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.

 Belated Birthday News — Posted Wednesday, 29 July 2015 On several occasions I've alluded to an interesting birthday cake that was presented to Einstein on his 70th birthday (14 March 1949). A number of close colleagues attended the party, including Hermann Weyl and Kurt Gödel, who presented Einstein with an unusual birthday gift — an innovative solution to the Einstein gravitational field equations representing a rotating universe in which time travel to the past is possible! Anyway, I've hunted all over for a photo of the cake, which I knew I'd seen somewhere but was never able to remember where I'd seen it. Today, while looking at Abraham Pais' excellent 1982 scientific biography of Einstein Subtle is the Lord: The Science and the Life of Albert Einstein (republished in 2005) for the umpteenth time, I came across the photo: The equations, written in frosting (I presume by his secretary Helen Dukas), summarize Einstein's final field equations for a non-symmetric metric $$g_{\mu\nu}$$ and connection $$\Gamma_{\mu\nu}^\lambda$$, which the great physicist hoped might prove to be the successful unified theory for which he'd searched intently for the last thirty years of his life. Sadly, the theory was a total waste of time but, as Pais notes in the book, Einstein had to follow his Muse wherever it took him. Whoever that Muse was (and I suspect it was that bitch Melpomene, the Muse of Tragedy), she must have had a wicked sense of humor.
 Tears in Rain — Posted Wednesday, 22 July 2015 Last week, noted Caltech physicist Sean Carroll and his colleagues posted a paper entitled "How to recover a qubit that has fallen into a black hole." The paper is also discussed in a non-technical New Scientist article. I'm oversimplifying, but essentially Carroll talks about a woman named Alice who has a qubit — say, a particle with spin — whose properties she knows nothing about. She drops it into a black hole, whose angular momentum she has carefully measured beforehand. After the particle falls in, she carefully measures the hole's angular momentum again. By taking the difference, she's able to ascertain the spin of the particle, thus obtaining a piece of information from the interior of the black hole. We can compare this situation to an electron-positron pair that is created just outside the hole's event horizon. Such quantum fluctuations happen in a vacuum all the time, where the particle pair exists for but a brief time before self-annihilating, but on occasion one could expect one partner to fall into the hole while the other flies off. This is of course the process behind Hawking radiation — the escaping partner, by conservation of energy, carries away a tiny bit of the hole's mass, which appears to an outside observed as radiation. Over time (zillions of years), a black hole of modest size would radiate away all of its mass (and all of its angular momentum), thus disappearing from existence. This raises the famous question (the black hole information paradox) of what happens to all the information that fell into the black hole. Rocks, sticks (and Republicans) don't carry much information, but if a zillion books fell in, it seems a pity that all their information would be irretrievably lost. However, the principle of quantum unitarity demands that information must somehow be preserved, and most physicists today believe the information is stored in Hawking radiation or some other process. Carroll's paper is short (five pages) and not too mathematical, but what I cannot understand is that his Alice is making a measurement that is essentially no different from making a measurement of the infalling particle's properties at the start. I also don't understand how she could measure the tiny difference in the hole's angular momentum, which is surely restricted by the Heisenberg uncertainty principle. Or maybe I just don't understand the paper. Those of you who take an interest in such things might want to read the paper and email me what your take is. I've seen things you people wouldn't believe ... attack ships on fire off the shoulder of Orion ... I've watched C-beams glitter in the dark near the Tannhäuser Gate. All those moments will be lost in time, like tears in rain ... time to die. — Replicant Roy Batty in Blade Runner, 1982
 Weyl in the News — Posted Friday, 17 July 2015 Dirac discovered his relativistic electron equation in 1928, which describes electrons (with up and down spin) and antielectrons (positrons) with up and down spin. The solution to the electron equation is therefore a four-component object called a Dirac spinor, which superficially resembles a four-vector. A year later, Hermann Weyl discovered that the Dirac equation for massless particles could be split into two equations, each consisting of two components, which described what became known as Weyl spinors. When a massless particle now known as the neutrino was proposed by Wolfgang Pauli in 1930, it was naturally assumed that the neutrino was the particle described by Weyl's spinors. Following the experimental discovery of the neutrino in 1956, there was talk that the neutrino might have a small but non-zero mass. This notion was largely dismissed until 1998, when researchers verified that the neutrino did indeed have a non-vanishing mass. This opened the question: if neutrinos are massive, what do Weyl's spinors represent? Yesterday, MIT News reported that a team of researchers at the Massachusetts Institute of Technology had discovered experimental evidence for physical solutions of Weyl's equations: they represent massless photons exhibiting distinct energy bands in certain kinds of photonic crystals. These energy bands are related to what are known as Weyl points in the crystal. MIT's Ling Lu and his colleagues posted a paper earlier this year called Experimental observation of Weyl points. It's a short paper (four pages) with very little math, but I sadly cannot follow it, having never taken a single course in condensed matter (solid state) physics. But numerous sites (see here, here and here) are hailing the discovery as the solution to an 86-year-old puzzle first proposed by Weyl, back when the neutrino hadn't even been dreamt of. Meanwhile, a related discovery was also reported today by Princeton University researchers, who announced confirmation of the existence of Weyl fermions, massless neutrino-like particles that promise faster and more efficient electronics. Offhand, I don't see any essential difference between the two discoveries (despite the fact that the photons of one discovery cannot be the fermions of the second, since photons are not fermions), based on preliminary results that were posted in April. I'm looking for the latest Science article and will cite it when I can track it down.
 Experimental Evidence v. Theory — Posted Thursday, 2 July 2015 We've been given these incredible clues from Nature and we're failing to make sense of them. In fact, we're doing the opposite: theory is becoming ever more complex and contrived. We throw in more fields, more dimensions, more symmetry — we're throwing the kitchen sink at the problem and yet failing to explain the most basic facts. — Neil Turok, Director, the Perimeter Institute Paul Dirac, arguably the greatest theoretical physicist who ever lived, once said he liked to play with equations to see where they led him. He was particularly fascinated by "beautiful equations," asserting on numerous occasions that a theory's mathematical beauty was more important than its adherence to reality. If a beautiful equation did not describe reality, he noted, then it would only be a matter of time before experiment proved it to be true after all. Dirac's relativistic electron equation, which he discovered in 1928 at the age of just 25, is perhaps the most beautiful equation ever devised by mankind. While it provided the first complete explanation of electron spin, it seemed to also predict a strange new kind of particle having the exact same properties as the electron but with a positive charge. Because of this prediction Dirac was openly ridiculed by several notable physicists of the day, but Dirac stuck to his guns. Then in 1932, Caltech physicist Carl Anderson noted the presence of a new particle in his cloud chamber experiments that looked just like an electron but rotated the wrong way in the chamber's magnetic field. It was the positron, Dirac's strange new particle, for which Anderson received the Nobel Prize in physics in 1936. Dirac's equation was subsequently recognized as describing not just particles, but antiparticles as well. In the 1 July edition of New Scientist (you'll need a free registration to access the article), "Battle for the Universe" describes how some physicists have grown weary of the deluge of theoretical papers detailing theories that may or may not be beautiful but, unlike Dirac's electron equation, have yet to make any predictions that can be tested experimentally. These include theories of higher dimensions, black hole entanglement, strings and a plethora of related ideas that probably can never be tested, either due to the energies required or the unavailability of objects (like black holes) that can be examined under controlled conditions. At the same time, the article notes that some beautiful theories have been shown to be of questionable value precisely because recent tests have been conducted that appear to rule them out (the Large Hadron Collider should have easily detected the particles predicted by supersymmetry, but saw none, neither has the LHC seen any evidence for dark matter or extra dimensions). But diehard theorists are not dismayed. They can always go back to their theories and make plausible corrections and adjustments or, as Turok has noted, produce ever more complicated and contrived explanations for why their theories are correct in spite of their inability to reflect experimental reality. Even a casual glance at some of the newer papers appearing in the online physics repository arXiv.org bears out what Turok is lamenting. While many of the papers are readable, far too many exhibit such a bewildering degree of mathematical complexity that it is doubtful any but the most mathematically sophisticated reader can truly comprehend them. And even when the papers are readable, they tend to exhibit the same proliferation of fields, dimensions and symmetry that Turok mentions in the above quotation. As an engineer in a previous life, I did a lot of curve fitting of experimental data to so-called "textbook" equations. Such data reduction is often necessary, because experimental data are invariably subject to some error, either systematic or human. While imperfect, some empirical evaluation of experimental data is required to get anywhere. But curve fitting of data in the absence of any theoretical model is an exercise in pure foolishness. Indeed, the famous mathematical physicist John von Neumann once remarked on the subject that "With four [curve-fitting] parameters I can fit an elephant, and with five I can make him wiggle his trunk!" Are extra fields and dimensions simply a means of curve fitting in theoretical physics? I sometimes wonder. At any rate, I tend to agree with Turok in the article that theory can be both beautiful and nice, but there is no substitute for empirical evidence. If string theory, p-branes, inflation and all other such theories cannot be tested then they're really no different from religious belief, and should be treated as such.
 Black Holes, Shallow and Deep — Posted Monday, 22 June 2015 Every astrophysicist knows that the idea of a black hole goes back centuries, back to the time of Newton, Laplace and a guy named John Michell, who took Newton's formula for escape velocity $$v^2 = 2GM/r$$ and wondered if there was anything strange going on at the point where the escape velocity is the speed of light itself. From Newton's formula, that radius occurs at $$r = 2GM/c^2$$ which, for the known masses of the Earth and Sun at the time, were ridiculously small distances (for the Earth, it's about that of a child's marble). Amazingly, this grade-school formula is the same as the one predicted by Einstein's 1915 theory of general relativity, (which requires a bit more schooling), where it's known formally as the Schwarzschild radius. In her new book Black Hole: How an Idea Abandoned by Newtonians, Hated by Einstein, and Gambled On by Hawking Became Loved, MIT writing professor Marcia Bartusiak describes how the very thought of speed-of-light escape velocity was ignored or rejected for nearly two hundred years. Similarly, the closely related concept of black holes was rejected by Einstein and many others until 1930, when the Indian physicist Subrahmanyan Chandrasekhar mathematically demonstrated that a star would collapse to a white dwarf if its mass exceeded 1.4 solar masses ($$1.4 \,M_{\odot}$$). Partly because the neutron was not discovered until 1932, Chandrasekhar's work was ignored as it implied that a sufficiently massive star would undergo runaway collapse to an infinitesimally small point of infinite density (which today is called a black hole). The very notion of an object collapsing to zero size by any process was considered anathema to Einstein and many others at the time. However, by 1939 Hans Bethe had worked out the details of the nuclear reactions that power the stars, allowing physicists to study the nature of highly-dense stars from a renewed theoretical standpoint. It was in that year that noted Berkeley physicist and future Manhattan Project director J. Robert Oppenheimer wrote (with colleagues Hartland Snyder and George Volkoff) two seminal papers investigating the gravitational fields around and within somewhat idealized stars. The first paper, On Massive Neutron Cores, dealt with the mass limit of a star undergoing collapse to densities approaching that of neutronic matter ($$10^{10}$$ to $$10^{18}$$ kg per cubic meter). Their calculations somewhat erroneously predicted that the star would undergo further collapse if its mass exceeded about $$0.7\, M_{\odot}$$ (improved equation-of-state models have since refined that figure to $$3 \,M_{\odot}$$. The second paper, On Continued Gravitational Contraction included considerations of the stellar interior, but nevertheless concluded that for a star of sufficient mass the contraction would continue indefinitely. However, the notion of complete gravitational collapse to seeming nothingness was still not being seriously considered. Bartusiak's book goes on to describe how the situation rapidly changed upon the discovery of pulsars and other exotic astronomical objects in the 1960s culminating at last with serious consideration of the existence of black holes (a term introduced by John A. Wheeler in December 1967). It's a pretty good non-technical book that is more of a black-hole history lesson than an exposition on black holes themselves, but I recommend it. On the other end of the black-hole spectrum is the current research being conducted on these objects, which is getting progressively more difficult to follow. This may be due to the fact that black holes are being increasingly considered as fundamental to the development of a quantum gravity theory, but there's still the more classical notion of what a black hole represents in terms of the fate of information falling into the hole (including entanglement), the ultimate fate of the universe and the nature of the event horizon itself. Some of this research involves some pretty weird stuff, like extra dimensions, black hole holography and firewalls, but it seems that nothing to date has really been resolved. It's really a pity that we don't have a black hole nearby to experiment with, but perhaps that's a blessing in disguise (a true black hole orbiting the Sun in our solar system might act up in some way, swallowing the planets). Physicist Samir Mathur at Ohio State University is a leading researcher into the idea that objects and information falling into a black hole never actually get past the event horizon (at least according to an external observer) but pile up on the surface as a kind of stringy fuzz having a thickness on the order of the Planck length, giving rise to the notion that a black hole is nothing but a "fuzzball" of cosmic string. Others have attacked this idea, believing instead that the surface of a black hole is a firewall that destroys anything impinging on it. A readable article on all this can be found here, while the paper it references can be found here, for those of you brave enough to tackle its 45 pages. Mathur believes the firewall idea has a loophole via the creation of an object/information hologram that is created every time something falls into the hole. How all of this will stand up to a consistent, workable theory of quantum gravity (assuming it ever arrives) is anyone's guess.
 Another Reason Why a Theory of Quantum Gravity is Needed — Posted Tuesday, 16 June 2015 The website Physics.org posted an article today on some interesting research being conducted at the universities of Queensland, Harvard and Vienna, in which the researchers posit the possibility that quantum superposition is destroyed by gravitation, specifically the gravitational time-dilation effect. In most quantum theories, the stage on which particles and fields play is assumed to be an empty special-relativistic background (or simply empty spacetime) on which the particle and fields are placed. Many theoretical physicists suspect today that the non-emptiness of spacetime is inextricably linked to the effects of gravity, which in effect becomes those articles and fields, including the vacuum or zero-point energy of spacetime itself. In this view, spacetime with gravity is never simply a background on which the actors play their parts, it is those actors. I would be interested to know, as many scientists wish to know, whether the collapse of the wave function is somehow mediated by gravity. But gravitational fields are ubiquitous, and physicists cannot perform any experiments in which gravity is totally absent. Nevertheless, wave functions do appear to collapse, so if it's not due to gravity then something else is going on.
 Kaluza and Weyl — Posted Monday, 15 June 2015 Here's a fairly recent post from the website of Czech physicist Luboš Motl, providing a few additional glimpses into the theories of Theodor Kaluza and Hermann Weyl. I am somewhat familiar with the work of Motl, who taught at Harvard for a number of years, but despite being an undoubtedly brilliant string theorist he's staunchly conservative with an apparent reputation for being difficult to work with. Nevertheless, some of his remarks regarding Weyl, Kaluza and Einstein are informative and worth reading.
 Weyl and Kaluza — Posted Saturday, 6 June 2015 Dr. Daniela Wünsch is a science historian who has taught at the University of Göttingen on a variety of subjects, but her specialty is Theodor Kaluza, the German mathematical physicist who "discovered" the fifth dimension in 1919. Her PhD thesis was based on Kaluza and his life and work, and I was surprised to learn from it that before about 1980 Kaluza was virtually unheard of. Today, modern string theory is often referred to as a Kaluza-Klein theory, with the Swedish mathematician Oskar Klein having gotten his name tacked on based on related work done in 1926. Despite the fact that Kaluza and fellow German mathematical physicist Hermann Weyl were born on the exact same day (9 November 1885), I was disappointed to learn that the men apparently had nothing to do with one another. Weyl's unified theory came out in 1918 and Kaluza's a year later, and both corresponded extensively with Einstein regarding their theories, but I have been unable to shed any light on any direct correspondence between Kaluza and Weyl. That's a great pity, because the work of both men went on without them in succeeding decades to form different cornerstones of modern quantum theory. Weyl's 1918 theory became the indispensable gauge invariance principle of quantum mechanics, while Kaluza's notion of five dimensions got bumped up a tad to the 11 dimensions of M-theory. While gauge invariance is today of enormous importance to modern physics, Kaluza's notion of extra dimensions is a fascinating but as yet completely undemonstrated idea. Nevertheless, it holds a far better chance than Weyl's original 1918 theory of providing an ultimately successful "theory of everything." Wünsch's relatively non-technical but informative essay "Theodor Kaluza and His Five-Dimensional World" can be found in its entirety on Google Books under the title Proceedings of the Tenth Marcel Grossmann Meeting on General Relativity, and I encourage you to read it. The Kaluza-Klein model is simplicity itself. It takes the 5D Einstein-Hilbert free-space action Lagrangian and, after some straightforward but laborious math, out pops the 4D Einstein gravitational field equations along with Maxwell's electrodynamics. I used to view this as a lucky happenstance, but now I see it as almost magic. There simply has to be something profound underlying the formalism. In addition to being proficient in mathematics, physics, biology, law, world literature and astronomy, Kaluza was fluent in 17 languages, with a preference for Arabic. What was it with these German guys, anyway?!
 Noether Again — Posted Monday, 18 May 2015 We now come to one of the most profound observations in theoretical physics, namely Noether's theorem, which states that a conserved current is associated with each generator of a continuous symmetry. — Anthony Zee, Quantum Field Theory in a Nutshell, 2nd Ed. We're approaching the 100th anniversary of Einstein's general theory of relativity, but there are two other important discoveries that were made in the same time frame that may not be as famous but are in some ways of equal importance. I would count Hermann Weyl's discovery of the notion of gauge invariance to be one of them, the other being Noether's theorem, discovered by the remarkable mathematician Amalie (Emmy) Noether. Like Einstein, Weyl and Noether were German, and both published their findings within a few years of Einstein. Yet what Noether (pronounced nuhr'-teh) discovered lies at the foundation of the discoveries that her male colleagues made. Born in 1882, Emmy was the daughter of noted University of Erlangen mathematician Max Noether. While initially planning to study French and English at university, she opted instead to follow her father's footsteps and went into mathematics. In an age when women were greatly discouraged from studying what was then perceived as strictly a man's field, Emmy nevertheless managed to earn a PhD in mathematics from Erlangen in 1907 by a combination of formal and informal study. While not allowed to teach formally, she often assisted her father in his work while going unpaid and (at first) largely unrecognized. Although her primary field of interest was abstract algebra, she also developed an uncanny proficiency for differential invariants. Her work eventually attracted the attention of great mathematicians like Hilbert, who invited her to teach at the University of Göttingen in 1915, where she remained unpaid in an unofficial teaching position for another four years. But it was also in 1915 that she developed her famous theorem (which was not published until a few years later), which had (and still has) an enormous impact on theoretical physics. Stated succinctly, Noether's theorem says that for every mathematical symmetry there is a corresponding conservation law. For example, the fact that Nature's laws do not change in time means that energy is a conserved quantity. Similarly, the fact that physical laws do not change from one place to another means that momentum is conserved. It was Weyl who subsequently showed that local gauge symmetry is responsible for the conservation of electric charge, which is also predicted by Noether's theorem. Drexel University physicist Dave Goldberg has a neat article in last month's New Scientist in which he bemoans the fact that Emmy Noether, whom Einstein, Weyl and many others considered the greatest female mathematician of all time, remains largely unknown today. Noether, a progressive Jew, fled Nazi Germany in 1933 and took a job teaching mathematics at Bryn Mawr University (at reduced pay, of course), where she died two years later at the age of only 53 (malignant ovarian cyst). Weyl tearfully gave the eulogy at her graveside, where he considered her to be his mathematical "superior," an astounding admission in view of Weyl's own standing as one of the preeminent mathematicians of the 20th century. Yet today, like Paul Dirac (who I consider to be the greatest physicist who ever lived), Emmy Noether, while hardly forgotten by today's physicists and mathematicians, remains unrecognized by the general public. Anthony Zee also notes in his book that "As is often the case with the most important theorems, the proof of Noether's theorem is astonishingly simple." Indeed, it is. I derive her theorem in my amateurish little write-up of a few years back, but Zee's is even briefer. Dear God, what a beautiful and far-reaching theorem. Mathematical symmetries exhibit an almost divine nature to me, and I see only two plausible explanations for their existence: One, we live in a multiverse where the consistent application of mathematical symmetries to Nature's laws is a one-in-a-zillion happenstance, the other one being the result of intentional design. As a post-Christian, I really don't know which is the correct answer, although the first explanation would appear to be the more plausible. (We could also be living in a computer simulation, but that's just another 'design' argument.) Something to think about. An English translation of Noether's original 1918 paper (Invariante Variationsprobleme) detailing the theorem can be found here. It is interesting to note that Noether considered the word "invariance" to be preferable to Einstein's use of the term "relativity" which, if it had been adopted at the time, would have blocked 100 years of "it's all relative" jokes. A common question posted on many physics discussion sites is "What conservation law is associated with Lorentz invariance?" The answer is rather uninspiring considering the importance of the Lorentz group of transformations itself (the center of mass of particles moving at relativistic velocities is preserved). How about conformal invariance? Well, its phase invariance counterpart in quantum mechanics ($$\Psi \rightarrow e^{i\theta(x)}\, \Psi$$) is indeed responsible for the conservation of electric charge, but the conservation law associated with the group of conformal variations ($$g_{\mu\nu} \rightarrow e^{\theta(x)} g_{\mu\nu}$$) has, in the opinion of Fritz Rohrlich ( Classical Charged Particles, 2007, 3rd Edition), "no simple physical interpretation." Guess they can't all be gems.
 QFT Again — Posted Sunday, 19 April 2015 Quantum field theory isn't an easy subject, but it's getting easier. Thanks to Anthony Zee's breakthrough 2004 book Quantum Field Theory in a Nutshell there has been a widening recognition that QFT is so beautiful and important that it shouldn't be consigned to the experts, but opened up to the masses — at least, the interested and motivated masses. Zee's book was a noteworthy effort in making the subject understandable, but now there is an even better text by British physicists Tom Lancaster and Stephen Blundell called Quantum Field Theory for the Gifted Amateur. At over 500 pages it still requires some effort, but having read the first 300 pages I haven't lost my mind yet, and I've already learned more than I did from the entire Zee book. In the end, we're all amateurs, so give it a try. Although the math is a tad more demanding than Zee's (why do the British insist on calling it "maths"?), it actually makes the book more readable. If there is a God, he undoubtedly uses quantum field theory on a routine basis. It's truly magical (Zee calls it "sorcery"), but it lies at the foundation of everything, and it's something we should all have some familiarity with.
 Doubly Special Relativity — Posted Friday, 27 March 2015 Paul Dirac, arguably the greatest physicist who ever lived, surprisingly left his post as Lucasian Professor of Mathematics at England's Cambridge University in 1970 and joined the faculty at Florida State University, presumably because the aging Dirac and his wife preferred a warmer climate (kind of like Einstein going to work at a community college). Not to demean FSU or anything, but outside of Dirac I've never given FSU much consideration physics-wise. But there's a guy there who's written a neat paper that's currently making the rounds in the physics community. Ahmed Farag Ali and colleagues explore the limits of special relativity in the context of quantum gravity in extra dimensions with their upcoming paper Absence of black holes at the Large Hadron Collider due to gravity's rainbow, which you can read online or download from the link. Along the way, they try to explain why black holes weren't created at the LHC using an innovative approach to something I'd read about but also never given much thought to — something called doubly special relativity, or DSR. You may recall that Einstein's theory of special relativity is grounded on two assumptions: the principle of equivalence (all inertial frames of reference are equivalent) and the constancy of the speed of light for all observers, regardless of their motion. No one seriously doubts the equivalency principle, but recent studies involving gamma-ray burstars indicate that the speed of light may not be limited to $$c$$, at least not in the very early universe. Several decades ago a few brave physicists dared to explore the possibility of light speeds greater than $$c$$, and came up with the theory now known (rather unfortunately) as DSR (the tachyon theory being a completely different topic). But there is another motivation for the theory, which involves what is known as the energy-momentum dispersion relation, a concept every high school student is familiar with. It's simply $$E=p^2/2m$$ in non-relativistic physics or $$E^2=m^2c^4 + c^2p^2$$ for the relativistic case. Note that in either expression the energy is unconstrained if the momentum $$p$$ is unlimited. For light, $$m=0$$ and $$p=\hbar \omega/c$$, where $$\omega$$ is the frequency, but again the energy is unconstrained because the frequency is unlimited. But what if there is a maximum energy that cannot be surpassed by a light ray? The likely candidate for such an energy would be the Planck energy, which is given by $$E_p = \sqrt{\frac{\hbar c^5}{G}} = 2 \times 10^9 \textrm{ joules } = 1.2 \times 10^{28} \textrm{ eV}$$ The most energetic cosmic rays do not even approach such energies, and so it is reasonable to assume that the Planck energy may define a physical limit of some kind. But if that is the case then Einstein's special relativity theory must be wrong, at least at the extreme-energy limit. Similarly, the concept of length may also be constrained by the Planck length, given by $$L_p = \sqrt{\frac{G \hbar}{c^3}} = 1.6 \times 10^{-35} \textrm{ meter}$$ which is a fantastically small distance. Perhaps, say some physicists, that contrary to the Lorentz-FitzGerald contraction law of special relativity observers in any inertial frame would all agree that the Planck length is not only the smallest possible size of an object, but also a true invariant. If that is the case then again we would have a violation of special relativity, since the length of an object shrinks without limit to zero as its velocity approaches the speed of light. DSR represents a rational attempt to modify special relativity based on an idea known as "gravity's rainbow," a term taken from a popular 1973 fiction novel by Thomas Pynchon. The name reflects the idea that, like a prism that bends light along angles depending on its wavelength (or frequency), the energy of light can affect its speed as well, with higher-energy light rays being able to move at higher velocities. Like the limiting speed of light in ordinary special relativity, DSR also presumes limits for length and energy that are absolute invariants. One consequence of DSR is that, at the instant of the Big Bang, nearly infinite amounts of energy were available, so that light was able to travel infinitely fast, thus eliminating the need for the theory of cosmological inflation. Anyway, the gist of all this is that Ali and his colleagues have taken DSR into the realm of higher dimensions, perhaps as high as 10, thereby giving the DSR idea an apparent link to superstring theory, while touching on the possibility that DSR has application to quantum gravity. I'm still reading the paper, and have even gone back to my copy of Lee Smolin's The Trouble With Physics: The Rise of String Theory, the Fall of a Science, and What Comes Next to understand all this (Chapter 14 of the book provides a layperson's overview of the DSR concept, but I highly recommend your reading the entire book). But Smolin, a brilliant physicist at the Perimeter Institute in Canada, is an avowed atheist with regard to string theory, although he himself has contributed substantially to DSR. "Doubly special relativity" and "gravity's rainbow." Can't they come up with better names?! A much better overview of all this is provided by LHC researcher Don Lincoln over at Huffington Post.
 Dead Sea Scrolls — Posted Thursday, 26 March 2015 When Charlie Chaplin was accused of being a Jew, he replied "No, I'm afraid I don't have that honor." I don't have that honor either, but it was a privilege to see a sampling of ancient Jewish heritage recently at the California Science Center's Dead Sea Scrolls exhibit in Los Angeles. I've long been fascinated with the history of the area from ancient times up to the 1st century CE and, barring the opportunity to see them someday in Jerusalem, this is probably the closest I'll ever get. To prepare myself for the visit, I watched the 24-lecture series given by Rutgers University's Gary Rendsburg, which I highly recommend watching if you plan to see the exhibit. The collection consists of a large number of pottery pieces, ossuaries, jewelry, coins, textiles and numerous fragments of the actual scrolls, which were discovered in Qumran in 1947 just two years after the almost equally momentous discovery of the Nag Hammadi library in Egypt. While the library proved to be a collection of gnostic books dealing primarily with early Christianity, the Dead Sea scrolls are mostly somewhat earlier books dealing with the Hebrew Bible. The scroll fragments are hermetically sealed and maintained under very low light levels, but they're readable (assuming you can read ancient Hebrew, which sadly I cannot). But they're accompanied by word-for-word English translations that are remarkably similar to what you'd read in the Old Testament today. Also on display are several pottery shards found at Masada inscribed in ink with Hebrew names. Could they have been part of the suicide lottery the Jews conducted while under attack by the Romans in 73 CE? Absolutely fascinating. I was, however, disappointed that photography was forbidden, otherwise I'd have posted some pictures of the shards here. The exhibit reminded me of the Merneptah stele I saw at the Cairo Antiquities Museum a few years ago. Merneptah, the son of the great pharaoh Ramses II, recorded Egypt's destruction of a people called Israel on the stele in 1207 BCE — the first and only recorded mention of that nation from ancient times. Although the reported destruction of Israel can be taken with a grain of salt (ancient Egypt invariably exaggerated its military victories), there is little doubt that the common image of ancient Israel as described in the Old Testament is nothing more than an elaborate myth, the result of Jewish scribes lamenting Israel's actual destruction in 586 BCE by the Babylonians while bent on revenge and the hope of a future restoration of the state. And, as the Dead Sea scroll exhibit correctly notes, the notion of Israel as a monotheistic culture in ancient times is also incorrect — it was predominantly polytheistic until much later in its history. Sorry, all you faithful Christians, Jews and Muslims, but the Bible is neither historically correct nor an accurate science text — it's mostly just myth, legend and wishful thinking. Still, a fascinating exhibit not to be missed.
 Divine Gravity — Posted Wednesday, 25 March 2015 I would like to suggest that it is possible that quantum mechanics fails at large distances and for large objects. Now, mind you, I do not say that I think that quantum mechanics does fail at large distances, I only say that it is not inconsistent with what we do know. If this failure of quantum mechanics is connected with gravity, we might speculatively expect this to happen for masses such that $$Gm^2/\hbar c =1$$, of m near $$10^{-5}$$ grams, which corresponds to some $$10^{18}$$ particles. — R. Feynman I've been reading Physics Meets Philosophy at the Planck Scale: Contemporary Theories in Quantum Gravity (a collection of papers by noted quantum physicists, along with a few knowledgable philosophers), which I mentioned previously in my 6 March posting. The last paper in the book is entitled Why the Quantum Must Yield to Gravity (a kind of turn-around from the expected notion that the exact reverse must be the case), and it is in that paper that Oxford physicist Joy Christian includes the above Feynman quote from the 1995 book Lectures on Gravitation (written by the very last physics professor I ever had). Christian's article makes a nice summary for the Physics Meets Philosophy book, which deals with the problem of quantum gravity and how everyone today is struggling to twist gravitation into a quantum form of Einstein's 1915 theory. As a number of the book's articles infer, electrodynamics and the weak and strong nuclear forces have all been quantized and, since they all involve charged particles and their associated fields, so gravity — assumed to be mediated by the hypothetical spin-2 graviton particle — must also be quantizable. Not so fast, says Princeton's Steven Weinstein in his article Naive Quantum Gravity, noting that conventional quantum theories necessarily assume a background "stage" (usually flat Minkowski spacetime) on which particles and fields do their thing as "actors," while gravity is not only an actor but the stage itself. Weinstein points out that a particle in a gravitational field will experience gravity and thus move along a geodesic line, but a co-moving observer constrained to move along the same line will not notice any force whatever on the particle. Weinstein notes that the presence of force fields in quantum theory necessarily involves a deviation from geodesic motion, and this deviation is missing from gravity for a co-moving observer (at least locally). Weinstein implies that this difference may be a consequence of the fact that the conservation of mass-energy principle does not hold (or is at least ambiguous) in general relativity. This view is in strict contradiction with quantum theories, which say that a particle will deflect in reaction to a given quantum field; if no field, then no deflection. In gravity, however, no deflection does not necessarily mean the absence of a gravitational field since the field can be transformed away locally by a suitable change in the coordinate system. Alternatively, then, Christian's article suggests, in accordance with Feynman's remark, that instead of trying to quantize gravity (which has proven to be a bitch of a problem to date), we should be gravitizing quantum theory. One very tantalizing aspect of this, which Christian does not go into, is that all the usual dimensionless Planck-scale constants (Planck length, Planck time, Planck charge) are insanely tiny, while the Planck mass (at roughly $$2 \times 10^{-5}$$ grams) is insanely large (about that of a small sand particle). Indeed, it would appear that gravity simply does not want to play in the quantum sandbox. Then there's Sir Roger Penrose's article On Gravity's Role in Quantum State Reduction, a favorite topic of his in which he discusses how gravity may be involved with the collapse of the wave function. In other works (including several of his books) Penrose has also looked at quantum collapse in the human brain, and thus he infers a possible intimate connection between the mind, quantum physics and gravitation. Whether all this is physics or metaphysics has yet to be seen but, since neurologists still have no real clue as to the foundations of human consciousness, perhaps there is indeed a relationship between mind and gravity, phenomena that appear just as intractable as ever. Lastly there is the article by Oxford University's Harvey Brown and Oliver Pooley, who include a nice summary of Hermann Weyl's 1918 gauge theory discussed in the context of gravitational and non-gravitational forces in Lorentzian frames of reference, along with the topic of non-metricity, which exemplifies Weyl's geometry. Since the book is a collection of articles by various authors, the book's introduction serves also as its closing remarks. There, the editors express their hope that $$\ldots$$ the reader, equipped with these ideas, will now have sufficient context to tackle the chapters of this book, seeing how they relate to the broad physical and philosophical issues that surround the topic; if so, he or she will surely find them as exciting and illuminating as we have. We look forward to the debates that they will spark! It is indeed a thought-provoking book that has the added attraction of being accessible to undergrads and the curious layperson. But above all it points to the very enigmatic mystery of gravity itself, a phenomenon that has all the earmarks of being divine (pardon my awful prose).
 Passwords — Posted Saturday, 21 March 2015 Americans of a certain age will remember the old television game show You Bet Your Life, hosted by Groucho Marx from about 1950 to 1960. The show featured a series of questions posed to a pair of contestants, but the one joker in the format was the password, which was always a common word that one of the contestants might utter during the questioning, such as "vacation" or "kitchen" or "electrodynamics." When the word came up, the show's iconic pasteboard duck would pop down and the contestant would win money or a prize; attractive female contestants might even get a kiss from old Groucho himself. The rise of the Internet made passwords obligatory for users, and the widespread appeal of online banking and purchasing sites like PayPal also made tough passwords obligatory (it's hard to believe, but there are still many people who use "123456" as the password for their bank accounts). But tough passwords usually mean lengthy or complicated passwords like "frlR3wn ! 3P e enlq4*(", which might flummox a hacker or the NSA but cannot be easily memorized by the user. There are combination password generator-storage programs available online for free, but then you need a password to get into it as well. John Clements of Cal Poly San Luis Obispo has taken note of this problem. He has suggested that passwords containing 56 bits of entropy are probably the minimum needed to positively prevent breaking, but he conjectures that passwords such as 11101101110001000001011001011110000111000101100001011101 might be a tad difficult to memorize. So he's written a clever paper entitled Generating 56-bit passwords using Markov Models (and Charles Dickens) in which the user can use a book like Dickens' A Tale of Two Cities to generate unbreakable but memorizable (or at least quickly recoverable) user passwords. I use a password system based on a $$9 \times 9$$ matrix code I devised that I assumed was unbreakable, but after looking into it I discovered that the NSA breaks such codes without much difficulty. So I'm going to take Clements' advice and use his book-based suggestion to change my passwords. And what book will I use? I'm not saying, but it won't be one I've ever mentioned on this site.
 New Taylor Book — Posted Thursday, 12 March 2015 My father used to tell me about the movies he went to in the 1920s, and since then I've been in love with silent films and the era that spawned them (one of my prized possessions is a short film he shot on 16mm film of the 1933 World's Fair in Chicago, including the circa-1920s projector that he left behind). I'm thankful that he lived just long enough to see the Sony Betamax video recorder that I bought in 1979 — he was absolutely fascinated with the technology. In the few years following I had collected a number of films on Beta, mostly Laurel and Hardy comedies, and my father couldn't get over watching films he hadn't seen for more than 50 years. I've since amassed a collection of about 1,000 silent films spanning the period 1909 to 1929, and I never get tired of watching them. Maybe it's just nostalgia, but the idea of photographic images captured from long ago, of people and places that no longer exist, fascinates me to no end. The 1920s was a time of American innocence and decadence. It was really the beginning of what we might call today celebrity worship, although unlike today the film people of that time were little more than has-been stage actors, uneducated (and usually illiterate) vaudevillians, rodeo stars and pretty but untalented Hollywood wannabes whose careers were often destroyed by alcohol, drugs, sex parties and the inevitable scandals that accompanied their misbehavior. Compounding it all was the money involved with the movie industry — in an era when $50 a week was a substantial wage for most people, incomes of$2,500 a week and higher were not uncommon for many actors. Lacking traditional educational backgrounds and often without any social or cultural skills to speak of, people who hadn't advanced beyond grammar school suddenly found themselves living in 75-room mansions, driving luxury automobiles, hobnobbing with the political elite and going to all-night drinking parties. But what really crystallizes the silent-movie era for me is the 1922 murder of noted Hollywood director William Desmond Taylor. To date, some half-dozen books have been written about his still-unsolved murder, along with innumerable articles written from 1922 until the present day. There are websites devoted to the case, the most notable and comprehensive being Taylorology.com. But today, with all the principals involved long dead and buried, the case of Taylor's murder is as intractable (and fascinating) as ever. Someone clearly got away with killing the 49-year-old director but, despite innumerable leads and an ocean of circumstantial evidence, we'll likely never know who did the killing. In 2014 writer William J. Mann published an interesting and well-written new book on the case that probably comes as close as possible to solving the murder as any previous effort has come. At nearly 500 pages, with comprehensive end notes (but regrettably lacking an index), Tinseltown: Murder, Morphine, and Madness at the Dawn of Hollywood advances yet another theory about the identity of the killer. Yet, even with a seemingly unending cast of possible perpetrators (Mary Miles Minter, Mabel Normand, Charlotte Shelby, valets, chauffeurs, drug dealers, blackmailers, jealous homosexual lovers, betrayed family members, nervous motion picture executives, etc.) and plausible associated motives and scenarios, writer Mann still cannot satisfactorily identify the murderer. A previous investigation, Sidney D. Kirkpatrick's appropriately-titled 1986 book A Cast of Killers was until recently my favorite text, having been based on a secret 1967 inquiry by the famed Hollywood silent film director King Vidor (The Big Parade, The Crowd, Show People, The Champ, Stella Dallas, The Wizard of Oz, War and Peace), who had known and worked with Taylor some years before his murder. Vidor had compiled an enormous amount of notes and files on the case, thinking he would produce a movie based on the killing. But he then scrapped the idea, fearing that still-living individuals involved in the case would sue him. After Vidor's death in 1982 Kirkpatrick discovered Vidor's hidden notes, on which his book is based. In spite of Vidor's first-hand if posthumous involvement, however, Kirkpatrick's identification of the murderer is still far from conclusive. And so it goes. By the way, one of the earliest suspects in the killing was actor Mabel Normand, the star of many early silent Mack Sennett comedies, and the best friend of William Desmond Taylor. She was unquestionably the first great comedienne of the movies and as a director she was, more than anyone (other than Sennett), responsible for Charlie Chaplin's career. One of her earliest films was 1912's The Water Nymph, a typical (and forgettable) farce involving a randy husband and mistaken identity. Here's an extremely rare outtake from the film (most of which ended up on the cutting room floor), showing Normand's acting (and diving) ability. Pronounced at the time to be curvaceous and extraordinarily pretty, I find her to be chubby and flat-chested with rather bad teeth (but then, so was the case with nearly everyone else in those days). But what really puts me off is Normand's obvious underarm hair in the film though, to be honest, in those days women didn't bother dealing with such things (and I shouldn't care, either). A chronic cocaine addict and notorious abuser of alcohol and tobacco, Normand got off the stuff with the dedicated help of Taylor, but her close involvement in the murder destroyed her career (she visited Taylor in his bungalow just 15 minutes before he was shot). She died of tuberculosis at the age of 37 in February 1930 at a sanitarium in Monrovia, California which, by coincidence, was also the home of Taylor's secret sister-in-law, to whom he'd been furtively sending support payments. It turns out Taylor had quite a past of his own, and that's putting it mildly. A final note: I recently played tennis with a childhood friend in Duarte, California (where I grew up), and afterwards we visited the town's historical museum, located in the same area (I was pleased to find it had some photos of my parents from the 1950s there). The center is housed in the former home of Buron Fitts, California's 29th Lieutenant Governor in the 1920s and later the District Attorney for Los Angeles. Many say that Fitts was criminally involved in the Taylor murder investigation (he reportedly destroyed evidence key to the case in return for money and a certain woman's, um, favors), and in 1973 he committed suicide with a .38-caliber revolver reputed to be the same weapon used to kill Taylor (and thought to be missing since 1922). A tangled, bloody mess, indeed, but fascinating.
 LHC Starting Up Again — Posted Monday, 9 March 2015 This week's edition of New Scientist anticipates the startup of the Large Hadron Collider which, following its momentous discovery of the Higgs boson in July 2012 (or something very much like it), has been shut down for refitting. Its startup (scheduled for this month) will bring it up to its full power of 14 TeV, double what it had been operating at. With that much energy, particle physicists are hoping for all kinds of neat stuff, like extra dimensions and confirmation of supersymmetry and string theory. But the LHC could also reveal a particle desert — a vast wasteland stretching beyond the Higgs to a region near the Planck energy level, equivalent to that of the Big Bang itself. Even the most optimistic physicist knows that can never be achieved, so if the larger, harder LHC finds nothing, then it could spell the end of particle physics as we know it. The field of particle physics has come full circle since Caltech's Carl Anderson (the discoverer of the positron, or anti-electron, in 1932) found the muon in 1936 (for which Isidor Rabi famously quipped "Who ordered that?") Then, after a slew of new particles was discovered in the following years, Enrico Fermi remarked, with real frustration, "If I could remember the names of all these particles, I'd be a botanist." By 1955, when Willis Lamb won the Nobel prize in physics, he noted that there were so many particles flying about that "The finder of a new elementary particle used to be rewarded by a Nobel Prize, but such a discovery now ought to be punished by a 10,000-dollar fine." Today, with the Standard Model of physics essentially complete (absent gravity), physicists are again hoping to see — if not confirmation of their theories — at least some new particles to play with. Like many other mediocre physics students, I personally found the special and general theories of relativity to be easy; worked hard to master quantum mechanics; struggled with quantum field theory; was perplexed by string theory; and now am hopelessly adrift trying to follow supersymmetry and loop quantum gravity. Let me correct that — I'll never be able to understand supersymmetry, if only because its notational system alone is beyond comprehension. But its essential prediction — new particles that mirror the ones we already know about — should have already been confirmed by the LHC, as the energies involved are well within that of the Higgs (125 GeV). So why should I bother to learn something that probably will never pan out? That's a cop-out, I know, but there are many notable physicists who do understand supersymmetry who don't think it's real, just as there are many notable physicists who don't think string theory is valid, giving me hope that my laziness can be masked as prescient wisdom and conservation of intellectual energy. For that reason, I'm hoping the LHC confirms extra dimensions, but kills off string theory and supersymmetry once and for all.
 Physics Meets Philosophy — Posted Friday, 6 March 2015 Weyl's publication of a stunning though doomed unification of gravitational and electromagnetic forces raised a number of intriguing questions about the meaning of spacetime structure which arguably deserve more attention than they have received to date. — Harvey Brown and Oliver Pooley I was first exposed to Hermann Weyl back in 1975, and for forty years now I have been fascinated with his ideas of space and time. And, unlike his friend and colleague Einstein, Weyl made the "quantum leap" that the great scientist was unable or unwilling to make, allowing Weyl access to the then-emerging ideas of modern quantum theory while Einstein wallowed in more classical but less productive pursuits. Weyl was also a mathematician of first rank, comparable to the greatest mathematicians of the 20th century, but he was also a noted philosopher familiar with numerous important philosophies, both of his own day (Husserl and Heidegger)) and those of the great German philosophers Goethe, Kant, Hegel and Nietzsche. By comparison, British mathematical physicist Paul Dirac (probably the greatest physicist who ever lived) avoided philosophy, believing it to be unnecessary for an understanding of the physical world. Regardless of where you stand on philosophy, I strongly encourage you to read the 2004 book Physics Meets Philosophy at the Planck Scale: Contemporary Theories in Quantum Gravity, a collection of semi-technical papers (edited by Craig Callender and Nick Huggett) written by physicists like Chris Isham, Carlo Rovelli, John Baez, Ed Witten, Roger Penrose, William Unruh and Julian Barbour, names that will be familiar to anyone interested in quantum theory. But this book is particularly special because it focuses on quantum gravity, arguably the greatest problem of physics today. The book has a really nice summary of Weyl's 1918 effort to derive gravity and electrodynamics from a single, unified formalism, doomed as it was. Weyl's effort had little to do with quantum gravity, as quantum theory was in its infant stages at the time, but the theory anticipated a more formal unification of gravity with the other forces of Nature. One of the book's articles (written by Harvey Brown and Oliver Pooley) includes an interesting discussion of Weyl's affine connection $$\Gamma_{\mu\nu}^\lambda$$ and the notion of non-metricized (or metric compatible) geometry and its relationship with the establishment of local inertial reference frames. The writers also touch on the ramifications of the non-vanishing of the metric tensor (fundamental to Weyl's 1918 theory), which I personally view as being key to a true understanding of the structure of spacetime. As I noted in a brief paper, Schrödinger derived a connection coefficient that represents a far better formalism than the one Weyl proposed, but it still suffers from a grievous problem: we simply don't know what the hell the metric covariant derivative $$g_{\mu\nu||\lambda}$$ is. Weyl suggested it was proportional to the electromagnetic 4-potential $$\varphi_\lambda$$ (as in $$g_{\mu\nu||\lambda} = 2 g_{\mu\nu} \varphi_\lambda$$), but that doesn't work on a number of levels that I describe in the same paper. Schrödinger made no such suggestion, but he didn't follow up on the non-Riemannian geometry that he literally stumbled upon. Like the editors of the above book have noted, theories like those of Weyl and Schrödinger deserve far more attention than they have received to date. Central to the idea of general relativity is the vanishing of the (covariant) divergence of the Einstein tensor $$G^{\mu\nu}$$: $$(R^{\mu\nu} - \frac{1}{2} g^{\mu\nu} R)_{||\nu} = G^{\mu\nu}_{\,\,\,||\nu} = 0$$ which tells us that the Einstein tensor is "conserved" in some sense, and since we believe $$G^{\mu\nu}$$ is proportional to the mass-energy content of spacetime, we also believe that the vanishing of $$G^{\mu\nu}_{\,\,\,||\nu}$$ is responsible for the conservation of mass-energy. But I know of only two ways to derive the Einstein tensor: by adjustment of the Bianchi identities (see my February 4 posting) or by variation of the Einstein-Hilbert lagrangian quantity $$\sqrt{-g} \, R$$, and each requires that the metric covariant derivative be zero. If it is not zero, then we have nothing, which is one reason why I believe that $$g_{\mu\nu||\lambda}$$ is so critical to general relativity. But issues such as non-metricity are small potatoes compared to the much larger problem of quantum gravity, and it is in this regard that the book really shines. In it you can learn at a relatively accessible level how efforts such as string theory and loop quantum gravity are bringing us tantalizingly close to a theory of everything, while at the same time why space and time themselves may be nothing but artificial constructs of the human mind. I believe it was that idea that originally sucked me into physics so long ago — the notion that time doesn't really exist and that things such as length, space and dimension are similarly meaningless. Weyl's 1918 theory essentially proposed just that: that time and space can be redefined (or re-gauged) at every point in a physically invariant manner; in other words, space and time are effectively irrelevant in physics. My goal? Right now, I want to be the "all-being," master of time, space and dimension. Then I want to go to Europe. — Steve Martin
 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, just like Elizabeth's Merrie Old England. 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 either, 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.