Index photos courtesy ETH-Bibliothek,
Zurich Bildarchiv

Why I'm No Longer a Christian

Who Was Hermann Weyl?

Wheeler's Tribute to Weyl (PDF)

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

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. Schrodinger
Why Did Weyl's Theory Fail?
Weyl and the Aharonov-Bohm Effect
The Bianchi Identities in Weyl Space
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
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
A Brief Look at Gaussian Integrals
Particle Chart

Uncommon Valor

Sophie did not forget Jesus!
Hans: "Long live freedom!"

 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.

 MOND — Posted Monday, 1 February Frankfurt theoretical physicist Sabine Hossenfelder has an interesting article in this week's online Aeon magazine that talks about the dark matter problem. As for what that problem is and why it's so important to physicists today can be gotten from the article itself, but it raises the potentially greater problem of what gravity is and how we've looked at it over the last 100 years since Einstein brought it into its modern, post-Newtonian form. Hossenfelder notes that the failure to date for experimentalists to irrefutably detect a single dark matter particle may be due to the fact that it's not a particle problem but a gravity problem. She cites a recent paper by Lasha Berezhiani and Justin Khoury of the University of Pennsylvania pointing to the possibility that dark matter is essentially a superfluid that exists thanks to the extreme coldness of interstellar space. The lengthy (44 pages) paper infers that it's undetectable because our Solar System is basically too warm for its gravitational effects to be noticed in our neighborhood, although its effects are noticeable in the dynamics of other galaxies. In this respect, Hossenfelder claims that the superfluidity of dark matter allows it to mimic gravitational effects, thus making the Berezhiani/Khoury idea a kind of modified Newtonian dynamics (MOND) theory. I've skimmed over the latter's 2015 paper, and while I haven't fully digested it yet I am confused — dark matter either exists or it doesn't, as Hossenfelder also notes, but if it does then I don't see it as being any kind of MOND theory at all. In 2009, John Moffat of the Perimeter Institute for Theoretical Physics in Canada published Reinventing Gravity: A Physicist Goes Beyond Einstein, in which he posits a true MOND theory that makes certain adjustments to Einstein's 1915 gravity theory to fit observations. However, like other theories that impose various tensor, vector, spinor and scalar components into a suitable action Lagrangian density for gravity, Moffat's theory fits the observational data well but only at the cost of introducing a lot of parameters that reduces it to an exercise in mere curve fitting. And while I haven't completely read the Berezhiani/Khoury paper, I suspect it might be something along the same lines. Conversely, I still prefer the approach first proposed some years ago by Mannheim and Kazanas, which to me represents a true MOND theory, although it utilizes Hermann Weyl's conformal tensor as the starting point. You might also want to look at this paper by the same folks (to see the entire paper, click on 'print this article', which will bring up the entire paper. I dare you to check the calculations in Appendix A, which are undoubtedly the work of some poor grad student). I hope time will tell who's right in the end.
 Being Number Two Ain't So Bad — Posted Friday, 15 January Japanese astrophysicists believe they've found the second largest black hole in our galaxy. They deduced the presence of a black hole from the range of observed velocities in the gas cloud known as CO-0.40-0.22, estimating the hole's mass to be that of about 100,000 Suns (100,000 $$M_ \odot$$). The cloud is located only about 200 light-years from the galactic center, almost a literal stone's throw from the biggest known black hole in the Milky Way (called Sagittarius A*), a monster of some 4 million $$M_\odot$$. Number Two is not in the cloud itself, but was deduced from observed extreme Doppler shifts of the emission spectra of gaseous hydrogen cyanide molecules. The data are consistent with a nearby black hole attracting the gases and then flinging them past the hole at high velocity in accordance with standard orbital "slingshot" physics. Interest in black holes has grown substantially over the past few decades. Originally dismissed by Einstein and others in the 1920s, by the 1960s it was apparent that no known physical or nuclear force could prevent their formation given enough "starting" mass. More recent observations appear to show that a massive black hole lies at the center of all galaxies, and there seems to be a very fundamental connection between black holes and the evolution of their host galaxies from nebulous gas clouds. Recent studies also indicate that there is a strong correlation between the mass of a galaxy and its central black hole, and just last month a group of astrophysicists calculated that black holes may have a theoretical limit of about 50 billion $$M_\odot$$. Earlier I posted a video link to a lecture Leonard Susskind gave recently on the connection of black holes with quantum physics, entropy and information theory. It is entirely possible that the birth, evolution and fate of the universe are tied to black holes in some fundamental way we don't yet understand. In addition, according to Susskind and others, the preservation of the principle of quantum unitarity (specifically, that information cannot be destroyed) may depend on black holes in one universe serving as portals to other universes. Fascinating.
 Waiting for Einstein — Posted Thursday, 14 January During a visit to the optometrist yesterday I took along my copy of Anthony Zee's Einstein Gravity in a Nutshell (unquestionably the best book of its kind), intending to read the chapter on Kaluza-Klein theory while waiting for the pupil-dilation drug to take effect. I was delighted to discover that the optometrist was a physics and math major in college, and he immediately recognized the book I was reading. A nice discussion followed, and we both noted that finding a kindred spirit in such an accidental manner is a rare but welcome occurrence. Anyway, in the chapter Zee discusses how Einstein reacted to the work of Hermann Weyl and Theodor Kaluza (shown) regarding their ideas on the unification of gravity and electromagnetism, and he quotes from some of the Einstein letters that Kaluza's son had preserved. April 21, 1919: Einstein writes "The idea [of unifying electromagnetism with gravity] has also frequently and persistently haunted me. The idea, however, that this can be achieved through a five dimensional cylinder-world has never occurred to me and would seem to be altogether new. I like your idea at first sight very much. From a physical point of view it appears to me more promising than the mathematically so penetrating ansatz of Weyl, because it concerns itself with the electric field and not with the, in my opinion, physically meaningless four-potential." (Zee notes that Einstein was completely wrong in this last sentence, as the four-potential is of far more fundamental importance than the electric and magnetic fields.) April 28, 1919: Einstein starts with "I have read through your paper and find it really interesting" but then adds "the arguments $$\dots$$ do not appear convincing enough." Zee goes on to note that here Einstein starts hedging, as he will ultimately (and unnecessarily) delay the publication of Kaluza's paper for two years. May 29, 1919: "It is true that I made a blunder with [some remark Einstein made in a previous letter] $$\dots$$ I see that you thought the matter over quite carefully. I have great respect for the beauty and boldness of your idea. But you will understand that I cannot take side with it as originally planned given the present factual doubts." More delays. October 14, 1921: Einstein again writes to Kaluza, saying "I am having second thoughts about having restrained you from publishing your idea on a unification of gravitation and electricity two years ago. Your approach seems in any case to have more to it than the one by H. Weyl. If you wish I shall present your paper to the academy after all, provided you send it to me." At last Einstein condescends to advise publication, but what I find remarkable is that Einstein seems to have misplaced Kaluza's original paper! Considering the gravity (no pun intended) of Kaluza's highly original and influential idea (today it's the basis of string theory and all higher-dimensional field theories), it's amazing that Kaluza, a fellow German, didn't tell Einstein to just go to hell and find someone else to promote his paper. And I find the "If you wish" part of Einstein's remark in his last letter to be particularly infuriating. What I find also remarkable is that Kaluza was born on the same day as Weyl, 9 November 1885, although he died a year earlier in 1954.
 Rambling Sunday Thoughts — Posted Sunday, 10 January People have written me to ask if I really believe in the multiverse, entangled black holes, holographic universes and all that stuff. I tell them that I try to keep up with current theories in modern physics, but at times the sheer volume of bizarre ideas that are out there today (just look at the number of papers that are being posted on arXiv.org) makes me wonder if things are getting out of hand. Twenty years ago the pinnacle of crazy, math-dense theories was considered to be M-theory, with its 11 spacetime dimensions, while until recently supersymmetry (with its bosonic-fermionic, partner-particle idea) was not only popular, but considered perhaps even true. Out of these came superstring theory and supergravity, and all the while modern physics was looking as if the mathematicians had not only taken over all of physics, but had gone mad as well. With the Large Hadron Collider scheduled for renewed power-up in March, scientists will have access to unprecedented energies (approaching 14 TeV) for particle research. While the LHC produced nearly unequivocal evidence for the Higgs boson in 2012, a series of negative results nearly hammered the last nail in supersymmetry's coffin. The collider's new power will hopefully send SUSY completely back to the mathematicians, who may then choose to make a religion out of it, but variants of string theory are likely to survive, especially if evidence for extra dimensions is found. And while I have absolutely no justification to say it, I think it's also possible that the LHC will provide nothing more than additional confirmation of the Standard Model. Beyond that may lie a virtual desert, devoid of any exciting new particles, forces and fields. If then even more powerful machines are built that produce the same non-results, would physics be essentially "finished" at that point, with nothing left for scientists to do but tack on more decimal places to the Standard Model's predictions? It's not out of the question. At the other end, perhaps at its deepest level physical reality lies at the Planck scale, where the so-called reduced Planck energy, roughly $$10^{19}$$ GeV, is needed to fully resolve what the hell's going on. But that energy is unimaginably greater than anything the LHC could ever produce (Susskind sees Planck-scale accelerators having the dimensions of the observable universe), meaning that humans will never be able to do Planck-scale physics. So, discounting a few new discoveries that the LHC may provide, there really is an ultimate limit to all this. Both string theory and loop quantum gravity (LQG) posit the existence of Planck-size vibrating strings and interconnected networks of spacetime, but again it is doubtful these ideas can ever be tested experimentally. In my opinion, LQG is more beautiful than strings, as it does away with the concept of time altogether, with space becoming granular or foamy, with quantized areas and volumes making up the resulting 3D network mesh of space. There used to be a draft version of Carlo Rovelli's 2014 book Covariant Loop Quantum Gravity available on the Internet, but it's gone now. It's hardly elementary, but the beautiful ideas it presents make me think that if I were the Creator, I might have resorted to that kind of thing. Ask your local public library if they have an interlibrary loan program that can get it for you, and check it out. And speaking of Creators, my mind goes back again to the notion of the simulation hypothesis, which is making more and more sense to me as I get older and even stupider. Unlike the "life is just a dream" idiom, it posits that in the distant future humans (assuming they're still around) will possess computer capabilities far in excess of anything we can imagine today (just extrapolate Moore's Law out a few hundred or thousand years and you'll understand what I'm talking about). Advanced humans (or what Oxford University's Nick Bostrom calls "post-humans") will be able to simulate nearly anything using computers, even computerized life forms having sentience and free will, all residing on a hard drive possessing incomprehensible capacity and speed. Bostrom believes that our own existence may in fact be a computer simulation which, if nothing else, would neatly explain the problem of theodicy. Those of you interested in this seeming lunacy fascinating idea might want to read Daniel F. Galouye's 1964 science fiction book Simulacron-3, or watch the excellent 1999 film The Thirteenth Floor based on the book. Other films, notably Inception, Dark City and World on a Wire deal with the same topic (Tom Cruise's Vanilla Sky also uses the idea, but, sorry to say, the film really stinks). From The Thirteenth Floor © Columbia Pictures and Centropolis Film Productions, 1999 Something we never could have expected?! Hell, that's the first thing I'd suspect. Perhaps the simulations go down forever, and go up forever. Perhaps their ends even link up, so that there are no simulators to begin with, and the universe we know creates itself out of nothing.