email:wostraub@gmail.com

AfterMath

Index photos courtesy ETH-Bibliothek,
Zurich Bildarchiv

Who Was Hermann Weyl?

Wheeler's Tribute to Weyl (PDF)

Old Stuff
2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016

Math Tools
Weyl's Spinor and Dirac's Equation
Weyl's Conformal Tensor
Weyl Conformal Gravity
Weyl's 1918 Theory
Weyl's 1918 Theory Revisited
Weyl v. Schrodinger
Why Did Weyl's Theory Fail?
Did Weyl Screw Up?
Weyl and the Aharonov-Bohm Effect
The Bianchi Identities in Weyl Space
Conformal, Parameter-Free Riemannian Gravity
Gravity Wave Tutorial
Conformal Kerr-de Sitter Gravity
A Child's Guide to Spinors
Levi-Civita Rhymes with Lolita
Weyl's Scale Factor
Weyl's Spin Connection
Weyl and Higgs Theory
Weyl & Schrodinger - Two Geometries
Lorentz Transformation of Weyl Spinors
Riemannian Vectors in Weyl Space
Introduction to Quantum Field Theory
A Children's Primer on Quantum Entanglement
Veblen and Weyl
Electron Spin
Clebsch-Gordan Calculator
Bell's Inequality
The Four-Frequency of Light
There Must Be a Magnetic Field!
Non-Metricity and the RC Tensor
Curvature Tensor Components
Kaluza-Klein Theory
The Divergence Myth in Gauss-Bonnet Gravity
Schrodinger Geometry
A Brief Look at Gaussian Integrals
Differential Forms for Physics Students
Particle Chart

Uncommon Valor

She did not forget Jesus!
"Long live freedom!"

Visitors since 11-4-2004:

2005 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 like 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.
 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.