the "Ur well" of the oil age). It's hard to believe, but humans really have burned about half the oil that was formed in the earth over the past few billion years. [Side note: I once asked a new-earth creationist friend how all that oil got formed in just 6,000 years, since no chemical or physical process known to man could have done it in that short of time. Her answer: "God put it there for our use." May the Lord preserve us from this incredible ignorance!] While curve fitting is great fun, I'm trying not to take things too seriously, at least not yet. Still, if there's any truth to this at all, it portends a terrible future for mankind. The worst part of it is that it may not be more than a few years away. Either way, we're not doing much about it. Dr. Albert Bartlett, Professor Emeritus of physics at the University of Colorado at Boulder, has been warning us of the peak oil issue for many years. He claims (and I believe he is correct) that one of mankind's greatest failures is his unwillingness to appreciate (or even understand) the exponential function. Because we tend to use untapped resources initially at an exponential rate, we naively adopt the misconception that unrestrained growth is always good and can be sustained indefinitely. To me, that's one of the stupidest aspects of human beings -- we think only in the short term and believe that God, technology or luck (or that old standby, the "indomitable human spirit") will somehow bail us out when things go to hell. I'll put up what I have so far in a few days and you can decide for yourself if world oil production is peaking.

 Pauli -- Posted by wostraub on Monday, July 4 2005 I finally finished reading Penrose's book (The Road to Reality), which is a remarkable text in terms of the sheer amount of material it covers. It doesn't go into a lot of detail, but if I were stuck on an uninhabited island somewhere I would probably like to have it with me. Alas, I never quite got through Zwiebach's A First Course in String Theory, despite a rather gallant effort on my part. The math is not too difficult (remember, this is a very introductory text), but the physical models it presupposes are simply beyond my comprehension. Yes, strings are actually strings, but they have this peculiar habit of attaching themselves to membranes in a God-awful number of dimensions. What the hell are these membranes other than highly-abstract boundary conditions? It's a right brain/left brain thing, I believe, and I've been forced to grudgingly accept the very serious limitations of my little grey cells, as Poirot puts it. Again, I emphasize that this is an introductory text. Lord, is there any hope for me? So in utter defeat this evening I pulled down my crumbling Dover copy of Pauli's Theory of Relativity, which always holds something that I had overlooked the last time I got it down. Although I am enamored of Weyl, his writing style (or at least the German translations of his writing) very much leave something to be desired. In short, Weyl's ideas are beautiful, but his writing is not, at least in my opinion. Pauli, on the other hand, is a joy to read, at least the stuff I understand, and this is especially true for his relativity book. I may be stretching things here, but the book actually covers about 35 years of progress on basic general relativity. Pauli wrote the first version in 1921 as a lengthy German encyclopedia article, then appended it in the mid-1950s with supplementary notes. The book includes a section on Weyl's theory of the combined electrodynamic-gravitational field, and as such was only the second book I acquired that provided details on Weyl's theory. The book is a pleasure to read, from Pauli's clear exposition of special relativity to general relativity and beyond. I was absolutely dumbstruck when I learned that Pauli had written the book when he was only a 21 year-old graduate student. Talk about grey cells! An oft-told anecdote about Pauli concerns his admittance to the hospital for cancer treatment in 1958. His lifelong fascination with physics included a similar fascination for the fine-structure constant of quantum mechanics, which is very nearly the pure number 1/137. He always wondered why God had created such a number. In quantum mechanics, constants tend to be truly microscopic (Planck's constant is about 6 x 10^-34, for example), so the appearance of a number that is about 0.008 boggles the mind. What also boggles the mind is that Pauli, who passed away in the hospital at the relatively young age of 58, died in Room 137. Don't ever think that God doesn't have a great sense of humor! Every high school student gets introduced to Pauli through his Exclusion Principle in chemistry. But the man was such a gigantic figure in the field of physics that he deserves so much more. He was an irrascible and impudent curmudgeon who was famous for his crushing verbal put-downs of lesser physicists who dared to expose their ignorance, but he could also be caring and supportive. He was fond of Weyl and truly loved Einstein, despite the great scientist's ill-fated rejection of quantum mechanics. The Dover book is still available as a paperback for maybe $10. I heartily recommend it.  Weyl and Chalabi -- Posted by wostraub on Saturday, June 25 2005 You cannot apply mathematics as long as words still becloud reality. -- Hermann Weyl I don't know what context Weyl intended in this quote, but I'm tempted to think that he saw empty rhetoric as the enemy of truth and reason. You cannot lie with mathematics because you will quickly be found out. It is far easier to lie with words, because until someone can check out what you're saying (which may not even be possible), people have to assume that you're telling the truth. Mathematics and words both come from the heart, but only one is required by its own nature to be true. It is true that one can lie with statistics, but the lie is sold through the interpretation of the meaning of the numbers, which gets us back to words again. Jesus Christ warned us to be careful about what comes out of our mouths, but it wasn't mathematics he was concerned with. Most people are not aware that years ago, the designated Iraq Minister of Oil Ahmad Chalabi was a professor of mathematics at the American University of Beirut, Lebanon. The son of a wealthy banker, Chalabi studied at MIT and the University of Chicago, where he received his PhD in mathematics in 1969 (I believe his specialty was ring theory). Of course, you're certainly aware that Chalabi, an Iraqi Shi'a Muslim, is a notorious liar who stuffed Bush's head full of lies (as if it wasn't already full of them) about Iraq's non-existent weapons of mass destruction. He is also under warrant for arrest in Jordan for embezzlement and money laundering. After all of his "disassembling," Chalabi still managed to wrangle the job as oil minister on the new Iraqi cabinet, largely on the basis of his ongoing connections with Bush, the CIA, and the Pentagon. He's their kind of people! This serves to show that while mathematics doesn't lie, mathematicians certainly can. My guess is that Chalabi will be assassinated when the Bush administration begins to siphon off large quantities of oil from the Iraqi oil fields to supply all the military bases we're constructing in that country. He certainly doesn't have the interests of the Iraqi people at heart, and his role as a Bush oil puppet is certain to get him into trouble. Before he dies, I hope the last thing that goes through his head (other than a bullet) will be the sincere regret that he didn't stay in mathematics. Sorry that I mentioned Weyl and Chalabi in the same breath; Weyl deserves better.  Grace S., 1924 -- Posted by wostraub on Friday, June 24 2005 Here lies a most beautiful lady: Light of step and heart was she; I think she was the most beautiful lady That ever was in the West Country. But Beauty vanishes; Beauty passes; However rare -- rare it be; And when I crumble, who will remember This lady of the West Country? Epitaph, Walter de la Mare, 1873-1956  Weyl and Philosophy -- Posted by wostraub on Friday, June 24 2005 I have been having great difficulty lately understanding Weyl. Not his physics (which is pretty straightforward) nor his math (which can be exceedingly difficult for a non-mathematician like me), but his extensive philosophical writings. During his life, Weyl went through various stages of philosophical speculation. Each was important to him in its own time, as Weyl aged and became wiser, from phenomenology to what might be called religious existentialism. He consequently devoted an enormous amount of time and effort to philosophy, no doubt a result of his deep reflections on the interconnectedness of mathematics, physics and the human mind. Unfortunately for me, I'm having one hell of a time understanding Weyl's philosophical musings. I'm inclined to state that he is very deep, but at times it all seems like a bunch of mumbo jumbo. The same thing happened when I tried to learn category theory, which has been described as both the fundamental basis of all profound mathematical theories and "generalized abstract nonsense." Being trained neither in formal mathematics nor philosophy, I'm at a distinct disadvantage to criticize (never mind fully comprehend) Weyl's efforts in either field. But I keep trying. In 1954, near the end of his life, Weyl reflected on what he had learned over the years in physics and philosophy, as necessarily colored by two world wars in which his native country, Germany, had participated in rather shamefully: "I did not remain unaffected either by the great revolution which quantum physics brought about in natural sciences, or by existentialist philosophy, which grew up in the horrible disintegration of our era. The first of these cast a new light on the relation of the perceiving subject to the object; at the center of the latter, we find neither a pure "I" nor God, but man in his historical existence, committing himself in terms of his existence." This is the philosophical Weyl that I can relate to.  No Weyl in Pasadena -- Posted by wostraub on Monday, June 13 2005 Well, I tried. This was the response I received from the Institute for Advanced Study: Dear Dr. Straub: Thank you for your inquiry to the Archives of the Institute for Advanced Study. I have searched the documentary evidence that we have for mention of any visits by Professor Weyl to Caltech. I'm sorry to report than I find none, though there is other travel documented, including west to Colorado, where Professor Weyl apparently went to escape allergies that plagued him in New Jersey. From my review of the literature, he seems to have been a reliable presence on the Institute campus during the academic year, and regularly gave lectures here. Of course, that does not preclude a brief trip here or there, and his summers were his own. I have searched for literature you might consult to advance your research, but don't find anything to add to what your website indicates you've already seen. I'm very sorry not to be able to be of more help, but I will keep your inquiry in mind, and be in touch if I find anything that might be of interest to you. Regards, Erica Mosner Library Assistant Historical Studies-Social Science Library Institute for Advanced Study Einstein Drive Princeton, New Jersey 08540 Thanks, and God bless you, Erica!  Weyl in Pasadena? -- Posted by wostraub on Saturday, June 11 2005 I recently contacted Dr. Judith Goodstein at Caltech to see if Hermann Weyl had ever visited the school. Goodstein is the University Archivist and author of Millikan's School, a history of Caltech, and co-author (with husband and fellow Caltech professor David Goodstein) of Feynman's Lost Lecture, so if anyone can help me, I thought she could. Although Weyl went to the Institute for Advanced Study (IAS) in Princeton when he left Germany in 1933, I figure that his wanderings over the the years must have brought him to Pasadena at least once. Unfortunately, Goodstein told me that Caltech has no record of any visits by Weyl. She suggested that I contact the IAS to see if anyone there keeps a listing of Weyl's domestic travels. I'm in the process of doing that, and will pass along whatever I find.  Jesus on Truth and Lies -- Posted by wostraub on Thursday, June 9 2005 From John 8: 42Jesus said to them, "If God were your Father, you would love me, for I came from God and now am here. I have not come on my own; but he sent me. 43Why is my language not clear to you? Because you are unable to hear what I say. 44You belong to your father, the Devil, and you want to carry out your father's desire. He was a murderer from the beginning, not holding to the truth, for there is no truth in him. When he lies, he speaks his native language, for he is a liar and the father of lies. 45Yet because I tell the truth, you do not believe me! 46Can any of you prove me guilty of sin? If I am telling the truth, why don't you believe me? 47He who belongs to God hears what God says. The reason you do not hear is that you do not belong to God." Why doesn't America truly follow Jesus? Why are we embracing the torture and imprisonment of innocents? Why are we spending half a trillion dollars annually on weapons of death and destruction? Why are we throwing away our Constitutional rights? Can't we recognize hypocrisy when it stares back at us in the mirror? Why are we following the Devil?  Science and Patriotism -- Posted by wostraub on Monday, June 6 2005 Johannes Stark was a great German scientist who won the 1919 Nobel Prize in Physics for his discovery of the "Stark effect," the splitting of atomic spectral lines by electric fields. He was a prolific researcher who published over 300 scientific papers in his lifetime. He was also a fanatical German patriot who early on embraced the Nazi belief that Jews were inferior human beings. He became a member of the Nazi Party in 1930. Stark was a strong proponent of "Deutsche physik," or Aryan physics, which be felt should be used solely for the purpose of advancing national defense and prestige. By comparison, he scorned what he termed "Judische physik" (Jewish physics) on the basis that non-Aryan physics was not scientifically objective (by this I suppose he meant that physics was not objectified unless it had a nationalistic purpose). In 1934, Stark wrote a book, "National Socialism and Science," in which he explained his views (I am going to read that book). He hated Einstein and was no friend of the loyal (but not rabid) German physicist Werner Heisenberg, whom Stark referred to as a "white Jew." In 1947, a court sentenced Stark to four years in prison for his contributions to anti-Jewish hatred before and during World War II. Stark's is a classic case of scientific inquiry gone mad. Pure science and mathematics are completely objective when their pursuit involves discovering the truth. I will take that statement one step further by adding that objectivity cannot exist when a political agenda is attached to the research. The most heinous example I can think of involves the research and development of weapons of mass destruction for purely military and/or political purposes. But a more common example would be the selective and deliberate misrepresentation or skewing of scientific data for the purpose of convincing someone that something is true when in fact it is not. But Stark, Phillip Lenard and other noted German scientists first had to convince themselves that Einstein's theories were wrong before they could convince others. How did they do that? Einstein wasn't right about everything, but his special and general relativity theories were thoroughly tested and found to be valid. Also, these theories were, as Paul Dirac once put it, mathematically "beautiful" (and they are). I believe that this is where ethnic and political hatred made their way into the picture. Stark believed that Einstein was of an inferior race, so his ideas had to be wrong. This was no small effort -- he almost had to convince himself that 2+2=5 in order to erase the truth of relativity from his mind. Fortunately for Stark, he easily found others that shared his Nazi mindset. Einstein's works quickly found themselves among the thousands of other papers and books that the Nazis burned during Hitler's reign. My younger son and I discussed a related topic today. I asked him why a seemingly-disproportionate amount of funding is being spent on HIV/AIDS research today. My straw-man argument was that AIDS is primarily a behavior-related disease while, say, malaria threatens everyone, so why not stress the preventive aspects of HIV. His response is that the human immunodeficiency virus is a threat to mankind simply because it now affects so many people. He felt that dwelling on issues like behavior-based prioritization of funding is too closely tied to moralizing, which is subjective. Subjectivity is the enemy of science and mathematics. It is also, sadly, a very human trait. I see the same thing happening to science today, and it is truly frightening. HIV/AIDS, evolution and cellular research are all being attacked on the basis of subjective moral and political arguments that have nothing to do with the scientific method. The Dobsons, Frists, and Falwells of this country fervently believe that HIV/AIDS is a punishment from God designed to strike down immoral people. They have forgotten that when God warned us "the wages of sin is death" he was referring to all sin, not just homosexual sin (and yes, I do believe it is a sin). If I look at a woman the wrong way, I have committed a sin that can put me in hell along with every other unforgiven sinner; will I then feel somehow more "sanctified" than the other lost souls? Because they demand logical, organized and rational thinking, science and math are giving Americans fits these days. Although we have some great (and objective) expositors like Weinberg, Kaku, Lederer, Davies, Hawking and Penrose around to explain things, we also have idiots like Dr. Frist whose subjective pseudo-science represents an enormous threat to America. I'm sure he's already compiled a long list of books that he plans to have burned when he's president. God save us! I hope he and his ilk can be stopped in time, but I truly fear for the scientific future of this country.  "Rods from God" -- Posted by wostraub on Thursday, June 2 2005 "Full-spectrum dominance." That's the term the US military is using to describe a proposed planet-wide system of nuclear-tipped intercontinental ballistic missiles and orbiting weapons utilizing what insiders call "Rods from God." RFGs are heavy metal cylinders that would be fired from orbiting space platforms to take out enemy fortifications on earth's surface. The rods would be made from dense metals like depleted uranium or tungsten and fired at such high velocities that they could penetrate many meters of soil and concrete. However, a half-dozen distinguished scientists, including Nobel prize winners Steven Weinberg (physics) of the University of Texas at Austin and John Polyanni (chemistry), professor at the University of Toronto, claim that the proposed defense system would be a "criminal" waste of hundreds of billions of dollars that would be better spent on public welfare programs. They go on to say that the system would be unworkable anyway and would offer only the illusion of absolute security if deployed. Weinberg is the author of an excellent book on the general theory of relativity and a three-text series on quantum field theory. The latter is a tough read, but Weinberg's response to the RFG proposal is easy to understand -- assuming that you're sane. RFG represents only one facet of Bush's "space exploration program." Of course, conservative faith groups are ecstatic over the proposal because it has the word "God" in it. Also, they don't generally trust math and science, because it's hard to understand and promotes stuff like evolution and all, but "Rods from God" has a nice Christian ring to it. Just the thought of evil doers being righteously blasted to smithereens warms their hearts. But the proposal has another, equally ominous aspect. Imagine you're the leader of a nuclear country like Russia or China (or even France). You see the United States being led by dangerous "Christian" fascists determined to take over the earth and enslave countries owning the resources needed by the United States to maintain its preposterous standard of living. What have you got to lose? In coordination with the other nuclear members, you fire off everything you've got, and hope for the best. It's madness, of course, but it would at least guarantee the destruction of the United States as a functioning society. Is this the American Taliban's real game plan -- to provoke a worldwide nuclear Armageddon and thus force Jesus Christ to make an early return? http://www.nuclearpolicy.org  Early Unified Field Theory and the Quantum -- Posted by wostraub on Wednesday, June 1 2005 I’ve been reading lately about the efforts of Einstein, Weyl, Rainich, Eddington and others around 1925 to find a unified theory of gravitation and the electromagnetic field. To my mind, these guys were the flip side to Bohr, Dirac, Pauli, and Fermi, who of course were almost solely focused on quantum physics at that time. In my opinion, Einstein’s discovery of general relativity in 1915 came at a really bad time. When the theory was brilliantly confirmed by the explanation of the perihelion shift of Mercury and the solar eclipse expeditions of 1919, there was no doubt whatsoever that general relativity was a valid description of spacetime physics. In those simple days, the only known forces of nature were gravitation and electrodynamics, and the only known particles were electrons, protons and photons. Following the brilliant but failed “near miss” of Weyl’s theory in 1918 and Rainich’s subsequent discovery of the algebraic similarities of gravity and electromagnetism, Einstein and his colleagues must have felt that a consistent unified theory was imminent. However, the theory stubbornly resisted discovery. In hindsight, we know that Einstein and the others were doomed to failure. Nature is not as simple as Einstein had presumed; instead, it hosts a dizzying array of elementary and composite particles, forces and fields requiring a much more sophisticated physical and mathematical approach. I get a real kick out of reading Einstein’s correspondence to the other early field theorists of that time. One idea after another is proposed – distant parallelism, bivectors, n-beins, a generalized (and traceless) Einstein tensor – and each one is subsequently tossed aside. Einstein constantly refers to the sublime secrets of nature and “the Old One,” and on occasion waxes quite philosophic about the nobility of the search. In spite of the failures he is not discouraged, and continues to press on. Most of his colleagues, however, begin to realize that it is probably a waste of time, and so they move on. But even as late as 1929, rumors spread that Einstein had finally achieved his goal. Several newspapers even printed the theory with all its mathematics for their undoubtedly puzzled readers. It was all a big fuss over nothing. By comparison, the development of quantum mechanics in the 1920s was met with one astonishing success after another. I’m inclined to feel that Einstein’s eminence in those years ultimately hurt physics, because his unification efforts were sidetracking himself as well as the talents of numerous great colleagues. Remember that when Einstein’s general theory appeared, about the only quantum theory that existed was that of Bohr’s hydrogen atom, which even then was seen as a hodgepodge of classical and quantum ideas. It wasn’t taken seriously until Heisenberg’s matrix mechanics and Schrödinger’s wave equation arrived in 1925, and it was about that time that real interest in unified theory was on the wane. Today, physicists are hard at work on unified theories that Einstein couldn’t have comprehended. String theory made its appearance in the 1970s, followed by supersymmetry, supergravity, superstrings, M-theory, and loop quantum gravity. Recall Einstein’s initial support of Kaluza-Klein theory in the 1920s, which sported a total of five spacetime dimensions. Would Einstein have been equally enthusiastic about ten, eleven and even twenty-six dimensions? I wonder. To my mind, God displayed a wonderful sense of humor when he let Einstein discover general relativity theory in 1915. God knew that scientists would initially think they were close to knowing everything. At about the same time he opened our eyes to quantum physics, and then probably watched with much amusement as we tripped over ourselves trying to sort things out. But eventually we did -- thanks to these wonderful, curious minds God gave us. Einstein once famously remarked that “God is subtle, but not malicious” (Raffiniert ist der Herrgott, aber boshaft ist er nicht). Recently, a noted physicist (darn it, I just can’t remember the the guy’s name offhand) dared rephrase Einstein’s remark as “God is not malicious, but he is subtle.” This makes much more sense to me. The maddening complexities of modern unification theories are all too real, but they follow a kind of simplicity involving spacetime symmetries and their evil cousins, the internal symmetries. Surely, this is what God had in mind for mankind – we’ll try our damnedest, but, even if we never find the true unified field theory, we’ll have glimpsed God’s glory along the way. And this would have surely pleased Einstein.  A Lagrangian for Evolution? -- Posted by wostraub on Thursday, May 26 2005 I was talking to my son the other day about Lagrangians and the action principle in physics. Lagrangians are mathematical quantities (usually integral scalar densities) that, when extremalized, define the actual dynamical path that a particle or wave function takes under a given set of conditions. These paths generally result when something like energy or time is minimized. William Hamilton was the first to formalize the mathematics (Hamilton's Principle), although Fermat knew about the minimum-time principle of ordinary light propagation. Because minimal principles represent the most efficient or "best" ways that Nature can conduct her business, many early scientists saw this as direct evidence of God's existence, and many still do. Lurking behind the Lagrangian formalism are mathematical symmetries. A symmetry is simply a modification in a Lagrangian quantity that leaves the quantity unchanged. For example, translation symmetry (the requirement that physics be the same on Earth as it is on some extragalactic planet) leaves the Lagrangian mathematically unchanged. I won't go into it, but this symmetry is also responsible for the conservation of linear momentum. There is a very powerful theorem (by Emmy Noether) which states that for every Lagrangian symmetry there is a corresponding conservation law. It was Weyl, in 1929, who showed that the conservation of electric charge is due to gauge symmetry, which was a brand new kind of symmetry in those days. Since symmetry is a form of beauty (and it may even be a definition of beauty), one may indeed argue that God is behind all of this. Now switch from physics to molecular biology. Is there a minimal principle behind biological processes? Most certainly, because the behavior of biologically-important molecules (proteins, enzymes, etc.) is governed by quantum mechanics, and QM itself follows Lagrangian principles. But what about large-scale biological processes such as genetics and evolution (or, if you prefer, random mutations over large time scales)? What minimal principle could possibly result in the formation, adaptation and maintenance of complex living systems? Given a supply of simple organic compounds, is life inevitable? Was the formation of the first RNA or DNA molecule the result of God or Nature minimizing something? And if so, what was the driving force or symmetry behind it? My argument with my son was that living systems today are so unbelievably complex that concepts such as driving force and symmetry are totally hidden from us. When physicists conduct particle collision experiments, they now have a large collection of theoretical tools that they can use to design the experiments and interpret the results. By contrast, for living systems scientists can only watch and maybe make educated guesses. There are no readily-apparent symmetries or driving forces that can be utilized to interpret what they see. No one asks "Why are there proteins?" or "Why did Nature decide to develop this kind of enzyme for liver function?" When my son conducts PCR experiments or sticks mutated plasmids into cells, all he can do is say "Let's see what happens." I happen to believe that God created life. I also believe that he created evolution so that life could adapt to changing environmental conditions. But how he did all this is a great mystery (maybe so is why he did it). But seeing God's penchant for symmetry in physical laws, my guess is that he employed a similar approach when he designed life. When I was very young, I remember asking my father why a flower grows. What does it think it's doing? Why doesn't it just fall apart? Why should it make other flowers? What's the purpose behind all this? In later years I learned about things like Gibbs free energy, the equilibrium driving force behind Newton's law of cooling, statistical mechanics, and the action principle. But these revelations told me nothing about why God did what he did, or why he used the approaches he did. At the same time, I very strongly believe that our striving to answer these questions is one of the principal reasons why God put us here in the first place. When we finally figure it out, we'll know for sure what a great guy he is. The computational physicist Kent Budge has a blogsite (Trolling in Shallow Water) that includes a rather off-beat look at God, Lagrangians, and why Jesus was needed. Odd, and probably not what God actually had in mind, but it's worth a look (it's near the bottom of the first page): http://shallows.blogspot.com/2005/02/u1xsu2xsu3-part-2.html  John Baez's Website -- Posted by wostraub on Sunday, May 22 2005 I've added a link on the menu for the website of John Baez, professor of mathematical physics at the University of California at Riverside. He specializes in quantum gravity, but the guy seems to know about everything. His site has a lot of neat stuff ranging from very easy to way over my head. His enthusiam is contagious. Give him a look.  Einstein on Nationalism -- Posted by wostraub on Saturday, May 21 2005 Einstein hated militaristic nationalism. He would have undoubtedly deplored living in America at this time, and would have certainly detested our pathetic Cowboy President, his love of war, and his pro-torture position on innocent foreigners and "enemy combatants." In reaction to his views, the Daughters of the American Revolution (which until recently excluded all minorities from its nobel ranks) told Einstein to get out of America. The House Committee on Un-American Activities (McCarthy's little band of Nazis) similarly denounced Einstein, and the cross-dressing transvestite J. Edgar Hoover had his FBI compile a huge dossier on the scientist. The attached Word file is a copy of a hand-annotated speech that Einstein gave in May 1947 (I believe to the Emergency Committee of the Atomic Scientists). Read it and ask yourself if these are the words of a dangerous mind. http://www.weylmann.com/einsteintalk.doc  Weyl and Petrarch?! -- Posted by wostraub on Friday, May 20 2005 Did you ever glimpse someone, perhaps only for a moment, but it changed your life forever, and for the better? Did you ever have one of those “Aha!” experiences or a “Road to Damascus” moment that had the same effect? In the following I present some random thoughts I’ve had about Weyl and Beauty; it might even make for a passing grade on a high school composition. Her name was Laura de Noves, and she lived and died almost 700 years ago. She was exquisitely beautiful, and when he was twenty-three the great Italian humanist Francesco Petrarcha (better known as Petrarch) caught sight of her at the Church of St Clare in Avignon, France. Although he saw her for only a few moments, he fell into lifelong love with her. This was reportedly the only contact he ever had with the woman. Yet she was his livelong inspiration and the true force behind all of his great writings, including his Canzoniere, his barely-concealed lyric poems in praise of Laura. In the great Orson Welles classic, Citizen Kane, the elderly attorney Bernstein has a similar story to tell to the shadowy Reporter: “One day back in 1896, I was crossing over to Jersey on the ferry. And as we pulled out, there was another ferry pulling in. And on it there was a girl waiting to get off. A white dress she had on. She was carrying a white parasol. I only saw her for one second. She didn't see me at all. But I'll bet a month hasn't gone by since that I haven't thought of that girl.” And one cannot even mention the great Italian renaissance poet and writer Dante without conjuring up the memory of his beloved Beatrice. It is this Beatrice, who Dante saw once and remained forever in love with, who guides Dante to Paradise from the depths of Inferno and Purgatorio. Like Laura’s influence on Petrarch, Beatrice was the inspiring force of Earthly beauty that compelled Dante to seek out truth, God and salvation. His Divine Comedy is considered by many to be the greatest literary work ever. Finally, the noted Austrian physicist Erwin Schrödinger, while on an extramarital fling in the mountains in 1925, came up with his greatest discovery, the aptly-named Schrödinger wave equation, for which he shared the 1933 Nobel Prize in physics. Who was the unnamed lady (if she can be called that), and just how did she inspire Mr. Schrödinger? Nobody knows. What does all this have to do with Hermann Weyl? Well, as a young man he too glimpsed Beauty, and the experience affected him for the rest of his life. I suppose I risk appearing to be the most blatant of intellectual snobs if I admit that I see a parallel between Petrarch and Weyl. In his book of self-revelation, "My Secret Book," Petrarch has a long and intense imaginary dialog with St Augustine over Petrarch’s sufferings as an errant human being who, though he has seen the truth and glory of God, cannot get the idyllic but still very fleshly memory of Laura de Noves out of his head. It is a wonderful and profound dialog, comprising about 100 pages, in which Petrarch argues with Augustine that his love for Laura is based in purest admiration of beauty, and that this love has been the inspiration of all his noteworthy achievements in life. I won’t go into it, but Augustine will have nothing to do with Petrarch’s imaginings. “It cannot be denied that the most beautiful things are often loved dishonorably,” says Augustine. But, Petrarch replies, “Loving her has increased my love for God.” In the end, Augustine wins out handedly, and Petrarch admits his folly. It is an amazing debate, all the more remarkable because it was written in 1347, at the dawn of humanistic thought. Similarly, Weyl glimpsed Beauty in 1918 in his theory of metrical gauge invariance, and then struggled to maintain his love against a disbelieving and chiding Einstein. In a series of written correspondences that stretched from 1918 to 1921, Weyl and Einstein debated long and hard over Weyl’s 1918 theory. Try as he would, Einstein could not get Weyl to acknowledge the theory’s fatal flaw. Weyl sent a proof of the first edition of his book "Space-Time-Matter" to Einstein for review. Along with the proof he boldly tells Einstein that he has “succeeded in deriving electricity and gravitation from the same source.” While Einstein is initially ecstatic, he spots the flaw, for which there is no cure, and replies “Regrettably, the basic hypothesis of the theory seems unacceptable to me [although] the depth and audacity of which must fill every reader with admiration.” Weyl counters with “Even if this theory is only in its infant stage, I feel convinced that it contains no less truth than [your] Theory of Gravitation.” In his book, Weyl rhapsodizes rather poetically: “…One Light and Life of Truth comprehends itself in Phenomena. Our ears have caught a few of the fundamental chords from that harmony of the spheres of which Pythagoras and Kepler once dreamed.” But Einstein holds firm. Weyl weakens a bit: “Your rejection of the theory for me is weighty [Weyl is all too aware of Einstein’s renowned insight and scientific wisdom] … But my own brain still keeps believing in it.” Like Petrarch and his flawed love for Laura, Weyl is at last forced to face the fact that his gauge theory is also flawed and, like Petrarch again, Weyl tries to fix it up by a rather unsound rationalization of what’s real and unreal in spacetime. For a time this isolates him somewhat from Einstein, Pauli and others who are all too aware that Weyl is grasping at straws. But finally, and happily, Weyl concedes to the fatherly and caring Einstein. Like Petrarch, Weyl picks up his life and moves on, and in 1929 he discovers the true gauge invariance principle, which lies not in generalized Riemannian geometry but in quantum mechanics – a profound and lasting discovery that represents Weyl’s reward for having recognized at last the real truth behind the Beauty he had glimpsed one day in 1918. I suppose it would be the acme of naivete to compare the chiding Einstein with Augustine, but I think the basic idea holds up. But if you think I’m nuts, then go read "My Secret Book" and "Space-Time-Matter" and decide for yourself.  A Final Word on Majorana -- Posted by wostraub on Friday, May 13 2005 In 1986 the German-born Italian director Donatello Dubini released a film entitled "Das Verschwinden des Ettore Majorana" (The Disappearance of Ettore Majorana), which starred Jean Seberg. Well, Blockbuster Video didn't have this one as it turns out, so I'm giving up. If you premultiply a conjugated Dirac spinor with the purely imaginary gamma matrix$\gamma^2$, you get a charge-conjugated Majorana spinor. If this is set equal to the Dirac spinor, then the object describes fermions that act as their own antiparticles (some physicists believe the Majorana spinor provides an accurate description of massive neutrinos). Well, I guess you could have learned this from anybody, but it's about all I have on the guy. I guess the story about him jumping into the Tirrenian Sea in 1938 was the best part after all. No doubt you're foaming at the mouth for more, but until I learn Italian I'm going to have to pass on Mr. Majorana. If you come across the movie, I'd appreciate an email.  More on Ettore Majorana -- Posted by wostraub on Thursday, May 12 2005 Earlier I mentioned the brief life of Ettore Majorana, the brilliant Italian theoretical physicist who mysteriously disappeared while on a short boat trip in 1938. Off and on, for perhaps two years now, I have begun to study supersymmetry theory, only to fall flat on my face. The material is not that difficult; Dirac and Weyl spinors move in and around the theory, so there's a feeling of comforting familiarity. But the stuff gets so compounded and interwoven with so many details that I always give up. Interspersed in this mess are references to "Majorana spinors," which are very similar to Weyl spinors, but even closer to the concept of neutrinos. I always figured Majorana was just some inconsequential guy who happened to come across a neat kind of spinor. Now I'm beginning to see how unappreciative and stupid I've been. Almost all of the biographical material I can find on Majorana has to be translated from Italian websites, and the Google translations just aren't very good. Here's what I found today: Majorana was born in 1906 in Catania, Italy. He and his family moved to Rome in 1923, where he studied engineering until 1928. He switched to theoretical physics, obtaining a PhD in 1929 with a dissertation entitled "Quantum Theory of Radioactive Nuclei." Many considered Majorana to be brighter than Enrico Fermi, who worked with him. Immediately prior to his disappearance on 25 March 1938, a note was found in his handwriting that included the plea "Do not condemn me, for you do not know how much I suffer." Colleagues noted that on occasion he regretted the knowledge he had acquired regarding nuclear fission and the possibility of making an atomic bomb. Since Majorana was not a sickly person, or in debt, or even lovesick, it was assumed that he had committed suicide by throwing himself into the sea. His mother did not buy this; she never mourned, but awaited his return until the day she died. A neat mystery! Why can't television produce a drama like this, instead of the tripe it continues to air?  Who the Hell Was Ettore Majorana? -- Posted by wostraub on Wednesday, May 11 2005 I've been reading a fascinating account of the life of the late Italian physicist, Ettore Majorana. A colleague of the great Enrico Fermi, Majorana also worked on spinor theory and came up with a type of spinor very similar to Weyl's. Majorana was one of the first scientists to recognize the role of the neutron in nuclear physics, especially nuclear fission. At the age of 32, Majorana was appointed the Chair of Theoretical Physics at the University of Palermo in 1938. However, he either took French leave, committed suicide or was washed overboard on a boat trip prior to taking up residence at the school. Since he was privy to the inner secrets of nuclear fission, rumors abound to this day that he was abducted or killed by the Nazis. His body was never found. There have been unsubstantiated reports over the years of Majorana being sighted in Italy and in South America. If he did bail, he did a good job of covering his tracks. Fermi noted that Majorana was an exceptionally gifted physicist who was also exceedingly eccentric and severely lacking in common sense. Majorana was therefore just your typical scientist. There's a biography on the guy that I'm trying to locate. If I find anything interesting on him, I'll put it up.  Radioactive Decay Rates -- Posted by wostraub on Friday, May 6 2005 This will be my last word on the creationism vs evolution issue, as my sanity depends on it. A creationist friend of mine once asserted that radioactive carbon dating is subject to error, because the rate of C-14 creation in the upper atmosphere depends upon the rate of cosmic ray influx from the sun, which is not constant in time. She was RIGHT. The concentration of C-14 in Earth's atmosphere has varied with time, so the rate of uptake of C-14 by living organisms has not been constant. However, the variation has not been overly significant. That's one of the reasons why organic samples dated by C-14 methods are qualified with +/- figures. This means that a human bone dated by C-14 to be 33,500 years old plus or minus 1,500 years is really about that old. It does NOT mean that the bone can be post-Diluvian (younger than about 4,000 years). Live with it -- there are fossils that are undeniably human that predate Noah and his ark by many tens of thousands of years. Other creationists have used similar arguments regarding potassium-argon dating, saying that the rate of radioactive decay of unstable isotopic elements can vary with time. These arguments are WRONG. The radioactive half-life of an unstable elemental isotope is a fixed, unchanging constant. To argue otherwise is akin to saying that the probability of a fair coin coming up heads after an infinite number of trials is 25%, or that the value of the transcendental number PI depends on the day of the week. Live with it -- when an Archaeopteryx fossil is dated at 150 million years, it really is about that old. It cannot be 4,000 years old. Noah did not stash a pair of Archaeopteryx dino-birds on the ark. To deny this is to deny reality. If you deny reality, you are either insane or a Republican. The difference is not spacious. An excellent article on radiometric dating from a Christian physicist's perspective can be found at http://www.asa3.org/ASA/resources/Wiens.html  The Newtonian Moment, One More Time -- Posted by wostraub on Friday, May 6 2005 In an earlier post I described the New York Public Library's exhibit on Isaac Newton, "The Newtonian Moment." It must have been popular, because it followed me home. The exhibit is now on display at the Huntington Library in San Marino. Part I of the exhibit, "All Was Light," is on display until June 12; Part II, "The Making of Modern Culture," will open here on July 23. Admittance is$15, a little steep to see some six or seven displays of Newtoniana, but then there's the rest of the Huntington Library itself, which is a must if you're in town. I've been going there for 40 years, and I never tire of the place.

 Weyl's Spin Connection -- Posted by wostraub on Wednesday, May 4 2005 In his 1929 paper on quantum mechanical gauge invariance, Weyl derived the spin connection for Dirac spinors in curved manifolds (roughly akin to the affine connection that is used in general relativity). Because spinor transformations are limited to the SU(2) symmetry (that is, they are neither scalars nor vectors), Weyl's spin connection, to the best of my knowledge, is the only route we have to analyze the behavior of spinors in spaces warped by gravitational fields. In case you haven't noticed, I'm rather enamored of both Weyl's 1918 theory and the basic concept of particle spin. Particle spin is just so damned fascinating to me! Some time ago I read a book that examined the spin connection in what is known as a "Weyl space," which is the manifold that fell out of Weyl's 1918 effort. I noticed that the spin connection could be described in two ways, depending upon one's preferences for simplicity. Like Weyl, I keep looking for ways to get his $\phi_\mu$ field into things; being retired, it's a source of amusement to me. [Physics is amusing?! Maybe I've got Alzheimer's.] Anyway, on the menu to the left of this site I've got a very rough draft of a write-up on this subject. I'll add to it and fix it up as time permits.

 The Accidental Scientist? -- Posted by wostraub on Monday, May 2 2005 Tim Russert talked to New York Times columnist and author Thomas L. Friedman on Meet the Press yesterday. Friedman was there mainly to push his new book "The World is Flat," which sounds a wake-up call to an America that seems to be sleeping while the world undergoes radical globalization. I don't always agree with what Friedman has to say, but the views he promoted on the show are very close to my own. Friedman's flat-world scenario really refers to the fact that new technologies are allowing countries like India, China and Ireland to compete with the United States on a playing field that is getting leveled more and more to their advantage every day. He believes that America's leadership in science and engineering has become eroded due to an attitude of arrogant "entitlement." Meanwhile, other countries are producing far more PhDs in technical fields, and they're showing signs of leaving us in the dust. Up until perhaps 10 years ago, foreign students came here to study science and engineering, but the enormous rise in the quality of foreign universities (coupled with American travel restrictions imposed due to 9/11) is keeping them in their own countries, which then get the primary benefit of their efforts. As a result, Friedman is seeing an steady and alarming erosion in America's ability to compete with foreign markets in technological fields. To get America back on track, Friedman recommends that we develop a "Moon Shot" program similar to what JFK initiated in the early 1960s (I would prefer something more akin to another Manhattan Project, but perhaps it's all just symantics). And Friedman believes that this program should be focused on energy self-sufficiency -- kicking the oil addiction once and for all (because Peak Oil is almost certainly going to happen), and developing cost-effective alternative energy sources (including solar, wind, and nuclear). He even went so far as to recommend that the President impose a $4 per gallon price on gasoline, beginning in 18 months, with the generated revenues going to finance the program. Friedman even told Russert that if Americans want to drive Hum-Vees, they should go to Iraq, implying that gas-guzzlers like Hummers have no economic or moral place in the world. Right on, Tom! Friedman sees global warming as a bigger threat to the world than terrorism. I personally see the biggest threat to be waning energy supplies and the increased militarism of countries to secure whatever oil resources remain after Peak Oil kicks in. Many scientists believe that we will run out of fossil fuels before global warming becomes an acute threat to the planet. Regardless of who's right, Friedman's plan is a giant step in the right direction. Energy self-sufficiency would require an enormous investment in scientific research and development. America is probably the only country that has the financial capability of undertaking such a task, and it consequently offers an ideal opportunity for America to retake the lead in science and technology. Where I disagree with Friendman lies my belief that we should not trust in technology alone to fix our problems. If you've ever studied Lagrangians in math and physics, you already know that Nature has cooked things up so that conjugate variables like ENERGY-TIME are minimized. This means that humans should make every effort to conserve resources and minimize waste. Also, we should minimize the amount of energy we throw away into ENTROPY. This simply means that it is far better to not make a mess (like an oil spill or air pollution) than to make it and then clean it up. Speaking system-wise, a cleaned-up mess is even worse than one that is left alone. Of course, all this flies in the face of that great god, Capitalism, so our current attitudes will necessarily have to be adjusted -- and quickly. As Friedman implies, all of this will require a hell of a lot of scientific education. My belief is that when people are adequately educated in science and math, they see the way things really should be, and they make changes in their lifestyle. Friedman's thinking is an example of this; his own background is in Mediterranean studies, but he has gotten himself educated in science to the point where he sees the truth about things. Science and math will do that to you. Lastly, Friedman expressed his hope that President George W. Bush will read his book and show the one thing that Friedman claims has been missing in America to date -- true LEADERSHIP. Lots of luck, Tom.  Weyl Articles -- Posted by wostraub on Monday, April 25 2005 Norbert Straumann, Professor of Theoretical Physics at the University of Zurich, kindly sent me a German reprint of his paper "On the Origin of Weyl's Gauge Theories." I've been looking all over for a copy of this article, as it gives an especially clear overview of Weyl's 1929 paper on quantum-mechanical gauge invariance. [If I can get permission, I'll translate it and post it as a pdf document on this site.] The paper includes a reproduction of the postcard that Einstein sent to Weyl regarding Weyl's original 1918 gauge theory. After I've translated Einstein's comments, I'll post a photo of the postcard along with the translation on the menu to the left. Straumann (with Lochlain O'Raifeartaigh) has also written a very readable overview of early gauge theories, including the 5-dimensional Kaluza-Klein theory. The article is available online at http://xxx.lanl.gov/abs/hep-ph/9810524  When Einstein Lived in Pasadena -- Posted by wostraub on Monday, April 18 2005 KPAS (Cable 55 in Pasadena) has got a great 45-minute video titled "When Einstein Lived in Pasadena." It's shown frequently, but to date I've only seen about half of it. It includes a neat story about how Nobelist Robert Millikin (remember the Millikan oil drop experiment in school?), Caltech's president in the 1920s and 1930s, enticed Einstein and his wife Elsa to visit Pasadena. They made three visits (all in December, I believe) in 1931, 1932 and 1933. During Einstein's first visit, he lived at 707 S. Oakland Avenue in Pasadena. The house is still there, looking very much today as it did then. It's a fairly simple, unassuming house in a neighborhood of nice, well-maintained homes built in the 1920s and 1930s. I stopped by Caltech this morning, and thought I'd also swing by and take a picture of the Einstein house. You can download it from the menu on the left (the file is about 550KB, so I hope you have a decent Internet connection). Of course, Millikin's plan was to get Einstein to join the Caltech faculty, and he nearly succeeded. But Einstein was lured away by Princeton's Institute for Advanced Study. When Einstein left Germany for good in 1933, that's where he went. He died there on April 18, 1955, exactly 50 years ago today. I used to have a next-door neighbor, Seth Baker (who sadly passed away four years ago at the age of 92), who was a communications professor at USC. He had lived in Pasadena since the 1920s, and he saw Einstein during one of his visits (I think it was 1931). Einstein gave a speech at the opening of Pasadena Junior College's then-new telescope facility, and Seth snapped his photo (which unfortunately got lost over the years). Too bad. I haven't inquired about it, but you may be able to get a copy of the Einstein video from Caltech. The website (which has lots of other neat stuff) can be found at http://www.archives.caltech.edu  Darfur -- Posted by wostraub on Thursday, April 14 2005 I rarely agree with anything that neocon Max Boot (Council on Foreign Relations) writes, but in today's Los Angeles Times he has hit on something that should touch every American. That something is the genocidal situation in Sudan's Darfur region. Over the past two years, religious and ethnic differences in that country have resulted in over 300,000 deaths and more than 2 million displaced refugees, who literally have nowhere to run. But the worst of it is the torture and rape of hundreds of thousands of people, including children. The Sudanese government, which itself is complicit in the crimes of radical Islamic militias, is otherwise helpless to stop the crimes against humanity that are occurring every day. Boot bemoans the apparent lack of American will in mounting any serious humanitarian assistance (military or otherwise), noting that we are too bogged down in Afghanistan and Iraq to be of any real help. But that does not absolve us of our Christian responsibility to help out when a human crisis of this magnitude arises. There are numerous charitable groups who are sincerely trying to alleviate the suffering in Sudan. One of the best, in my opinion, is Doctors Without Borders (DWB), which is already providing a wide range of medical and humanitarian services to the region. However, like many other such groups, its resources have been overstretched by the Indonesian tsunami disaster. You don't have to mortgage your house to help out; even a few dollars can be put to good use -- whatever you can spare. This is a chance for self-loathing liberals and hypocritical right-wingers alike to do something right for a change! :) DWB can be reached at: Doctors Without Borders www.doctorswithoutborders.org 1-888-392-0392 toll free "Leave wringing of your hands: peace! Sit you down, And let me wring your heart" -- Hamlet  Fifty Years After Einstein and Weyl -- Posted by wostraub on Monday, April 11 2005 Believe it or not, I can just barely remember Mrs. Webster, my Kindergarten teacher at Northview Elementary School in Duarte, California, mentioning to her otherwise oblivious little charges that Albert Einstein had just died. I remember this only because she made a big deal about how smart the guy was and how important it was to do well in school. I also remember that I didn't know who the hell Albert Einstein was. Now, if it was Sheriff John or Bozo the Clown that had died, I would have have really taken notice. That was fifty years ago this week. It's odd that I can remember stuff that happened to me thirty, forty and even fifty years ago (or at least I THINK I remember), but I can't recall what I did last week to save my soul. Chalk it up to advanced middle age. Einstein died in Princeton on April 18, 1955. Weyl was to follow him in death in December of that year. I certainly DON'T remember Mrs. Douglas (my first-grade teacher at Northview) telling us that Hermann Weyl had passed away!  Weyl and Vierbeins -- Posted by wostraub on Saturday, April 9 2005 While reading Weyl's 1929 paper for what seems to be the umteenth time (there are still parts of it that I find puzzling), I began to wonder if it was Weyl who came up with the VIERBEIN (or TETRAD) concept. Because there are no finite-dimensional representations of spinors in gravity, the only way of tying a flat-space spinor field to the curved spaces of gravitation is through vierbeins. A vierbein is just a quantity having a Lorentz (flat space) index (usually denoted by a Latin letter like a,b,c...) and a general coordinate or tensor index, which is denoted by a Greek letter. Vierbeins are also used to express flat-space tensor quantities into curved-space forms. The vierbein is written simply as$e^a_\mu$, although either index can be up or down (or even juxtaposed with the other). We raise or lower Latin indices with the flat-space metric$\eta_{ab}$, while the curved-space metric tensor$g_{\mu\nu}(x)$is used to raise and lower Greek indices. We can therefore express the curved-space metric tensor using the vierbein formalism with the Lorentz metric:$g_{\mu\nu}(x) = e^a_\mu(x) e^b_\nu(x) \eta_{ab}$(1) Spinors can be viewed as flat-space fields that inhabit local tangent spaces. They transform in general-coordinate spaces like scalars, but in Lorentz space they transform using a certain unitary 2X2 matrix (see my write-up on Weyl spinors on the menu to the left). This matrix involves the Dirac gamma matrices$\gamma^\a$, which also live in a flat Lorentz space. To get a spinor representation in a curved manifold, we use the unitary transformation matrix as usual but with the gamma matrices expressed in curved-space form,$\gamma^\mu(x)$. This is where the vierbein comes into the picture:$\gamma^\mu(x) = e^\mu_a \gamma^a$Weyl used this vierbein approach in his 1929 paper. What puzzles me is whether Weyl was the first to do this. I know that a year earlier, Einstein (and maybe also Wigner) had used vierbeins, but in a completely different application. I do believe that Weyl was the first to derive the connection term for the covariant derivative of a Lorentz vector. Using this term, one can calculate the total covariant derivative of a mixed Lorentz-coordinate tensor quantity. However, the covariant derivative of the vierbein vanishes. In Weyl's 1918 theory, the covariant derivative of the metric tensor is not zero, and this would also require a non-vanishing derivative for the vierbein as well. I wonder if Weyl ever considered what happens to the vierbein when the metric tensor in (1) is rescaled via$g_{\mu\nu} -> \exp(\pi(x)) g_{\mu\nu}$. Do the vierbeins get rescaled, or does the flat metric$\eta_{\ab}$eat the scale factor? Had Weyl completely given up on his earlier theory by the time he wrote his 1929 paper? I don't think so, because gravity was still very much on Weyl's mind at the time. Indeed, the title of the paper (Elektron und Gravitation) would have likely been Elektron und Wellenmechanik if Weyl had completely sworn off his earlier effort. It seems a shame to me that God called Weyl home in 1955 at the relatively young age of 70, because so much neat physics was to arise in the 15 years that followed his passing. I've often wondered what role Weyl might have played in the development of this new physics, because his gauge principle lies at the root of so much of it.  "DEAD WRONG" -- Posted by wostraub on Thursday, March 31 2005 The cover letter of the now-released report from the President's Commission on the Intelligence Capabilities of the United States Regarding Weapons of Mass Destruction claims that the government was "DEAD WRONG" in its findings and that the intelligence information it dumped on the American people and the world was either "WORTHLESS" or "MISLEADING." Analysis of the 618-page report (and I will read every word of it) by the world's press has only just begun, but many are already saying that 1,533 American servicemen and women died in vain in Iraq, while more than 100,000 Iraqi civilians (innocent men, women and children) died as "collateral damage." As Defense Secretary Donald Rumsfeld has said, "Stuff happens" in war. I am outraged by this report, and every American should feel the same! PRESIDENT GEORGE W. BUSH IS A WAR CRIMINAL, AND SHOULD BE TRIED FOR HIGH CRIMES AND MISDEMEANORS AGAINST THE AMERICAN PEOPLE AND FOR CRIMES AGAINST HUMANITY. His cohorts in evil, CHENEY, RUMSFELD, RICE, POWELL, WOLFOWITZ, PERLE, FEITH and MYERS should also stand trial for treason. The accessories to these crimes, in my opinion, are the AMERICAN PEOPLE. You and I stood by and let a half-dozen monsters take over this country, suspend the rights of American citizens, authorize the seizure, imprisonment, torture and murder of innocent people, all for political and corporate power. In doing so, these people lied and lied and lied, and we let them. And we call ourselves Christians! God forgive us! In his recent book "The Sorrows of Empire," Chalmers Johnson reluctantly and sadly admits that the only options available to the American people to change the current administration are probably through radical means. If nothing changes, America will become a fascist empire that will enslave us all. The Commission's report pulls at the curtain hiding the President and his co-conspirators. What will it take to rip it away completely? Will we wake up then, or will we go back to sleep? THIS IS OUR COUNTRY, DAMN IT!! Post Script: Notice how closely the announcements of the death of Terri Schiavo and the release of the above report followed upon one another. Is it a coincidence? Today's TV news is devoted almost 100% to Schiavo's death, while the WMD report has been consistently absent from the airwaves. I see this as more evidence that our beloved leaders are relying upon the American public's addiction to triviality as a means of avoiding culpability in issues that really matter.  On Bioethics and Related Matters -- Posted by wostraub on Tuesday, March 22 2005 I apologize for the following rambling, unstructured diatribe, but I have other things to do so I'm making it quick. The Op-Ed section of today's Los Angeles Times has several interesting articles involving the interplay between science and religion. One is "Why Science Can't Show Us God" by Margaret Wertheim, the author of "Pythagoras' Trousers" (which I have not read), which won a book prize funded by the John Templeton Foundation (see my previous entry). Another article is by Jeremy Rifkin, the noted scientist-ethicist and author of "The Biotech Century" (which I have read). A third article is from Robert Scheer, the political gadfly and notable Bush critic, whom I happen to admire a lot in spite of the fact that (or perhaps because) he is consistently hated by rightwingers because he happens to have a mind and the courage to speak out against the mind-numbing hypocrisy of our times. These articles jumped out at me because not long ago I read "The Beginning of Wisdom: Reading Genesis" by Leon R. Kass, MD, PhD, who in 2001 was appointed by President George W. Bush to the Chair of the President's Council on Bioethics. This 700-page book is fascinating; it analyzes the first book of the Old Testament in almost excruciating detail, and thus provides much insight into the mind of God at the beginning of creation that I think many people (myself included) were never aware of. For example, the order that God chose to create the sun and moon, light, the Earth, and mankind reflects much profound subtlety, while God's oft-repeated pronouncement "...and He saw that it was good" was NOT made when he created Adam. Very interesting, deep, neat stuff. However, Kass elects to minimize the importance of several things that reveal the author's shortcomings, which I believe spring from his conservatism. For example, when Cain is sent away following Abel's murder, he goes to Nod where "he knew his wife." Where did the Land of Nod come from, and more importantly, where did his wife come from? These are not new questions, and have in fact been asked for many centuries, but Kass brushes them off with a single footnote citing incest as the possible answer. He also admonishes the reader to not think about these questions too much, but instead says "That he [Cain] had a wife (and descendants), not where she came from or who she was, is what we here need to know." In view of the meticulousness with which Kass has developed his arguments (after all, he has written 700 pages on the Book of Genesis ALONE), these kinds of brush-offs are most annoying. The Cain story and footnote appears on Page 144, and I was so annoyed at Kass' attitude at this point that I almost stopped reading the book. It's almost as if Kass, when faced with inconsistencies that he cannot explain within the context of his own belief system, is saying "Let's not dwell on this, because I don't want you to use your own mind. Just accept my dogma without question." While Kass does not address bioethics in his book at all, I feel that to dismiss the possible early biology of humanity as inconvenient IS unethical. God did not fix the time scale in Genesis (he did not create the Sun until Day Four, so millions of years could have been involved), and in my opinion it's very possible that Cain's wife was an advanced australopithicine or other early human biped that was capable of mating with Cain. But this view presupposes evolution, which to any red-stater is a no-no. Kass states that "None of these biblical teachings needs to be retracted because of the findings of evolution." If Kass can acknowledge evolution in this manner, why can't he simply acknowledge that evolution is just a tool God invented to ensure that his creation can adapt to changing environmental conditions? The denial of evolution is, to me, a sin against bioethics because it denies biological reality, and a sin against God because evolution is his creation. On a separate but related note, Scheer points out the hypocrisy of the Bush sycophants in their rabid determination to save the "life" of a brain-dead patient (Terri Schiavo) for purely political purposes, in spite of the fact that Governor George W. Bush himself championed a Texas law permitting spouses and significant others to OK the withholding of extraordinary life-saving measures in just these kinds of cases. This is most CERTAINLY a case of bioethics that has been perverted by politicians for their own selfish ends. The state courts have already weighed the Schiavo case to the nth degree, and declared that Terri should be allowed her appointment with God (and I have prayed that her soul is saved). Bush and his fellow maniacs would rather Terri spend another few decades in a brainless, lifeless limbo to further their own despicable causes. And the fact that Congress jumped in to vote on this SINGLE case tells me that our Constitution is in deep, deep trouble. Meanwhile, Rifkin chimes in to voice his opposition to genetic engineering. He reminds us that researchers are trying to create hybrid creatures ("chimeras") using spliced human and animal DNA so that they'll have more human-like laboratory animals to experiment with. And why would such research be tolerated? So that pharmaceutical companies can make billions by developing new drugs tested on creatures like "humanzees," a truly horrendous cross between humans and our closest genetic relative, the chimpanzee. It seems that America's bioethicism can be stretched without limit provided there's money to be made for the CEOs and the shareholders. Here, I'm 100% behind Rifkin and presumably even Mr. Bush. God created every living species to be separate and distinct, and by literally monkeying around with this set-up we risk disaster. Last but not least is Wertheim's otherwise excellent article, although I really don't agree with what she says. She states that "rational inference can never substitute for personal experience of the divine," and claims that God should not be equated with the structure and function of nature. I agree -- God is not nature. But this avoids the real issue, which is whether science and religion complement one another or must be kept separate. I believe Wertheim is a proponent of the latter. Any person can justify anything by simply claiming that "I received a message from God," which is absolutely not scientifically verifiable. I defy anyone to take away one's right to believe they have received divine instructions from God, but this simply cannot be proved. As it stands, people like this are locked up, unless they're the President of the United States. What Wertheim does not address is the fact that God gave us superb reasoning organs called brains. The capabilities of our minds far surpass the need to simply acknowledge God and the wisdom behind his creation. I believe God gave us the ability to think because he wanted to challenge us, to wonder about the physical world and to figure out how he did it. This included questioning where we came from, where we're going, and what our purpose is in life. It even included wondering whether or not God exists. Otherwise, he could given us minds like sheep -- then we wouldn't have sinned, and we'd all spend our eternal ovine afterlives peacefully munching grass. Wertheim seems to want to maintain the wall between religion and science. That's why biology texts in red-state schools are being rewritten to emphasize allegorical creation over evolution, and that's why when Bush tells us that 2 plus 2 is 5, we'll unquestionably accept it as divine revelation. On a purely personal note, I confess that I was not able to accept the existence of God until I studied science and math. Then God's existence became a rational certainty to me. My belief in Jesus Christ and personal salvation, on the other hand, is more faith-based, but even there I see a sound scientific reason based on the constraints imposed by God's gift of free will: Did God want man to have free will? Yes. Did he know that free will would cause us to know evil (because we have to know both good and evil to have a choice)? Yes. Therefore, if mankind was to be saved from himself, God had to provide a Savior. Without free will, there would be no need for Jesus, but then we'd all be incapable of intelligent thought. The Sciavo case is way overblown, of course. The war in Iraq and a host of other ills is far more important, and I suspect our government is just using Schiavo as a screen to keep the sheep from looking behind the curtains. The apostle Paul admitted that he would rather be dead and be with Jesus Christ than continue to live and be subjected to the world and its temptations. I cannot speak for Terri Schiavo's parents, who must truly love their daughter. But in this case I say let Terri be with God, and I believe any compassionate Christian bioethicist would agree with this.  More About Growth -- Posted by wostraub on Thursday, March 10 2005 I was a civil engineer for many years, and one thing I did was try to predict water demand using population growth rates. The idea was to extrapolate population levels using regional economic and socio-demographic data, then assign a per capita water demand (typically 150 gallons per day per person). Multiplying one by the other gave us the total water demand. The only trick is to get the future population right. Curve fitting is fun! Well, there are two ways to look as this type of planning. One assumes that the population growth will occur no matter what, so the water purveyors had better be ready for it. But the other way says that there might be a cause-and-effect issue involved -- by planning for growth, the growth occurs because we made it possible. No civil engineer ever said to the Mayor, "This is all the water we'll ever have available, and when that's allocated, the city must stop growing." Civil engineers are a lot like politicians -- they'd rather be employed than be out of work. I forget who said that humans will always take a resource and use it up as quickly as possible, regardless of how much was originally available. If this causes a problem, they will form blue ribbon committees consisting of experts to study the problem, then produce a lengthy report that no one will read or take action on. It's just human nature. Take the infrastructure issue, for example. The American Society of Civil Engineers (of which I've been a member for 29 years) recently published a report on the state of America's infrastructure (that's water systems, dams, treatment plants, roads, bridges, etc.), and I sure as hell hope people read this one. It gave the country a grade of "D" based on calculated estimates of infrastructure deterioration, and added that it will take$1.6 trillion over the next five years to fix it nationwide (the grade is slightly better than the "D-" ASCE gave it in 2001). Unfortunately, repairing a bridge is not as "sexy" as designing and constructing one -- after all, that's what we engineers went to school for! In my opinion, the infrastructure problem will persist because it is a reminder of the problems associated with growth, and people just don't want to face it. Well, eventually we will have to face these problems, and there's a tried and true method that the government has historically resorted to, and always will. It's this -- ignore the problem until you can't anymore (a disaster, etc.), then raise taxes on the middle class to pay for a partial fix. And believe me, with the Bush cartel in power today, the middle class is gonna get slammed pretty hard this time around. ASCE's a great organization that's trying to do the right thing, and you can help by at least getting educated. If you want to see the infrastructure report, go to their website at ASCE.

 The Peak Oil Issue -- Posted by wostraub on Thursday, March 10 2005 Since retiring four years ago, I've been following the Peak Oil issue with increasing interest and concern. It began in 1999 after I spoke with Dr. Albert Bartlett of the University of Colorado at Boulder, who is a vocal proponent of the issue (and, alas, a Cassandra because the issue's being ignored by the vast majority of humans). Since then I have read Dr. Kenneth Deffeye's book, "Hubbert's Peak," and I don't know how many other articles that have come out. Dr. Bartlett is fond of quoting his Third Law of Experts, which is "For every PhD there is an equal and opposite PhD." By this he means that if an expert comes out for or against something, there is another expert who will counter those claims. So I decided to look at the data myself; there are several web sites that have summarized the data. There is no question that oil production is following a Gaussian (or bell) curve. The oil gusher that began modestly enough in Titusville, Pennsylvania back in 1859 grew into millions of oil wells around the world, and the production of oil grew exponentially. In spite of a few blips (the North Sea oil discovery in the 1970s was a kind of unexpected gift), the world's production rate does indeed look Gaussian. The production curve for the United States fits a Gaussian curve almost perfectly (and it peaked in 1970, which is why we're importing most of our oil nowadays). Unless you believe that the world is flat and infinitely two-dimensional, the world's supply of oil is indeed finite and will be exhausted at some point in the future. But this is not the real problem; the problem is that when world oil production hits the peak of the Gaussian curve, production can only go in one direction -- down. When that happens, there will be no such thing as CHEAP OIL anymore, and then really bad things will start to happen. Imagine -- every now and then some politician suggests raising the gasoline tax a few pennies a gallon to pay for some pet project, and he/she gets crucified for it because opponents scream that it'll hurt people and businesses. After Peak Oil kicks in, and it will at some point, the base cost of gasoline might be $10/gallon. What will we do then? Who wrote the law that says gasoline can never go higher than$2/gallon? Another thing that Bartlett says is that one of the saddest facts about us humans is our unwillingness to understand the EXPONENTIAL FUNCTION. This is a mathematical function that describes the rate of growth or decay of something. Growth can be good or bad -- nobody wants a malignant tumor to grow, for example -- but so-called "good" growth can be bad, too. My observation is that no one really seems to know what "good" growth is. A small town with a population of, say, 10,000 people might be desperate for growth, as it would attract more goods and services as well as tax revenues to pay for the public works infrastructure needed to sustain the citizens. After a period of growth, the town's citizens might start to say "That's enough growth, things are just right now." But that is never the case; growth continues whether people want it or not. People are still flocking to Los Angeles, where growth has truly destroyed the lifestyle that once existed in the 1930s and 40s. Now it's just smog, hellish traffic congestion, and high taxes. This is not what folks wanted for LA, but it happened anyway. I think most people believe there's an "ideal" growth rate at which nothing bad ever happens. There are currently several ads running on TV featuring Erik Estrada hawking residential lots in Florida (and Arkansas, of all places). The ads say that growth is explosive, with the Florida community expected to increase its population by 33% by the year 2010, so come on down! Good Lord, who would want to live in a place that's growing like that?! I believe people tend to think of growth as they do their bank accounts; nobody wants an annual rate of return of only 2%, they want more. This is understandable -- everyone wants more money, more goods and services. But we humans do not know how to stop growth when it's not wanted anymore. Housing developers now say that "people have to have a place to live," and go on building. Or they talk about "sustainable growth," which is a lame excuse to keep on building because it sounds as if some smart person has a plan that will fix everything. The term is also an oxymoron. Growth is growth; something that grows at only 1% per year doubles itself in about 69 years. A community experiencing 10% population growth (a "nice" figure for bank accounts) will double in size in only 7 years. Do you want your city to grow that fast? How would you like your doctor to tell you that your cancerous tumor is growing only a few percent per year? Peak Oil, if true, will abruptly stop growth, and it will stop it all around the world. Unfortunately, it has the very real potential to rapidly create chaos and human suffering of the kind not seen since Noah's flood. Oil has been described as the nearest thing to a free lunch. Its energy density surpasses all other sources of cost-effective energy. We should have been using it to develop a truly sustainable energy source, like large-scale solar power or safe nuclear reactors, rather than burn it in 12-MPG automobiles. Forget fusion -- it will never happen in our lifetime, and if it does its development will require an enormous amount of energy just to get it built and distributed. To me, the only solution now is to conserve like we never have before. But conservation is anathema to growth, and as long as humans believe growth is good, we're doomed. That's one reason why I drive a hybrid car. It's a drop in the bucket, but I'm trying to do my share. Bartlett has a wonderful term for the point in time that Peak Oil proponents are talking about; it's called the "Dirac Delta Function in the Darkness." Try to imagine the history of mankind over the past 10,000 years or so. A space visitor looking down on Earth at night during that time would see mostly darkness, because until Titusville came along there were only scattered campfires and the like to serve us and keep us warm. Then, a brilliant flash of light for maybe 150 years or so, representing the Oil Age. After that, darkness again. The Dirac function, which represents a sudden and intense "spike" of activity on the axis representing time, is a very appropriate analogy. I urge you to Google "peak oil" and look at some of the many websites that address this issue. Politically, environmentally and economically, it might very well be the defining issue of our age -- and it's going to happen very quickly if it's true. A good site to check out is www.fromthewilderness.com, which should open your eyes. You don't have to believe everything you read, but at least get yourself educated so you can decide for yourself.

 Einstein at the Skirball Museum -- Posted by wostraub on Friday, February 18 2005 Today my older son and I visited the Einstein exhibit at the Skirball Museum in West LA. Lots of neat stuff -- Einstein's grade school report card; Einstein's handwritten reproduction of his 1905 special relativity theory; original letters to and from many famous scientists and statesmen; his brass refractor telescope; his magnetic compass; his smoking pipes and classical 78-rpm records; his erroneous light-deflection calculation from 1912; and the "Holy Geometry Book," an elementary text in German, given to Einstein as a child by his uncle, which Einstein cherished as the source of his interest in science and mathematics. Also on display are letters to extra-marital paramours, of which he had many, and the list of demands he handed to his estranged first wife, Mileva, instructing her to silently deliver hot meals to his room and to not expect any tokens of intimacy from him. This is Einstein, warts and all. To me, the best part of the exhibit focused on his later years, in which he became increasingly involved with nuclear weapons control and human rights. He was caught up only marginally in the McCarthy trials, but a paranoid American government nevertheless considered Einstein to be a socialist subversive. The FBI spied on him and compiled a 1,500-page file on his activities. Reading some of these reports shows what a bunch of dangerous morons Americans can be when they are frightened. That was 50 years ago and, alas, it is happening in this country again. Einstein lobbied strenuously for African Americans, whom he felt were being disenfranchised of their civil and human rights. He was a friend of the black actor Paul Robeson and the scholar-activist WEB Dubois. And when she was barred from staying at a Princeton hotel following a performance in 1937, Einstein put the great black soprano Marian Anderson up in his own house. Einstein defended Robeson and Dubois against McCarthy, and noted that the only place they were referred to as "niggers" was in their own country. The exhibit is a testament to a great scientist who hated militarism and anti-intellectualism. Einstein reluctantly urged FDR to move forward with the Manhattan Project, but was devastated when the weapon was used on Japanese civilians. Because of his pacifist and progressive beliefs, the Daughters of the American Revolution wanted Einstein kicked out of America! If he were alive today, I believe Einstein would be horrified and disgusted by the current mad-dog militarism and pro-stupidity movement that is prevalent in America now. May God save us! The Einstein exhibit ends May 29, 2005. Admittance is only $12, and it's well worth it, even if it has nothing on Hermann Weyl!  Beauty and Truth -- Posted by wostraub on Thursday, February 17 2005 I died for Beauty - but was scarce Adjusted in the Tomb When One who died for Truth, was lain In an adjoining room - He questioned softly "Why I failed"? "For Beauty," I replied - "And I - for Truth - Themself are One - We Brethren are," He said - And so, as Kinsmen, met a Night - We talked between the Rooms - Until the Moss had reached our lips - And covered up - our names - One of my favorite poets, Emily Dickinson wrote this poem in 1862, and I have long wondered about its exact meaning. The above version represents the way she actually wrote it, with those exasperating hyphens and a tendency to capitalize words that apparently didn't need any emphasis. Her poems are usually presented in a "cleaned up" format in modern anthologies of her works, and are sometimes even almost rewritten. The poem tells us that truth and beauty are the same thing, and that they are worth dying for. But it also implies that they can fail -- Dickinson depicts Truth and Beauty lying powerless in the grave. This bothers me a great deal, because to me truth and beauty, at least from the scientific and mathematical point of view, transcend the human experience. I doubt very much if Ms Dickinson thought about it from that perspective. The poem reminds me very much of the Old Testament, which describes our own righteousness as filthy rags fit only for burning. It is mainly because of this that I see Truth and Beauty in their noblest aspects as coming from God, who is perfect. In a very real way, righteousness is truth and beauty because it represents the way things really are according to God. It is an unfortunate fact that many of today's premier scientists are atheists or agnostics, and I have never been able to understand this. Perhaps they see themselves as the creators of truth and beauty, rather than the holders of minds that have been awakened by God. The book of Ecclesiastes notes that wisdom is meaningless unless God is acknowledged as its true author. I would go so far as to add that human wisdom is less than meaningless -- quantum mechanics has given us a glimpse of God's mind, but it has also been used by humans to build the most awful weapons imaginable. And maybe this is what Dickinson was trying to tell us.  Beauty and Symmetry -- Posted by wostraub on Tuesday, February 8 2005 Needing a respite from Zwiebach's string text, I read "Symmetry and the Beautiful Universe," a new book by Leon Lederman and Christopher Hill. It presents a very readable introduction to the various types of mathematical symmetries that give rise to the physical laws we all know and love. Although the book's authors mention the earth-shattering achievements of Christina Aguilera, they more appropriately focus on the work of a far more notable female, the mathematician Emmy Noether. Noether is the wunderfrau whose 1918 theory revealed the deep connection between mathematical symmetries (like gauge invariance) and conservation laws. The book offers a substantive narration of Noether's life and work and is worth reading only for this. The authors do not overlook the sad truth that the public is almost totally ignorant of Noether's achievements as both the greatest female mathematician who ever lived and the travails she bravely faced as a Jewish intellectual at the dawn of the German hate machine in 1933. Perhaps more importantly for a science book aimed at the general public, the authors move past the usual grammar-school description of symmetry (oh look, what a pretty snowflake!) and talk in relative depth about how physical and mathematical symmetries lie at the core of our understanding of matter and energy and their interactions. I was disappointed that the book treats Hermann Weyl primarily as a historical character who interacted with Noether, Einstein, Hilbert and others. However, the authors appropriately put local gauge invariance (Weyl's discovery) at the top of the symmetry list in terms of importance. There's a nice description of the Higgs mechanism which, if proved (and it will be, in my opinion), owes as much to Weyl as Peter Higgs himself. Unfortunately, the authors neglected to summarize the symmetries and conservation laws we're currently aware of. To the best of my knowledge, they consist only of these: translational and rotational symmetry (conservation of linear and angular momentum); time evolution and translation (conservation of energy); time-reversal symmetry (conjugation of charge); space reflection (conservation of parity); gauge symmetry (conservation of electromagnetic and other types of charge); and permutation symmetry (invariance of quantum statistics). There are several other symmetries (like Lorentz invariance) and conservation laws (like conservation of baryon and lepton number) that I am unable to place in specific categories like the others. Gauge invariance was the last continuous symmetry to be discovered, and that was in 1929. Are there any others? Time will tell. All of these symmetries and laws (either singly or in combination) are absolutely inviolate. When fully understood and appreciated, they constitute the best proof we have of the existence of a wise and benevolent God. When we learn math and physics, we learn something about how God's mind works. When we read the New Testament, we learn how to live from Jesus Christ. It disturbs me greatly to know that my country is now being run by a bunch of dangerous fools who understand neither science nor the philosophy of Christ.  Weyl and String Theory -- Posted by wostraub on Thursday, January 27 2005 I'm nearing the end of Zwiebach's book, and it has been rough going at times, but I'm beginning to see what all the fuss is about. The theory is really quite beautiful (at least the parts I understand), but some of the math is still very hard to swallow. It remains to be seen if string theory is anything more than just a pretty mathematical construct. I was heartened to find that Weyl's gauge symmetry idea has found a place in string theory as well. Like the rescaling of the metric tensor in ordinary 4-space, a Weyl transformation in string theory comes about by rescaling the Polyakov world-sheet metric$h_{\mu\nu}(\tau\rho)$with an arbitrary function of the surface parameters \tau and \rho. The Polyakov action is invariant under such a transformation, so there must be a conserved quantity associated with this symmetry (but I haven't read that far yet). What would Weyl and Noether thought of all this? I'm very grateful that Zwiebach put this book out. It's clearer than anything else I've seen, and I highly recommend it as a self-study text.  Strings Attached -- Posted by wostraub on Wednesday, January 19 2005 I bought Barton Zwiebach's book "A First Course in String Theory," and am about a third of the way through. If you're anything like me (relatively mathematically adept but a klutz nevertheless), you might want to invest in this book, as it's just about the most readable text of its kind. Almost all of the "Popular Science" kind of stuff that has been written on string theory is a total waste of time -- golly-gee stuff that is nothing more than handwaving and gushing about tiny vibrating strings in multiple dimensions. However, the only other alternative is to actually do the math, which can be excruciatingly difficult. Zwiebach's book was intentionally written for people in between, and as far as I can seen he has succeeded admirably. Could the world really be a ten- or eleven-dimensional place connecting multiple (or infinite) universes? Is this what Jesus meant when he said "In my father's house are many mansions ... I go to prepare a place for you"? The theory seems to be mathematically plausible, but there also seems to be no way of actually demonstrating anything experimentally. Our particle accelerators can now "see" down to about 10^(-18) meter, but this is a long way from the so-called Planck scale of 10^(-35) meter where those hidden extra dimensions may live. At this time, string theory represents the only real candidate we have for a unification of nature's fundamental forces. One of the brightest aspects of the theory is that gravity falls out of it naturally (indeed, it is actually necessary). The noted quantum physicist Michio Kaku has expressed his hope that one day string theory (or its modern variant, M-theory) will allow us to write down a single, inch-long equation that describes how the physical world works in its entirety. If this happens, I will see it not only as a fantastic intellectual achievement in its own right, but also as the ultimate proof of an Intelligent Designer.  Just Reminiscing -- Posted by wostraub on Thursday, January 6 2005 Not long ago, I visited the grave of Richard Feynman, Caltech physicist extraordinaire, who's planted in the Mountain View Cemetery right next door to me in Altadena, California (coincidentally, his grave's about a stone's throw from the house I was born in). My parents moved here from Missouri in 1944, no doubt to take advantage of the fantastic riveting and welding opportunities at Lockheed during the war years. The house is still there, though a tad worse for wear. I mention this because my father would have been 100 years old on January 11 of this year, and I guess it's starting to get to me (I just turned 56 myself, so my own threescore and ten years are about 80% gone). In October of last year, I visited the midwestern house my father was born in, the streets I had played in during annual boyhood visits, and other sights and stuff. Earlier, I mentioned my viewing of Newton's death mask in New York. So, maybe I'm just feeling a little more mortal than usual. On the return flight from New York, I stopped over in Missouri to do some genealogical wandering. My parents and relatives are all from the northeastern and southwestern parts of the state, and it took me some time to find all the cemeteries they're buried in. I found them all, but seeing their moss-covered headstones (some of them go back to the late 1700s) was yet another reminder of my mortality. To make matters worse, it rained constantly during my visit, and I was often ankle-deep in mud. It reminded me of the joke in the PBS Civil War series about Tullahoma, Tennessee -- "Tulla" is an Indian word meaning "mud," while "Homa" means "more mud." But there's good news, too. A side trip to a little German tavern/restaurant in Springfield, Illinois brought my attention to a brew called Straub's Beer, which was served with the meal. I don't drink beer as a rule, but this was an earned exception. Anyway, it turns out that Straub's operates a regional brewery in St. Mary's, Pennsylvania, and it was founded by a guy named Peter Straub from Baden-Wuerttemberg, Germany. Well, that's where my Dad's family is from! As I can trace my father's family there back to the 1400s, I have resolved that I will visit that area this year (though I will try to stay away from the cemeteries). Now, wouldn't it be neat if I found that Weyl had lived there at one time ... Newton's Face -- Posted by wostraub on Wednesday, January 5 2005 I stopped by the New York Public Library recently, which is exhibiting a small but neat collection of original publications and memoirs by Isaac Newton. "The Newtonian Moment" includes three copies of Newton's Principia and numerous scientific notebooks, some of which have handwritten comments and pre-publication editorial corrections by the man himself. One describes an experiment he conducted in which he thrust a metal probe along the side of his eyeball; he noted that it produced some interesting optical effects! The exhibit included what appeared to be one of the three original plaster death masks taken of Newton within hours of his passing in 1727. I could not get over how small and delicate the man's features were. He was elderly when he died, but he still looked remarkably like the paintings I've seen of him as a younger man. The mask faithfully captured facial artifacts like pockmarks, scars and other defects that he had accumulated during his 84 years on earth. Looking on Newton's face reminded me that no matter how great one's achievements may be, things always end up this way. Newton, a devout Christian whose prodigious religious writings far eclipsed his extensive scientific output, would likely be amused by all this! This excellent exhibit ends on February 5, 2005.  The Conformal Tensor and Weyl's Gauge Theory -- Posted by wostraub on Tuesday, December 19 2006 Some time ago I wrote about Weyl’s conformal tensor. It has some neat properties, but it usually crops up only in a gravitation/cosmology context, and hardly ever in differential geometry. But it was in that sense that the conformal tensor was used by Einstein to get around his primary objection to Weyl’s 1918 gauge theory, which was that the line element ds is not invariant with respect to a metric gauge transformation (also known as a conformal transformation of the metric). Recall that an infinitesimal local gauge transformation of the metric gμν → (1 + ε π) gμν regauges the lengths or magnitudes of vectors under physical transport, where π(x) is the gauge parameter. Consequently, the line element ds2 = gμν dxμdxν is also regauged in accordance with ds → ½ ε π ds. Einstein’s argument was that ds can represent time as well as distance, so time-independent processes such as the spacings of atomic spectral lines can be invariant only if the line element is gauge invariant. Since it is not, Einstein thought Weyl’s theory had to be wrong. But later, Einstein took up the problem once more. He felt that ds could be made gauge invariant if the line element were revised to ds2 = J(x) gμν dxμdxν, where J is a scalar function of the coordinates whose gauge variation goes like δ J = -ε π J (that is, J must be of gauge weight -1). This would cancel out the gauge change in the metric tensor and leave the line element invariant. Try as he could, Einstein could not come up with an appropriate scalar. Finally, he noticed that the Weyl conformal tensor Cαμνβ was exactly what he needed, for the combination √ Cαμνβ Cαμνβ is of gauge weight -1 in a Riemannian space. Unfortunately, the Weyl conformal tensor vanishes in the absence of a gravitational source, leaving a null line element (ds = 0) whose gauge invariance is now trivial. Furthermore, the counterpart of the foregoing expression in a Weyl space is unknown. What Einstein apparently overlooked is the scale factor from the Weyl theory itself, which considerably simplifies things. Consider the integral quantity k ∫ φμ dxμ, where k is a constant and φμ is the Weyl vector (which he identified as the electromagnetic four-potential). Under a metric gauge transformation, the Weyl vector varies in accordance with δ φμ = λ ε ∂μπ, where λ is another constant. Gauge-transforming the above integral puts the gradient ∂μπ under the integral, which is easily integrated. We can now set the Weyl scale factor to J via J = ek ∫ φμ dxμ, which, by appropriate selection of the constant k, will have gauge weight -1. This seems like a better approach than that provided by the conformal tensor, because in the absence of the electromagnetic potential φμ the exponential term is identically 1. Thus, the line element can be made gauge invariant only in a Weyl space containing a non-zero electromagnetic field! I haven’t found any evidence that Weyl resorted to this counterargument to Einstein’s objection, but by that time Weyl had moved on, anyway. In 1929, Weyl applied the gauge concept to quantum theory, which was a huge success. One has to assume that he never looked back.  Hermann Weyl and Dimensional Reduction -- Posted by wostraub on Monday, December 18 2006 In his neat little book The Dawning of Gauge Theory, Dublin physicist Lochlainn O’Raifeartaigh writes 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. Historically, the most famous example of this statement comes from Kaluza-Klein theory. In 1919, the German physicist Theodor Kaluza postulated the existence of a fifth dimension which was hidden from observation because it was too small to be seen. Kaluza thought that the electromagnetic four-potential of Maxwell’s electrodynamics resided in this dimension, but that its effects were observable only in the more familiar four-dimensional world we humans reside in. Kaluza assumed that the true metric tensor gμν(x) was five-dimensional. Viewed as a 5x5 symmetric matrix, it has a 4x4 subblock representing ordinary four-dimensional spacetime, while the g0μ "boundary" elements include the potential Aμ by way of the identifications g0μ = g55Aμ (μ = 0,1,2,3) and g55 is a constant. Thus, the four-potential Aμ lives in the fifth dimension. The potential is brought down into our world via dimensional reduction. Kaluza took as his action quantity the integral ∫ √ –g R d5x where the metric determinant g and the Ricci scalar R are the old familiar ones, but now in five-dimensional form. Using Kaluza’s above formulas for the g0μ quantities, this five-dimensional integral can be reduced to four-dimensional form, which is ∫ √ –g (R – FμνFμν ) d4x This, amazingly, is the familiar expression for the combined gravitational-electrodynamic action! (Physicist Ian Lawrie considers this result a minor miracle. It isn't, because God just made it that way!) I find it remarkable that Kaluza was able to deduce this way back in 1920, because the calculation (while straightforward) is not trivial. (Kaluza excitedly sent his paper to Einstein in 1919 to get a recommendation for publication. Einstein, though quite impressed, was nevertheless uncomfortable with a five-dimensional world, and so suppressed publication until 1921. Kaluza was not particularly happy about this!) The Swedish physicist Oskar Klein published a subsequent paper in 1926 that made numerous important improvements to Kaluza’s idea in the context of the then-emerging quantum theory. Hence the theory's present Kaluza-Klein moniker. Interestingly, in 1953 the great Austrian physicist Wolfgang Pauli took Kaluza-Klein theory one step further -- that is, one dimension further, to n = 6. This resulted in the very first non-abelian approach to non-gravitational (particle) physics. Several years later, using a similar approach, Yang and Mills developed the first consistent theory for the strong interaction. You might note that, in accordance with O’Raifeartaigh’s assertion, the coordinate-invariant form of Kaluza’s five-dimensional action results in a fully gauge-invariant term (√ -g FμνFμν) following dimensional reduction, while the original action is not gauge invariant at all. We got a gauge-invariant term by reducing the dimension by just one; imagine the possibilities if one started with, say, an eleven-dimensional action! This is the so-called M-theory of string physics, which promises great things (but has delivered nothing to date except beautiful mathematics). Note, however, that Kaluza-Klein theory, while interesting, eventually lapsed into obscurity because it did not predict any new observable phenomena – it was just a pretty theory. String theory is now finding itself in the same boat, and if the legions of brilliant physicists now grinding away (and maybe wasting their precious talents) at this theory cannot produce anything predictive from it (like explaining the magnitudes of the gravitational and electromagnetic coupling constants), it may also be forgotten. Did Hermann Weyl play around with dimensional reduction? Did he ever consider the possibilities of a higher-dimensional gauge theory? I’ve seen no evidence that he ever did. By dying in 1955, Weyl missed Yang-Mills and a lot of other neat stuff he would have undoubtedly contributed to. Weyl was taken from us too soon.  Louise Brooks: Lulu Forever -- Posted by wostraub on Saturday, December 16 2006 Peter Cowie's new book Louise Brooks:Lulu Forever is out, and at long last. Finally we have a large-format book with hundreds of rarely-seen photos, motion-picture production stills and first-person accounts of 1920s actress-flapper Louise Brooks, who would have turned 100 years old last month (she passed away in 1985). I probably would not care so much for this actress if it were not for the fact that I first saw her signature film Pandora's Box (filmed in Germany as Die Büchse der Pandora) as an impressionable young college student in 1970. At the same time, I was taking an elective course in literature (very odd for a chemistry major), where I was also reading Vladimir Nabokov's irreducible masterpiece Lolita for the first time. It was in Chapter 6 of the novel that I encountered Monique, Professor Humbert's French girl-whore, the predecessor of one Ms. Dolores Haze. To me, Louise and Monique were one and the same at the time, and I have forgotten neither in all these years. Of course, as a Christian I have mixed feelings about all this now, but literature is literature, and life itself isn't squeaky clean. Humbert, Monique and Lulu all paid dearly for their shortcomings (as did Louise Brooks), and so I will let it go at that. Cowie's book can be purchased from Amazon for about$35. If you're interested, you might also consider buying Lolita* which, in my humble opinion, is the third greatest book ever written (right behind Hamlet and the New Testament). Exceedingly well-written, hilarious, disturbing and heart-breaking at the same time, it's all the more amazing that it was written by a Russian who picked up the English language later in life (much like Joseph Conrad, who also ranks right up there). * Five points to the person who figures out the identity of John Ray, Jr. PhD, credited as co-author of the book

 Bombs Bursting in Air -- Posted by wostraub on Tuesday, December 12 2006 Several weeks ago, I was driving through San Raphael near San Francisco and happened to stop by Autodesk, the company founded by AutoCAD's creator, John Walker. Coincidentally, Walker's name popped up on an Internet search with that of John von Neumann, the great mathematical physicist and close friend/colleague of Hermann Weyl (see my December 9 post). Von Neumann worked on the Manhattan Project, where (among many other things) he discovered that an atomic bomb would be much more destructive if detonated high above the target area (something involving shock wave pressures, which I know nothing about). It turns out that John Walker is also interested in such things, if only academically (unlike me, he is extremely wealthy and has even more time on his hands). He has a website that explains the effects of nuclear weapons on human populations, something we should all get familiar with as long as President Bush is running the world. Anyway, Walker's site includes print-out materials and instructions for making a nuclear effects calculator. It's basically a circular slide rule that will allow you to ponder (in a very quantitative way) the death and destruction that a nuclear device can have on your least favorite city (Crawford, Texas, for example). Well, I made one, and it's very neat. It's one way to personally experience the practical aspects of the complicated science that folks like von Neumann, Oppenheimer and Teller turned into godless, immoral sin. [Note: Optimum burst height = maximum resultant death and destruction] Walker warns that his calculator won't be of much use in a post-nuclear war world. But that may not be that far off -- I'll bet you anything that one of the alternatives Mr. Bush is considering for the New Way Forward© in Iraq is to nuke Iran, in which case all bets are off.

 Weyl and von Neumann -- Posted by wostraub on Saturday, December 9 2006 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."

 Louise Brooks at 100 -- Posted by wostraub on Sunday, December 3 2006 [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) [More pics of the exhibit can be found here]

 Units -- Posted by wostraub on Thursday, November 16 2006 One of the more appealing aspects of Hermann Weyl's metrical gauge theory deals with the concept of "units." Humans measure length in terms of meters and feet and, in ancient times, cubits -- different, but all interchangeable, and therefore the same thing. But in the presence of a strong gravitational field (or when dealing with velocities approaching the speed of light), the lengths of physical objects can become ambiguous -- the length of a physical measuring rod, for example, can depend on the observer. In Weyl's original gauge theory, length can be continuously redefined as one goes from one point in spacetime to another. The basic units of length, time, mass etc. used to be based upon physical objects or anthropological effects (all called "artifacts") that were explicitly defined to represent the units they stood for. For example, the meter used to be defined as 1/10,000,000 of the distance from the equator to the North Pole (via Paris). Similarly, the second was once defined as 1/86,400th of a day. In these examples, the physical earth was a measurement artifact. All of these artifact-based units (except the unit for mass) have since been replaced by non-anthropological representations. For example, the meter is now defined by a specified number of wavelengths of the emission spectrum of a certain cesium isotope. Similarly, the speed of light in vacuo is now fixed at exactly 299,729,458 meters per second. The second itself has a specific definition based on isotopic transitions. But to date the kilogram has resisted all such conversions. Officially, it is still defined as the mass of this platinum-iridium alloy cylinder having equal dimensions of length and diameter (39 mm) maintained near Paris: But this object is not entirely stable. It has been observed to change on the order of 50 parts per billion per year (Corrosion? Sublimation? Old age?). Now scientists are attempting to revise the definition of a kilogram to a non-artifact basis. But it has been a difficult road. The December issue of Scientific American describes the most recent attempt. It is based on a nearly perfect, ultra-pure sphere of crystalline silicon-28 having a number of atoms that is very nearly that of Avogadro's number (roughly 6 x 1023), which is defined itself as the number of anything in one mole of a pure elemental substance. But to my mind, this just replaces one artifact with another. Furthermore, Avogadro's number is another "unit" having an anthropological basis. Is there no way to define the unit of mass that is free of some kind of human subjectivity? Quantum physicists long ago realized that their equations could be greatly simplified by setting Planck's constant and the speed of light to unity. But this is really nothing more than a convenience, as these simplifications only establish yet another set of units that is no better than any other now in use. My suggestion? Define the kilogram as the mass of one atom of hydrogen and be done with it.

 Weyl and Einstein, Again -- Posted by wostraub on Friday, November 10 2006 Taken aback by Hermann Weyl's insistence that his gauge theory was valid despite the physical evidence, Einstein wrote to his friend on 1 May 1918 with this remarkable correspondence: 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. You may not always agree with Einstein, but he just nails it here.

 Stupid Notation -- Posted by wostraub on Friday, November 10 2006 From September 1918 until late November of that year, Hermann Weyl and Einstein corresponded repeatedly, with the main topic being Weyl's geometrical gauge theory. Einstein loved the basic idea, but was upset over the fact that the line element in the theory was not gauge invariant. This unsettling little fact ultimately doomed Weyl's idea. But the two also bickered over Weyl's expression for the equation of the geodesics, which is obtained by extremalizing the related integral expression . Weyl's result was where Einstein vehemently stated that this was wrong. Weyl disagreed, and for three months the issue came up again and again. The two men never resolved it, and Weyl persisted in using his expression in all five editions of his book Space-Time-matter. So who was correct? Well, Einstein was right after all, but the whole thing was trivial, and the two great scientists should have known better. The correct expression is What's the difference? It's in the partial derivative term for the metric tensor: Weyl used a covariant term for x in the denominator when he should have used the contravariant term. It's no big deal, but it serves to show how important it is to maintain consistency in your tensor notation. Few areas of mathematics have displayed such a wide and bewildering range of notation as has tensor calculus in its 150-year history. In the years immediately following Einstein's general relativity theory, it seems that everyone was using a different notation (even Einstein). Contravariant and covariant indices were constantly being intermixed, and that is really what lies at the bottom of this little Weyl-Einstein disagreement.

 Pandora's Box on IFC -- Posted by wostraub on Tuesday, October 17 2006 Shot in Berlin in the waning years of Germany's Weimar Republic, the late silent film classic, "Pandora's Box" (Die Büchse der Pandora) is airing on the Independent Film Channel (IFC) at midnight tonight and tomorrow at 7:45 am PST (October 18). The film is shown uncut and uninterrupted, with both its original German and English subtitles. German filmmaker Georg Wilhelm Pabst's 1929 classic stars the hauntingly beautiful American actress Louise Brooks (1906-85) as the libertine but curiously innocent dancer/vamp Lulu. Now widely regarded as a cinematic masterpiece, the film received surprisingly scathing reviews because of its (then) shocking sexuality (but there's no nudity, parents). Sickened by the excess and amorality of Hollywood (though hardly an ingenue herself), and stuck in a series of profitable but brainless "flapper" roles, Louise Brooks left to further her career in Germany, where she starred as Lulu in "Pandora" and Thymiane in "Diary of a Lost Girl" (Das Tagebuch einer Verlorenen, also 1929). Following another starring role in the 1930 French film Prix de Beauté ("Beauty Prize"), Brooks returned to the states. She grudgingly made several more films in the 1930s, but she was essentially blacklisted by the film industry because she refused to play by its rules. She left Hollywood for good in 1939 and went to New York, where she lived a rather impoverished, hand-to-mouth existence as best she could until her death in 1985. A victim of childhood sexual abuse and gross parental neglect, Brooks ironically and tragically became a hedonistic abandonee herself, and by the mid 1950s was, in her own words, "a questionable East Side dame." But about that time she started writing about her life and the many stars she had known personally (often very personally) during her acting days. While her work was not prolific, her writing demonstrates a remarkable talent for intelligent exposition. Her 1982 book, Lulu in Hollywood, reflects a truly brilliant mind. A chronic drinker and smoker, Brooks succumbed to emphysema on August 8, 1985 after a long struggle with the disease. Brooks led an absolutely amazing life, which is chronicled in Barry Paris' excellent 1989 book, Louise Brooks: A Biography. Silent film fans around the world will celebrate Louise's 100th birthday next month (on November 14), at which time Criterion Collection Films will release a digitally remastered DVD of Pandora with many extras, including a rare filmed interview of the actress from 1979.

 Melvin Schwartz Dead at 73 -- Posted by wostraub on Wednesday, August 30 2006 1988 Nobel Prize winner Melvin Schwartz has died at 73. He shared the prize with Leon Lederman (The God Particle) and Jack Steinberger for their work on the weak interaction and their discovery that neutrinos come in different flavors. But what appealed most to me about Schwartz was his approach to electromagnetism. Like many other befuddled graduate students in the 1970s, I had the great misfortune of being forced to learn electrodynamics from J. Jackson's Classical Electrodynamics, arguably the most difficult text on the subject (the third edition was presumably "dumbed down" in the 1990s in belated response). It's a shame that Schwartz' Principles of Electrodynamics, first published in 1972, didn't achieve the same (inexplicable) popularity as Jackson's book, because Schwartz' approach is much clearer. It's even entertaining -- he starts it off with the statement Electrodynamic theory is beautiful! What a wonderful way to start a book! Schwartz was one of the few physicists who insisted that electric and magnetic fields, which are essentially the same thing, share the same units. This in itself represents a tremendous simplication of the subject, as the "units problem" in electrodynamics has caused no end of troubles for students. He similarly simplifies the understanding and calculation of the Lienard-Wiechert potentials, another chronic stumbling block for mediocre students like myself. The subject is laid bare in a wonderful chapter entitled Let There Be Light!, in which the author unashamedly shares his enthusiam for and appreciation of God's scientific and mathematical wisdom. Indeed, Schwartz' writing style is peppered with statements like At this point when the laws were being written, God had to make a decision ... God naturally chose the antisymmetric tensor as His medium of expression (Chapter 3). I love it! Fortunately, Schwartz' book is available as a Dover reprint and can be had for about $10, so you have no excuse for not buying it. No physics library should be without it.  Dark Matter Discovered? -- Posted by wostraub on Tuesday, August 22 2006 On August 15, a group of astrophysicists announced they had seen indirect evidence for the existence of dark matter. What does this mean? For decades, astronomers have noticed that the rate of rotation of galaxies does not jive with the amount of matter contained in them – that is, there is not enough gravity contained in the galaxies to keep them from flying apart. Astronomers therefore believe that there must be a form of matter unlike normal matter that keeps the galaxies together. This matter have been given the name dark matter. It is optically invisible because it does not interact with ordinary matter. Scientists have no idea what dark matter is composed of. Since it cannot be made of ordinary stuff like protons and electrons, other, more exotic forms of matter have been proposed (axions, anyons, etc.). But to date, all this conjecturing has been purely theoretical. Now a team of scientists (including members from the Stanford Linear Accelerator Center and the University of Arizona) have announced indirect evidence of dark matter in the Bullet Cluster, two groups of over one thousand small galaxies that collided about 100 million years ago in the constellation Carina, forming the shock wave shown in the above photo (which is a composite of visible and x-ray photographs). As the galaxies collided, the ordinary matter slowed down as one would expect in any physical collision. However, the dark matter component, which is immune to any kind of physical (mainly electromagnetic) interaction, kept right on going. The scientists were able to deduce this by measuring the amount of gravitational lensing caused by the dark matter on more distant galaxies seen in the photo's background (dark matter may not interact directly with ordinary matter, but it can still affect it gravitationally). Thus, a cosmic collision event can serve as a means of "filtering out" dark matter from its ordinary counterpart. Indeed, there is speculation that past events have generated "dark matter galaxies," whose presence can only be deduced by gravitational lensing effects. [Interesting Question: Is intelligent dark matter "life" possible, or does it require the usual quarks and leptons? Maybe God is not quarkic/leptonic at all!] Who cares, you might be tempted to say. But astrophysicists have estimated that dark matter makes up about 25% of the total matter in the universe, whereas ordinary matter accounts for only about 5%. The remaining 70% is thought to consist of dark energy, a hypothetical energy field (called quintessence by some scientists) that permeates the entire universe. Thus, the visible universe you and I know and love accounts for only 5% of physical reality. This concept is truly mind-boggling. Hermann Weyl and others postulated that what we today call dark energy is nothing more than a mathematical artifact of Einstein’s general theory of relativity called the cosmological constant (I tend to agree, as "quintessence" sounds a tad like the old "ether" idea of the early 1900s). The cosmological constant is simply a term in Einstein’s gravitational field equations which, depending on its sign, can either act with or against the usual attractive force of gravity. Many scientists believe that a non-zero, repulsive cosmological constant exists and is responsible for the observed large-scale repulsion effect that is forcing the universe to expand at ever-greater velocities. If true, the universe will eventually expand at the speed of light, resulting in a rather bleak future for all existence. The relationship between dark matter and dark energy has not been established. If the human race can keep from blowing itself up over petty tribal conflicts (which I find highly doubtful), we may have a chance at someday understanding the fantastic universe that God has made for us.  String Theory Unraveling? -- Posted by wostraub on Tuesday, August 15 2006 The great Austrian physicist Wolfgang Pauli once remarked "What God hath put asunder, let no man join." He was referring to the seemingly-intractable problem of unifying quantum theory with general relativity, two theories that work just fine by themselves but which have, since Hermann Weyl's time, resisted all attempts at unification. It's the one great open problem of physics, and every physicist worth his salt worries about it. Why bother with unification when the theories work fine on their own? Because only through unification can we simplify our world and begin to grasp the mind of God. Recall James Clerk Maxwell, who in the 1860s discovered that the electric field and magnetic field are the exact same thing. Instead of a hodgepodge of unconnected, complicated vector equations, we have the four Maxwell equations expressing the unified electromagnetic field (which, I might add, are among the most beautiful mathematical expressions mankind has ever gotten its hands on). The unification problem can be traced to the fact that the general theory of relativity is not renormalizable (which means that infinite probabilities invariably arise during the perturbation step), so efforts to describe gravitation using a quantum-perturbative approach fail miserably. Thirty-odd years ago, string theory promised a way around this problem. The theory's early developers noted that it demands the existence of a massive, spin-two particle, and it was assumed that the as-yet undetected graviton feld would fit the bill. Initially, it looked good on paper. But the most advanced version of string theory requires that spacetime consist of 11 dimensions (1 time dimension and 10 spacial dimensions) and not the usual 4. In order to demonstrate the existence of all these extra dimensions, physicists would need access to energies that are many magnitudes beyond those currently produced in the most powerful particle accelerators. Indeed, these energies rival that of the Big Bang itself, making it almost a certainty that string theory can never be tested. The theory's critics insist that any untestable theory is unscientific, and therefore has no place in science. Some have even gone so far as to say that it is akin to religious faith. Anti-evolutionists (and Republicans) might be crazy about this, but not physicists. My personal problem with string theory is far simpler -- I just can't follow the mathematics. Back when Einstein first announced his general theory of relativity in November 1915, it was said that only a dozen people in the world could understand it. That was simply not true -- relativity is pretty straightforward, and while the math at the time was unfamiliar, it wasn't difficult at all. Same thing with Heisenberg's matrix mechanics in 1925 -- physicists just weren't all that familiar with matrices, which even today's middle school kids can understand. String theory, by comparison, is nothing less than a convoluted maze of unbelievably complicated mathematics that seems beautiful only to the relatively few people who can work with it. And in their own words, even they don't really understand what they're doing! So now we get the August 21 issue of Time magazine, which has an article entitled The Unraveling of String Theory. It reports that two new books by respectable physicists (Lee Smolin and Peter Woit) are heralding a renewed criticism of string theory that might just catch on. The criticism advances the now decades-long suspicion that string theory, which provides absolutely no testable predictions, may be nothing but mathematics after all. If this can somehow be demonstrated, it would serve to free up the minds of some pretty smart people (like Ed Witten at the Institute for Advanced Study at Princeton) who currently are obsessively pursuing M-theory, which is the 11-dimensional version of strings I mentioned earlier. In my mind, it's entirely possible that God considered basing physical reality on string theory, but gave it up because it gave even him headaches -- and a theory with headaches lacks beauty, and God's way of thinking always involves beauty. But if not strings, then what? Is there no way to unify gravity with quantum mechanics? Was Pauli's admonition correct after all? If string theory bombs, then we're back to where Weyl, Einstein, Pauli and many others were 80 and 90 years ago. To be sure, we know a lot more than those folks did, but one thing remains the same -- our intellectual curiosity is simply not matched by our intellectual ability.  Straumann Again -- Posted by wostraub on Thursday, August 3 2006 Here's a new article from Norbert Straumann (University of Zürich), which was the basis of a talk he gave in 2005. Some new stuff on Hermann Weyl and early gauge theory, along with some reflections on the gauge principle in quantum electrodynamics. Article  Persistence -- Posted by wostraub on Monday, July 31 2006 Edison once said that discovery is 1% inspiration and 99% perspiration. Einstein asserted that persistence trumps intelligence. Weyl's efforts to bail out his 1918 metrical gauge theory certainly represents a classic example of persistence in the face of withering criticism. Weyl persisted because he believed he was in possession of the truth. Recall that Mr Einstein rejected Hermann Weyl’s original gauge theory on the basis that it did not preserve the invariance of the line element ds under a gauge transformation. In spite of the simplicity of Einstein’s argument, Weyl tied himself in knots desperately looking for a way out. As far as I know, he tried three escape routes. One was to assume that the ds of measurement was not the same as the mathematical ds. That is, what we measure as ds is a true invariant, whereas the mathematical version is not. This almost metaphysical option was quickly dismissed by Einstein, Pauli, Eddington and others. Weyl then moved on to a line element that replaced the metric tensor gμν with the Ricci tensor Rμν, a quantity that is a true gauge invariant in Weyl’s geometry. This was an interesting dodge, but it too was thrown out. Weyl’s last gasp was to make ds invariant by multiplying the metric tensor with a scalar J(x) of gauge weight –1, so that the line element now goes like ds2 = J gμν dxμdxν. After considerable thought, Weyl realized that the only reasonable J-quantity had to be the square root of Cμναβ Cμναβ, where the C-quantity is the Weyl conformal tensor (see my pdf article on this tensor on the menu to the left). This time Einstein was impressed, though to this day no one knows if Weyl’s J-quantity has any relevance in physics. It is straightforward, if rather tedious, to calculate the equations of the geodesics associated with Weyl’s J-invariant. I did the calculation many years ago, and found that they’re completely nonsensical. I’m sure Weyl did the same calculation, and maybe that’s when he finally tossed in the towel.  Riemannian Vectors in a Weyl Space -- Posted by wostraub on Sunday, July 16 2006 I've posted the final write-up on Riemannian Vectors in a Weyl Space, which tries to address a mathematical inconsistency in the original Weyl theory (and which has nothing to do with the conformal aspects of the theory). Fixing the inconsistency leads to simple derivations of the Klein-Gordon and Dirac equations. I've also included lots of other junk as food for thought. In this paper I've tried to include all the reasons why I think Weyl was really close to a unified theory of the combined gravitational-electrodynamic field, but please believe me when I say I have no illusions that this will ever be rigorously demonstrated -- certainly not by my overly-simplistic treatment. Feel free to criticize. Riemannian Vectors in a Weyl Space  Houston, We Have a Problem -- Posted by wostraub on Saturday, July 15 2006 The disturbing events in the Middle East and the recent hoopla over the space shuttle mission made me think about that old science fiction movie in which astronauts take off for the moon or Mars or someplace only to learn that a world war has destroyed earth's population (along with everything else) and they have nothing to return to. I'm also reminded of the scene in Planet of the Apes (the 1960s version, that is) in which one of the returning astronauts plants a little American flag in the lifeless soil, while Charleton Heston laughs hysterically. Now that the shuttle missions have been reduced to meaningless public-relations trips designed to see whether the ship's insulation is still intact, I am again forced to unveil the true stupidity behind America's "space travel" experience. Here it is: 1. The shuttle orbits at an altitude of about 210 miles. At that height, the force of gravity is still more than 90% of what it is here on earth's surface. It's just like an astronautic flea who "soars" above the surface of an onion by jumping onto the nearest outer skin layer. The astronauts are not in "outer space." 2. The weightlessness of the astronauts is induced solely by the orbital centripetal force. If they increased the shuttle's speed by only 2.5%, they could orbit the earth at a height of one foot. 3. Shuttle missions are incredibly boring. There's little for the astronauts to do other than maintain their life-support equipment. All meaningful on-board experiments have been done to death, including observing how crystals grow and ants propagate in microgravity. In their free time, the astronauts can look out and see the curvature of the earth. That's about it. 4. Without the protection of earth's atmosphere, dangerous levels of cosmic rays constantly permeate the shuttle and its inhabitants. Do you recall the "flashes of light" reported by Armstrong, Aldrin and Collins during their interplanetary trip on Apollo 11? Those flashes were caused by "Z particles" (cosmic ions with atomic weight around that of carbon) piercing the astronauts' corneas. Meanwhile, microgravity induces rapid loss of bone density and muscle tone. 5. There are literally thousands of pieces of space junk now orbiting the earth, from grain-size ejecta particles to car-sized failed satellites. And it's all flying around at many thousands of miles per hour -- many times faster than a speeding bullet. NASA mission planners have to keep track of every known chunk to avoid a catastrophic collision with the shuttle. One of these days they'll lose track of something, with disastrous results. All it takes is a particle the size of a grain of salt. 6. In spite of all the high-tech you see in the orbiter, its boosters and the tracking equipment, the shuttle is really nothing more than a fancy Chinese rocket utilizing 2,000-year-old chemical technology. Chlorine- and nitrogen-containing pollutants spewed out from each launch measurably impact the earth's biosphere (chlorine is about the worst thing you can imagine for the ozone layer). 7. President Bush says we're going to Mars! The round trip will take years, cost more than$100 billion, and if cosmic rays don't kill the astronauts the boredom will drive them insane. But it's a trip Americans may have to buy into whether they like it or not, as it's rumored that Mars has weapons of mass destruction. We can't just stand here and wait to be killed! They don't call it the Red Planet for nothing! The real reason why Americans support the "space" program is that they don't know a solar system from a galaxy, a mile from a megaparsec. I constantly hear people say things like "Our brave astronauts are out there among the stars and galaxies, blazing the trail to discovery." No, they're 200 miles above your head, idiot. And if I ever hear that poem again about "touching the face of God" (in a nuclear-armed fighter jet, yet), I'll scream. And "spin-offs"? Please, did we have to spend a trillion bucks for Tang and freeze-dried ice cream? All this nonsense, and for what? A trifling trillion dollars or so to date, and counting! The real purpose of these missions today? That's a national secret, but I can tell you that it involves looking down on you and me and everyone else, and not looking out on this wonderful universe God made for us. My suggestion? Let's first get our planet in order, make our resources and human, animal and plant life sustainable, find a way to deal with our aggressions, then reach for the heavens. Wouldn't this please God more than what we're doing now?

 Weyl and Dirac -- Posted by wostraub on Monday, June 26 2006 Someone asked me for a copy of a 1973 paper by P.A.M. Dirac today. I got it out of the garage to email, and read it again for the first time in years. In it, Dirac uses Weyl's gauge theory in an attempt to explain why the gravitational constant G should be decreasing with time. In the paper, Dirac reveals a fondness (if that's the right word) for Weyl that I had missed earlier. He even provides a counter-argument for Einstein's famous objection against Weyl's theory. But then he goes on to describe how a non-moving charged particle in a Weyl field can be used to break charge symmetry while maintaining CPT symmetry. Dirac's argument is simple: a vector associated with a particle in a Weyl field changes magnitude according to dL = φμ L dxμ. If the particle is at rest, vector length still changes with the flow of time according to dL = φ0 L dx0, where φ0 is the Coulomb potential and dx0 = cdt. If the change in length is positive with increasing time, then it must shrink with decreasing time (and vice versa). Regardless of the convention you choose, the change in length is effected by the sign of the particle's charge. Thus, symmetry is broken between positive and negative charge. It's so beautiful. Dirac, who won the 1933 Nobel Prize in Physics at the age of 31, was once asked if there was anyone who was so smart even he couldn't understand. "Weyl" was Dirac's answer.

 Hermann Weyl Resources Online -- Posted by wostraub on Sunday, June 25 2006 Among the papers, books and articles I have collected on Hermann Weyl are a number of contributions made by the German mathematical historian Erhard Scholz of the University of Wupperthal. Scholz has written extensively about Weyl's mathematics (from a primarily historical perspective), although I find his English difficult to follow for some reason. Nevertheless, his online materials are well worth acquiring. Just Google "erhard scholz, weyl" and you'll finds lots of stuff, mostly in pdf format. You might also want to Google "john l. bell, weyl" (presumably no relation to the John S. Bell of Bell's Theorem fame) regarding several online papers he's written on Weyl and his philosophical leanings. I really don't "get" philosophy, but it's worth checking out. Another resource that I have not yet acquired is "Hermann Weyl -- Mathematics and Physics, 1900-1927," a 1991 Harvard University PhD dissertation by Skuli Sigurdsson. I haven't found it on any of the online dissertation libraries, so it's probably not out there. I suppose I'll have to get it directly from Harvard for much more than I care to pay. I'll let you know if I find it. [God bless the Pasadena, California Library! It acquired a set of Einstein's collected writings (German and English translations) after a loan request I made several months ago. The collection includes many references to Weyl and his gauge theory and is just plain fun to read.]

 Albert, Mileva and the Noble Engineering Profession -- Posted by wostraub on Thursday, June 22 2006 I've been reading the letters that Einstein and his first wife, Mileva, wrote to each other in the period 1914-19. This was a period of increasing estrangement between the two of them following their split-up around 1913, and the correspondence ranges from cordial to angry. The letters take on a decidedly monetary tone after 1916, when it became apparent that Einstein would eventually win the Nobel Prize. Mileva was constantly asking for money, and Einstein provided it, often grudgingly. Indeed, the letters from 1916 to 1919 seem to be nothing but arguments over money. But Mileva was hard up, unemployed, and looking after two young children, while Einstein, not yet famous, was himself just getting by. (Einstein got the Nobel in 1921, and all the prize money, as he promised, went to Mileva. It amounted to 121,572 Swedish krona. Worth roughly $20,000 back then, it's not much today, and it wasn't that much even in 1921. Nowadays, the prize is about$1 million.) Mileva seems to have used their two boys, Hans Albert and Eduard (nicknamed "Tete"), as a means of coercing funds from her estranged husband, but the real villain of the story is Einstein himself, who was never really cut out to be a husband or father. In the letters, Einstein frequently apologizes for having to cancel out on planned visits and such, and he seems content to simply blow kisses at them while coolly blowing off Mileva's demands for money. Later in the decade, we see letters to and from Einstein and his soon-to-be second wife, Elsa. It's almost disgusting to experience Einstein's kissy-kissy attitude with Elsa in these correspondences, especially when one knows that this marriage was also a colossal failure. Mileva was no beauty queen, but I could never understand Einstein's attraction to that pudding of a woman, Elsa. Anyway, I got mildly ticked off when I read Einstein's letter to Mileva dated November 9, 1918 (also Weyl's 33rd birthday!), in which he impuned all us noble engineers: 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! The nerve of the guy! PS: Einstein's insult to the engineering profession backfired on him. Hans Albert Einstein went on to become a noted professor of civil engineering at UC Berkeley. Ha! PPS: The letter, sent by Einstein without a return address, was opened and read by a Berlin government censor, who threatened to withhold future deliveries unless the address was clearly marked. Sounds very similar to what's going on in this country today.

 Looking for Lulu -- Posted by wostraub on Tuesday, June 20 2006 The other night Turner Classic Movies reaired the 1999 documentary Looking for Lulu, a great one-hour overview of the life and works of American silent film actor Louise Brooks (1906-1985), whose character Lulu in the 1928 German classic Die Büchse der Pandora (Pandora's Box) is said to have enraptured Adolf Hitler long before Marlene Dietrich or Eva Braun came along. I'll bet anything Hermann Weyl and Albert Einstein for once agreed with Hitler on something (however, Hitler subsequently denounced the film itself as "degenerate art"). I saw the film years ago at the old Vagabond Theatre in Los Angeles and fell head over heels for this lady, whom I consider to be easily the most beautiful film actor of all time. But she wasn't just a pretty face -- she was a child prodigy, educated in classical literature from an early age, and a gifted classical dancer with an equally brilliant mind. In her early films she played a typical 1920s "flapper," but soon left for Germany to seek more demanding roles. In Germany she was known as Schwarze Sturzhelm (Black Helmet) because of her unusual coiffure. Amazon sells the documentary DVD for $90. I burned it on DVD from the TCM airing and will send it out for a nominal fee if you're interested, provided I don't get inundated by hundreds of requests. Drop me a line. PS: Pandora's Box is currently unavailable in Region 1 (USA) DVD format, and Kino Video does not plan to release it anytime in the near future. If you live in the UCLA area, you can attend a free screening of the film at the Armand Hammer Museum at 8 pm on July 7, 2006. Update: The Criterion Collection will release the digitally remastered Pandora's Box on American region DVD on November 10, 2006. It will include four different musical soundtracks, the Looking for Lulu documentary, stills, an interview with Brooks, and other extras.  Feynman's Wheels -- Posted by wostraub on Tuesday, June 20 2006 While purging files from my Powerbook today, I came across a couple of pictures having to do with Caltech physicist Richard Feynman (I don't remember where I got them, but they must be fairly old, as the guy died in 1988). Anyway, here is his license plate (I would have gotten quanta for my plate, but we can excuse Feynman for the bad spelling). This next shot of Feynman's van is interesting because it was obviously taken while parked at the Dorothy Chandler Pavilion in downtown Los Angeles. How do I know that? Because the tiered building in the background is the home of my old employer, the Department of Water & Power! It doesn't show up very well, but Feynman's van is covered with paintings of (appropriately enough) Feynman diagrams. I wonder which auto detail shop in Pasadena did that (I might get something Weylian for my Prius). Today's Factoid: The DWP has a really neat engineering library on the fifth floor, and I looked out from that vantage point one day many years ago to see then-Mayor Tom Bradley standing with Queen Elizabeth in the Pavilion right across the street. You don't see that every day!  Fourth Order -- Posted by wostraub on Friday, June 9 2006 One of Einstein's objections to Weyl's theory of the combined gravitational/electrodynamic field was that Weyl's field equations were of fourth, not second, order in the metric tensor gμν and its derivatives. However, variation of the fourth-order Weyl action with respect to the metric tensor for empty spacetime gives where the subscripted bar and double bar notation indicates partial and covariant differentiation, respectively. It is relatively easy to show that this differential equation has a non-trivial solution only when the Ricci scalar R is a non-zero constant; the second term then vanishes, and R can be divided out of the remainder, leaving a term of second order. The surviving term can be solved (it's almost the same as Schwarzschild's solution), giving the familiar expressions for the advance of Mercury's perihelion, the deflection of starlight, etc., provided R is taken as a small constant. The great Austrian physicist Wolfgang Pauli was aware of this calculation as far back as 1921 (when he was just a 21-year-old kid), and noted that Weyl's theory was just as capable of explaining the perihelion shift and light deflection as was Einstein's theory. The Schwarzschild-like solution includes a small repulsion term (proportional to R) that might have something to do with the observed accelerated expansion of the universe. Numerous researchers have linked this term to the cosmological constant. It is also interesting that Weyl's theory gives an Einstein tensor with a 1/4 term (rather than 1/2). This makes it automatically traceless, a desirable feature that Einstein himself searched for in vain. No wonder Weyl thought he was really on to something!  Ray Davis Dead at 91 -- Posted by wostraub on Saturday, June 3 2006 Raymond Davis, the Brookhaven/University of Pennsylvania physical chemist and 2002 Nobel Laureate who devised accurate neutrino detection and counting methods, has died at his Blue Point, New York home. He was 91. Davis, whom I wrote about on this site a few months ago, showed conclusively that the number of neutrinos (elementary particles first anticipated by the work of Hermann Weyl) reaching Earth from the sun is only one-third the number predicted by the standard solar model developed by the late astrophysicist John Bahcall (and a close friend of Davis). It was learned in the 1980s that the three types of neutrino can morph into one another, so out of 100 solar neutrinos emitted by the sun, only one-third will still be solar neutrinos by the time they reach Earth. In an amazing case of theoretical/experimental jousting, both scientists were proved to be right! At the time of Davis' Nobel Prize in Physics, Bahcall said of his friend 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. Davis is survived by his wife of 57 years, Anna.  Hermann Weyl and Imaginary Length -- Posted by wostraub on Saturday, June 3 2006 Mathematical symmetries, like Hermann Weyl's gauge symmetry, are essentially undetectable aspects of action Lagrangians. This is the essence of all mathematical symmetries. For example, the electromagnetic four-potential Aμ has no absolute value -- an arbitrary gradient can be added to it without changing Maxwell's equations. Before the advent of the gauge revolution in physics, it was thought that the four-potential therefore has no intrinsic meaning, and that the electric and magnetic fields E(x) and B(x) represent the only true reality. Nowadays we know better; E(x) and B(x) are themselves composed of various derivatives of Aμ which, though "undetectable" in a real physical sense, is the true underlying reality. To paraphrase Columbia University's Brian Greene, trying to determine the absolute value of Aμ is tantamount to trying to figure out if the number 9 is happy. Those of you who studied complex analysis in school may recall the theory of residues, which provides a means for evaluating certain improper integrals by integrating around the singular pole in the Argand plane. Probably the first problem you solved involved the "single pole" integral where z is a complex quantity and i is the imaginary number (-1)1/2. It is interesting to note that Einstein's objection to Weyl's gauge theory can be avoided by an appeal to this pathetically simple equation if we identify z with the (variable) length of a vector L under parallel transplantation in a Weyl manifold. In fact, the German mathematical physicist Fritz London used this equation in 1927 to derive the quantized radii of orbital electrons for the Bohr atom in a Weyl space. The only downside is that quantities such as vector length L and the four-potential itself become essentially imaginary quantities in Weyl spacetime. This observation has interesting consequences, and perhaps the most profound consequence is that Weyl's theory has validity only in quantum mechanics (where imaginary quantities are de rigeur), not geometry. If you have followed this site at all, then you already know that in 1929 Weyl successfully applied his gauge concept to quantum theory, where it now represents one of the most profound ideas in all modern physics. But there are some researchers (and they keep emailing me!) who insist that imaginary vector length is ok provided the square of the length L2 always comes out real (reminiscent of the probability interpretation of the square of the wave function Ψ2, which is real). Well, I still don't know about all this, but I keep thinking about it. If one always gets L2 when doing a physical measurement, its complex or imaginary aspects are totally hidden from us because we always just take the square root, thinking that it, too, is real. For example, the square root of the real quantity |z|2 is not +/- z, but a +/- ib, where a and b are real numbers. I'm an idiot, it's true, but you have to admit it keeps one's mind off the moronic (and criminal) antics of President Bush, whose mind (and legitimacy as a human being) are pure imaginary but whose crimes are all too tragically real.  Hermann Weyl and Steve Martin -- Posted by wostraub on Saturday, June 3 2006 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. (Weyl wrote this in English, not German, and it has always been one of my favorite quotes of his.) Anyway, back to Ryckman, who in January 2001 gave a lecture at Berkeley on the influence of Husserl on Weyl's gauge idea. I will not pretend that I understand the philosophical part, as my brain is not really wired for it (and it's not a chronic "senior moment" thing for me, either; like pure math and mathematical logic, it just plain escapes me). But Ryckman's talk did provide a pretty good introduction to Weyl's gauge principle, and you just might understand the rest of it as well, especially if you have ever studied transcendental phenomenology or logical empiricism (whatever the hell they are). Here is Ryckman's lecture in Microsoft Word format: Article  Absolute Truth in an Age of Lies -- Posted by wostraub on Tuesday, May 30 2006 In a letter to Einstein dated 19 May 1918, Hermann Weyl asserted 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 Fμ 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. Einstein himself once stated that if his theory of general relativity (gravitation) was not correct, he would have pitied the Lord for having overlooked such a beautiful idea. This is what sheer truth and beauty does to a person -- it is so compelling that it takes on almost divine qualities, even to scientists who are otherwise devoid of any religious faith. In the purest of examples, it is completely objective, overriding any issues of ego or self-righteousness. Another case in point: I am rereading The Physics of Immortality: Modern Cosmology, God and the Resurrection of the Dead by the noted astrophysicist Frank Tipler (he's the same guy who proved that an infinitely-long rotating cylinder could be used as a time-travel device). I am looking at it again only for the mathematics, which may or may not be relevant to the author's central thesis -- that religion is actually a branch of physics, and that we will all be resurrected to eternal life by God when the universe reaches the so-called Omega Point some umpteen zillion years from now. As a newly-minted PhD in 1976, Tipler was a diehard atheist until he experienced an epiphany of sorts while playing with Einstein's gravitational field equations. Whether one completely agrees with Tipler or not is beside the question (as a Christian, I do not, but the stuff's interesting nevertheless). The main point is that mathematical and physical truth has a beauty to it that transcends much of what one experiences in day-to-day living. Part of that truth (at least for me) is the realization that God exists and had a purpose for putting us here in the first place (either as Adam and Eve or as a couple of enlightened Australopithecines). I'm not always sure he did the right thing, considering the mess we've made of the world, but that's another story. Weyl's own Road to Damascus occurred in 1918, when the concept of gauge symmetry sprang into his mind. Einstein's was in the period 1913-15, when he realized that another symmetry -- spacetime invariance -- could be used to develop a theory of gravity. Both men were absolutely convinced that they were in possession of the truth, and it changed their lives forever. I often ask myself what inspires or moves other people. Is it absolute truth, or what they themselves believe to be the truth based on what others have told them? How can we recognize absolute truth, and not be fooled by others (or ourselves) that that truth is not in fact a lie? To me, the only path is math and science, in combination with the teachings of Jesus Christ, because these things cannot lie. But not everyone finds math and science to be very interesting. Can truth be found in accounting, economics, politics, American Idol or auto mechanics? Can truth be found in the New Testament if mathematics and physics are ignored? The answer is very clear to me, but who am I to impose my beliefs on others?  Gravitational Lensing of a Quasar -- Posted by wostraub on Thursday, May 25 2006 This amazing photograph, taken by the Hubble Space Telescope, shows a cluster of galaxies (about 7 billion light-years distant) splitting the image of a single very distant quasar (about 10 billion LY away) into no fewer than five images (the bright bluish-white points of light near the photo's center). The galaxies act as a gravitational lens that imperfectly reproduces the quasar's image in a circular arc about the galactic field of view. The photo also shows distorted images of galaxies near those responsible for the lensing. Quasars (quasi-stellar objects) are themselves the cores of galactic-sized objects containing super-massive black holes. The extreme luminosity of a quasar is powered by matter being accreted into the hole; as it spirals in, friction from the accretion heats the matter up to the point where intense x-ray and gamma radiation comes pouring out. Quasars were originally a great mystery to astrophysicists because their great luminosities didn't seem to agree with their extreme distances. Another example of God's miracle universe. Sadly, the Bush Administration, in its hatred and fear of legitimate science, has cut funding for the Hubble Space Telescope, whose orbit will eventually decay until it burns up in Earth's atmosphere. On the plus side, the money saved will be available to help fund new wars of aggression for oil and other dwindling resources, but in the name of truth and justice and liberty and Christian goodness. But hey, whaddya want, America -- buck-fifty gasoline or a geeky orbiting science project?  Reality v. Formalism -- Posted by wostraub on Sunday, May 21 2006 In the preface to his 1917 book The Continuum, Hermann Weyl tells us 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. Weyl goes on to say that the then-popular use of arithmetic and irrational number theory to solve the problem of the continuum should be stamped (sarcastically) with the title Pythagoras. Therein lies the seed of Weyl's thesis: much of real analysis at the time was based on circular logic; it was non-rigorous, and therefore corrupted by false or meaningless formalism. In his book, Weyl set out to make things right. Today we have a few people around who are brilliant in both modern mathematics and physics; Witten, Penrose and Baez come immediately to mind. But Weyl was the first of this kind to come upon the scene. Trained initially as a mathematician, he immediately ventured out into physics, where he made many profound and fundamental discoveries. Weyl was not afraid to attack what he perceived as either unintentional misrepresentation or outright lies. In his book he attempts to set straight issues that at the time contradicted unquestioned mathematical thinking that greats like Dedikind had established decades earlier. Later, when Weyl proposed his theory of the combined gravitational-electromagnetic field, he did not back down even when his theory was questioned by the great Einstein. Weyl's persistence, which was based on a firm conviction that what he had proposed was based in absolute truth and beauty, paid off when he applied his theory to the then-emerging field of quantum mechanics. Weyl's gauge principle stands today as one of the most profound tenets of quantum physics. Who today has the courage to go up against established authority? Today we are told lie after lie by our political leaders, and we swallow every one of them, hook, line and sinker. Who among us has the courage to tell Bush and Company that they are liars and false prophets? Weyl denounced Hitler, but had to flee his beloved Germany in 1933 because neither he nor all his brilliant contempories could stem the tide of nationalistic insanity that was sweeping the country. Not long afterward, magazines and posters were displayed with Einstein's photo and Noch ungehängt! (not yet hanged) all over Germany. Not surprisingly, Einstein too fled the country. A dedicated Nazi effort to discredit the theories of special and general relativity soon had the German people thinking they had been tricked into believing a lot of intellectual hokum. Books were burned, ideas themselves were banned, and the great edifice that was once Deutsche Mathematik und Physik was destroyed in favor of ignorant, arrogant nationalism. Sadly, this is America today. But the danger is heightened infinitely by America's possession of 10,000 nuclear bombs, spy satellites that can watch and record everything we do and say, a hatred of legitimate science and truth, and a crazed, fanatical, nationalistic Christian millennialism that wants desperately to hasten Armageddon through unilateral, preemptive war. It is ironic that today Weyl and Einstein would almost certainly be forced to return to Germany to escape the fascist madness that has overtaken our country. This is "circular logic" of a most disturbing kind. God save us all.  Einstein -- Collected Papers -- Posted by wostraub on Monday, May 8 2006 I spent several hours at Caltech today perusing its copy of the Collected Papers of Einstein (writings and correspondence, nine volumes, with a few English translation versions). I went there to copy Einstein's correspondence with Hermann Weyl, only to realize that I already have most of it. But I was unprepared for the sheer volume of Einstein's correspondence with other notables of the time. Letters in those days (this was 90 years ago) was the email of their time, and Einstein must have spent a fair amount of his free time just writing letters. Particularly interesting are the letters to Mileva (his ex-wife) and son Hans Albert, all of which show varying degrees of the man's emotions, including warmth, concern, impatience, intractitude, and even a little hostility. Though he genuinely cared for his two sons (in 1903 he and Mileva had an out-of-wedlock daughter, Liserl, who was given up for adoption), Einstein was not a family man, and his boys must have suffered for it. Hans went on to become a professor of civil engineering at Berkeley (he developed the Einstein bed load function in sedimentation theory), while Eduard had mental problems all his life and died at an early age. The fate of little Liserl is a mystery. The Collected Papers abounds with correspondence between Einstein and hundreds of notable scientists, mathematicians, philosophers and political scientists. It's well-organized and makes fascinating reading, if you've got the time. It's also available for purchase, but each volume runs around$100, which is far beyond my pocketbook. As for Weyl, Einstein and my favorite mathematical physicist wrote to each other dozens of times, discussing many different topics, including Weyl's gauge theory and related/unrelated mathematics, gravitation theory, philosophy, German politics, the war, and what kind of salaries professors should be given. Weyl was also the frequent subject of Einstein's correspondence with others. It very much takes you back to a time when it appeared that Einstein's relativistic theories (and generalizations) would eventually solve all the standing problems in physics (this was before quantum theory, of course).

 Wilczek on Weyl -- Posted by wostraub on Friday, April 28 2006 In October 2005, MIT's Frank Wilczek, the winner of the 2004 Nobel Prize in Physics (for discovering the principle of quark asymptotic freedom), wrote a nice tribute to Hermann Weyl. Here it is in .PDF format.

 Einstein v. Weyl -- Posted by wostraub on Wednesday, April 26 2006 Partly as a means of ridding my mind of the preposterously immoral state of this country and the criminal actions of the Bush Administration, I'm writing a brief synopsis of the argument that Einstein and Weyl had regarding Weyl's early metric gauge theory. Several people have written in, asking what was behind Einstein's objection to the theory, was it valid, did they remain friends, etc. Hardly the appropriate subject matter for the general public, but the story itself is fairly interesting, and I hope I can do it justice. I'm getting ready for a trip, but I'll try to have it up in the next few days.

 Derivation of the Weyl Conformal Tensor -- Posted by wostraub on Wednesday, April 12 2006 Some time ago I mentioned the Weyl conformal tensor, which is fundamental to the understanding of gravitational tidal effects. Whereas Einstein's equation (which involves only the Ricci tensor and scalar) describes gravitational compression and compaction of matter (volume reduction via gravitational attraction), the Weyl tensor is responsible for the deformation of matter, with the initial volume of matter remaining intact. For example, if you ever happen to fall into a black hole, your body's volume will be retained but you'll be increasingly squished sideways and elongated in the direction of the hole. This rather unpleasant phenomenon, known to black hole afficionados and the cognoscenti as spaghettification, is due to the Weyl conformal tensor. Why God allows black holes to exist is anybody's guess (maybe just because they're fascinating). How did Weyl discover this tensor? I could never find out. He seems to have simply written it down (he was that good). Numerous people have asked me how the tensor can be derived. Since I've never seen the derivation, I'd never done the calculation, at least until now. It's much simpler than you might think. The file conformal.pdf is on the menu to the left.

 Weyl and the Question of Asymmetric Time -- Posted by wostraub on Thursday, April 6 2006 What really interests me is whether God had any choice in the creation of the world. -- Albert Einstein In the early 1920s, Hermann Weyl discovered a new tensor quantity (the Weyl conformal tensor) which is basically the Riemann-Christoffel curvature tensor with the contracted pieces (the Ricci tensor and scalar) removed. The resulting tensor is conformal (angle preserving) as well as metric gauge invariant. Weyl must have come across the tensor while investigating the consequences of his 1918 gauge theory and its presumed (but wrong) unification of gravitation and electrodynamics, but I have been unable to confirm this. The Weyl curvature tensor is zero for flat spacetime, but for curved manifolds it is non-zero, even in the absence of matter. The tensor is responsible for gravitational tidal effects, in which (say) a spherical collection of particles is contorted into a prolate ellipsoidal shape (although the tensor preserves the initial volume). In fact, Weyl curvature is responsible for the tidal bulge in the Earth's oceans caused by the moon's gravitational pull. By contrast, the Ricci curvature terms deform matter by gravitational compression, and volume is not preserved. In 1979, the British mathematical physicist Roger Penrose (also a gifted science writer) announced the Weyl Curvature Hypothesis, which essentially states that the Weyl tensor was precisely zero when the Big Bang occurred and will become infinite if and when the controversial Big Crunch occurs. On the basis of this hypothesis, Penrose believes that time must be asymmetrical; that is, time proceeds from the Big Bang to the Big Crunch in only one direction. This contradicts the CPT theorem, which basically states that physics is also valid for reversed time (that is, all equations remain valid if we replace t with -t ). The laws of physics may be time direction-invariant, but on a universal scale this might not be the case. Penrose believes that a consistent quantum-gravity theory (assuming we ever come into possession of it) will demonstrate that the direction of time is really only one-way. Whether the universe will end in a Big Crunch is debatable (current data indicate that the universe will continue expanding forever), but what is certain is that much of the matter in the late universe will coalesce into black holes. Spacetime curvature near the event horizon of a black hole is highly Weylian, so even if Penrose is wrong the totality of Weyl curvature in the late universe will undoubtedly be extremely high if not infinite. I've mentioned Penrose before. He has two excellent (and very readable) books out: The Emperor's New Mind (Penguin Books, 1989) and The Road to Reality -- A Complete Guide to the Laws of the Universe (Knopf, 2004). The latter book is a life-altering text that should be read by everyone who has even the slightest interest in physics, the universe, and God's role in it all. Buy this book, read it carefully, and then place it next to the Bible and Hamlet on your bookshelf; you will then be able to call yourself an enlightened member of the human race. The Weyl Curvature Hypothesis provides a direction for time's arrow, and is therefore intimately connected with the increase of entropy in the universe (as demanded by the second law of thermodynamics). Indeed, Stephen Hawking and others have proved mathematically that the surface area of a black hole's event horizon is proportional to the hole's entropy. Thus, in the late universe, the level of entropy contained in black holes will be enormous. By comparison, the entropy of spacetime at the time of the Big Bang was very low, if not exactly zero. Thus, the Big Bang and Big Crunch are distinctly different events. This calls into question the reality of a "cyclic universe," that is, one that comes into existence and then recollapses over and over. I think Weyl would have been pleased that his curvature tensor is today profoundly associated with the fate of the universe and related unsolved problems in modern physics.

 The Fly in the Cathedral -- Posted by wostraub on Wednesday, March 15 2006 I just finished reading Brian Cathcart's excellent 2004 book The Fly in the Cathedral (Farrar, Straus and Giroux, publishers), which describes in detail the discoveries of Rutherford (the atomic nucleus), Chadwick (the neutron), and Cockcroft and Walton (induced atomic fission). The book's title refers to a comment made by Rutherford, whose original atomic "plum pudding" model gave way to the correct view of a tiny, lone nucleus sitting in the vast empty space of the atom -- like a fly in a cathedral. But the bulk of Cathcart's book is taken up by the story of John Cockcroft and Ernest Walton, who in early 1932 bombarded lithium metal with accelerated protons. It is an intriguing tale of frustration, dashed hopes, personal tragedy, and ultimate victory. The scientists' apparatus, Neanderthal by today's standards, continually broke down, adding months to their efforts (and always overshadowed by lack of funds in those early days of worldwide depression). But their efforts were repaid many times over -- they ultimately found to their utter amazement and delight that protons could split lithium-7 -- a stable element -- into two helium atoms. The Cockcroft-Walton experiment was the very first experimental observation of man-made atomic fission, the transmutation of one element into another, the first splitting of the atom. The experiment also offered the very first practical test of Einstein's E = mc2 formula. The observed 8.5-MeV energy of each product helium nucleus balanced the books with respect to the reactant particle energies. Like Einstein had said in 1905, mass and energy are truly equivalent! In recognition of their work, the Nobel Committee awarded Cockcroft and Walton the 1951 Nobel Prize in Physics. The book's decription of Cockcroft/Walton's discovery is nothing short of heartwarming, but it also includes tragedy. At the time of their triumph, both men lost infants to childhood disease, tragedies that nearly destroyed the men and their wives in spite of their groundbreaking discovery. But Cathcart saves the best for last. Ernest and Frieda Walton had a long, happy marriage, and they went on to have four more children who all pursued careers in science (three in physics!) Meanwhile, John and Elizabeth Cockcroft went on to have five more children -- a scientist, an engineer, a priest, a nurse, and a teacher. God be praised!

 The Anthropic Principle -- Posted by wostraub on Wednesday, March 15 2006 For a long time I've been planning to put up an article on the Anthropic Principle, which purports to offer some proof that an intelligent, omniscient entity (let's call it God) really did engineer the universe for our benefit. There's actually quite a bit of evidence that such a principle is scientifically valid, but, until I get around to it, here's one of the more convincing arguments. The Big Bang, which occurred about 13.7 billion years ago, started out basically as a super-hot plasma of quarks and leptons. But within 2 minutes this plasma had cooled sufficiently to allow for the formation of protons and neutrons. Using Boltzmann's equation and the fact that neutrons decay into protons, electrons and antineutrinos (the life of an unbound neutron is about 15 minutes), it can be shown that the expanding fireball at age 2 minutes was composed of about 75% protons and 25% neutrons (with relatively minor concentrations of other stuff). Thus, three-fourths of the universe consisted of hydrogen, the basic fuel of star formation via nucleosynthesis. Gravity gradually coalesced clouds of matter into spheres and compressed them to the point where nuclear fusion began in their cores. Almost all of this fusion involves the creation of helium nuclei from hydrogen. This is the kind of fusion that mankind is now trying to duplicate for long-term energy generation. But after a few billion years, many stars burn up their supplies of hydrogen. A star begins to cool, contracts further under gravity, and then the core heats up again as a result of the increasing pressure. The temperatures eventually get so high that the star's helium can fuse into carbon, oxygen, neon and several other low-atomic weight elements. Some of these stars eventually explode as novas and supernovas, and their supplies of carbon, oxygen, hydrogen and other trace elements is what makes up all living things. All living things on Earth owe their existence to stardust flung out by ancient star explosions. Humans, for example, are composed of about 20% carbon and 65% oxygen, a reflection of the equilibrium battles in the sun involving these two elements. During the helium-burning phase of a middle-aged star, there are three primary reactions going on that affect the formation of carbon. One is the triple-alpha reaction, in which three helium nuclei fuse to form a carbon-12 nucleus. The carbon formed in this process, however, can be scavenged by another process in which a helium nucleus fuses with a carbon nucleus to form oxygen-16. At the same time, oxygen-16 can also be scavenged by a helium nucleus, yielding neon-20. It turns out that the rates of these competing reactions and the physical constants that determine their equilibrium points are very finely tuned; if the excited states of carbon-12 and neon-20 nuclei were only slightly different, all middle-aged stars would evolve atmospheres that are either oxygen-rich and carbon-poor or composed predominently of neon. The physics of these processes, including a few minor ones that take place simulaneously with those I've described, have been worked out by astrophysicists over the years to a gnat's eyebrow. If the fundamental physics of the elementary particles making up carbon, oxygen, helium and neon were displaced by only a few parts per million, life could not exist anywhere in the universe. Thus, the universe as we know it must have been designed, or we wouldn't be here. However, some physicists have argued that, if the many-worlds interpretation of quantum physics is valid, then our universe is just one of an infinite number of possible universes in which the physics happened to be "just right." In the vast majority of these other universes, the physics is "off" and life does not exist. In this way of thinking, a designer God would not be necessary because life exists here because of a statistical fluke. But this argument is flawed. The processes I described above represent only a single example of the Anthropic Principle. There are many other processes that are also finely tuned for the existence of life, and the totality of these processes makes the statistical argument really hard to accept. I have estimated that, given the totality of "life-favorable" physical processes and constants, on average only one such universe would arise out of 1040 universes. While I suppose this is still statistically possible, it seems much easier just to admit the existence of God. In my mind, these numbers imply that the non-existence of God is statistically about one in 10-40. To me, that's essentially zero. So get over it -- God exists! Whether he/she/it is Jehovah, Allah or Brahma is purely a matter of faith. And I'll leave it at that for now. Post Script -- Speaking of faith, where do a lot of Americans derive it? I recall about a year and a half ago a grilled cheese sandwich that sold on eBay for $28,000. Why? Because it had what appeared to be the figure of Jesus Christ on it (the pan it was cooked in was subsequently auctioned off for$20,000). Since then, I've heard of trees, pancakes, highway overpass stains, and hotcross buns that appeared to depict Christian figures. All of these events made a big splash in the media, and literally thousands of people made pilgrimages to these "holy" sites. Yet the average American knows nothing at all about the scientific Anthropic Principle, which to me represents the only rational way of looking at physical "proof" of God's presence. Americans truly are idiots.

 "Ich fahre nach Pasadena ..." -- Posted by wostraub on Sunday, March 5 2006 Here's a brief letter (along with its English translation) that Einstein wrote just prior to his second visit to Pasadena in 1931. In it, he lauds America's love of science and its ability to balance production and consumption. How times have changed. Americans now prefer superstition, video games and celebrity worship to math and science, while our gluttonous material appetite threatens to consume the entire world. Exactly where and when he wrote this (and to whom) is anybody's guess. Like it? You can have it for only $23,000.  More on Neutrino Oscillations -- Posted by wostraub on Sunday, February 26 2006 Many people have written to say that they were fascinated by last week’s PBS program on neutrinos, The Ghost Particle. It is interesting to note that Hermann Weyl also made fundamental contributions to our understanding of these particles, which may be the most numerous things in the universe. In his seminal 1929 paper, Electron and Gravitation (Zeitschrift f. Physik, 330 56), Weyl was the first to recognize that the treatment of spin-1/2 particles (like the neutrino) in a gravitational field requires a covariant derivative that is appropriate to fermionic fields. Weyl’s development of the spin connection ω abλ and the associated spin covariant derivative emerged from this work, as was his recognition that the zero-mass version of the Dirac relativistic electron equation allowed for a description of particles that violate parity (this is practically the de facto definition of the neutrino!) While Weyl’s paper preceded Pauli’s 1930 neutrino hypothesis by a year (and it is doubtful that Weyl had any inkling about the existence of this particle at the time), his work nevertheless provided a sound basis for the neutrino’s subsequent mathematical elucidation. Weyl also was totally unaware of the existence of three types of neutrino or the possibility of neutrino oscillation, which was the subject of the PBS program. Whatever the physical process behind a neutrino’s penchant for converting itself into any of the three types, it is abundantly clear that a successful description will involve the dynamics of fermionic fields against a gravitational background, and this will by necessity involve Weyl’s spin connection and derivative. Not bad for a mathematican who was once scolded by Pauli for straying into the physics community! Several people have asked me about more advanced yet readable information on neutrino oscillation. I’m the wrong person to ask, because I know practically nothing! But there are several papers I’ve collected that have helped me understand the things a tiny bit: Dieter Brill and John A. Wheeler, Interaction of Neutrinos and Gravitational Fields (1957). Reviews of Modern Physics, 29 465. [This is probably the first article you should seek out.] C.Y. Cardall and George M. Fuller, Neutrino Oscillation in Curved Spacetime: A Heuristic Treatment (1997). Physical Review D, 55 No. 12, 7960. Xin-Bing Huang, Neutrino Oscillation in de Sitter Spacetime. arXiv:hep-th/0502165 v1 (12 February 2005). Victor M. Villalba, Exact Solutions to the Dirac Equation for Neutrinos Propagating in a Particular Vaidya Background (2001). International Journal of Theoretical Physics, 40 No. 11, 2025.  The Ghost Particle on PBS -- Posted by wostraub on Tuesday, February 21 2006 I hope you all caught The Ghost Particle on PBS tonight. The ghost particles are, of course, neutrinos, first postulated by Wolfgang Pauli in 1930. Nearly massless and traveling close to the speed of light, approximately 100 trillion pass through your body every second, and "pierce the lover and his lass," to quote the famous John Updike poem. The program chronicles the search for the solar neutrino, and focuses on the initially-contradicting data between experiment and theory. First came the theory, proposed by John Bahcall in 1964, that electron neutrinos would be produced by the sun at a rate of X per second. Then Ray Davis and colleagues built an apparatus that could actually measure the things. Strangely, their observations indicated that the sun was producing neutrinos at the rate of only X/3. Scientists around the world couldn't figure out just who was right (if either). The Standard Model of particle physics says that electron neutrinos are massless and travel at the speed of light. But in the 1970s and 1980s two more neutrinos showed up -- the muon neutrino and the tau neutrino. Today, the family consists of electron neutrinos, muon neutrinos and tau neutrinos, along with their antimatter counterparts. Most physicists believe that no new neutrinos will ever be found. Anyway, to make a long (but very fascinating) story short, it was later discovered that the three kinds of neutrinos can randomly oscillate from one kind into another. Thus, the number of solar electron neutrinos reaching Earth is reduced by a factor of two-thirds. Bahcall's theory was vindicated, as were Davis' experiments. Both physicists were right! Because neutrino oscillation requires that these particles have a non-zero proper time measure, neutrinos cannot travel at the speed of light, so they must have a tiny but non-zero mass. Consequently, there was early conjecture that neutrino mass might account for the "missing mass" in the observed universe (the total number of neutrinos in the universe is almost unimaginable, so even a tiny mass would add up to something truly significant). However, it is now believed that other, more exotic forms of non-baryonic matter make up the vast bulk of the known universe's mass-energy. Oddly enough, the ordinary matter that you and I know and love (protons, neutrons, electrons, hamburgers, etc.) accounts for only about 5% of the "stuff" of the universe. The rest is "dark matter" and "dark energy." Very odd, indeed. Although Davis (now in his 80s) is suffering from Alzheimer's disease, he was fully cognizant back in 2002 when, to the delight of his family and fellow researchers, he won the Nobel Prize in Physics for his neutrino work. He was accompanied in Stockholm by no fewer than 23 ecstatic family members -- wife, children and grandchildren. God be praised! I recorded this excellent PBS program on DVD-R. Let me know if you'd like a copy.  Weyl's Take on the Gravitational Energy-Momentum Tensor -- Posted by wostraub on Friday, February 17 2006 Shortly after Einstein's November 1915 announcement of his general theory of relativity, Weyl attempted to derive a coordinate-invariant form of the energy-momentum tensor that expressed conservation via an invariant divergence formula. His failure to find a fully-covariant expression of this tensor puzzled many physicists at the time. And despite repeated attmepts over the years by many scientists, no one has discovered a satisfactory form of the tensor. This is very odd, because general relativity is practically the de facto definition of invariance theory, yet something as conceptually simple as gravitational energy-momentum conservation continues to elude us. Weyl's attempt is documented in the first edition of his 1918 book Space-Time-Matter (Raum-Zeit-Materie). It's a mess, if only because of the inconsistent index notation he used in those days. But, yes, a divergenceless energy-momentum tensor can be written down (it looks like T μν + t μν) but the quantity t μν is really only a pseudotensor -- it's not invariant with respect to a change in the coordinates, and it's not even symmetric with respect to the indices. This is very frustrating! Long ago, I thought it might be possible to use Weyl's φ-field to derive a truly covariant form of the energy-momentum tensor. I failed in this attempt, but I'm little better than a total idiot, so it's still possible that this approach is valid. Something to think about on an otherwise cold and rainy night.  Spin Connection -- Posted by wostraub on Saturday, February 11 2006 I rewrote The Spin Connection in Weyl Space (a somewhat pretentious title, I know) and included an elementary overview of vector parallel transfer and covariant differentiation. The .pdf file is posted on the menu to the left. The typos are all fixed now (I think), but I'm washing my hands of the whole thing, as it still doesn't read the way I wanted it to. Enjoy it if you can.  DPGraph -- Posted by wostraub on Monday, January 30 2006 Years ago when I was teaching, I used a simple but powerful program for creating static and animated computer graphics called DPGraph. If you have ever attended university, chances are the program can be downloaded for free (the site has a list of hundreds of site-licensed universities, and all you have to do is click on yours [be honest, now] and the program is automatically downloaded gratis). You can also buy the thing outright for just$10. Either way, it's a bargain, and it's a blast. Just enter a 2-D or 3-D algebraic expression on the command line, toss in a few parameters, and the program gives you fantastic graphics. It's used by countless schools (including elementary, middle and high schools), but with a little imagination you can easily create exceedingly-complex graphics and animations that look like output from a graduate institution. The program, which only takes up about 500 KB on your hard drive, will run on any Windows OS (it also runs on my Mac Powerbook via Virtual PC, but Virtual PC is such a dog that very high-resolution graphics [especially animated graphics] take a long time to generate). The program's documentation is minimal; there's no manual, but the HELP menu should provide all the information you'll ever need. Also, the DPGraph website has many hundreds of free downloadable graphics files that serve as useful examples of the required programming. DPGraph is a far cry from professional programs like Wolfram's Mathematica, but that's overkill for most people, anyway (it's also a far cry from my old Keuffel & Esser sliderule, but that's another story, and I don't want to date myself too badly here).

 Deep Down Things -- Posted by wostraub on Friday, January 20 2006 Seduced by the glowing reviews of UC Santa Cruz physicist Bruce Schumm's new book, Deep Down Things (Amazon, about $19) and, curious over the fact that the book devotes almost 80 pages to an elementary elucidation of gauge theory, I bought the thing and read it. First off, the book is pretty much aimed at the motivated lay reader who wants to understand the non-mathematical particulars of the Standard Model of the electromagnetic and strong and weak nuclear forces (he wisely left gravity out of the loop because it's clearly beyond the scope of a book of this kind). The book includes rather extensive (though elementary) expositions on topics such as Lie groups and Lie algebra, SU(2) and SU(3) isospin and hypercharge symmetries, the weak interaction and the quark model, and by book's end I had regained much of what I invariably tend to forget about this stuff. However, the book is inconsistent at the levels with which it treats (or should treat) complex numbers, quantum mechanical phase invariance, and related topics. For example, Schumm writes down Schrodinger's one-dimensional wave equation on numerous occasions, explaining what all the parts represent, but he doesn't feel that the reader is quite up to understanding the exponential version of complex numbers. This lack of confidence extends to his description of phase invariance, in which nary a e i θ appears in the book. This is a shame, because anyone who has even a smidgen of knowledge about z = a + ib knows that the exponential form (known as Euler's relation), which is ubiquitous in quantum mechanics, is easier to use and more intuitive. You cannot explain to someone what a unitary operator is with z! Schumm's description of gauge invariance, Weyl's brainchild, is particularly muddled. The principle of gauge symmetry is easy to understand, but not if you leave the basic math out of the discussion. The book's last chapter, Into the Unknown, discusses a few advanced opics, along with the Higgs field and physicists' hopes to discover it with the European Large Hadron Collider, which is scheduled to go into operation in 2007. And while Schumm plays down the role of gravity in all this, he hints at the possibility that a unified theory of all four forces will radically change the way we think of everything. The problem with Schumm's book is the same one that plagues all popularized expositions of modern physics theories -- there is precious little middle ground that these writers are willing to explore between a non-mathematical, golly-gee treatment and a higher-level textbook-like approach. In my opinion this is not Schumm's fault, but rather that of a dumbed-down reading public coupled with a rather cynical attitude of the publishers.  What a Waste -- Posted by wostraub on Monday, January 16 2006 Here's a morality play masquerading as a physics problem. You no doubt know that controlled nuclear fusion would solve the world's energy problems for all time. Fusion is really very simple -- just get a deuterium atom and a tritium atom (these are available or easily-made isotopes of ordinary hydrogen) close enough, and they'll fuse to form helium-4 (along with a left-over neutron), with the release of lots of energy. How hard can it be to get two tiny nuclei close to one another? Well, it's deceptive -- the Coulomb repulsion between the particles is so great that only truly enormous confinement temperatures and pressures can get them to fuse. While this has actually been done in gigantic experimental machines (like the Tokamac), the energy expended in the experiments far outweighs the energy derived from fusion. Scientists are still trying to achieve "breakeven," and as a result practical nuclear fusion is still many decades away. But there's another way that doesn't require huge pressures and temperatures. Take a deuterium-tritium (DT) ion with a single shared electron. Fire a muon into the ion (the muon is an elementary particle that is identical to the electron, but about 207 times as heavy). The muon knocks the electron out of the DT pair and begins to orbit the nuclei, just like its electron counterpart did. But because of its greater mass, the mean orbital radius of the muon is 207 times smaller than that of the electron. This causes the deuterium and tritium nuclei to move very close to each other. The muon's small orbital radius also effectively shields the positive Coulombic repulsion of the DT nuclei, which gets them in even closer. Within a few thousandths of a nanosecond the nuclei undergo fusion, with the release of about 17.6 MeV of energy. The muon is unharmed during the fusion event and leaves the helium-4 in search of another DT pair, where it can do its trick all over again. Because the muon comes out unscathed, this process is called muon-catalyzed fusion. It has been demonstrated many times in laboratories over the past three decades. So what's wrong with this picture? Nothing, but there are a few technical problems that have to be overcome. One, muons have to be created, you can't buy them in stores. They come from decaying negative pi-mesons (or pions, π-), and you need a linear accelerator to get the necessary pions. Second, the muon is itself unstable and decays into an electron, a muon neutrino and an anti-electron neutrino (a muon has a typical life of only 2 microseconds). And third, once a muon catalyzes a fusion event, it often develops the habit of hanging around the helium-4 nucleus once it has formed. In view of the muon's short lifetime, this "stickiness" of the muon wastes valuable time. However, the first and second difficulties are not all that critical -- they can be dealt with. The most critical problem is the muon's tendency to loiter around and be unproductive. A means for making unsticky muons would represent a truly profound discovery and a wonderful gift to mankind's future welfare. There's even another particle just like the electron and the muon called the tau (τ-), which is about 17 times heavier than the muon. Tau-catalyzed fusion might someday be demonstrated, although the tau's lifetime is about ten million times shorter than the muon's. So what's the upshot of all this? Every year, the world's nations spend nearly$1 trillion for weapons of war (about half of this amount is spent by the United States). Recent estimates (by the 2001 Nobel prize winner in economics, no less) of the actual out-of-pocket costs of the Iraq war total about $2 trillion Article. Forgetting annual US defense expenditures, what do you think we could have done with$2 trillion? Develop practical muon-catalyzed fusion, maybe? It boggles my mind to think that the US might be able to develop non-polluting nuclear fusion energy generation if it would only get its head out of its ass!! This is just another of President Bush's outrageous legacies -- at a time when Peak Oil is rapidly approaching, and the country is in desperate need of an alternative energy source, Bush decides that what we really need to do is monopolize (i.e., steal) the world's remaining oil resources! This cannot save us, because even if Europe, China, India and the other developing countries can be held at bay, the resulting destruction of the world's economies will also destroy ours. And my guess is that the other countries of the world wouldn't stand for it -- remember, Russia still has 6,000 nuclear weapons and the missiles to deliver them. Make no mistake about it -- the lunatic US Emperor George W. Bush is the most dangerous man in the world, and we tolerate him at our extreme peril.

 Cosmic Landscapes -- Posted by wostraub on Sunday, January 15 2006 In his new book The Cosmic Landscape: String Theory and the Illusion of Intelligent Design (Amazon, about 16), Stanford physicist Leonard Susskind suggests that the universe we inhabit is only one of a nearly infinite number of "megaverses," perhaps as many as 10500. Each of these possible universes is based on a different set of fundamental physical constants, so one universe may permit life while another does not. Susskind, a leading string theorist, does not necessarily imply that intelligent design is wrong, it's just that in his multiverse theory there's no need for it. Given an almost infinite number of possible universes, it is inevitable that at least one universe will look just like the one we live in. And there we are! For the same reason, Susskind feels that things like beauty and elegance are also inevitable, especially when a universe contains thinking creatures. This would seem to imply that there is no such thing as "absolute truth," which is abhorrent to me, but it's still something worth thinking about. Is it possible to flip an "honest" coin 10500 times in a row and have it come up "heads" each time? Of course! The probability is very small, but it's not zero. If there are a similar number of universes out there, all orthogonal to the one we inhabit, the chances are good that just about any kind of otherwise implausible event or condition will be observed. Are these universes the "many mansions" that Jesus spoke about? Susskind would probably disagree, but I argue that it's equally possible that it is. Meanwhile, purely for your enjoyment, here's the Sombrero Galaxy (M104), located about 50 million light years away from us in the constellation Virgo. This beautiful galaxy is just one of the hundreds of billions known to exist in our one universe. God be praised!  Embracing Lies as Truth -- Posted by wostraub on Saturday, January 14 2006 I haven't read James Frey's A Million Little Pieces, and I never will, but I have to comment on Frey's appearance on Larry King Live three nights ago. The fuss centers around accusations that Frey's "redemption" is based on the many lies and half-truths contained in his book, and the fact that Oprah Winfrey had championed his book on her book club. The book sold well as a result, but the allegations are pushing sales beyond Frey's wildest dreams. Anyway, Winfrey herself called in to King's show to defend Frey and the book (and her own reputation). Instead of admitting that she had made a mistake by trumpeting a liar's work of fiction, she made two incredible claims -- that the book's publisher was to blame for any untruths in the book, and that she and Frey had apparently created a "new genre" of legitimate literature, a pleasant blend of fact and fiction. The ancient adage about turning a sow's ear into a silk purse comes immediately to mind. By the way, Frey also brought his mommy along to the King show, no doubt to throw added weight onto the lies he was spinning. Anyone who believes Frey's nonsense is a boob, and I guess that includes Oprah as well as most Americans, many of whom have actively campaigned to have Winfrey run for president. Winfrey herself seems to be self-delusional, thinking that her billions are proof that she's infallible. But the real blame falls on a celebrity-intoxicated American public that cannot differentiate between truth and lies anymore. It's no wonder Bush is president.  Teach Quantum Physics in the Churches? -- Posted by wostraub on Thursday, January 12 2006 Today, two opposing members of the Ohio Board of Education appeared on Lou Dobbs Tonight to present their respective cases for and against the teaching of intelligent design in Ohio public schools. Although ID suffered a stinging defeat in Pennsylvania last month, its adherents are regrouping and taking their arguments to school boards in numerous states -- Ohio, Georgia, Missouri, Kansas, and even California. Dobbs listened to both sides, and at one point actually suggested teaching quantum physics alongside evolution in the schools! He then asked if it would be proper to teach comparative religion in public school. The pro-ID guest disagreed, and implied that religious education had no place in the schools. My read on this is that ID supporters would try to quietly introduce Christian education into public schools via the teaching of ID. Apparently, IDers are sold on the idea that intelligent design belongs solely to the Christian faith; comparative religion be hanged. I have yet to see a Jew, Muslim or Buddhist demand that ID be taught as a scientific discipline in public schools. PBS' Frontline is currently running a series (Country Boys) on the problems of rural youth -- lack of jobs, premarital sex, methamphetamine abuse, depression, etc. Last night's episode took us into a rural Kentucky high school classroom where the teacher ridiculed evolution as anti-God pseudo-science: "Did Jesus Christ look like an ape? Do you think you came from a bunch of monkeys? That's not what I believe!" or words to that effect. I really felt sorry for the school's students, who undoubtedly have enough problems in their lives. Now they can add ignorance to the list. I really like the idea of teaching quantum mechanics (at the appropriate level) in churches, because QM is undoubtedly one of the tools God uses to run his universe. Like religion, QM is based on numerous postulates that cannot be scientifically proven, and so have to be taken on FAITH. Everyone believes in QM, because modern life could not be possible without it. Why can't the IDers look at evolution the same way? Evolution is just one of God's tools to ensure the perpetuation of the planet's species. But, as Lou Dobbs surmised today, ID ain't going away, and it will continue to itself evolve until it is absolutely disproven as a science and banned outright or legitimized and made a mainstay in the public schools. Hopefully, the legitimization of ID, like evolution, will also take millions of years.  Fascinating Physics -- Posted by wostraub on Saturday, January 7 2006 A hydrogen atom is just an electron bound to a proton. What if you replace the proton with some other positively-charged particle? If that particle is an antielectron (or positron), you get something called positronium, or Ps. Physics Today rightly calls it "nature's simplest atom." Electron/positron pairs are created near charged particles, but they invariably annihilate one another, resulting in photon pairs. But under the right circumstances, and for intervals on the order of 100 nanoseconds, they join to form Ps. There are two bosonic species of Ps: ortho-Ps, in which the particle spins are aligned (spin one), and para-Ps, in which the spins cancel (spin zero). The latest issue of Physics Today (January 2006) reports that researchers at the University of California at Riverside have found indirect evidence that two Ps atoms can join to form a diatomic molecule (Ps2). If confirmed, the researchers believe that they can form a Bose-Einstein condensate at a temperature of around 15 degrees K. This in turn might then lead to gamma-ray lasers in which each photon has an energy of about 0.5 MeV. Amazing! I imagine even George W. Bush will be interested in this, but for another reason -- gamma-ray space lasers to zap the evil-doers (set your weapon to deep fat fry, comrade!)  To The Mall, Patriots! -- Posted by wostraub on Saturday, January 7 2006 I laughed out loud when I read Jesse Eisinger's WSJ article about neoconsumerism under the Bush Reign of Terror. He calls it zombie consumerism -- for a reason you'll need to read the article to understand. Eisinger points out that for the first time since the Great Depression, Americans spent more money in 2005 than they earned. This negative cash flow greatly increased the nation's debt, as Americans flocked to take out home equity loans to pay for all their SUVs, gadgets and credit card installments (I guess they see this as "found money" that doesn't have to be paid back). I peeked in on the Commerce Department's website back in November, and it confirmed that Americans were indeed saving nothing but spending nevertheless. Eisinger warns that with the flattening of the real estate market, record bankruptcy rates, usury-like credit interest rates, the spiraling differential between workers' pay and lavish CEO pay, and the very real connection between stock market performance and the availability of consumer cash, the future bodes ill for Bush's wundereconomy. WSJ Article Can't afford food? Eat EQUITY! EQUITY  Weyl on Time Travel -- Posted by wostraub on Saturday, January 7 2006 While slaving away at the gym this morning, I started to think about time travel again. This is one of my favorite topics, in spite of the fact that I really don't think it's possible, at least for any massive object. If you check out the New Testament, you'll see numerous references to God and light. I believe this connection is more than just hyperbole, because if God is purely spiritual then he undoubtedly moves on a null geodesic, which is to say that he is free to move around just like a photon of light. A photon lives is a very strange world, indeed. Because its line element vanishes (ds2 = gμν dxμdxν = 0), it exists everywhere in the universe at all times -- past, present and future. This fact is paradoxical to us humans, because when we snap on a light, photons are created at that instant, and when they are absorbed by our cornea, they are annihilated. Clearly, then, light can be created and destroyed in a short time interval. But a photon's own existence is much different -- to a photon, it is people whose lives are sedentary and fleeting. This is nothing more than an extreme example of Einstein's so-called twin paradox, which of course is not a paradox at all when you've understood special relativity. So the saying "God is light" is probably closer to the truth than one usually imagines. In 1994, the noted Caltech physicist Kip S. Thorne published his wonderful book Black Holes and Time Warps -- Einstein's Outrageous Legacy. Mostly a non-mathematical look at time travel through wormholes and the like, it's a fascinating read that investigates various time travel possibilities along with their inherent problems and paradoxes. Following one of his promotional lectures (I think it was one of the Leon Pape Lectures), I got the chance to talk to Thorne about time travel, quantum field theory, relativity, and life in general. But when I asked him if he himself believed in time travel, I got an elusive answer. [By the way, I took along a copy of Thorne's 1965 first book Gravitational Theory and Gravitational Collapse. He signed it for me, and told me that he still gets a miniscule royalty check from the publishers each year from the half-dozen or so copies that they sell.] Thorne and his British pals and colleagues Stephen Hawking and Roger Penrose are arguably the world's foremost authorities on time travel. But many years ago, our good friend Hermann Weyl also speculated on the time-travel possibilities associated with a rotating universe: It is possible to experience events now that will in part be an effect of my future resolves and actions. Moreover, it is not impossible for a world line (in particular, that of my body), although it has a time-like direction at every point, to return to the neighborhood of a point which it has already once passed through ... In actual fact the very considerable fluctuations of [the metric tensor gμν] that would be necessary to produce this effect do not occur in the region of the world in which we live. Although paradoxes of this kind appear, nowhere do we find any real contradiction to the facts directly presented to us in experience. In fact, Weyl took the rather extreme view that the very concept of time is illusory and an inherent debilitation of the human mind. He once famously remarked The objective world simply is, it does not happen. Only to the gaze of my consciousness, crawling upward along the world-line of my body, does a section of the world come to life as a fleeting image in space which continuously changes in time. Before you dismiss these words as overly metaphysical, consider the remarks of St. Thomas Aquinas (or was it St. Augustine?), who "knew what time is except when asked what it is." What really is time? Einstein treated it as the fourth dimension, but it's obviously not just another coordinate. My guess is that we will never really know until we stand before God. But will time still exist then? The concept of time not existing (which must have been the case "before" the Big Bang) is not so far-fetched because if time doesn't exist there can be no causal loops or other time-related paradoxes. A universe without time might not be such a bad place. At least we wouldn't get old and decrepit! On a perhaps more realistic note, consider this. Let's say that you want to travel back to Northside Square in Bolivar, Missouri at exactly 5:00 pm CST on November 5, 1955 (like Professor Brown in Back to the Future). You get into your Way Back machine, set the dials on the flux capacitor, and hit the "go" button. An instant later, you materialize in the vacuum of empty interstellar space, where you quickly decompress and die. What went wrong? The problem is due to the fact that Earth was not at the location you've traveled to in time. Because the Earth is rotating on its axis and moving around in the solar system (which is also moving in the Milky Way Galaxy, which is also moving relative to the Local Cluster), you don't want a time machine. What you want is a spacetime machine, a device that will take you both when and where you want to go. This means that you have to know the precise spacetime coordinates of your destination, otherwise you're likely to end up in airless space or physically embedded in a tree or mountain. To get these coordinates, you need to have the world line (4-D history) of the travel-to location. But where are the world lines of all physical objects and locations in the universe maintained? I don't believe that UFOs are Little Green Men; I prefer to believe that (if they exist at all) they are time-traveling historians or scientists who, for whatever reason, prefer to remain undetected. If way-back time travel is at all possible, future travelers in their spaceships will know that it is far safer to approach Earth from space, where any imprecision in their spacetime coordinates won't matter (unless they happen to materialize within an asteroid, but then space travel is an adventure, isn't it?) The best reference I've found on the subject is the 1993 book Time Machines by Paul J. Nahin, a professor of electrical engineering at the University of New Hampshire. It's still available in paperback, and it's excellent.  Hermann Weyl and CPT Symmetry -- Posted by wostraub on Saturday, December 15 2007 As I noted earlier, I managed to locate a copy of Hermann Weyl's Zaum — Zeit — Materie: A General Introduction to His Scientific Work, and I'm still making my way through it. A considerable portion of the book is in German which — contrary to popular belief — I am only moderately fluent in, so it's taking some time. Weyl seems to have been in love with group theory, especially the continuous Lie groups (SU(2), SU(3), and all that) and he appears to have concentrated the latter part of his life on this topic, along with other subjects in pure mathematics. His earlier interests in general relativity, cosmology and philosophy seem to have waned during this time of his life, and it leads me to wonder how his immigration to America in November 1933 might have affected his professional inclinations. At any rate, I find Weyl's mathematics (unlike his physics) to be very difficult and hard to follow. But there is one thing that jumps out of the book at me, and that involves the following questions: Why does mathematics describe the physical world so well? Why should Nature obey the laws of mathematical symmetries? Why does group theory govern so much of what goes on in the world? These are hardly new questions, but the answers seem to be just as far from us today as they were in Weyl's time. Even more intriguing is the fact that Weyl, in 1929, was able to deduce that Nature should also obey the discrete symmetries described by charge, parity and time (CPT) invariance. Even today, we haven't the slightest idea why God decided that these symmetries should carry so much power and influence over Nature. In a later edition of his 1928 book Group Theory and Quantum Mechanics, Weyl wrote The problem of the proton and the electron is discussed in connection with the symmetry properties of the quantum laws with respect to the interchange of right and left [parity invariance], past and future [time invariance], and positive and negative electricity [charge invariance]. At present, no acceptable solution is in sight; I fear, that in the context of this problem, the clouds are rolling together to form a new, serious crisis in quantum mechanics. Weyl's analysis of the Dirac and Maxwell equations in the context of combined CPT symmetry led him to the correct conclusion that the mass of the then newly-discovered positron (the anti-electron) should be identical to that of the electron. The "crisis" that Weyl referred to involved the then-prevailing opinion that the positron should be nothing more than the familiar proton, whose mass exceeds that of the electron by a factor of almost 2,000. Of course, we now know that there was no crisis at all. But the reason why CPT symmetry mandates these kinds of physical consequences remains a total mystery.  God and the Many-Worlds Interpretation of Quantum Mechanics -- Posted by wostraub on Monday, December 3 2007 I apologize for this overly-long post. This month’s Scientific American (it’s becoming an oxymoron, isn’t it?) has a fascinating article about Hugh Everett III, the late physicist whose 1956 PhD dissertation formalized the idea of parallel universes. And once again we see the hand of the noted physicist John Archibald Wheeler (see my earlier post), who was Everett’s academic advisor at Princeton University. Wheeler, apparently enthralled by the multi-universe concept, went to Copenhagen to discuss the idea with the great Niels Bohr, no less, who, unfortunately, didn’t like the theory. Wheeler returned to Princeton and coerced Everett into paring down his dissertation to make it more conventionally acceptable. The PhD, which was finally awarded to Everett in 1957, was a shadow of its former self, but it nevertheless succeeded in bringing the idea of parallel universes into the scientific world. Hugh Everett, 1930-1982 Everett’s idea (which is now called the many-worlds interpretation of quantum mechanics) is pretty bizarre. But for reasons I will explain shortly, it deserves serious consideration. The basic idea is very simple. The Schrödinger equation says that the wave function Ψ of an object represents a superposition of possible physical states (rather like a complicated musical sound wave that consists of a linear combination of individual waves). The wave function is a complex-valued quantity, which means that it is essentially unobservable until a measurement is made. Then it collapses probabilistically but uniquely into one of its allowable states; the resulting state is a real-valued quantity, and is the one we humans can actually observe with our eyes and ears. Wave function collapse is one of the central pillars of what is known as the Copenhagen interpretation of quantum mechanics. But since its development in the late 1920s, the Copenhagen interpretation has dogged physicists with the whole collapse idea. How does observation (which can be a very “gentle” process) bring about collapse? Does it take an intelligent entity to collapse a wave function? Can a mouse collapse a wave function? Why can’t we see a quantum-mechanical superposition of states? Why just one? These aren’t just idle musings; they involve the very foundation of that slippery thing we call reality. Everett was also bothered by these questions, and he considered what might happen if a measurement doesn’t collapse a wave function. The conclusion was inescapable — incredulous as it sounds, if you flip a coin and get heads, then the universe must split off another universe in which a parallel YOU gets tails. Similarly, if you measure the energy state of a free particle (which has an infinite number of superposed energy states), an infinite number of universes is created, one for each possible observation. In both cases, the wave function remains intact and uncollapsed, and the only price we have to pay for this is our sanity! (Everett went even farther than what I've sketched above. His original PhD thesis dealt with a continuously-evolving wave function for the entire universe, and included the observer as an intrinsic quantum participant in the overall observation process.) The main reason this SciAm article aroused my interest is because I just finished reading Frank Tipler’s latest book, The Physics of Christianity. Tipler, a highly respected (and apparently even sane) mathematical physicist at Tulane University, wrote a similar (and more mathematical) book called The Physics of Immortality, and his latest effort is a slightly more refined follow-up. Basically, Tipler’s thesis is this: the Trinity of God, Jesus Christ and the Holy Spirit is all that Christianity says it is, and Everett’s many-worlds interpretation (MWI) proves it. Assuming you are a relatively sane person, even a casual glance at Tipler’s writings will lift your eyebrows. I don’t for a minute accept a lot of what he says, despite really, really trying to follow his mathematical logic. But here’s something to consider: the majority (60%) of the world’s notable physicists believe (often grudgingly) that the MWI is either certainly true or probably true (see this reference). A few of their comments: I think we are forced to accept the MWI if quantum mechanics is true. — Richard Feynman, Physics Nobel Laureate I don’t see any way to avoid the MWI, but I wish someone would discover a way out. — Leon Lederman, Physics Nobel Laureate I’m afraid I do [believe in the MWI]. I agree with John Archibald Wheeler, who once said that it is too much philosophical baggage to carry around, but I can’t see how to avoid carrying that baggage. — Philip Anderson, Physics Nobel Laureate The MWI is okay. — Murray Gell-Mann, Physics Nobel Laureate The MWI is trivially true. — Stephen Hawking For what it is worth [I prefer the MWI over the Copenhagen interpretation]. — Steven Weinberg, Physics Nobel Laureate To this list, Tipler adds another authority: Jesus answered, “My kingdom is not of this world.” — John 18:36 Tipler could have also added: “In my Father’s house are many mansions …” — John 14:2 I need to shut this down now, but lastly consider this: Tipler believes that God is not a magician, only (only!) an eternal and very clever physicist and mathematician who has figured everything out. If we can believe that God fashioned Eve out of a rib bone he yanked from Adam, we can surely believe in Everett’s many-worlds theory. I urge you to read The Physics of Christianity and decide for yourself.  Weyl, Wheeler and Wormholes -- Posted by wostraub on Sunday, December 2 2007 I've been reading Hermann Weyl's Raum-Zeit-Materie and a General Introduction to his Scientific Work, a neat collection of articles by noted Weylophiles Erhard Scholz, Skuli Sigurdsson, Hubert Goenner, Norbert Straumann, Robert Coleman and Herbert Korte. Having recently re-read Kip Thorne's book on wormholes, I was struck by a comment made by Coleman and Korte regarding Weyl's supposed "discovery" of the wormhole idea. On Pages 198 and 199 of the book the writers provide a short list of Weyl's accomplishments, including his "invention of the wormhole concept in connection with his analysis of mass in terms of electromagnetic field energy." Since Thorne does not even mention Weyl in his book, I pulled Weyl's Space-Time-Matter off my shelf and went through it with a fine-toothed comb. Yes, Weyl talks at length there about electrodynamics and the problem of matter (and there's some discussion of "world canals" in Section 36), but I'll be damned if I can find anything remotely related to the wormhole concept. Thorne has demonstrated that wormholes almost certainly cannot exist but, if they do, they would require a kind of negative-pressure exotic matter to keep them from collapsing. Nowhere in Thorne's book do I see any primary role for electromagnetism in relationship with this exotic matter. I can imagine that, when Karl Schwarzschild wrote down the first exact solution to Einstein's gravitational field equations in 1916, the concept of a black hole (a term coined by John Wheeler in 1967) may have crossed his mind. However, black holes were quickly dismissed in those early days, and it is not hard to suppose that the idea of a wormhole (a term also coined by Wheeler in 1957) had not even been dreamed about. I give Weyl credit for many wonderful ideas, but I don't think wormholes can be included on that list.  Not Even Wrong -- Posted by wostraub on Monday, November 5 2007 Wolfgang Pauli (1900-1958), the Austrian physics Wunderkind of the early-mid 20th century, often intimidated younger, inexperienced physicists by declaring their ideas ganz falsch, or "utterly wrong." Those who he really zeroed in on suffered the rather more blistering comment nicht falsch, or "not even wrong." Not Even Wrong is the title of Peter Woit's poison pen-letter to string theory (and also the title of his fascinating website). Woit, a noted Columbia University physics lecturer who likens the untestable string theory to a kind of religion, feels that the theory's promise to unite the four fundamental forces of nature is nothing more than hope disguised as hyped progress. Although the Standard Model of physics successfully unifies all of quantum theory with electrodynamics, it does so at the expense of assuming all kinds of physical constants that it cannot account for. But its most glaring oversight lies in the fact that it cannot incorporate Einstein's gravitation theory into the mix. To date, the Standard Model is 100% accurate in terms of its predictions of experimental quantum results, but it can tell us nothing about gravity. Over the past 90 years, gravity has steadfastly refused to associate itself with quantum theory despite the efforts of literally thousands of physicists, including Einstein himself (who spent the last 30 years of his life in the effort). The curmudgeonly Pauli himself also tried in vain, and finally declared that "what God hath put asunder, let no man join." Woit's book is a great introduction to the Standard Model, including quantum field theory, but his description of the details of string theory is necessarily lacking, if only because the theory's mathematics is maddeningly difficult. But as simplistic as it is, Woit's book has made me wonder if the ideas of Truth and Beauty, which I have always assumed to be identical, truly hold up. Although my own understanding of the mathematical details of string theory is limited, the parts I do understand are truly beautiful, and like many others I have tacitly assumed that string theory is too beautiful a concept for God to have overlooked. But Woit warns us not to be overly impressed with Beauty alone, because it does not necessarily represent Truth. I had often suspected this, noting the concept of broken symmetry in quantum mechanics — if God's physical laws were perfect, then quantum symmetry could not be broken. It seems that although God started out with a great idea, he found it impractical — some imperfection is needed in the universe, if only to make things interesting. It goes without saying that God made mankind imperfect, but I believe he did this intentionally in order to give us free thought. Exactly why God gave us this gift or, for that matter, why he even gives a damn about us, is a profound mystery. Woit considers string theory to be an "ossified ideology," and recommends that scientists now move on toward a fuller understanding of quantum field theory and its relationship to mathematics. Will string theory prove to be a waste of time and effort? Even if it is, it at least has given us a glimpse into the mind of God, which probably cannot be understood anyway.  Insects and Worldlines -- Posted by wostraub on Sunday, October 28 2007 2005, the "Year of Physics," brought about the appearance of I don't know how many more books on Einstein, no doubt inspired by the 100th anniversary of Einstein's annus mirabilis, 1905, the wonder year in which the 26-year-old Swiss patent clerk cum world-renowned scientist produced four papers that would forever change physics. Now another book has appeared. Albert Einstein: The Persistent Illusion of Transience (edited by Ze'ev Rosenkranz and Barbara Wolff), is too slim (264 pages) to qualify as a coffee-table book, but its high-quality photographs of its subject more than make up for the book's brevity. I'm not sure that it really adds anything that we didn't already know about the man, but it's nice to see that people are still interested in him and his science. Einstein used to quip that his fame grew out of his awareness of something that had escaped most people (and insects): When the blind beetle crawls over the surface of a world globe, he doesn't realize that the track he makes [a "worldline" or geodesic] is curved. I was lucky enough to have spotted it. In their very comprehensive (and, at nearly 1,300 pages, very long) 1973 foundation text Gravitation, Misner, Thorne and Wheeler also spotted it, this time using the analogy of an ant crawling over the surface of a piece of fruit ("The Parable of the Apple"): It was the very first graphic in this book (above) that caught my eye one day in 1975, when I spotted the text on the shelves of the miniscule public library in Lone Pine, California. Widely viewed as the standard graduate-level text on general relativity, I wondered how in hell it had landed in a tiny town whose only claim to fame was that, as the portal to Mt. Whitney, it had hosted Humphrey Bogart and company during the filming of the 1940 classic High Sierra. The book was my companion on a day-long hike up the 14,000-foot mountain during that glorious summer that I discovered the miracle of general relativity. It also brought me closer to God, whose miracles and wonders I continue to marvel at.  The Fox and the Forest -- Posted by wostraub on Tuesday, October 2 2007 Roger and Ann Kristen are government scientists, developing leprosy bombs and other high-tech disease-culture weaponry for a war that never seems to end. Unlike most American patriots in the fascist country of the United States in the year 2155, they hate what they do. They hate the killing, the fear-mongering, the torture, the constant propaganda, the all-pervading culture of death. But they play along, and eventually they are rewarded with a vacation courtesy of the government-sponsored Travel in Time, Inc. Of course, they have to put up a bond and leave all their assets in government hands as assurance they will return after their time vacation. But where to go? They decide on New York City, 1938. But Roger and Ann have no intention of staying in New York, nor do they plan to return to their own time. They run off to Mexico City, where they adopt the names Bill and Susan Travis. They've managed to take with them a small fortune in travelers checks, and plan to live out their lives in total anonymity, away from the horrors of 2155. They carefully erase all evidence of their escape, hoping they'll find peace in another place, another time. Unfortunately for these little rabbits on the run, there's also a fox in the forest of time. This is the premise of Ray Bradbury's brilliantly disturbing 1950 short story, The Fox and the Forest, which is reprinted in his collection of short stories, The Illustrated Man. It may even be on the Internet somewhere. At any rate, you should read it. Bradbury is now 87 years old. I saw him frequently at the old Vagabond Theatre in Los Angeles in the late 1960s, where they used to run science fiction movies and silent films. He was a friendly, approachable guy who clearly loved his medium, which was mostly science fiction, fantasy and eccentric horror. Bradbury's The Fox and the Forest, The Lake, and The Small Assassin are irreducible masterpieces. I wish there were more writers of his caliber today. Or, at the very least, more writers with a moral sense. Where would I go? Probably Europe in the mid-1920s. Or Victorian England. Dear God in Heaven, anywhere but here and now.  Wolfgang Panofsky Dead at 88 -- Posted by wostraub on Thursday, September 27 2007 Stanford University's Wolfgang Panofsky is dead. The father of Stanford's linear electron accelerator and one of the discoverers of the neutral π meson, Panofsky was also noted for his abhorrence of nuclear weapons and their proliferation. I remember a soft-spoken, kindly, balding Panofsky at a 2004 lecture he gave in Los Angeles. His entire talk was about nuclear proliferation and ways of reducing or eliminating the spread of nuclear weapons, and I recall being touched by the compassion the man felt toward humanity and the underlying sadness he felt about the seeming inevitability of mankind's willingness to wage war and the role that nuclear weapons would ever play in that insanity. Panofsky was born in Berlin in April 1919. A family of intellectual Jews, the Panofskys left Nazi Germany in 1935 fearing for their lives. Wolfgang's father, a noted art historian, took up teaching at Columbia University and the Institute for Advanced Study. Wolfgang and his older brother were of high school age in Germany but were accepted at Princeton, where Wolfgang majored in physics. Graduating at the age of 19, he then went to Caltech in 1938 after receiving a personal invitation from the school's president, Nobel Laureate Robert Millikan. Panofsky received his physics PhD in 1942 but, being a native German, was declared an enemy alien under California's Alien Exclusion Law. Millikan came to his defense, however, and Panofsky was granted naturalized citizenship. He then went on to consult for the Manhattan Project in New Mexico, where he personally witnessed the Trinity bomb test from a B-29 bomber on July 16, 1945. During a stint at UC Berkeley during the McCarthy era, Panofsky abruptly resigned after being coerced into signing a loyalty oath. He subsequently made his way to Stanford, where he distinguished himself not only in groundbreaking accelerator physics but in world peace. He was instrumental in developing the Atmospheric Test Ban Treaty of 1963 and, in 1972, the Antiballistic Missile Treaty. He leaves his wife of 65 years, Adele, five children, eleven grandchildren, and two great-grandchildren. There are so few of the great physicists like Panofsky still alive today. God bless him, and may we all meet with peacemakers like him in heaven.  The Connection (Γ αμν) Again -- Posted by wostraub on Tuesday, September 18 2007 In 1918 Hermann Weyl tried to unify gravity and electromagnetism by a generalization of Riemannian geometry. He did this by eliminating the notion that the magnitude of a vector is invariant with respect to parallel transport. In doing so, he was forced to identify the electromagnetic 4-potential with a non-zero covariant derivative of the metric tensor. Subsequent to this effort, numerous other prominent physicists tried their hand at the unification game, which at the time was simplified by the fact that only two forces were then known — gravitation and electrodynamics. Einstein, Kaluza, Eddington, Pauli and Schrödinger each took their turns and, ultimately, their lumps. Weyl’s effort remains notable for the fact that the geometry that describes his unification is invariant which regard to a local gauge variation of the metric tensor; this idea failed, but in 1929 Weyl applied the gauge concept to quantum mechanics, where it found a home. But why does the gauge idea work for the wave function and not the metric tensor? The most obvious answer has to do with the fact that the wave function Ψ(x,t) is a complex-valued quantity whose meaning is clear only when its conjugate square Ψ*Ψ is taken. Even then, this square (though real) can only be understood as a probability. By comparison, the wave function by itself is at best a probability amplitude. The metric tensor gμν, on the other hand, is a purely real quantity that needs no “squaring.” Similarly, the invariant line element ds2 = gμν dxμdxν, which measures the interval between events in spacetime, is also a real quantity. On the basis of Einstein’s criticism that the line element itself should be invariant with respect to gauge variations (but isn’t in Weyl’s geometry), Weyl decided to adjust the metric tensor via an exponential scale factor gμν → exp [ k ∫φμ dxμ ] gμν where k is a constant and φμ is the Weyl vector (which he associated with the 4-potential). Weyl knew that in quantum mechanics this vector was a complex quantity; consequently, the adjusted metric tensor and the line element could be made gauge invariant by a suitable choice of the constant k. Thus, it is (gμν*gμν)1/2, and not gμν, that must be taken as real. The Weyl scale factor makes for some interesting physics, but its presence in Lagrangian actions introduces an integral term that is hard to interpret (it actually prevents the derivation of classical, tried-and-true equations of motion). Eddington was aware of this defect, and in response he decided that the metric tensor should not be taken as the fundamental quantity. Instead, he chose to develop a theory based on the affine connection, which defines the parallel transport of vectors (the concept of a connection was first proposed by Cartan, and later expanded by Weyl). In Weyl’s original theory the connection term has φμ embedded in it, which makes the connection complex-valued. (Indeed, the terms making up the connection are to a large extent arbitrary; the connection only collapses to the usual Christoffel definition when a Riemannian manifold is imposed.) This renewed focus on the connection term motivated Einstein and others to consider a connection that is non-symmetrical in its two lower indices. Indeed, the so-called theory of the non-symmetrical field occupied Einstein for the last decade or so of his life. Most physicists today consider the theory to have been a tragic waste of the great scientist’s time and effort. The connection term is still an open topic in mathematical physics and differential geometry. If we do not impose the demand of a Riemannian manifold, its precise makeup is largely arbitrary. Is this how quantum effects enter into gravitation, as Weyl and Einstein had hoped? Probably not, although it can be argued that a connection describing internal spaces, possibly in multiple spacetime dimensions obeying higher gauge symmetries, may yet find application in the description of a consistent quantum gravity theory.  Ramanujan -- Posted by wostraub on Saturday, September 8 2007 I just finished reading Robert Kanigel's award-winning 1991 book The Man Who Knew Infinity: A Life of the Genius Ramanujan. The book's great length stands in stark contrast to the very brief life of its subject, the largely self-taught Tamil mathematician Srinivasa Ramanujan, who died in 1920 at the age of 32. Ramanujan's genius was saved from obscurity by the noted British mathematician Godfrey Hardy, who brought the 25-year-old to Trinity College in 1913 and served as the younger man's mentor until Ramanujan's death by tuberculosis seven years later. Although devoted to Ramanujan, the book is almost equally a tribute to Hardy who, unlike many other noted scholars in his circle, saw Ramanujan as an equal and not as a talented but inferior person of color. The book does not overlook the profound tragedy of genius cut off at an early age, and the author ponders what heights Ramanujan might have attained if he had lived longer. Ramanujan was particularly adept at evaluating truly complicated improper integrals, and I could not help but wonder what luck the mathematician might have had with the infinite-dimensional path integral of quantum field theory, which can only be solved perturbatively. A devout Hindu, Ramanujan saw a divine hand in all mathematical expressions. "An equation for me has no meaning," he wrote, "unless it expresses a thought of God." UPDATE. There's a new book out based on the life of Ramanujan.  Why Gödel Thought US Dictatorship Possible -- Posted by wostraub on Tuesday, September 4 2007 In his 2005 book A World Without Time, Brandais University philosophy professor Palle Yourgrau writes Years later, asked for a legal analogy for his incompleteness theorem, [Gödel] would comment that a country that depended entirely upon the formal letter of its laws might well find itself defenseless against a crisis that had not, and could not, have been foreseen in its legal code. The analogue of his incompleteness theorem, applied to the law, would guarantee that for any legal code, even if intended to be fully explicit and complete, there would always be judgments "undecided" by the letter of the law. If this is indeed how Gödel felt, then he unequivocally predicted that an event like 9/11 could plunge the United States into a dictatorship, an outcome that the Founding Fathers simply could not have foreseen. [Gödel's later years were plagued by paranoia and hypochondria. Fearing that he would be poisoned by hospital doctors, he stopped eating and died in 1978 of self-imposed starvation. At the end, he weighed 65 pounds.] We Americans are always bragging about how brilliant the Founding Fathers were in drafting the US Constitution. But I believe that Gödel was absolutely right -- the Founders could not have foreseen that their country would utilize an event (largely brought upon by itself) as an excuse to give the president dictatorial powers. And this is exactly what has happened.  Kurt Gödel and the US Constitution -- Posted by wostraub on Friday, August 31 2007 I noted in my previous post that in 1949 the brilliant Austrian-American mathematical logician Kurt Gödel had discovered a solution to Einstein's field equations that allowed for time travel. His discovery was presented to Einstein on the occasion of the latter's 70th birthday party. (See my September 25, 2005 post for more info.) Kurt Gödel and friend, early 1950s I neglected to mention that a year earlier Gödel believed he had discovered a logical inconsistency in the US Constitution that allowed for the establishment of a dictatorship in America -- and told a federal judge about it! The story, which is true, has Gödel traveling by car with his Princeton colleagues Albert Einstein and economist Oskar Morgenstern to Trenton, New Jersey, where Gödel was to be sworn in for his US citizenship. During the drive, Gödel expressed his concern that an inconsistency in the US Constitution allowed for a dictatorship to be imposed on the American people. Einstein and Morgenstern told him not to worry about it. The attending federal judge had earlier sworn in Einstein, and he invited the distinguished trio into his chambers for a pre-swear-in chat. The judge happily informed Gödel that, unlike war-time Germany, a dictatorship could never happen in America. At this point an agitated Gödel blurted "Yes, it can! I've discovered a loophole in the Constitution that allows for a dictator to take over the country!" or words to that effect. Einstein and Morgenstern were able to defuse the situation, however, and Gödel was duly sworn in. I've heard this story many times, but I've never heard the basis for Gödel's argument. Some think it's Article 5, which allows for amendments. Others think it involves the establishment of executive powers. But I'm not a lawyer, and despite a careful reading of the Constitution I can't even imagine what might have concerned Gödel. But I fear he was right all along. (I'll omit my usual anti-Bush rants, as you all probably know of which I speak.) Anyone know more about this story? If you're an armchair Constitutional theorist, I'd be happy to hear from you. UPDATE: Several readers directed me to this: New Yorker Article, but it still doesn't explain why Gödel thought the Constitution was flawed.  Good Bye to Clocks Ticking -- Posted by wostraub on Thursday, August 30 2007 I can't go on. It goes so fast. We don't have time to look at one another. I didn't realize. So all that was going on and we never noticed. Take me back — up the hill — to my grave. But first: Wait! One more look. Good-by, Good-by, world. Good-by Grover's Corners ... Mama and Papa. Good-by to clocks ticking ... and Mama's sunflowers. And food and coffee. And new ironed dresses and hot baths ... and sleeping and waking up. Oh, Earth, you're too wonderful for anybody to realize you! Do human beings ever realize life while they live it? — Every, every minute? ... I'm ready to go back ... I should have listened to you. That's all human beings are! Just blind people. — Emily Webb to the Stage Manager in Thornton Wilder's Our Town The newly-deceased Emily got her wish to travel back in time to witness her 12th birthday. Did Weyl ever wonder about time travel? Indeed, he did. Thirty years before Kurt Gödel's 1949 discovery that a rotating universe could enable travel backward in time, Weyl wrote It is possible to experience events now that will in part be an effect of my future resolves and actions. Moreover, it is not impossible for a world-line (in particular, that of my body) — although it has a time-like direction — to return to the neighborhood of a world-line point which it already once passed through. The result would be a spectral image of the world more fearful than anything the weird fantasy of E. Hoffmann [an eccentric 19th-century German writer] has ever conjured up. In actual fact the very considerable fluctuations of the components of the metric tensor needed to produce this effect do not occur in the region of the world in which we live. Although paradoxes of this kind appear, nowhere do we find any real contradiction to the facts directly presented to us in experience. No doubt, Weyl (like Einstein) did not believe in super-luminal velocities, so that mode of time travel to the past was verboten. Also, Weyl probably never heard of wormholes, so that idea was out, too. That left motion about the spacetime surrounding a rotating massive body. Although Weyl died eight years before the physicist Roy Kerr discovered the exact metric describing a spherical, chargeless rotating mass, he was aware of the theoretical work of Lense and Thirring, who in 1918 were able to deduce the approximate field of a rotating body. Today, this effect is called frame-dragging. Weyl knew that the field of a sufficiently massive body undergoing a high rate of rotation would cause the light cones of a test particle moving in the direction of rotation to tip over in the same direction, thus creating what is known as a closed timelike curve. Timelike, because the body never travels faster than light, and closed because the rotating field brings the particle back into its own past light cone. The net result — backward time travel (maybe). Weyl thus realized, as far back as 1918, that matter not only warps spacetime, but that rotating matter "drags" spacetime along with it. Gödel's discovery only confirmed this effect. But this is just science fiction, right? Many physicists today don't think so. The dynamics of an object in free-fall within the dragged spacetime of a massive spinning black hole are now well-known, and they are bizarre. What is not known is what ultimately happens to the object. Does it emerge from the black hole's ergosphere into another place and time? Or does it eventually fall into the singularity, to be crushed out of existence? University of Connecticut physicist Ronald Mallett thinks that he might have a clue as to how a table-top time-travel device could be constructed using a circular rotating beam of laser light, which theoretically produces dragged spacetime within its interior (to see his short and very readable paper, go here. ). Mallett with prototype device, circa 1960! Mallett, whose father died at the age of 33 due to a heavy smoking habit, decided at an early age to become a physicist so he could go back in time and save his father. Mallett no longer believes this is possible, but his fascination with the concept of time travel has continued to this day unabated. So it is with many of us! Since black holes result from the collapse of spinning stars and the accretion of rotating matter, it is hardly an overstatement to say that all black holes spin and so have angular momentum (neutron stars, the closest cousins of black holes, can have measured spin rates of hundreds and even thousands of revolutions per second). Therefore, frame dragging (and all its associated odd phenomena) is the rule rather than the exception in this wonderful, strange place that God created for us.  Weyl Letter with Autograph -- Posted by wostraub on Tuesday, August 28 2007 If you're interested in getting your own autograph of Hermann Weyl (I have several), have a look at this offering on eBay. The letter was sent to Artur Rosenthal, a mathematician at Heidelberg University. Like other professors of Jewish descent, he was summarily fired by the Nazis in 1933. By 1938 he was probably desperate to get out of Germany. Weyl tried to get him a job at Princeton. I don't know what became of him. It's going for about40 now, but my guess is it will top $100 by auction's end. Good luck! [It sold for$158. Ouch.]

 Expanding Spacetime -- Posted by wostraub on Thursday, August 23 2007 Some time ago I was contacted by Johan Masreliez, who has developed a theory of expanding spacetime somewhat along the lines of what Hermann Weyl had proposed. But while Weyl assumed that the metric tensor could be appended by a non-integrable 4-dimensional scale factor, Masreliez' theory assumes that the metric involves a factor that instead involves a global time factor alone. General relativity is a classical theory, and one of its primary tenets says that there can be no global time marker. Nevertheless, cosmological models like the Robertson-Walker metric have provided theoretically important descriptions of the behavior, evolution and fate of the universe. So, I try to remain objective. However, Masreliez' theory predicts that black holes do not exist. While it is important to keep in mind that black holes have never been directly observed, a universe devoid of these objects deviates so radically from current cosmological thought that it really makes me doubt that Masreliez is on the right track (also, I've been in love with black holes for 40 years). Still, Masreliez' theory leads to some pretty interesting things. So again, I try to remain open minded. You can download Masreliez' book on his website. It's a fairly straightforward read, and I recommend it.

 Four Neutrino Flavors? -- Posted by wostraub on Tuesday, July 17 2007 Hermann Weyl was perhaps the first physicist to posit the existence of the neutrino. At first it was only a mathematical prediction. In 1930, Pauli proposed the neutrino in order to preserve mass-energy conservation. Twenty-five years later, it was found experimentally. Still later, two more types of neutrino were discovered following Weyl's death in 1955. In the 1990s it was discovered that the three types of neutrino can oscillate into one another or "mix." That is, a muon neutrino could be "caught" as an electron neutrino, and so forth. Because neutrinos are now known to have small but different masses, they can exist as a superposition of three mass eigenstates. That picture may now be changing. The July 2007 edition of Scientific American includes a summary of the efforts by Fermilab researchers and others to confirm very tentative evidence to date for a fourth neutrino. The Standard Model currently allows for only three -- the electron, muon and tau -- all of which participate in the weak interaction. But there is some leeway for a fourth species (dubbed the sterile neutrino), with the provision that it not interact with the weak force. If it exists, the sterile neutrino would interact only with gravity. This scenario is in line with current string theory predictions in which the sterile neutrino (like the graviton) can weave in and out of multi-dimensional branes. One result of this mobility allows the sterile neutrino to influence the flavor mixing of the other three, which are supposedly bound to the four-dimensional "braneworld" in which laboratory observations are made. For a relatively simple explanation of neutrino mixing and how the sterile neutrino might fit into the scheme of things, see this article by Fermilab researcher B. Kayser.

 Smolin on String Theory -- Posted by wostraub on Friday, July 13 2007 I just finished reading Lee Smolin’s The Trouble with Physics Amazon Books, in which the renowned quantum physicist bewails the impending failure of string theory. As a string-questioner myself (actually, I don't get most of the theory's math at all), I think it's a wonderful book! Smolin is quick to point out that it’s not technically a theory, because it cannot be tested. It’s more like a hunch. Meanwhile, theoretical physics now finds itself in a desert, its greatest achievements well behind it, with little more than string theory to cling to. And it all started with Hermann Weyl, to whom this often-annoying website is devoted. Smolin credits Weyl as the originator of the “unified theory” craze that caught up Einstein, Pauli, Heisenberg, Schrödinger and many others from 1918 until about the 1960s. String theory then picked up where the old unified theories left off, and it has been just as unsuccessful. The so-called Standard Model of physics, known more affectionately as SU(3)×SU(2)×U(1), reached its zenith in the 1980s and 1990s, when the predicted weak-interaction particles Z0, W+ and W- were discovered (1985) and the top quark was finally detected (1995). Since then: very little, with the possible exception of the notion (Smolin calls it a discovery) that neutrinos have mass. No wonder, he notes, that the world’s smartest physicists are hitching their stars to string theory. Smolin, late of Yale and Pennsylvania State and now at the Perimeter Institute, is no less an accomplished string theory expert himself. But he sees little beyond the theory’s beautiful mathematics and the allure of extra dimensions (seven at last count, not including the 3+1 of good old spacetime). Without experimental verification, it’s really nothing more than a religion without even any Gospels to back it up. He quotes physics Nobelist Gerard t’Hooft: Imagine that I give you a chair, while explaining that the legs are still missing, and that the seat, back and armrests will perhaps be delivered soon. Whatever I did give you, can I still call it a chair? But Smolin isn’t just complaining. He points out that there are other ideas out there that might beat out strings as understandable and experimentally verifiable unified theories: loop quantum gravity, spin networks (see my post of earlier today) and various spacetime-background-independent approaches to quantum gravity. So there’s optimism to be had, but Smolin nevertheless regrets the thousands of physicists and untold academic resources that are currently being expended in the (possibly futile) search for strings. Jesus Christ once said that there are many mansions in his father’s house (John 14:2). I still think he was referring to the many-worlds interpretation of quantum physics, in which there are an infinite number of universes awaiting us after death. I don’t personally see a need for many dimensions, and until string theory is completely played out (hopefully in my lifetime), I will side with Smolin.

 Spin Networks -- Posted by wostraub on Friday, July 13 2007 For those of you who are interested in an easy introduction to spin networks, John Baez has posted a write-up by Roger Penrose on some of the simpler details. You civil engineers out there who have done finite-element modeling (structural dynamics, groundwater transport, pipe networks, etc.) should find this easy going. Spin networks involve combinatorial methods that preserve certain quantities at each vertex, although the details are more complicated. Here's a somewhat related problem for you engineers. If you can solve it, you will become famous and probably very well-off. Large finite-element grids involve very sparse admittance or coefficient matrices whose components are based on the way the grid nodes are numbered (sparse matrices have many zeros in them). Sparse matrices are good, as they reduce computer storage requirements and computational effort. All solution algorithms involve some method of inverting these matrices in an efficient manner. If you take a square sparse matrix and invert it, chances are it will no longer be sparse. But by simply renumbering the grid, you can increase the sparseness of the inverted matrix. The sparseness of the inverted matrix will always be equal to or less than that of the coefficient matrix. Example: the following graphs show the results of before-and-after vertex renumbering. The renumbering increases the sparseness of the inverted coefficient matrix by a factor of three (trust me): Problem: develop an algorithm that produces the optimal renumbering of the grid nodes such that the sparseness of the inverted matrix is as large as possible. Hint: Based on my playing around with the problem many years ago, the solution likely involves combinatoric extremalization of the pure number N = , where A is a square coefficient matrix (aij = 1 if nodes i and j are connected, 0 otherwise) and x is the numbering vector, which starts out as [1, 2, 3 ...]. Warning: discrete extremalization is much more difficult than continuous extremalization. You can't just take a derivative and set it equal to zero! Those of you who have investigated the "traveling salesperson" problem will see a parallel here. Bell Labs has worked on this problem for many years, as it's involved in how digital communications are routed efficiently. Thousands of brilliant scientists and mathematicians have not been able to come up with an optimal solution. Indeed, it is not known whether such a solution even exists. But maybe you can do it. Oh, and yes, I have no life to speak of.

 "We do not know what death is ..." -- Hermann Weyl -- Posted by wostraub on Monday, July 2 2007 Last week, Peter Roquette, Professor Emeritus of the University of Heidelberg, posted a comprehensive and very moving description of the personal and professional relationship between Hermann Weyl and Emmy Noether (whom you can read about in my Weyl-Higgs write-up). Roquette, a mathematician, has written extensively about the mathematical correspondence between Noether and the German mathematician Helmut Hasse. You can Google him if you want more information. Noether and Weyl Article Included in Roquette's insightful article is the full text of Weyl's funeral dedication to Noether on April 18, 1935, which contains We do not know what death is. But is it not comforting to think that our souls will meet again after this life on Earth, and how your father’s soul will greet you? Has any father found in his daughter a worthier successor, great in her own right? [Note: Noether's father was himself an esteemed professor at the University of Erlangen, and justifiably proud of his daughter's substantially greater mathematical abilities.] It also includes Weyl's moving but fruitless petition to have Noether retained as a professor in Germany in the summer of 1933, when the Nazis summarily fired all scholars of Jewish descent or heritage. In coming to Princeton as a German emigre himself in late 1933, Weyl selflessly endeavored to obtain a position for Noether at the Institute for Advanced Study as well. This was denied (possibly because of the school's antisemitic attitude), although she did find a position (at reduced salary) at Bryn Mawr.

 Veltman on Elementary Particles -- Posted by wostraub on Wednesday, June 27 2007 A non-scientifically-minded friend of mine recently pointed out to me that, in accordance with Einstein's E = mc2, the energy available to mankind must be nearly infinite. He reminded me of the scene in Back to the Future where Doc Brown replenishes the power source of his flying Delorean with a few banana peels and a shot of stale beer, throwing in the beer can for good measure. I had to explain to him that Brown's act violates all kinds of conservation laws, not to mention the fact that nobody knows how to convert the energy of ordinary matter into pure energy. Instead, I asserted, Einstein's famous equation is useful mainly as a mass-energy accounting tool, not a prescription for free energy from trash. By far the best book I've seen to date that explains all this in a straightforward and (mostly) non-mathematical manner is Martinus Veltman's 2003 book Facts and Mysteries in Elementary Particle Physics, admittedly not the kind of book my friend would be picking up at Barnes & Noble anytime soon. The 1999 Nobel Physics Laureate, Veltman (curiously, his Christian name is the same as that of my late aunt's!) is Professor Emeritus at the University of Michigan, although he originally hails from Utrecht University in the Netherlands, where he worked on weak-interaction physics. Veltman was the PhD advisor of Gerardus t'Hooft (co-recipient of the 1999 Nobel with Veltman) who, as a lowly Utrecht graduate student in 1971, proved that all gauge theories are automatically renormalizable. This would have made Hermann Weyl very proud, indeed. I have one other book by Veltman, 1994's Diagrammatica: The Path to Feynman Diagrams (paperback). Mathematically, it's a readable, mid-level text that introduces canonical quantization from first principles using creation/annihilation matrices whose properties are so neat, they're actually fun. The Almighty Creator (who undoubtedly knows these matrices intimately), is not only the greatest physicist but is also entertainingly practical in the extreme. Anyway, if you're interested in modern elementary physics and want the best resource available on the subject at the layperson's level, you can't go wrong with Veltman's book. It explains everything from quarks and gluons to hadrons and their antiparticles on up, their interactions and their conservation principles, along with brief but fascinating sketches of many famous physicists. Equally enjoyable is Veltman's rather strange and often hilarious use of the English language.

 Krauss on Extra Dimensions -- Posted by wostraub on Tuesday, June 26 2007 Case Western Reserve University's Lawrence Krauss is a leading particle physicist and cosmologist, and he has written a number of excellent books (including The Physics of Star Trek, which I thought was rather silly, but that's another story). His most recent book, Hiding in the Mirror, discusses the subject of extra dimensions and why they hold so much allure nowadays. Krauss with friend Krauss ends his book with Hermann Weyl's "Truth/Beauty" quotation, and he graciously credits Weyl as the guy who essentially started the entire extra-dimensions craze. Krauss seems to be not so crazy himself about string theory, which proposes that we live in an eleven-dimensional "membrane" world. Krauss feels that, because string theory cannot (as yet) be demonstrated experimentally, it is really no different than a religious belief. I do not know what faith (if any) Krauss practices, but he is also a leading proponent of reason over nonsense (he is especially critical of early creationism and the right wing's continued attacks upon evolution), although that, too, is another story. It is true that Weyl's 1918 geometric gauge theory, like Einstein's general relativity theory, involved only four dimensions, but his work provided the stimulus for Theodor Kaluza's five-dimensional theory, which was worked out in 1919. But all of these guys owed a tremendous debt to the German mathematician Bernhard Riemann, who in the 1860s developed the mathematical basis for all their work. Krauss' book does not mention Riemann, a curious oversight in a book dealing with extra dimensions. As perhaps the greatest mathematician of the 19th century, Riemann was no stranger to multiple mathematical dimensions. Riemannian geometry, perhaps Riemann's greatest achievement, is the basis of modern geometrodynamics and, if Riemann had lived a few more years, he might have trumped Einstein and everybody else. Sickly for most of his life, Riemann was born in 1826 and died of tuberculosis at the tragically-young age of 39, not long after developing his geometry. He was convinced that his was the "true geometry of the world," and believed it could be used to describe all physical phenomena. His initial efforts failed, but it was only because Riemann was stuck in three dimensions. If he had only been gifted with Einstein's foresight to view time as the fourth dimension, the general theory of relativity (gravitation) would have undoubtedly appeared around 1870, 45 years earlier than Einstein's opus of November 1915.

 Adieu to Schrödinger -- Posted by wostraub on Wednesday, May 23 2007 I finished Moore's book on Schrödinger and found it to be a fascinating account of not just Schrödinger's life and work but a glimpse of how the physicists of his day struggled to make sense of the emerging quantum theory of the mid-1920s. It's interesting to note that Schrödinger initially wanted to believe that the wave function Ψ was a purely real quantity, despite the fact that it was embedded in his complex wave equation (actually, it's a diffusion equation, but what the hell). It's also notable that Hermann Weyl, Schrödinger's best friend, helped enormously with the mathematics. In my opinion, it should have been called the Schrödinger-Weyl equation. In early 1927, Erwin and his wife Anny were invited to Cal Tech in Pasadena. Anny found Pasadena "unbelievably beautiful, like a great garden." The sentiment was echoed by Schrödinger, who loved the Southern California climate. The great Dutch physicist Henrik Antoon Lorentz (Einstein's idol) was also visiting at the time. It's neat to think that these great scientists might very well have driven down my street (Orange Grove Boulevard) exactly 80 years ago. Schrödinger remarked to his host, the noted Cal Tech Nobel laureate Robert Millikan, that he wished Pasadena were populated by Italians or even Spaniards, not Americans, although he felt they were considerate to a degree quite unknown in Germany at the time. This is understandable, as Schrödinger, who had recently visited New York City, hated the place and thought Americans to be uncultured.

 Schrödinger: Life & Thought -- Posted by wostraub on Tuesday, May 22 2007 I managed to find a library copy of Walter Moore's Schrödinger: Life and Thought and am in the process of reading it. Erwin Schrödinger was Hermann Weyl's best friend (from their days together at the ETH in Zürich until Weyl's death in 1955), and I thought this book would provide additional information on Weyl. Yes, it did. Moore relates the notoriously open relationship that the otherwise devoted Schrödinger and Anny (his wife of 41 years) practiced, which was due primarily to Erwin's predilection for extramarital affairs. Schrödinger's intellectual abilities seems to have been matched only by his libido, and he had many lovers, even into his old age. Anny herself had her share of paramours, including Weyl (whom she called Peter): Anny would find in Hermann Weyl a lover to whom she was devoted body and soul, while Weyl's wife Hella was infatuated with Paul Scherrer [another ETH physics professor]. This relationship was confirmed in, of all things, a friendly letter from Anny to her husband Erwin in 1936: Even if the love between Peter and me should sometime come to an end, I would always be blessed that it had formerly existed, as I know that fate has given me the greatest happiness that a person can ever be given. But the best part of the book (so far) is the story behind Schrödinger's famous wave equation, and how he came across it late in 1925 (and even this story involves an amorous romp between Erwin and an unknown former love in Arosa, a secluded Alpine resort). Amazingly, Erwin and Hermann remained best of friends until Weyl's death in 1955. And when Schrödinger's heart finally stopped at age 73 on 4 January 1961, Anny was there to give him a farewell kiss. Go figure. If I find anything else interesting in the book, I'll report on it later.

 Biggest Supernova Ever Seen -- Posted by wostraub on Tuesday, May 8 2007 A team of astronomers from the University of California at Berkeley has discovered an enormous supernova in the galaxy known as NGC 1260. It exploded in September last year, producing the most massive outpouring of energy ever witnessed. (Actually, because this galaxy is 240 million light years away, the star blew up 240 million years ago.) The supernova, designated as SN 2006gy, had an estimated energy output of 1045 joules, enough to outshine the star's entire galaxy of perhaps 200 billion stars. It's bigger than anything ever seen, and its output has been remarkably persistent: The pre-nova mass of SN 2006gy is estimated to have been about 150 solar masses. That's truly enormous, because stars that big are notoriously unstable and have extremely short lives. But most supernovas blow off only a fraction of their total mass into space, leaving a neutron star or black hole behind. Scientists believe SN 2006gy blew up completely, which would explain why the explosion's energy was so great. Any chance of such a supernova occurring in our Milky Way? Well, there's an unstable, 100-solar-mass star known as Eta Carinae about 7,500 light years away from us that scientists say will probably do the same thing. Its light output would be so great that the supernova could be seen during the day, but it would pose no hazard to life on Earth. On the other hand, star explosions known as gamma-ray burstars are far more dangerous; if one went off within several hundred light years, most life on Earth would be extinguished (President George W. Bush was recently overheard saying "We just gotta get one a them things fer the Department of Defense"). Here's the Berkeley paper. It's about 10 pages long and somewhat technical, but very readable. Eta Carinae underwent a colossal false nova event in 1843 which almost destroyed the star. It survived, but remains the best candidate to date for a SN 2006gy-like explosion. The dumbbell-shaped ejecta cloud streaming out of the region now dwarfs the central star itself. Frightening.

 Feynman's Thesis -- Posted by wostraub on Wednesday, May 2 2007 I finally got around to reading Laurie Brown's book Feynman's Thesis, which, to the best of my knowledge, is the only publicly available version of Feynman's 1942 PhD dissertation. I was astonished to find that Feynman's thesis, which details his discovery of the path integral, is understandable, fun to read, and short -- incredibly, the document's only 68 pages long. (Mine was the exact opposite -- at 225 pages, it was incomprehensible, boring, and long.) I have long been fascinated by Feynman's idea, which represented an entirely new approach to quantum mechanics. In fact, it represents another quantum theory altogether. Basically, Feynman said that a particle goes from Point A to Point B along an infinite number of different paths, or "histories." It can travel forward and backward in time, at any velocity, do loops, interact with virtual particles, and cross the entire universe an infinite number of times. Every path, no matter how improbable or illogical, is just as important as any other path. Each path is assigned a probability amplitude* that by itself says nothing. But when you combine these amplitudes, you find that the infinite number of paths available to the particle shrinks enormously. In classical physics, the path that a particle settles into is the classical path -- a straight line in flat space, or a parabolic arc in the presence of gravity. Several years ago I tried to explain this in my write-up Introduction to Quantum Field Theory. I wrote the path integral part off the top of my head, and today I was pleased to find that it was remarkably similar to Feynman's treatment in his dissertation. Now if only I could understand all that Feynman did in the 46 years following his Princeton thesis! * The probability amplitude of an event is a complex number whose norm (its "square") is a real number. The physicist Nick Herbert describes things this way: the wave function Ψ is a probability amplitude, also called a "possibility," while its square Ψ*Ψ is a real number, the "probability."

 The Most Beautiful Thought -- Posted by wostraub on Saturday, April 28 2007 Right after his discovery of special relativity in 1905, Einstein was looking for a more general application of his theory -- one that included gravitation. I think it was in 1908 or thereabouts that he had what he later described as the most beautiful thought of his life: if a person were to to fall off the roof of her house, she would not feel her own weight during the fall. Einstein instantly realized that the phenomenon known as gravity could be transformed away by a suitable change of coordinates, and he began to look for a description of relativity that was independent of any particular coordinate system. This led him ultimately to his 1915 theory of general relativity, which is cast in the coordinate-invariant language of tensor calculus. I was instantly reminded of this little story of Einstein's when I read about Stephen Hawking's recent experience with weightlessness. The 65-year-old Lucasian Professor of Mathematics at Cambridge University boarded the Vomit Comet and undertook half a dozen parabolic "plunges," each of which rendered him weightless and floating for 20 to 25 seconds. He remarked later that he greatly enjoyed a brief chance to escape his wheel chair and experience Einstein's beautiful thought first hand. Although I believe that purely recreational flight (like billionaires paying millions to go into orbit) is a ludicrous waste of resources, I'm happy for the guy (and I'm sure Einstein was with him in spirit).

 Weyl's 1929 Paper -- Posted by wostraub on Tuesday, April 24 2007 I once remarked that I planned to review Weyl's 1929 paper "Electron and Gravitation," which formally presented Weyl's gauge idea in the context of the then still-emerging quantum theory. While I've always found Weyl's physics to be straightforward, his mathematics tends to be rather obtuse and difficult to follow (at least for me). Complicating the matter is the fact that Weyl's notation is unfamiliar to many physicists (for example, Weyl expresses the tetrad as ea(β) rather than the purely tensorial index form eaβ). Maybe this is no big deal, but I have a hard time following stuff when the notation is odd. Lochlainn O'Raifeartaigh's excellent 1997 book The Dawning of Gauge Theory includes a translation and detailed clarification of Weyl's paper (with modernized notation), but it's still a tough read. Prof. Wulf Rossmann of the University of Ottawa recently contacted me on a different matter but then put me onto his own take of Weyl's paper. He sticks with Weyl's tetrad notation but to his credit clarifies Weyl's paper to a greater extent than O'Raifeartaigh. The interested reader will definitely want to visit Rossmann's website (which includes his book on differential geometry in downloadable pdf format, along with other online papers). Rossmann also points out a reference in Weyl's paper that has always puzzled me. In the the paper's first section, Weyl notes that his 2-component spinor ψ is unable to accommodate left-right parity; he correctly surmises that parity would therefore require another, independent 2-component spinor (Weyl even suggests that this spinor describes the proton). Weyl cryptically notes that he will address this issue in a "Part II," but he never mentions it again. Rossmann believes that Weyl never wrote Part II because Anderson's 1932 discovery of the positron (not to mention Chadwick's discovery of the neutron in the same year) compromised Weyl's ψ-connection in its relationship with the Weyl current j μ=ψ*σ μψ (σi are the Pauli matrices; σ0 is the 2×2 unit matrix). Thus, Weyl's hope for a theory of everything (which in those days was just gravity and electrodynamics) was quashed for good. Finally, Rossmann offers the following quote from Weyl's book Space-Time-Matter, which is all too true (but gives non-mathematicians like me little solace): Many will be horrified by the flood of formulas and indices which here drown the main idea of differential geometry (in spite of the author's honest effort for conceptual clarity). It is certainly regrettable that we have to enter into purely formal matters in such detail and give them so much space; but this cannot be avoided. Just as we have to spend laborious hours learning language and writing to freely express our thoughts, so the only way that we can lessen the burden of formulas here is to master the tool of tensor analysis to such a degree that we can turn to the real problems that concern us without being bothered by formal matters.

 Weyl's Pedestal Shaken a Bit -- Posted by wostraub on Tuesday, April 17 2007 In 1970 I took three courses in physical chemistry at university. The book we used was Walter Moore's text, appropriately entitled Physical Chemistry. Now a classic, it was, and still is, a tough book, especially the chapter on quantum chemistry. As I recall, we undergrads didn't like Moore very much. Over thirty years later, in 2001, Moore wrote a biography of the great Austrian physicist Erwin Schödinger called Schrödinger: Life and Thought, which was reviewed for the New York Times by Richard Teresi, the author of The Three-Pound Universe. Here is an excerpt from that review: Schrödinger's wave equation, published only a few weeks later, was immediately accepted as "a mathematical tool of unprecedented power in dealing with problems of the structure of matter," according to Mr. Moore. By 1960, more than 100,000 scientific papers had appeared based on the application of the equation. Schrödinger lavishly thanked his physicist friend Hermann Weyl for his help with the mathematics. (He was perhaps indebted to Weyl for an even greater favor: Weyl regularly bedded down Schrödinger's wife, Anny, so that Schrödinger was free to seek elsewhere the erotic inspiration he needed for his work.) Ouch. I'm not a prude, but this aspect of Weyl's personal life troubles me. I have read in numerous places that post-World War I Germany was sexually liberated, and I already knew that Einstein had more than a few sexual skeletons in his closet, but this hits close to home. I have not yet read Moore's Schrödinger book, primarily because a) I don't want to pay Amazon \$50 for the thing; b) it's not available from my library's interlibrary loan program; and c) I'd already had enough grief from Moore almost 40 years ago. But if I can lay my hands on a copy I'll you all know what I think of it -- and whether my opinion of Weyl changes any as a result.

 Arrhenius and Climate Change -- Posted by wostraub on Monday, April 9 2007 Svante Arrhenius was a Swedish chemist who, around 1895, was the first to quantify the relationship between chemical reaction rates and temperature. He won the Nobel Prize in Chemistry in 1903 for his work on electrolyte theory. Every undergraduate chemistry major learns how to derive Arrhenius' rate equation, but what most don't realize (as I didn't realize at the time) was that Arrhenius was the world's very first climate modeler. In 1894 he derived a remarkably accurate relationship between atmospheric carbon dioxide levels and global temperatures. Using only a slide rule, he calculated that a doubling of CO2 concentrations would raise the Earth's average temperature by 9.9oF. By comparison, today's modern supercomputers and vastly more accurate climate models predict an increase of 10.4oF. Arrhenius' story is recounted by environmental writer and journalist Fred Pearce in his new book With Speed and Violence: Why Scientists Fear Tipping Points in Climate Change. An erstwhile skeptic of doomsday climate-change scenarios, Pearce looks at all the evidence from all the angles, and comes up with a prediction: we're all in very big trouble. Even Arrhenius could not have foreseen the day when humans would be pumping out 8.2 billion tons of CO2 into the atmosphere annually, an amount far in excess of the planet's ability to absorb without global climatic consequence. From CO2 buildup and the breakdown of the Atlantic Conveyor to deforestation and the unavoidable release of tens to hundreds of billions of tons of frozen Siberian methane, Pearce paints a bleak picture for 21st century Earth and its inhabitants. While reading the book, I was at once struck by the words of Chris Hedges in a recent edition of New Statesman, where he notes that the real motivation of today's fundamentalist Christians is not religiosity but despair: ... The danger of this theology of despair is that it says that nothing in the world is worth saving. It rejoices in cataclysmic destruction. It welcomes the frightening advance of global warming, the spiraling wars and violence in the Middle East and the poverty and neglect that have blighted American urban and rural landscapes as encouraging signs that the end of the world is close. Those who cling to this magical belief, which is a bizarre form of spiritual Darwinism, will be raptured upwards while the rest of us will be tormented with horrors by a warrior Christ and finally extinguished. The obsession with apocalyptic violence is an obsession with revenge. It is what the world, and we who still believe it is worth saving, deserve. If true, then may God forgive us all.

 Hermann Weyl in Göttingen -- Posted by wostraub on Monday, April 2 2007 Here are some photos of Weyl talking at two separate colloquia taken in the early 1930s. The original caption on the first photo indicates that the lecture hall was filled to capacity, as the students at Göttingen did not know how much longer Weyl would be staying in Nazi Germany (Weyl's wife was Jewish, which jeopardized Weyl's entire family; they ultimately emigrated to America in November 1933). The second shows Weyl with the great German mathematician Richard Courant (far left). Permission to use photos courtesy of the Los Alamos Laboratory, Emilio Segrè Visual Archives (American Institute of Physics Photo Archive).

 Relativity for the Masses -- Posted by wostraub on Saturday, March 17 2007 The act of Taylor & Wheeler (MIT's Edwin F. Taylor and Princeton's John A. Wheeler, that is) has produced two books that you need to read if you want to learn special and general relativity quickly. The first is 1992's Spacetime Physics, a 300-page fun read that covers pretty much everything on Einstein's special relativity. The book's side bars feature two iconic characters (Rodin's The Thinker and a wise-cracking black crow) who help guide the reader through special relativity's often confusing concepts (these guys are much like Simplicio and Sagredo from Galileo's Dialogues). This book will also show you how to do some amazing calculations using only the flat-space metric ds2 = c2dt2 - dx2. For example, if you want to go to the Andromeda Galaxy in one year, all you have to do is build a spaceship capable of traveling at 0.999999999999875 times the speed of light*. Of course, your friends on Earth will be 2 million years older when you get there, but you can always make new friends in Andromeda. Also, President Bush will almost certainly be dead by the time you arrive. Break out the champagne! The second book is 2000's Exploring Black Holes: An Introduction to General Relativity. It follows the same format but deals with general relativity, the theory of warped spacetime and gravitation. The book focuses primarily on simplified presentations of the Schwarzschild and Kerr metrics (which respectively describe static and rotating black holes), which are really all you need to know about gravity. Thinker and Crow are back again to make sense of the mathematics, which requires a knowledge of elementary calculus. Wheeler, who will be 96 this year, is one of the few still living great physicists from the days of Einstein, Dirac, Wigner, von Neumann and Pauli. He knew them all personally, and he also knew Weyl well (you can read Wheeler's tribute to Weyl elsewhere on this site). Wheeler coined the term black hole in December 1967. Why he never won a Nobel Prize is a great mystery to many people. Interesting tidbit: Wheeler graduated from Johns Hopkins University in 1933 with a PhD in physics. And that's all he got -- he skipped getting his BS and MS. Talk about being focused! * The energy needed to bring every pound of your spaceship up to this speed is equivalent to approximately one billion Hiroshima-size atom bombs.

 The Natural Gauge of the World -- Posted by wostraub on Friday, March 16 2007 Sometime ago I wrote about Eddington and his unified field theory of 1921, the one that Weyl called "not worthy of discussion." But it was Eddington who came up with the phrase "natural gauge of the world," and in was in the spirit of Weyl's 1918 theory that he proposed it. In Eddington's theory the Ricci tensor Rμν, like Weyl's version, is constructed from a symmetric metric gμν and affine connection Γλμν. He then separates the Ricci tensor into its symmetric and antisymmetric components; the symmetric part is then made proportional to the metric tensor using a gauge scalar that, for all intents and purposes, is the cosmological constant. This gauge, asserted Eddington, represents the natural gauge of the world. The great Austrian physicist and all-round curmudgeon Wolfgang Pauli was appalled. He denounced Eddington's theory as having "no significance to physics" and expressed his resentment of having a mathematician poke his nose into the realm of the physicists (actually, Eddington was both), a criticism that Pauli later directed at Weyl regarding the latter's application of the gauge concept to quantum mechanics. But Weyl won his argument, with Pauli apologizing and admitting that Weyl had been right all along. Still, Eddington raised an important question: if the world permits (or demands) that metric spacetime be conformally rescalable from point to point, then what is the nature and consequence of that symmetry? Indeed, the British mathematical physicist Roger Penrose is very much enamored of conformal invariance, and he asserts that the Weyl conformal tensor Cλαμν (which lies at the heart of the Weyl Curvature Hypothesis, or WCH) is essentially responsible for entropy and the assumed asymmetry of the arrow of time. Penrose even goes so far as to say that the WCH must be an essential element of a successful quantum gravity theory, which itself must be time asymmetric. This results from the assertion that the Big Bang was a topologically distinct event in the history of the universe -- the Weyl curvature tensor was identically zero at the time, whereas it is positive now and will remain positive (or even become infinite) regardless of whether the universe continues to expand forever or eventually falls back on itself in a Big Crunch. You can learn a little more about Weyl's conformal tensor (and how to derive it) in my write-up on the menu to the left.

 Hermann Weyl -- The Centenary Lectures -- Posted by wostraub on Saturday, March 10 2007 I just finished reading Hermann Weyl, 1885-1985: The Centenary Lectures, three talks that were given by C.N. Yang, R. Penrose and A. Borel on the 100th anniversary of Weyl's birth. I've hunted all over for this book. Caltech didn't have it, or Berkeley, or UCLA, or Stanford; it finally turned up at Cal State Fullerton! Anyway, it's interesting because it includes a full bibliography of Weyl's works, which total 167 scientific and mathematical papers, 17 books, and a dozen or so lecture notes. Also interesting is the fact that he published only a handful of physics papers after his move to the Institute for Advanced Study at Princeton in 1933, the year he left Germany (by his own admission, Weyl's interests changed as a result of his emigration). The book includes transcripts of some of the speeches that were made at the Centenary Dinner held on October 24, 1985 at the Swiss Federal Technical Institute (the university where Weyl taught from 1913 to 1930). I was struck by the depth and breadth of Weyl's literary, poetic and philosophical interests, subjects that he knew intimately from the likes of Democritus, Leibniz, Kant, Mann, T.S. Eliot, Husserl, Russell, Kierkegaard, Nietzsche and Heidegger. How the guy ever had time to read all this stuff, while maintaining a reputation as one of the world's leading mathematical physicists, is simply beyond me. Best of all, the book includes the speech given by Weyl's son Michael, also a PhD, whom I've been trying to locate for some time now. It is a touching and heartfelt reminiscence of a son who fully appreciated having a father who was not only a "mathematician-father with the soul of a poet" but a truly learned man who passed on his passions to Michael and his mathematician brother, Dr. Joachim Weyl. In his speech, Michael included this poem by the minor poet Anna Wickham (1884-1947): God, Thou great symmetry, Who put a biting lust in me From whence my sorrows spring, For all the frittered days That I have spent in shapeless ways Give me one perfect thing. Hermann Weyl included this poem in his last book, Symmetry, in which he remarks that the sphere in space represents true perfection -- so perfect, in fact, that it inspires not only awed admiration but sorrowful longing as well, because it is a reflection of the perfect symmetry and unattainable perfection that is God himself.

 Weyl's Gauge Factor, Again -- Posted by wostraub on Monday, February 26 2007 Several people have written me asking how Weyl’s geometry might be modified using the nonintegrable gauge factor that I mentioned in my December 19 post. There I suggested that the line element ds2 could be made gauge invariant (or conformally invariant) via a “phased” metric tensor: gμν → exp[k∫φλdxλ] gμν where k is a suitable constant. In addition to the line element, all tensor quantities constructed from the new metric (the metric determinant, the Riemann-Christoffel tensor and its contractions, the Christoffel symbols, etc.) would then be automatically gauge invariant, as would all quantities raised and lowered using the new metric. Well, I had never considered this possibility, so I did a few calculations to see if it leads to anything interesting. For one thing, we can now define an action Lagrangian that is linear in the Ricci scalar density to get the free-space field equations: S = ∫√(-g) R d4x where the Ricci scalar R is constructed solely from the new metric. Here we are on familiar ground again, as the above quantity is the old Einstein-Hilbert gravitational action we all know and love. Thus, we neatly kill off the two objections Einstein held against Weyl’s theory: a non-gauge-invariant ds, and a fourth-order action Lagrangian. However, if we try to append the Maxwell action terms Fμν Fμν and sμ φμ (where sμ is the electromagnetic source four-vector) to the above Lagrangian density, we immediately run into a problem: sμ φμ is not gauge invariant! Of course, it’s not gauge invariant in any other theory, either, but here it’s particularly problematic. Making matters worse is the fact that the Weyl vector φμ does not exhibit a true gauge weight (instead, it's a gradient, making an integration by parts necessary, which really messes things up). This brings up an issue I’ve thought about for a long time: Just what the hell is the source vector, anyway? Classically, it’s just sμ = ρ(x) dxμ/ds where ρ is the electromagnetic 3-density. It seems like the only way to make this quantity gauge invariant is to consider its interaction with φμ and some quantum field, along the lines of Ψ*sμΨ, etc., and then impose gauge conditions on the wave function. I'll think about it, but I don't believe it goes anywhere. Suggestions?

 Squid Memories -- Posted by wostraub on Thursday, February 22 2007 Is it my imagination, or are giant squids (not Superconducting Quantum Interference Devices, but the scary ones with tentacles) showing up more frequently nowadays? New Zealand fishermen looking for fish in Antarctic waters accidentally hooked a Colossal squid weighing almost 1,000 pounds and measuring over 30 feet in length. Article The colossal squid, known even by elementary school children as Mesonychoteuthis hamiltoni (just kidding), is the larger (!) cousin of the more familiar giant squid (Architeuthis), which may grow to longer lengths but is rather more slender. I've had a life-long fascination with these things. When I was five, my father took me to see Disney's 20,000 Leagues Under the Sea (that was in 1954), and several years later my older sister took me to see It Came From Beneath the Sea. By then I was hooked, so to speak. Disneyland used to have a full-sized model of a giant squid in its 20,000 Leagues Exhibit in Tomorrowland. It was so realistic that as a child I had a really hard time going in there: In the 1980s, while scuba diving in extremely murky water off the Coronado Islands, some enormous, shadowy thing came up behind me. I turned, and as I did it quickly swam off, leaving me spinning head over heels in the backwash. The ascent back to the surface (I had been down about 100 feet) seemed to take forever. I never really got a look at it, but on the boat the divemaster told me it was only a harmless grey whale that had been seen swimming nearby. Nevertheless, that was the last dive I made that day.

 Lorentz Symmetry -- Still Asking -- Posted by wostraub on Friday, February 16 2007 It was Hermann Weyl who showed that quantum mechanical gauge invariance is the continuous symmetry responsible for the conservation of electric charge. Lorentz invariance (invariance of the Lagrangian with respect to Lorentz transformations) is also a continuous symmetry, so what conservation principle does it represent? I've addressed this problem before, but nobody was able to help me with it. Baez has been asked this same problem, but his answer is less than illuminating. In his excellent book The Road to Reality: A Complete Guide to the Laws of the Universe, British mathematical physicist Roger Penrose provides the answer. In non-relativistic mechanics, the quantity N = pt - mx where t is time, p is the 3-momentum and x is the position vector, is (obviously) invariant with respect to time. Penrose notes that Lorentz invariance is responsible for making the center of mass of a particle (or that of a collection of particles) move in a straight line, with velocity p/m. Pretty simple, isn't it? But I'll be damned if I know how to derive it! (short of using the Lorentz generators, as this guy does it). Penrose doesn't derive it either, but only says that Lorentz symmetry is a tad less common than rotational symmetry. Well, that's a big help. Penrose obviously doesn't take into account idiotic readers like myself.

 Juliet or Esmerelda? -- Posted by wostraub on Wednesday, February 7 2007 Unlike her happier fate in motion pictures, the gypsy Esmerelda in Victor Hugo's classic novel The Hunchback of Notre Dame dies by hanging. Her lovelorn admirer, the hideous hunchback Quasimodo, dispatches the corrupt Frollo but is unable to save the love of his life. Heartbroken, he disappears. Years later, grave diggers accidentally unearth the skeleton of Esmerelda, which is inexplicably embraced by that of a grossly deformed man. [And] when they tried to detach the skeleton which he held in his embrace, it fell to dust. Cut to February 2007. Archaeologists working in Mantua, Italy (about 25 miles south of Verona) uncover the skeletons of two Neolithic people embraced in death. The remains, those of a young man and a young woman, are estimated to be from 5,000 to 6,000 years old. Neolithic burials nearly always involve only a single skeleton. Evidently, there was something very special about these two people from long ago. Sadly, their story is lost to us. Article Coincidentally, Shakespeare placed his Romeo and Juliet in Verona, Italy. The ending to that story wasn't very pleasant, either. Something about a happy dagger that finds its sheath ... Touching.

 Famous Papers -- Posted by wostraub on Wednesday, January 31 2007 The website Trivial Anomaly provides links to a dozen or so sites where you can read or download seminal papers of famous scientists. Want to see a translation of Einstein's original 1915 paper on the general theory of relativity? How about Schrödinger's 1926 paper announcing his discovery of wave mechanics? It's pretty neat stuff. I'm also pleased that it includes my write-up of Weyl's 1918 gauge theory, perhaps because it's more detailed than the paper Weyl originally wrote (and because you don't have to know German to read it).

 Hermann Weyl in Exile -- Posted by wostraub on Wednesday, January 24 2007 I just finished reading Forced Migration and Scientific Change: Emigre German-Speaking Scientists and Scholars After 1933 (1996), a collection of articles edited by Mitchell Ash and Alfons Söllner. Of particular interest is the chapter Physics, Life and Contingency: Born, Schrödinger and Weyl in Exile by Sküli Sigurdsson, whose 1991 PhD dissertation on Hermann Weyl I'm still trying to run down. The book is not so much an overview of how emigrating German and Austrian scientists dealt with Hitler's rise to power in 1933 but a brief history of how their views on science, mathematics and philosophy were altered in the years immediately following the end of World War I, up to the time they left Germany. Sigurdsson's article provided me with a little more information on Weyl's state of mind in the years 1918-1933 than I'd seen previously. For example, his decision to accept the mathematics chair in 1930 at Göttingen when the great David Hilbert retired was not an easy one. He seems to have accepted it more out of nationalistic pride than for any other reason, believing that he needed to help Germany maintain its "thought collective" and promote its tradition of highest-quality science. Nevertheless, he announced his resignation in October 1933 after becoming depressed and disillusioned with the Nazis, the overall political and economic climate of Germany, and the resulting restrictions on scientific inquiry. During his years at Göttingen, Weyl's productivity had waned considerably and he was dissatisfied with the quality of science at the school. At his previous post at the Swiss Technical University in Zürich (where he was Mathematics Chair from 1913 to 1930), Weyl was the highest paid professor. His salary was increased at Göttingen, but was soon cut because of the school's rapidly deteriorating finances, the result of Nazi-imposed state cutbacks in higher (and mostly theoretical) education. In January 1933 Weyl received an offer from Princeton's new Institute for Advanced Study. But he was so depressed that he could not muster the strength to make the decision to accept (he seems to have had difficulty throughout his life making life-altering decisions, an observation that was later confirmed by the great Göttingen mathematican Richard Courant). However, Weyl's wife was Jewish, which placed both her and their two sons in jeopardy with the Nazis. This forced Weyl to accept Princeton's invitation, and the family left for America in November 1933. For Weyl, Born and Schrödinger, their forced emigrations (Born was Jewish and thus barred from teaching, while Schrödinger quit in protest over Born's firing; both men emigrated to England) brought about significant changes in their attitudes and philosophies regarding science and mathematics. Weyl retreated more and more into pure mathematics and pretty much abandoned his earlier interest in mathematical physics, particularly unified field theory. His book Classical Groups came out in 1939, while his earlier interest in physics and philosophy waned. And when he finally returned to Europe after his retirement in 1952, Weyl went to Zürich, not his beloved Germany. Sigurdsson's article is rather dry and humorless, but there is one funny anecdote worth repeating. Weyl suffered from asthma and hay fever, and Sigurdsson notes that Weyl's decision to go to the Institute for Advanced Study was compromised by the fact that he could not get affordable health insurance in America! Plus ça change, plus c'est la même chose ...

 Eddington -- Posted by wostraub on Tuesday, January 23 2007 Earlier I remarked that God cannot create a new integer between the numbers 1 and 10. Here's a little story about how one man tried to do essentially the same thing. Sir Arthur Stanley Eddington (1882-1944) was a British astrophysicist who, like Hermann Weyl, tried to develop a unified theory of gravitation, electromagnetism and quantum mechanics. His 1921 book The Mathematical Theory of Relativity (a copy of which I happen to own) praises the work of Weyl, whose ideas Eddington used to advance his own theory. But the physics community at the time roundly criticized Eddington's ideas; Weyl himself even went so far as to call them "not worthy of discussion" (undiskutierbar) in 1923. But it was also Eddington, who, on a solar eclipse expedition in 1919, took photographs of the sun and nearby stars and verified, rather sloppily, Einstein's prediction that gravity can bend starlight. Thus it was Eddington who made Einstein into an overnight scientific superstar. Anyway, as brilliant as he occasionally was, Eddington made one famous goof. There happens to be a fundamental, dimensionless constant in quantum physics known as the fine structure constant, which is defined as 2πe2/hc = ≈ 1/137 where e is the electron charge, h is Planck's constant, and c is the speed of light (the fact that this constant is very nearly the reciprocal of the prime number 137 has profoundly disturbed physicists for over 80 years). But in Eddington's day, uncertainties in the values of Planck's constant and the electronic charge made this number closer to 136. With characteristic aplomb, Eddington set out to prove that it was exactly the integer 136. By considering the magnitudes of certain quantities in an abstract phase space, Eddington came up with the number function ƒ(n) = ½ n2(n2 + 1) and, using some kind of reasoning, Eddington believed that ƒ(4) = 136 was the fine structure constant (Eddington used a similar argument to "prove" that the ratio of proton mass to electron mass was also an integer, 1836). Only several years later, it was determined that the fine structure constant was actually closer to 137. Not to be outdone, Eddington, admitting to an earlier algebraic oversight, revised the above formula by adding +1 to the right hand side, thus recovering the correct value for the constant. But the world's physicists were not to be taken as fools. They renounced Eddington's preposterous theory and, in mild rebuke, jokingly dubbed him "Sir Arthur Stanley Adding One." Note: Interestingly, the late, great German Nobel laureate physicist Max Born (who happened to be Olivia Newton-John's grandfather!) noticed that Eddington's formula reproduced two numbers from the New Testament Book of Revelation, Chapter 13: And I saw a beast coming out of the sea having ƒ(2) = 10 horns ... [and] his number is ƒ(6) = 666. I feel fairly certain that God did not use Eddington's formula when he inspired John to write Revelation! PS: Wolfgang Pauli (Nobel physics prize, 1945) was also fascinated by the fine structure constant, and he devoted much time and thought to its provenance. He passed away from cancer in 1958, and the number of the hospital room where he died was ... 137. Good one, God!

 Twenty Eighty -- Posted by wostraub on Monday, January 22 2007 My older son's girlfriend loaned me a copy of James Surowieki's book The Wisdom of Crowds. It's really interesting -- it pretty much destroys the idea that irrational mob rule is the norm, and shows how crowds of people may have differing points of view but, when the average is taken, it tends to be pretty close to the truth. The book's one caveat is that a crowd must have a fairly firm grasp of reality, otherwise mob rule does indeed take over, with disastrous results. You've probably heard about the 20/80 rule: 20% of the workforce does 80% of the work; 20% of people are difficult, while you can get along with the other 80%; 20% of the population is outright irrational, while 80% seem to have some grasp of what's going on, etc. Well, that seems to apply only to the rest of the world. Here in America, we have the 33/67 rule: 33% of all Americans are out of their friggin' minds. The most recent polls show that Bush's approval rating is now at 33%, the lowest for a sitting president since "I am not a crook" Richard Milhous Nixon occupied the White House. This is awful, but the flip side of the coin is that 33% of Americans still think Bush is doing a great job. I know such people. While perhaps not certifiably insane, they all seem to have the mindset for nonsensical and/or dogmatic thinking. They are also very suspicious of things they do not want to believe, while leaving themselves open to outright falsehoods that they do want to believe. And they tend to believe what they are told to believe. I once asked one such person (with two M.S. degrees in engineering, yet) if she believed it was possible for God to create a new integer between the numbers 1 and 10, or if scientists had somehow overlooked an undiscovered chemical element between sodium and magnesium in the periodic table. "Yes, of course," she replied, "because sin has blinded us from the truth." How does one begin to argue with such nonsense? The trouble is, I like a lot of these folks. Most are decent people, and share the same Christian values that I hold to. But many have allowed their values to become warped by political opportunists and liars. If Bush's popularity was 20%, he would undoubtedly be exposed as the monster he truly is. Impeachment and prosecution as a war criminal would probably follow. But at 33% he can hang on. 20/80 works, but 33/67 does not. America cannot survive when 33% of its people are crazy. And I fear that neither can the rest of the world.

 Weyl and von Neumann -- Posted by wostraub on Saturday, January 13 2007 There's a story that Hermann Weyl, when talking about his work at a conference or lecture hall, would become extremely agitated whenever colleague John von Neumann was in the audience. His nervousness was presumably due to the fact that von Neumann was widely viewed as a genius, and Weyl was afraid he'd make a fool of himself. A better word would be "respectful," because in actuality the two men were friends as well as colleagues. And while it is quite true that von Neumann was a mathematical genius, his brilliance extended into physics, economics and linguistics as well. For example, at the age of six he was fluent in Greek (along with his native Hungarian), and could divide two 8-digit numbers in his head within seconds. Later, von Neumann did pioneering work in computer science, and today is known as the father of the digital computer (see my December 12 post). There is a famous story involving von Neumann, apparently even true, that he was approached by the hostess of a party he was attending and given the following puzzle to solve: Two bicyclists on a road are 100 miles apart. At a predetermined time they begin pedaling toward each other, each with a uniform speed of 10 mph. At the moment they start out, a fly sitting on the wheel of one of the bicycles starts flying toward the other bicycle at a speed of 20 mph. Upon reaching the other bicycle, it instantaneously turns around and starts flying back to the first bicycle. It does this repeatedly until the bicycles meet in the middle of the road, squishing the fly between the tires. How many miles does the fly travel? [This story is so old that I am almost ashamed to repeat it.] There are two ways to solve the problem, but one way is immediate: the bicyclists meet at the midpoint after 5 hours of pedaling. The fly has been flying constantly during this time, so it flies a total of 5×20 = 100 miles. The second method involves calculating the infinite series L = 100/3 ∑ (2/3)n, where the sum is taken from n = 0 to n = ∞. Again, L = 100 miles. At the party, when asked for the answer, von Neumann instantly said "100 miles." The hostess smiled and said, "Oh darn, you know the trick." To which von Neumann replied, "What trick? I got it by doing the infinite series."