©William O. Straub, 2016

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Why I'm No Longer a Christian

Who Was Hermann Weyl?

Wheeler's Tribute to Weyl (PDF)

Old Stuff
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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

Einstein's 1931 Pasadena Home Today

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.

Turtles All The Way Down? — Posted Friday, 5 February
When I first heard the following story I thought it was attributed to Richard Feynman, but Stephen Hawking notes otherwise:
A well-known scientist (some say it was Bertrand Russell) once gave a public lecture on astronomy. He described how the Earth orbits around the Sun and how the Sun, in turn, orbits around the center of a vast collection of stars called our galaxy. At the end of the lecture, a little old lady at the back of the room got up and said: "What you have told us is rubbish! The world is really a flat plate supported on the back of a giant tortoise." Upon hearing this, the scientist flashed a superior smile and replied, "Well, then what is the tortoise standing on?" "You're very clever, young man, very clever," said the old lady. "But it's turtles all the way down!"
This story is mentioned by University of Irvine astronomer Virginia Trimble in an article she wrote in 2013 on physicist Julian Schwinger (shown) entitled Gravity Before Einstein and Schwinger Before Gravity. Schwinger was the co-recipient of the 1965 Nobel Prize in physics (for his contributions to the theory of quantum electrodynamics), along with Feynman and Sin-Itiro Tomonaga, and Trimble's essay notes on more than one occasion the man's eccentricities, the most noteworthy, to me, being Schwinger's fascination with Green's functions — and anyone who has studied Green's functions in detail knows they are to be avoided like the plague.

Schwinger's work was perhaps the most erudite, formal, well-written and mathematically thought-out of any physicist, but as Trimble notes his work was also almost unintelligible, even to his colleagues. I once owned a book, which I donated to Caltech, entitled Selected Papers on Quantum Electrodynamics, and the papers by Schwinger were indeed all erudite, formal and well-written, as well as completely unintelligible.

I stumbled upon Trimble's article because I was looking into Schwinger's ideas on gravity and also considering buying the book Fields of Color: The Theory That Escaped Einstein by retired Harvard physicist Rodney A. Brooks. Brooks was a student of Schwinger, and the main thing he came away with was Schwinger's belief that all of reality (or at least most of it) is described by either classical or quantum fields, with particles representing merely highly localized fluctuations of those fields. The "everything is a field" concept might indeed actually underlie all of reality, but our brains are too puny to conceptualize anything other than particles and the matter they're made of. (After watching several of Brooks' YouTube video lectures and reading reviews of his otherwise excellent book, I opted not to buy it because it's too elementary, though you might find it useful.)

Here's a neat photo. It's Schwinger's gravestone in Cambridge, Massachusetts, emblazoned with the reduced fine structure constant, which he used on many occasions in his quantum field theory calculations:

By the way, Schwinger came from a Jewish family, and it still astonishes me that so many of our greatest scientific thinkers were and are Jewish. In 2013 the noted astrophysicist Neil deGrasse Tyson was featured in a short YouTube video talk in which he notes how, in spite of the many profound physical and mathematical discoveries that were made by the early Arabs, in or around 800-1100 CE Islamic theologians came to believe that physics and mathematics were actually the work of Satan. With few exceptions, Tyson notes, progress in science and mathematics was then halted in Islam-ruled countries with the result that today, some 900 years later, there are no longer any significant discoveries being made by Muslims (I'm sure I'll get some angry emails in response to this). Tyson's talk is only about 10 minutes long —

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.
Pure Math is a Bitch — Posted Wednesday, 27 January
I occasionally get questions about some aspect of Hermann Weyl's mathematics. Although Weyl's favorite topic was group theory, a subject that has deep and profound applications to modern physics, much of his work involved pure mathematics. Unfortunately, I understand almost none of it. While I took numerous courses in graduate-level mathematics at university, it was all what might be called "physics math," often called applied math. I did take one class in pure mathematics dealing with the calculus of variations, but it was taught by a pure mathematician using a math language that I felt very uncomfortable with — lots of stuff about unions, intersections and categories, along with symbols I had never seen before, much less used. So while I can follow Weyl's physics math pretty well, his pure math has always been undecipherable to me. And that's why I never talk about it here! (And it's not just because I'm an idiot — Paul Dirac, arguably the greatest physicist who ever lived, never understood Weyl's math, either. To see the story behind this, see the first page of my paper Weyl's Spinor and Dirac's Equation.)

Before winning the Nobel Prize in physics with T. D. Lee in 1957 for the discovery of non-conservation of parity, C. N. Yang worked with Robert Mills on nonabelian extensions of Weyl's 1918 gauge theory, which in 1929 was found to underlie all modern quantum theories. Today, Yang-Mills theory underlies the Standard Model's explanation of electroweak unification and quantum chromodynamics, although in the early 1950s Yang had few notions as to how successful it would turn out to be, and he had no idea that Weyl had started it all. Anyway, Yang made a statement back in 1980 that endeared him to me forever:
"If you try to understand fibre bundles [an abstract view of modern gauge theory] by reading mathematics, you would probably not succeed, because modern mathematics is extremely difficult to read, and I believe that there exist only two kinds of modern mathematics books: one which you cannot read beyond the first page, and one which you cannot read beyond the first sentence."
Yang is 93 years old now, and lives in China. Interestingly, after Weyl retired from the Institute for Advanced Study in Princeton he sold his house to Yang, where the latter lived from 1957 to 1966. It's located at 284 Mercer Street (a few blocks from the house Einstein lived in), and you can see a larger photo of the house in my post dated September 17, 2011.
On the Beach, Again — Posted Wednesday, 27 January
Another beach photo, this time taken while on a trip to Skagen, a tourist town at the northern-most tip of Denmark. It was probably snapped in the summer of 1951 when Weyl was 65, about the time of his second marriage to the noted artist and sculptor Ellen Bär and around the time of his retirement from the Institute for Advanced Study in Princeton. Widowed in 1948, Weyl and Ellen split their time between Princeton and Zürich until his unexpected death in December 1955. Ellen was born in Switzerland in 1902 and passed away in 1998 at the age of 96.

(From the 2009 book Mind and Nature: Selected Writings on Philosophy, Mathematics and Physics by Hermann Weyl with the express permission of the University of Princeton Press. Please respect the copyright when sharing)
Black Planet, or Pluto's Revenge — Posted Friday, 22 January
Why do astrophysicists believe black holes exist, even though they've never been directly observed? Well, imagine that our Sun somehow collapsed into a black hole. The planets in our solar system would continue to revolve about the Sun as usual, only over time they'd get much colder, to be sure. An observer from a distant stellar system who happened to be tracking Jupiter would see something strange: a large planet orbiting a point in space where nothing appeared to exist. That observer would almost certainly deduce that the central object was a black hole. And that's essentially how scientists know that black holes exist, because they've observed exactly this behavior with extrasolar planets, stars and nebulae.

Researchers Konstantin Batygin and Mike Brown at Caltech recently observed similar behavior with observable objects in the Kuiper Belt beyond the orbit of Neptune. The slight variations in the orbits of these objects (mostly large, visible icy debris) intrigued the scientists, and after carefully accounting for the perturbative gravitational effects of the Jupiter, Saturn, Uranus and Neptune they deduced that there's another planet lurking out there that's roughly 10 times the mass of Earth and from two to four times its diameter. So why hasn't it been detected before? Batygin and Brown say it's due to the fact the the orbital eccentricity of the planet is so great that its orbit takes 10,000 to 20,000 years to complete one revolution around the Sun, far longer than astronomers have been around. Provisionally dubbed "Planet Nine," the scientists are pretty positive that it's really out there. Time, and further studies, should decide the planet's existence once and for all.

An important fact that the linked article does not mention is that more than 200 years ago astronomers observed a similar discrepancy in the orbit of Mercury, the planet closest to the Sun. After laboriously calculating the perturbative gravitational effects of all the other known planets (this much have been a real bitch, as all calculations had to be done by hand), they could not account for the slight observable discrepancy in Mercury's orbit using Newtonian physics. Despite repeated observations and calculations, the discrepancy persisted, leading scientists of the day to conclude that either Newtonian physics was wrong or there was another planet orbiting the Sun that had never been observed. Newton's law of gravitation at the time was sacrosanct, so astronomers began to look for a new orbital body they called "Vulcan." They calculated that if Vulcan orbited the Sun in a manner exactly opposite to Earth's orbit, it would forever lie behind the Sun and thus be unobservable (this exact problem came up on a physics final I took as a graduate student many years ago).

It wasn't until Einstein came along in 1915 that the discrepancy in Mercury's orbit was explained, thus killing off the Vulcan theory. In a calculation he made in 1916, Einstein discovered that his general theory of relativity exactly explained the discrepancy. In his theory planets do not orbit the Sun in perfect ellipses, as Newton's physics predicted, but in ellipses that precess over time. Shortly after he made this discovery, Einstein wrote to a friend that he was so deliriously happy that he was unable to sleep for several days.

Nevertheless, lay people at the time did not appreciate the enormity of Einstein's discovery. But in 1919 another of Einstein's predictions, the deflection of starlight by gravitating masses, was definitively observed, making him a scientific superstar overnight.

Upon hearing of the Caltech finding, my son Kris immediately dashed off this little cartoon, which provides a scarier aspect to Pluto's being downgraded from planet to run-of-the-mill planetoid:

Palin and Riefenstahl — Posted Thursday, 21 January
I posted this photo on my site back in 2010, but I still think it's appropriate considering how the political zombie Sarah Palin has risen yet again from the grave of irrelevance to plague America, this time thanks to her public support of GOP presidential candidate Donald "Two Corinthians" Trump.

"Momma, when I am prezdint I wanna nucular war this big!"

(Odd how Text Edit keeps wanting to spell it "nuclear." Must be one of them Democrat Party word processors.) No, Ms. Palin did not blame President Obama for Trig's Down syndrome, but she is blaming Obama for older son Track's recent arrest for domestic abuse. Similarly, she didn't blame Obama for her and hubby Todd's 2008 legal problems, presumably only because they occurred before Obama's election that year, and even Sarah has some concept of and appreciation for physical and temporal causality. Meanwhile, Sarah's daughter Bristol (the noted Christian abstinence spokesperson) recently had yet another child out of wedlock.

Well, as King Claudius put it in Hamlet, "When sorrows come, they come not single spies, but in battalions."

But I believe we can legitimately blame Obama for Sarah's recent resurrection from the dead if only because the GOP, having now gone full-blown fucking insane, would not even consider running the likes of Donald Trump, Ted Cruz, Marco Rubio, Bobby Jindal or Mike Huckabee as 2016 presidential candidates had not Obama thoroughly messed with their minds. Even worse, it's becoming ever clearer that the GOP will have to contend with Trump being their nominee, meaning they'll also have to somehow completely ignore everything Jesus supposedly taught them in the Gospel of Matthew regarding greed, pride and hypocrisy. We can blame Obama for that, too.

While Obama's to blame for everything wrong in the observable universe (and in several nearby parallel universes), the media somehow neglected to adequately cover candidates Cruz, Huckabee and Jindal in their appearance at one of those "Freedom and Faith" rallies that the Republicans are so fond of holding. Watch how noted Colorado pastor Kevin Swanson promotes his belief in the actual ritual killing of sinners with accomplices guests Cruz, Huckabee and Jindal in tow, who show no signs of discomfort over Swanson's insane rantings:

Somehow this reminds me of the 1935 film Triumph des Willens (Triumph of the Will) by Leni Riefenstahl, the gifted and beautiful German film director and Nazi propagandist whose work could legitimately be compared to what the equally beautiful but brainless Sarah Palin is doing for die Partei today.

And yet, nearly 50% of Americans today are Republican or Republican-leaning. Hell, they would nominate Heinrich Himmler if it meant defeating Clinton or Sanders. God help us.
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.
Mr. Cool — Posted Monday, 11 January
A few pictures. The first has Hermann Weyl on a seesaw at a Gasthaus in Nikolausberg, near Göttingen, Germany in 1932. In the second photo, Weyl and his wife Hella are at a beach in New Jersey in 1937, some four years after their emigration to America from the Nazis. I find it odd that anyone would visit the seashore dressed in a suit and carrying a briefcase, but times were different then (I can still remember neighbors mowing their lawns in the 1950s wearing slacks and dress shoes à la Ward Cleaver. Just what the hell was wrong with people in those days?!)

Weyl was a true German intellectual in the classical sense, and he loved teaching in Switzerland (1913-1930) and even in Göttingen (1930-1933), where he took over the mathematics chair when the great German mathematician David Hilbert retired. When offered a position at the then-new Institute for Advanced Study (IAS) in Princeton, Weyl initially accepted. But he spoke little English, and he detested the thought of leaving the cultured academic life he had known for a country he had visited several times but still felt uncomfortable in. At the same time, his wife was Jewish, and Weyl knew things would go downhill very rapidly in Nazified Germany. Still, he accepted and rejected repeated offers from the IAS over a number of months in 1933, at one point becoming essentially mentally and physically disabled from the stress of having to make a decision. After much pleading from colleagues and the IAS, Weyl at last accepted and came to America in November 1933.

(Photos scanned from the 2009 book Mind and Nature: Selected Writings on Philosophy, Mathematics and Physics by Hermann Weyl with the express permission of the University of Princeton Press. Please respect the copyright when sharing)
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.
Susskind Again — Posted Friday, 8 January
Now approaching his 76th year, renowned Stanford University physics professor Leonard Susskind just keeps on going. Here he is in 2013, lecturing to a group of mostly grad students on the transition of scientific reality from the 20th century (general relativity, quantum mechanics, the advent of accelerators, a little thermodynamics and even less information theory) to the 21st century (quantum entanglement, quantum information, black holes, entropy and how they might all relate to general relativity). The talk deals with quantum gravity and why is is so difficult to find a workable theory, but Susskind never really gets around to it, having to present a bunch of prerequisite material first which takes up most of the 102-minute lecture. The sound quality is uncharacteristically bad for a Susskind video lecture, and it's not helped by the noise generated by fidgeting students who constantly tap with their pencils, open and close doors and scrooch around in their desks, usually right next to the microphone. Worse, the position of the video camera is static, making the blackboard difficult to make out at times. On the plus side, you've got to admire an elderly man lecturing authoritatively in a pair of shorts, a tee shirt and running shoes.

One of the reasons for the transition to 21st century physics, Susskind explains, was the limitations imposed due to the increasing energy of particle accelerators, despite the fact that the LHC now has a resolution capability of some 14 TeV. The 20th-century concept of building bigger and more powerful machines was in fact limited by the tendency of such machines to produce new particles from the very energy put into them to see smaller and smaller things. Today, Susskind notes, much higher energies will not provide better resolution, but the creation of mini black holes instead. It is at this point that Susskind begins to talk about entanglement and its close relationship with entropy, black holes and relativity in general which, to be frank, he does a much better job at in some of his other videos. Susskind does give the listener the hope that a consistent theory of quantum gravity will spin out of all this, and perhaps even redefine our concepts of reality and existence.

I'm trying to imagine Einstein, Hermann Weyl or Paul Dirac lecturing in this fashion. Einstein was wont to wear sockless shoes, a rumpled pair of pants and a floppy sweatshirt in his later years, but earlier he and all other scientists I can think of were always immaculately dressed. Well, times have changed, and I think for the better.

Weyl and Sons — Posted Wednesday, 6 January
I know I've mentioned this before, but Hermann Weyl had two sons, Fritz Joaquim Weyl (1915 - 1977) and Michael Weyl (1917 - 2011). Fritz became a renowned mathematician like his father, while Michael was a high-ranking employee of the U.S. State Department who worked in various capacities, including interrogator of German prisoners during World War II. Both came to the United States when Hermann and wife Hella fled Nazi Germany in late 1933.

Interestingly, Fritz graduated Pennsylvania's Swarthmore College in 1935 with a BA in mathematics, although when he matriculated he spoke only German! Both he and his brother graduated Princeton University in 1937, Fritz with a MA in math (earning his PhD there in 1939) and Michael with degrees in German literature and art history. Here's a photo of Michael and his father taken in 1937, around the time he graduated:

And here are two photos of older brother Fritz, the first taken in 1932 at the age of 17 with his father and mother and another in 1970, at the age of 55:

(The arm of the woman clipped off on the right of the first photo is that of Emmy Noether, a close friend of the Weyl family and generally considered to be the most renowned woman mathematician of all time; I'll put up the entire photo at a later date.)

While Michael lived to the ripe old age of 93, his brother died at the age of only 62 (I don't know the cause, but it was unexpected).

Lastly, here's a beautiful photo of Weyl with his grandsons Peter and Thomas taken around 1954 with the Matterhorn providing a nice backdrop:

(Photos scanned from the 2009 book Mind and Nature: Selected Writings on Philosophy, Mathematics and Physics by Hermann Weyl with the express permission of the University of Princeton Press. Please respect the copyright when sharing)
Weyl and God — Posted Monday, 4 January

The mathematical lawfulness of Nature is the revelation of divine reason. — Hermann Weyl

This quote from Weyl's 1932 essay "The Open World" predates by nearly 30 years a similar (though more secular) observation made by Nobel laureate Eugene Wigner entitled The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Both papers reflect on the unusual relationship that mathematics shares with Nature's laws but, more importantly, on the profound connection that appears to exist between those laws and their mathematical description. There's really no reason why this should be the case; it's one thing to make a physical observation and subsequently tie it to a suitable mathematical expression, but to discover that this expression then holds in a completely different physical application makes one wonder exactly which is more fundamental — the physics or the mathematics. The noted MIT physicist Max Tegmark has gone so far as to assert that Nature is nothing but an enormous mathematical expression.

In all my readings of Weyl, I've never been able to precisely define his attitude toward God, though he wrote regularly about the divine nature of things. The same applies to Paul Dirac, who once noted that God is a brilliant mathematician, though Dirac was probably no more than an informed agnostic on the subject of Nature and its connection with religious faith. (Fun fact: Dirac married Eugene Wigner's sister in 1937. When presenting his wife to someone, he would often introduce her by saying "This is Wigner's sister.")

I myself am a confirmed agnostic who nevertheless sees a profound connection between Nature and a Higher Something (although I could be wrong). The problem is, we have no idea what that Higher Something might be, if it even exists. The Bible is too full of irreconcilable contradictions and silly nonsensical stories to have any real validity on the subject, and while I haven't studied the Koran or any of the Eastern religions to any great extent I suspect the same conclusion holds with them as well (I detest the idea of fear-based faith, which I see all around us). Consequently I believe that the so-called God of Einstein and Spinoza is the most correct — a Higher Something that simply created everything, set things in motion, then walked away to let us hapless humans try to understand it all.

While consolidating some files on my old (and dying) Mac Book Pro, I came across this photo of a contemplative Weyl with the title "Zürich, late 1955." I can't remember now where it came from. I have a few others, but I need to check out the copyright issue before posting them.

Update: Okay, this photo and others are from the 2009 book Mind and Nature: Selected Writings on Philosophy, Mathematics and Physics by Hermann Weyl. I have obtained permission to post these copyrighted materials from the University of Princeton Press via the Copyright Clearance Center and will put them up here when I have the time.

Uuo 118 and Friends — Posted Monday, 4 January
Uranium has the highest atomic number of all naturally occurring elements. With 92 protons and 92 electrons, it can exhibit a varying number of neutrons in the nucleus that characterize its more common isotopes. (Plutonium, with 94 protons, is also primordial but is no longer found in Nature.) Elements with higher atomic weights are all synthetic, being produced in accelerators by bombardment of heavy elements with other elements and particles, resulting in unstable nuclei that tend to decay quickly. We thus have elements like einsteinium and fermium, along with later elements with strange names like ununnilium (atomic number 110) and ununbium (atomic number 112).

A team of scientists from Japan, Russia and the United States now reports the creation of four new synthetic elements that stretch the atomic number up from 113 to 118, thus completing the seventh row of the periodic table. These species all decay almost instantaneously, exhibiting vanishingly small half lives because of the enormous numbers of protons and neutrons that are packed into their nuclei. The strong nuclear force may be very strong, but it can't hold these things together for very long.

As a former chemistry major and chemist, I can't for the life of me understand what practical use these experiments represent. Perhaps, like pure mathematics, pure chemistry does not concern itself with applications, just the pursuit of knowledge, which I suppose is okay, but then I could never appreciate pure mathematics, either.

An article in this week's New Scientist describes the findings, along with the scientists' hopes that an "island of stability" will be found within Group 18 in the periodic table (where the "noble gases" helium, neon, argon and krypton are found). However, Element 118 found by the scientists also falls into Group 18, but it doesn't appear to be particularly stable. Provisionally named ununoctium (Uuo), it might actually be what could be called a "noble solid," since quantum theory predicts it would be a solid, not a gas. In addition, the outer electrons of Uuo are moving close to the speed of light, making it truly a "relativistic element" whose physical and chemical properties should be very interesting.

No more than a relative (!) handful of Uuo atoms has been produced so far, however, making it highly unlikely that Uuo office paperweights will be showing up any time soon. The U.S. military will also have to wait for a Uuo replacement for its armor-piercing projectiles (which currently use depleted uranium).
Two Places at the Same Time? — Posted Friday, 1 January
Stanford University's Mark Kasevich and colleagues report the creation of a macroscopic object that exists in two different places at the same time. The object is a small but observable "cloud" of 10,000 rubidium atoms cooled to a temperature near absolute zero. Because the cloud is a Bose-Einstein condensate, each of the atoms exists in the exact same quantum state. A laser beam is then used to push this cloud up an evacuated 10-meter chamber, while also splitting the cloud into two distinct quantum states. At the top of the chamber the cloud represents a 50-50 superposition of these two states, but the atoms in these states are now separated by 54 centimeters. Falling back down, the laser beam converts the cloud back to the original, single quantum state. The differing arrival times of the atoms demonstrates that the cloud did indeed exist in two places at once at the top of the chamber. Because of the extreme sensitivity of the apparatus to extraneous interferences (including the rotation of the Earth), the results are subject to experimental confirmation, but Kasevich asserts the finding is solid.

I want to see the complete paper in the journal Nature before I can even begin to believe this. The 28 December 2015 edition of the magazine New Science has additional information on the experiment.