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

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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
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Weyl and the Aharonov-Bohm Effect
Weyl's Scale Factor
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Lorentz Transformation of Weyl Spinors
Riemannian Vectors in Weyl Space
Introduction to Quantum Field Theory
Electron Spin
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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)

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 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.

 Exaggerating Eddington — Posted Thursday, 29 January 2015 On 29 May 2014 I posted an article on the 2009 BBC film Einstein and Eddington, which dramatized Eddington's 29 May 1919 solar eclipse expedition to Principe, an island on the west coast of Africa. I included a screen capture of the "evidence" of starlight deflection around the Sun (right photo, above), thinking it was an enlargement of one of Eddington's few good photographic plates (shown on the left). This morning I learned from Delft University's Leo Vuyk that the photo I posted is a probably a fabrication created by the filmmakers to exaggerate the deflection effect. I've actually seen the real photographic plates, which show only a few stars in the Sun's field along with tiny deflections, and I have a several good reproductions of the photos, but the one from the film is so patently false that I'm ashamed I didn't suspect "artistic license" from the filmmakers from the beginning. The starlight deflections that Eddington actually measured were indeed tiny, and his results just barely proved out Einstein's general theory of relativity (later, more careful photographic examination showed that Einstein's gravity theory was indeed upheld). As this article on the film notes, Einstein was a relative unknown prior to 1919, and Eddington's Principe expedition made Einstein a worldwide scientific superstar. It's still a good film, and noted Shakespearean actor David Tennant (Hamlet) is excellent as Eddington. But I promise to be more careful in what I post in the future.
 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.