Who Was Hermann Weyl?
Wheeler's Tribute to Weyl (PDF)

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

email:bill@weylmann.com

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. Schroedinger
Why Did Weyl's Theory Fail?
Weyl and the Aharonov-Bohm Effect
Weyl's Scale Factor
Weyl's Spin Connection
Weyl and Higgs Theory
Lorentz Transformation of Weyl Spinors
Riemannian Vectors in Weyl Space
Introduction to Quantum Field Theory
Electron Spin
Clebsch-Gordan Calculator
Bell's Inequality
The Four-Frequency of Light
There Must Be a Magnetic Field!
Kaluza-Klein Theory
A Brief Look at Gaussian Integrals
Particle Chart (Courtesy CPEP)

Uncommon Valor

Sophie did not forget Jesus
Long live freedom!

2949689 visits since 11/1/2004.

 My work always tried to unite the Truth with the Beautiful, but when I had to choose one or the other, I usually chose the Beautiful. Hermann Weyl I died for Beauty, but was scarce Adjusted in the tomb, When one who died for Truth was lain In an adjoining room Emily Dickinson

Hermann Klaus Hugo Weyl (1885-1955). German mathematical physicist. In 1918, proposed an early form of gauge symmetry in an attempt to unify electrodynamics and gravitation. Subsequently applied a similar approach to quantum physics and discovered what is today considered one of the most profound and beautiful concepts in modern physics -- the principle of gauge invariance.

Shortly after Einstein announced his theory of general relativity (gravitation) in November 1915, Weyl began an intensive study of the theory's mathematics and was soon publishing related scientific papers dealing with its physical applications. In 1918 Weyl published his book Raum-Zeit-Materie (Space-Time-Matter), which provided the first fully comprehensive analysis of the geometric aspects of the theory and its relationship with spacetime physics. One of the topics covered in the book was Weyl's idea that gravity and electromagnetism might both be derivable from a generalization of Riemannian geometry, the mathematical basis from which Einstein had developed his relativity theory. Weyl's idea was based on a new mathematical symmetry that he called gauge invariance.

I came across Weyl's book in 1975, but it didn't impress me very much because I didn't know general relativity. However, in the summer of that year I stumbled across Misner-Thorne-Wheeler's massive Gravitation during a one-week work assignment in the microscopic rural town of Lone Pine, California (which then had a population of perhaps 500 people). Miraculously, the town's tiny public library somehow had this book, which is now regarded as a classic graduate text on general relativity. I checked out the book and brought it back to the hotel room to read in the off-hours. The book took immediate and total possession of me, and motivated me to learn everything I could about general relativity. (I spoke with co-author Kip Thorne about this in 1994, and he was quite amused to learn where one of his books had ended up.)

But Gravitation is not an easy read, and I had to look for more introductory texts. I soon came across Adler/Bazin/Schiffer's Introduction to General Relativity, which besides being easier had a chapter on unified field theory, including Weyl's 1918 theory of the combined gravitational-electromagnetic field. For whatever reason, the theory's mathematical beauty absolutely fascinated me. I had known about local and global phase invariance from my studies of quantum mechanics, but I was not aware that Weyl's theory was the origin of this powerful symmetry principle in quantum physics.

I have since read all of Weyl's books and many of his papers. Although today I believe that my interest is now based more on an appreciation of modern gauge theory (easily the most profound and beautiful concept of physics), I credit Weyl for having initiated the idea in 1918 and for his subsequent (1929) seminal application of the idea to the then still-developing quantum theory.

In his 2002 biographical memoirs, the great contemporary mathematician Sir Michael F. Atiyah praised Weyl as the discoverer of the gauge concept and as the driving force behind the current emphasis of gauge theories on modern theoretical physics:

The past 25 years have seen the rise of gauge theories--Kaluza-Klein models of high dimensions, string theories, and now M-theory, as physicists grapple with the challenge of combining all the basic forces of nature into one all embracing theory. This requires sophisticated mathematics involving Lie groups, manifolds, differential operators, all of which are part of Weyl's inheritance. There is no doubt that he would have been an enthusiastic supporter and admirer of this fusion of mathematics and physics. No other mathematician could claim to have initiated more of the theories that are now being explored. His vision has stood the test of time.

Weyl was an exceptionally gifted mathematician and physicist, but he was also a highly cultured man in the classical German tradition. He studied and wrote extensively on philosophy and was a serious student of German poetry and literature. His mathematical writing style could be exceedingly obtuse, but his other writings reveal a genuinely warm person who truly understood the human condition. Weyl was also very human himself; he could be overly thoughtful and cautious, often to the point of being unable to take action or make even basic decisions, and sometimes with the result that he became physically incapacitated. He was a devoted and loving husband and father, yet he could also be persuaded to stray, in accordance with the surprisingly liberal attitudes of post-World War I Weimar society.

This website is my feeble attempt to document (and in many cases expand on) Weyl's ideas and thoughts on gauge symmetry in a manner that will be accessible to anyone with a basic understanding of calculus. Not a lot has been written about the original theory's underlying mathematics, and I wanted to provide a fairly detailed and complete mathematical description for those who want to learn about Weyl's ideas and to appreciate the beauty of his gauge theory (I'm even of the opinion that much of Weyl's work can be understood and appreciated at the high school/beginning university level). As this site progresses, I will also include discussions of other topics in mathematical physics (as well as some related scientific philosophy) which exhibit a similar mathematical beauty and elegance.

 In Search of Elizabeth — Posted Wednesday, 4 March 2015 The film is set in the life of the poet in the turbulent times in which he lived. In Episode One, we are introduced to the dark side of Queen Elizabeth I's police state — in a time of surveillance, militarism and foreign wars. This week PBS has been airing the much-lauded four-part series In Search of Shakespeare, filmed in 2004 and hosted by British historian Michael Wood, and I finally sat down and watched the whole thing. I'm kind of a history buff, and a sometime Shakespeare fan, though my son Kurt is the real expert, having read all his works. I tend to quote Hamlet a lot on this website, as if I know anything, but it's one work of Shakespeare I actually have read — many times — and turn to whenever I need an understanding of this stupid species called humankind. At the same time I am quite illiterate when it comes to Elizabethan history, and I think I learned far more from the PBS series concerning 'Good' Queen Bess than I did about Shakespeare. It was turbulent times, indeed, when the otherwise Catholic Henry VIII, infatuated with Anne Boleyn and seeking separation from first wife Catherine of Aragon, abandoned Roman Catholicism when the Pope denied Henry's request for a divorce and turned the country toward Protestantism. Sometime after his death in 1547, 'Bloody' Queen Mary took the country back to Catholicism, and bloody she was, imprisoning, torturing and murdering all those who stood in her way, particularly if they were Protestants. But after she died Elizabeth took the throne, and the country went back to Protestantism. She carried on her predecessor's bloody ways, only this time against Catholics, and it was in these times of religious and political tumult that Shakespeare was born in 1564. In those days England literally had an official state religion, though one might be hard pressed to know whether being Catholic or Protestant was a good thing. At any rate, the official state religion was just that, whatever it might be at the time, and religious observation was not only encouraged but mandated — you had to attend and donate to the church whether you were Catholic, Protestant, agnostic or atheist, although being one of the latter two would have seen you being fined or getting thrown into prison. But why am I really telling you all this? It's because the State of Mississippi now wants to establish Christianity as a state religion. Yes, I know, that would violate the 1st Amendment to the Constitution, but the Magnolia State never really got over the 13th Amendment abolishing slavery anyway, and the right-wing notion of state's rights is as powerful there today as it was in 1863 (and, presumably, in Queen Elizabeth's time). Mississippi also wants to establish English as the official language (Liz would be okay with that too), adopt "Dixie" as the official state song, and make April the official "Confederate Heritage Month" (April 26th is already recognized in most Southern states as "Confederate Day"). Yee haw! Surveillance, militarism and foreign wars. Dear God in Heaven, it's been over 400 years since the time of Elizabeth. Do we never fucking learn? You know, for some reason plane turbulence is always at its worst when I fly over Mississippi. It never fails — either leaving Arkansas heading east or leaving Alabama heading west, the plane always acts up. I think it's because Mississippi knows how I feel about the state, and it's her way of getting back at me.
 Several Roads — Posted Wednesday, 4 March 2015 The close similarity of the inverse-square laws of gravity and electromagnetism and that of the electromagnetic and gravitational potentials themselves has been known for hundreds of years. All school children are aware of this similarity, and most wonder if electricity and gravity are somehow the same thing. I recently returned from a long visit to the east coast, where my younger son just bought a house. As I've never experienced weather like the severe cold snap that coincided with my visit, my outings consisted only of numerous trips to Home Depot for house repair materials and a single two-mile hike in 13-degree weather with snow flurries. Outside of that I did a lot of reading, which included an interesting paper that Hermann Weyl wrote back in 1944. You can read it yourself here, if you have the energy (and that's a pun). In Weyl's 13-page "How far can one get with a linear field theory of gravitation in flat spacetime?", Weyl looks at the linearized form of Einstein's field equations, their relationship with Einstein's notorious 'pseudo-tensor' (which is used to get a mass-energy conservation law out of the field equations), and the connection between Maxwell's electrodynamics and gravitational waves. I found it amusing to note that, in getting essentially nowhere in the paper, Weyl (primarily a mathematician) defers to "the attention of physicists" regarding some of the paper's more theoretical remarks. He also uses the French term faute de mieux, implying that crazy theories often result from a lack of any better alternative. Nevertheless, gravitational waves almost certainly exist, and as such they are a form of radiation, and radiation is described by Maxwell's equations. Consequently, one could naively say that gravitation and electromagnetism must be unified at some level. Currently, there seem to be three approaches: non-Riemannian geometry, which somehow embeds the electromagnetic 4-potential $$A_\mu$$ into spacetime, à la Weyl's 1918 theory; gravitoelectromagnetism, which uses Weyl's conformal tensor $$C_{\mu\nu\alpha\beta}$$ to derive variants of the electric and magnetic fields, albeit in a rather goofy way; and quantum theory, which seeks a much more formal, non-classical unification of gravity and EM. (There is yet another description that Nordström, Reissner and Weyl derived in 1916-18, which is to use the electromagnetic energy tensor $$T_{\mu\nu}$$ in the field equations themselves to derive a Schwarzschild-type metric, but this is not really a unification.) Anyway, what Weyl was getting at in his paper is that unlike classical electromagnetism, in which Maxwell's equations are linear and well-defined in terms of conservation of electromagnetic energy, the gravitational field equations are neither linear (because a gravitational field, possessing energy, can act on itself) nor well-defined in terms of energy conservation. To fix this, Einstein had to linearize his equations (which required a weak gravitational field via $$g_{\mu\nu} \rightarrow \eta_{\mu\nu} + \epsilon h_{\mu\nu}$$) while proposing a tensor $$t_{\,\mu}^\nu$$ that has a vanishing ordinary divergence (thus making the field conserved). But Einstein's tensor is only a tensor in special coordinate systems, and so it is not really a valid gravitational quantity. His first task was successful — he was able to show that the quantity $$h_{\mu\nu}$$ satisfies the wave equation, proving that in the weak-field limit gravitational waves exist and that gravity propagates at the speed of light. But Einstein's pseudo-tensor $$t_{\,\mu}^\nu$$ was a total bust, of some limited theoretical interest today but wholly unsuitable as a description of gravitational energy conservation. Einstein and Weyl both went to their graves in 1955, having never solved the problems of unifying gravity and electromagnetism or gravitational energy conservation. These issues still have not been resolved and now, 100 years after Einstein's unveiling of the general theory of relativity, exactly what gravity is, how it fits into quantum theory and what its relationship to electromagnetism might be continues to represent the most pressing and exasperating problem of physics today.
 If English Was Good Enough for Jesus $$\ldots$$ — Posted Wednesday, 4 February 2015 The gods have imposed upon my writing the yoke of a foreign language that was not sung at my cradle. —Hermann Weyl, 1939 If English was good enough for Jesus Christ, it ought to be good enough for the children of Texas. — Attributed to Miriam Ferguson, first female governor of TexasNo one knows what the first formal spoken or written language was, although it was probably some variant of cuneiform or ancient Egyptian, but there is little doubt that Greek was the first language used to express higher scientific and mathematical thought. Much later was Latin, used first by the Romans and then much later by Isaac Newton and his crowd to express their technical ideas; Latin was the lingua franca of Newton's day. By the turn of the 20th century, scientists and mathematicians alike routinely published their work in French, German, Russian, Italian and English, and the likes of Einstein, de Broglie, Schrödinger and many others were fluent in several languages (even Hitler knew French). But today it's mostly English that's used. Why is that? As noted by Princeton University science historian Michael Gordin, English is used today in about 99% of published technical papers. While it wasn't always that way, polyglotism has given way to monoglotism. Gordin traces that change back to World War I, although he also notes that Hitler's summary expulsion of Jewish professors and intellectuals in 1933, which resulted in their emigration to America or other English-speaking countries, also contributed to monoglotism. But Gordin infers that the refusal or inability of Americans to learn other languages probably was the greatest factor in the world's adoption of English as the lingua franca of science and math. I can read French and German, but I'm not truly fluent in those languages. My excuse is that I don't use them frequently enough to be good at them. But I have family members who speak Arabic around me all the time, yet in nearly 40 years I've learned very little Arabic. So I suppose it's an American thing — we're monoglotic simply because we're basically lazy and expect others to accommodate us. "In France, oeuf means 'egg' and chapeau means 'hat.' It's as if those French have a different word for everything!" — Steve Martin
 Moving On — Posted Wednesday, 4 February 2015 As I've noted repeatedly on this website, Hermann Weyl's 1918 theory continues to show up regularly in physics journals and papers, though I believe it's quite wrong in its present form. That's not to say that the notion of conformal invariance is wrong (it's beautiful, and may actually be a valid symmetry, at least cosmologically), but Weyl's idea that the electromagnetic field is derivable from geometry simply doesn't appear to work, Einstein's arguments against it notwithstanding. Here I will take an approach different from Einstein's and show that any non-Riemannian geometry is unworkable — and by that I mean any tensor formalism in which the non-metricity tensor $$g_{\mu\nu||\lambda}$$ (that is, the covariant derivative of the metric tensor) does not vanish. It's a real problem, and I wish researchers would look at ways around it. Consider any symmetric affine connection $$\Gamma_{\mu\nu}^\lambda$$ that is not identically the Christoffel symbol of Riemannian geometry. It is easy to show that the familiar cyclic symmetry condition $$R_{\alpha\mu\nu\lambda} + R_{\alpha\lambda\mu\nu} + R_{\alpha\nu\lambda\mu} = 0$$ still holds, where $$R_{\alpha\mu\nu\lambda}$$ is the lower-index form of the Riemann-Christoffel tensor. It is also easy to show that the Bianchi identity $$R_{\,\,\mu\nu\lambda||\beta}^{\,\alpha} + R_{\,\,\mu\beta\nu||\lambda}^{\,\alpha} + R_{\,\,\mu\lambda\beta||\nu}^{\,\alpha} = 0$$ is also valid. The only other symmetry property is the antisymmetry of the last two indices of the RC tensor itself. When space is Riemannian we have two additional symmetry properties: $$R_{\alpha\mu\nu\lambda} = - R_{\mu\alpha\nu\lambda}$$ along with the index-pair exchange property $$R_{\alpha\mu\nu\lambda} = R_{\nu\lambda\alpha\mu}$$ Using these expressions, it is a standard undergraduate exercise to show that the Bianchi identity leads to the covariant conservation property of the Einstein tensor: $$\left( R^{\mu\nu} - \frac{1}{2} g^{\mu\nu} R \right)_{||\nu} = G^{\mu\nu}_{\,\,\,||\nu} = 0$$ Ever since Einstein came out with his theory of general relativity 100 years ago, this expression has been viewed as the fully-covariant version of mass-energy conservation. But it's not really a true divergence in the traditional mathematical sense, and the question of mass-energy conservation in general relativity has long been (and still is) plagued by confusion. However, for brevity we'll pass over this issue for the time being. For a general non-Riemannian space (like the one Weyl came up with), the non-metricity tensor $$g_{\mu\nu||\lambda}$$ no longer vanishes. But we still have the definition $$g_{\mu\nu||\alpha||\beta} - g_{\mu\nu||\beta||\alpha} = -g_{\mu\lambda}R_{\,\,\nu\alpha\beta}^{\,\lambda} - g_{\lambda\nu}R_{\,\,\mu\alpha\beta}^{\,\lambda}$$ or $$g_{\mu\nu||\alpha||\beta} - g_{\mu\nu||\beta||\alpha} = - \left( R_{\mu\nu\alpha\beta} + R_{\nu\mu\alpha\beta} \right)$$ Consequently, the antisymmetry property of the first two indices in the RC tensor no longer holds. Furthermore, it is also easy to show that $$g_{\mu\nu||\alpha||\beta} - g_{\mu\nu||\beta||\alpha} = \left( R_{\nu\beta\mu\alpha} - R_{\mu\alpha\nu\beta} \right) + \left( R_{\mu\beta\nu\alpha} - R_{\nu\alpha\mu\beta} \right)$$ and so now the index-pair exchange property of the RC tensor is gone as well. But this is a disaster: without these properties, we can no longer derive a divergence/conservation expression for the Einstein tensor $$G^{\mu\nu}$$. Indeed, without a specific definition for $$g_{\mu\nu||\lambda}$$, it is impossible even to have a formal theory of gravity anymore. In short, the non-vanishing of the non-metricity tensor ruins everything. But this did not stop Weyl from trying (although he likely never derived any of these expressions). In fact, he was never able to derive a mass-energy expression for his theory. While it did result in a geometrical form for the Maxwell electromagnetic source vector density $$\sqrt{-g} \,s^\mu = \sqrt{-g}\, g^{\mu\nu} \left( R \varphi_\nu + \frac{1}{2} R_{||\nu} \right)$$ (where $$\varphi_\nu$$ is the Weyl vector), it is not divergenceless. Consequently, the Weyl vector cannot be the electromagnetic four-potential $$A_\nu$$ and Weyl's $$s^\mu$$ cannot be the Maxwell source vector. While Weyl's 1918 theory doesn't work in its present form, ten years later it led him directly to the notion of gauge invariance in quantum theory, where it was a profoundly successful idea. In fact, today Weyl's idea is arguably the major cornerstone of quantum physics, and without it modern physics would likely be completely lost. More Fun — Although I love Roger Penrose's books and writing style, he has a rather nasty habit of setting complicated mathematics into unintelligible graphical format. Here is his "explanation" of the Riemann-Christoffel tensor and its symmetry properties. taken from Chapter 14 of his otherwise excellent book The Road to Reality: I finally worked out what this thing means schematically, but I can't figure out what it's good for. It certainly doesn't teach anyone what the Riemann-Christoffel tensor is or what it does. Maybe you'll have better luck with it.
 Generalizing General Relativity — Posted Thursday, 1 January 2015 This year marks the 100th anniversary of Einstein's general theory of relativity (gravitation), and I still find it remarkable that efforts to generalize the theory remain in vogue to this day. The non-Riemannian theory of Hermann Weyl (1918) was the first of many such theories, notable today if only for the fact that its gauge aspects and application to quantum physics a decade later became the foundation for all modern quantum theories. Einstein himself resorted to non-Riemannian formalisms in his own 30-year-long effort to unify gravity with electromagnetism and quantum physics. And yet, sadly, the hoped-for unification of gravitation and quantum mechanics today remains as elusive as ever. Ten years after Einstein announced his theory, mathematician Luther Eisenhart of Princeton University published an interesting book on non-Riemannian geometries, which to this day represent perhaps the oldest, purely-classical approach to generalizing the theory. Such geometries necessarily involve vector and scalar additions to the Riemannian connection term $$\Gamma_{\mu\nu}^\alpha$$ and, while at times fascinating, the approach is best summed up by the noted physicist and Einstein biographer Abraham Pais, who wrote "the use of general connections means asking for trouble." Indeed, there seems to be no end of problems with generalized connection coefficients, which at the very least invariably wipe out many of the cherished symmetry properties of the Riemann-Christoffel tensor $$R_{\,\,\mu\nu\lambda}^{\,\alpha}$$ while providing no discernible benefits. Weyl's 1918 theory arguably came closest to describing a consistent, all-encompassing theory (although it involved only gravity and electromagnetism), but his death in 1955 preceded a perhaps more lucid approach to generalization, which was the 1961 theory developed by Princeton's Carl Brans and Robert Dicke, who used a combination scalar-tensor approach within a wholly Riemannian geometrical framework. While this avoided the non-Riemannian pitfalls previously alluded to, it was rather ugly and unable to successfully reproduce the predictions made by Einstein's original theory. Over the ensuing decades the Brans-Dicke theory itself has been generalized by numerous researchers to include scalar and spinor quantum fields, producing a variety of even uglier theories that have yet to produce any meaningful predictions. As many people noted in 2014, scientific progress seems to be stuck in a rut. One hundred years after Einstein gave us his profoundly beautiful theory of gravity, there has been little real progress made on unified field theory or on any theory that describes gravity within the equally beautiful and profound mathematical formalism of quantum physics. Despite the ongoing brilliant work of scientists like Edward Witten, Stephen Hawking and others, we're no closer to a truly comprehensive description of Nature than we were in the days of Weyl and his fellow generalists. Think about that for a minute — scientists have routinely cracked the mysteries of Nature, often only after long and arduous searching, but in the case of gravity all of man's ingenuity has yet to provide any real progress. We still do not know if gravity involves extra dimensions, extra forces, strings, branes, supersymmetry, quantum loops or other fanciful notions, while the Large Hadron Collider, which supposedly disclosed the Higgs boson (or something very like it) in 2012, provided no useful clues. Gravity, undoubtedly the first of Nature's forces that mankind became aware of (often painfully), has turned out to be far more mysterious and elusive than we might ever have imagined.