Who Was Hermann Weyl?
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Weyl's Spinor and Dirac's Equation
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Weyl v. Schroedinger
Why Did Weyl's Theory Fail?
Weyl and the Aharonov-Bohm Effect
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Lorentz Transformation of Weyl Spinors
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Introduction to Quantum Field Theory
Electron Spin
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The Four-Frequency of Light
There Must Be a Magnetic Field!
Kaluza-Klein Theory
A Brief Look at Gaussian Integrals

Kets, Bras and All That
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Particle Chart (Courtesy CPEP)

Uncommon Valor

Sophie did not forget Jesus
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 My work always tried to unite the Truth with the Beautiful, but when I had to choose one or the other, I usually chose the Beautiful. Hermann Weyl I died for Beauty, but was scarce Adjusted in the tomb, When one who died for Truth was lain In an adjoining room Emily Dickinson

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

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

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

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

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

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

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

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

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

 Solitary, Poor, Nasty, Brutish and Short (or Infinitesimal) — Posted Wednesday, July 30 2014 If men are left to their own devices, according to Hobbes, fear of one's neighbor leads to war, war leads to more fear, which in turn leads to more war. Under such conditions there is no point in investing in the future, and life is a misery. — Amir Alexander One of Zeno's famous Paradoxes involved a race between Hercules and a tortoise. Being obviously slower, the tortoise is given a head start, and at the shout of "Go!" (or whatever they yelled in those times) both the tortoise and Hercules race for the finish line. Zeno reasoned that it would take some finite time for Hercules to reach the half-way point of wherever the turtle might be during the race, then one-fourth, then one-eighth, then one-sixteenth, etc. Assuming that this halving of the distance could go on forever, Zeno concluded that Hercules could never reach the tortoise, much less win the race, since the process of taking one-half the distance could go on for an infinite number of steps. But Hercules won the race (and later enjoyed turtle soup for dinner). But how did he do it? Did Zeno not prove mathematically that a Herculean win was impossible? The basic idea of elementary differential calculus is that one can take smaller and smaller increments of distance, time or whatever while preserving the idea of a finite result when all is said and done. After all, the finite result of Zeno's paradox is that Hercules won the race! But the notion of letting things get smaller and smaller without end was a real problem for some people a long time ago, and their refusal to believe such a thing resembles something akin to what we see in this country today. Let's take the slope of some curve defined by $$f(x)$$, which is defined as $$f^\prime (x) = \lim_{dx \to \, 0} \frac{f(x+dx) - f(x)}{dx}$$ Four hundred years ago the focus would have been solely on the denominator: "Division by zero is impossible!" But the numerator also goes to zero in the limit, making it possible to have a finite result. Thus, the slope of the parabola $$f(x) = x^2$$ at any point $$x$$ is just $$f^\prime(x) = 2x$$. But four hundred years ago, the Church might have had you burned at the stake for advocating such a heresy. In his new book Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World, mathematician Amir Alexander describes how the battle over infinitesimal quantities such as $$dx$$ raged in the times of Galileo, Louis XIV and Thomas Hobbes. In 1632, Rome's influential Society of Jesus, an order of supposedly learned men espousing many of the beliefs held by the Jesuits and other religious and doctrinal groups of the day, declared the idea of infinitesimals "repugnant", "improbable" and therefore "forbidden in our society." Their reasoning? The universe was ruled by divine order of God, and the notion of infinitesimal quantities such as microscopic distances, areas, volumes and "atoms" violated God's order. I can only imagine how many scientists would have been burned at the stake if by some miracle quantum mechanics had been discovered in those days. As Kelly Bundy would say, "The mind wobbles." Meanwhile, the concept of "infinitesimal" is taking on a new meaning in Williamstown, Kentucky where, in conjunction with the Answers in Genesis folks and their magical Creation Museum, are currently constructing a full-scale "replica"of Noah's ark. In all fairness, the builders are not using "gopher wood" as God demanded of Noah (the wood does not and never did exist), but the dimensions of the ark will comply with biblical specifications: 300 cubits in length (450 ft), 50 cubits in width (75 ft), and 30 cubits high (45 ft). If one neglects interior volume taken up by bulkheads, flooring and miscellaneous storage areas, that gives the ark something like 1.5 million cubic feet of volume, into which Noah and his small family of eight had to cram up to 9 million species of terrestrial creatures, either two at a time ("unclean" animals) or seven ("clean" animals). And, don't forget, this had to include some pretty large critters: since the Answers in Genesis folks claim that the Earth and universe are only 6,000 years old, herds of dinosaurs and huge herbivorous megafauna like Paraceratherium would have also been housed on the ark, not to mention fresh-water crocodilians and all manner of fish and birds. In the 2014 movie Noah these animals were all put to sleep with a kind of aromatic sedative to make them more docile, but it sadly did nothing about the overcrowding problem. Many apologetic Jews and Christians believe God had a magical "shrinking ray" that squeezed the animals down to shoe-box or smaller size, thus explaining the storage issue. But most believers are willing to just "leave it to the heretics to figure it all out." A lot of people haven't progressed much over the past 400 years. Isn't it wonderful to be stupid?
 The Neanderthal Genetic Remnant — Posted Friday, July 25 2014 Scientists today have irrefutable fossil and genetic evidence that the modern human genome contains a small but measurable amount of Neanderthal DNA. The fact that any Neanderthal material is present in modern humans at all provides good evidence that at some point the two species must have interbred. Neanderthals existed from roughly 250,000 years ago to 30,000 years ago, when they rather abruptly vanished from the fossil record. Their decline and extinction coincided with the rise of Cro-Magnon or modern Homo sapiens, lending some support to the idea that the more lithe and mobile (and perhaps more intelligent) modern species wiped the Neanderthals out in the competition for resources. But there are in fact many factors that have to be considered. It is highly likely that global climate change influenced their demise, since their short, stocky muscular frames were more acclimated to a colder world. At the same time, it's possible that Neanderthals did not actually go extinct, but were assimilated into the larger human gene pool through interbreeding. But one thing is very clear: the Neanderthals disappeared, and modern Cro-Magnon man inherited the Earth. This observation alone gives creedence to the notion that whatever Neanderthal represented in terms of the overall human experiment, the species was inferior with regard to evolution, so now they're gone. Or are they? Social scientists for at least the past forty years have advanced the idea that there appear to be two distinct human personalities: those who tend to favor tradition, stability, security and avoidance of radical social change, and those who tend to favor innovation, reform, social experimentation and risk-taking. Scientists have, of course, labeled these two personality types conservative and liberal, though the idea (if not the actual labels) goes back hundreds of years. Like all social differences in people today, conservatism and liberalism have historically been attributed to factors such as parental influence, environmental conditions and, to some extent, happenstance. But decades of testing with identical and fraternal twins has revealed that there is also a heritable factor that can affect a person's conservative/liberal tendencies. And more recent studies indicate that these tendencies may be genetic as well. This month the Journal of Behavioral and Brain Sciences of the University of Cambridge published a lengthy scientific paper entitled Differences in Negativity Bias Underlie Variations in Political Ideology by Dr. John Hibbing of the University of Nebraska and his colleagues. The gist of the 54-page paper is that there is today a large body of scientific and experimental evidence that the conservative personality is highly sensitive to "threatening and aversive" stimuli and that this trait is consistent with a threat-oriented biology possibly genetically inherent in that personality type. The scientists conclude that resulting conservative political ideology is likely to result in strong security-seeking behavior, such as acquisition of multiple personal weapons, allegiance to and support of a strong military, aversion to uncertainty, unquestioning obedience to accepted authority figures, intolerance of ambiguity, and distrust or dislike of strangers and outsiders. Hibbing refers to these traits as "negativity bias," and notes that they would have been extremely useful for survival in the Pleistocene geological epoch (roughly 12,000 to 2.5 million years ago). Readers who frequent this website will naturally assume that I am going to identify conservatives with the Neanderthal genetic remnant. Yes, you're right, I'm going to do just that. I will go even further and compare the country's 60 million or so legitimately crazy Americans (20% of the country, and they're all Republicans) with the patient with a severe bacterial infection who neglected to finish his antibiotics. Most of the bugs were wiped out, but the few who survived developed an immunity to the drug and went on to kill the patient. This is exactly what the Republican Party is doing to America today. A good summary of the paper can be found here, but please download and read the paper itself. It's 54 pages, but main part is only 10 pages, followed by 44 pages of peer-contributed commentary. Have a great weekend.
 The Old Dog, New Trick Conundrum — Posted Thursday, July 24 2014 Geometry is the gate of science, and the gate is so low and small that one can only enter it as a child. — William K. Clifford Much has been said about Maxwell's equations since the brilliant Scottish mathematical physicist Jame Clerk Maxwell first wrote them down in 1861. What is not said is that Maxwell's original vector notation was nearly incomprehensible, and it wasn't until Oliver Heaviside and Josiah Williard Gibbs came along in the latter part of the 19th century to put the equations in the vector form we all know and love (or hate) today, along with their divergences, gradients, curls and other vector whatnots. The equations can also be cast in beautiful covariant form, which I presented in my post dated March 6. However, there is a sad twist to the whole vector notation story, which goes all the way back to the German mathematician Hermann Grassmann (1809-1877) who in 1844 discovered (or invented) an algebra involving mathematical quantities that anticommute under multiplication ($$ab = - ba$$). His work was largely ignored, but 100 years later anticommuting Grassmann numbers became the basis of fermionic quantum field theory. About the same time (1840s), the British mathematician William Rowan Hamilton sought to extend the idea of a complex number to three dimensions. He came up with the quaternions, which brilliantly anticipated the Pauli and Dirac matrix formalism of quantum mechanics in the 1920s. Hamilton's quaternions were also pretty much ignored. Then came the British mathematician William Kingdon Clifford (1845-1879), whose Clifford algebra resurrected many of the ideas of Grassmann and Hamilton. It has been said that if Clifford had known about quantum mechanics, he might have beat out Paul Dirac in his 1928 discovery of the relativistic electron equation (which, IMHO, is the most profound achievement of the human mind). Indeed, the Dirac or gamma matrices $$\gamma^\mu$$ of quantum field theory encapsulate much of Clifford's algebra. So what has all this dry math history to do with vector notation? As authors Venzio de Sabbata and Bidyut Kumar Datta describe in their slim and very readable 2007 book Geometric Algebra and Applications to Physics, what the likes of Grassmann, Hamilton and Clifford discovered was geometric algebra, which describes geometry and physics in terms of a purely coordinate-free, geometrical approach. It is ironic indeed that Gibbs' vector notation caught on, because not only is it vastly inferior to geometric algebra, it nevertheless became the standard notation for generations of students and scientists since he first developed it. (It's also ironic that most of Gibbs' scientific papers were famously grammatically obtuse. Outside of Gibbs' free energy, how many people have ever heard of the guy?) The sad twist to the story, at least for me*, is that trying to learn geometric algebra (in spite of its intuitive, powerful and very natural notation), requires simultaneously unlearning the standard vector notation that we've all used for the past 120 years. Consider the fact that the cross product of two vectors $$\mathbf{a} \times \mathbf{b}$$ or the curl of a vector $$\nabla \times \mathbf{v}$$ in the standard notation only makes sense in three dimensions. In GA (written as $$a \wedge b$$ and $$\nabla \wedge v$$ ), they make sense in any dimension. Not only that, but all vector quantities in GA are tied fundamentally to the geometry of points, lines, areas, volumes and higher-dimensional objects. Best of all, the notation is coordinate-free (and nearly index-free). The real guts of GA lies in the simple vector product $$ab$$. In two dimensions, a vector $$a$$ can be expanded in terms of its components $$a_i$$ and the orthogonal direction base vectors $$e_i$$, or $$a = a_1 e_1 + a_2 e_2$$. The base vectors as expected obey $$e_1 e_1 = e_2 e_2 = 1$$ and $$e_1 e_2 = - e_2 e_1$$, so that $$ab$$ can be written simply as $$ab = a \cdot b + (a_1 b_2 - a_2 b_1) e_1 e_2$$ or $$ab = a \cdot b + a \wedge b$$ Note that in GA, a vector product is the sum of a scalar ($$a \cdot b$$) and a bivector ($$a \wedge b$$), an unheard-of concept in ordinary vector algebra. Note also that the square of $$e_1 e_2$$ is $$(e_1 e_2) (e_1 e_2) = e_1 e_2 e_1 e_2 = -1$$ so that $$e_1 e_2$$ behaves as the imaginary number $$i$$, a characteristic that persists in any dimension $$n$$ ($$e_1 e_2 e_3 .. e_n$$). We can then write $$ab = |a| |b| \cos \theta + i |a| |b| \sin \theta$$ or $$ab = |a| |b| e^{i \theta}$$ If $$a,b$$ are unit vectors, the product $$ab$$ can be used as a vector rotation operator, a property of GA that immediately lends itself to countless applications in physics, including quantum theory. For almost fifty years, the leading proponent of GA has been Arizona State University's David Hestenes, who has heroically championed the overturn of standard vector analysis and calculus in favor of the GA approach. Today, many schools (from high school to Cambridge University) routinely teach GA to students. But will it take another 100 years for it to finally catch on? I found that the Doran and Lasenby book on GA is one of the best to learn the subject. You can also download a sample of the book here. It's worth looking into; give it a try. * I read several books on GA two years ago, and was surprised at how difficult it was for me to unlearn the notation I've used for many years. But I finally got it figured out. Then yesterday, turning again to GA to understand some of its applications to computer animation, I realized I had forgotten it all. I'm chalking it up to age, but you don't have to!
 Let's Get Small — Posted Friday, July 18 2014 It has been known since the 1930s that ordinary matter can be compressed to fantastic densities in stars, with the compressing force being provided either by impact or gravity. The primary forces opposing compression in a star's interior are generally the ordinary hydrostatic and electrostatic forces that accompany heat and outgoing photon pressure. But for very massive stars gravity can overcome these forces and squeeze matter until electron degeneracy (that it, the resistance associated with electrons being pushed into disallowed shared quantum states, thus violating the Pauli exclusion principle) takes over and halts continued contraction of the star; a white dwarf star is a common and well-studied example. However, if the star's mass is truly enormous gravity can overcome the degeneracy pressure and squeeze the matter down to where electrons and protons join, forming neutrons. This produces an even stronger kind of degeneracy pressure that characterizes a neutron star. Modern thermodynamic equations of state accurately describe the densities of even these stars, where matter is compressed to the order of $$10^{17}$$ kilograms per cubic meter. Ouch. But if a star has sufficient mass, gravitational collapse can continue unabated until the physical star itself literally "winks out" of existence, producing a black hole. In classical general relativity there is no known force or process that can prevent the total collapse of matter to a mathematical point of zero volume and infinite matter density. An important assumption in this classical point of view is that spacetime is smooth and continuous all the way down to zero volume. This week's issue of Nature reports on a theory in which spacetime is considered to be granular or foamy at distances near the Planck length ($$10^{-35}$$ meter); that is, the theory posits that spacetime is quantized at some sufficiently tiny scale, so that spacetime volumes below this scale do not exist. The underlying theory, known as loop quantum gravity, has actually been around for about as long as modern string theory. But the Nature article raises the possibility that gravity cannot "scrunch" matter down to zero volume because such a volume simply does not exist. A collapsing star would resemble a classical black hole only until the central portion reaches the fundamental "Planckish" volume limit, with the result that the matter rebounds off this volume. The black hole then explodes, producing a white hole. Interestingly, the arXiv paper on which the article is based reports that the time frame of this rebounding process could be many billions of years, making it unlikely that even a single event could ever be witnessed. Carlo Rovelli, one of the paper's authors, is a leading theorist in quantum gravity. He is publishing a new book called Covariant Loop Quantum Gravity: An Elementary Introduction to Quantum Gravity and Spinfoam Theory (which appeals to me mainly because it has the word "elementary" in its title!) The book is scheduled to come out in November, but if I can get my hands on a preview copy I'll let you know if it's worth reading. Update: I found what appears to be a legally-downloadable preview draft of Rovelli's book here at the Aix Marseille Université-Centre de Physique Théorique website (it's only 277 pages, but at 9 MB it may take a while to download). The book has numerous spelling and grammatical errors, none of which are critical to the subject matter (hopefully all the math is correct). I've only skimmed through it so far, but it does look to be fairly readable. Have a nice weekend.
 "Something Funny is Happening to the Sunset" — Posted Thursday, July 17 2014 Lemmings leaping to their deaths by the thousands. Whales beaching themselves in droves. Worldwide colony collapse disorder in bees. I don't believe in intentional animal suicide; it's not their nature. When the food runs out, or when there are few females to mate with, sure, somebody's gonna get hurt, maybe even die. But suicide? No. It takes higher brain functioning to even contemplate that. It takes the ability to actually think. "Momma, when I am prezdint I want a nucular war this big!" Okay, yes, it's a cheap shot, posting Sarah Palin's brain-damaged child Trig here, but I'm trying to make a point. It not only takes the ability to think to commit suicide, but you also have to be insane. And that seems to describe the world today. Israel initiated a ground campaign in Gaza this morning, a passenger jet carrying 295 people was shot down over Ukraine, Australia has abandoned its global warming mitigation efforts, extreme drought in America is expanding, the entire legislative body of the United States is dysfunctional, the Supreme Court thinks corporations are people, America and Russia are at each other's throats again, European countries are going hard right wing, and the Museum of Creation in Kentucky is building a full-sized replica of Noah's Ark to proselytize the notion that the Earth is only 6,000 years old and that humans lived with dinosaurs. If this isn't insanity, I don't know what is. I recall my older son Kris suggesting to me years ago the idea that while homosexuality has always existed, its apparent expansion in recent years may be Nature's way of trying to cope with 7 billion human beings on a planet designed for far fewer. In hindsight today, I think he was on to something. But it's also possible that it was no accident that Nature allowed a single species to develop the ability to annihilate itself. (BTW, my son and his wife will make me a grandfather for the first time next month, making me wonder what kind of world the kid will be coming into.) Many years ago I visited the birthplace and home of science fiction writer Robert A. Heinlein in Butler, Missouri, one of those tiny, nondescript, fundamentalist Christian burgs in western Missouri ("The Electric City," it calls itself) along the eastern edge of Tornado Alley. Heinlein's 1952 short story The Year of the Jackpot sticks out in my mind as a particularly prescient tale of what we seem to be witnessing today, although in that story Nature was a bit more direct in Her dealings with us. Trig Palin for President in 2040! Sorry, Trig, but I think someone will get to the planet long before you do.
 Amen — Posted Thursday, July 17 2014 Dear Devoted Reader: Is your corporation or 501(c)(3) washed in the Blood of the Lamb? How can you know for sure that your business is going to Heaven? Just send $19.95 cash, check or money order to William O. Straub care of this website to learn how you can spend eternity with your beloved corporation or for-profit tax exempt charity! Hurry, and God bless you!  Einstein and Unification — Posted Friday, July 11 2014 Jeroen van Dongen is an assistant professor at Utrecht University in the Netherlands and an editor at the Einstein Papers Project here at Caltech in Pasadena. I'm currently reading his 2010 book Einstein's Unification, which chronicles Einstein's efforts to find a common geometrical basis for gravitation and electromagnetism, which was essentially his life's work from 1925 until 1955 when, as biographer Abraham Pais eloquently put it, he laid down his pen and died. The book has a rather lengthy section on Einstein's involvement with Kaluza-Klein theory, which he worked off and on with as late as 1938, along with many other of his own theories, including teleparallelism, semivectors, spinors and his final work on non-symmetric unified field theory. The book includes numerous references to Hermann Weyl and his related work on unification, Einstein's attempts to reconcile his distain for modern quantum theory (inlcuding quantum entanglement), and the collaborations that Einstein enjoyed (and sometimes merely tolerated) with the likes of Weyl, Schrödinger, Heisenberg, Planck and Bohr, along with Wolfgang Pauli, who often served as devil's advocate for many of Einstein's more hairbrained ideas. Readers of Einstein have no doubt noted that many young physicists of the time (perhaps anxious to further their careers or just move on to more productive fields of research) seemed to exhibit a true disliking of Einstein, whom many viewed as a relic of an earlier, classical age despite Einstein's truly momentous contributions to quantum theory and relativity. And while von Dongen doesn't address the matter, I for one am puzzled by the fact that, during their twenty years together at Princeton's Institute for Advanced Study, Einstein and Weyl never collaborated on anything. At nearly$100 on Amazon, van Dongen's slim (213 pages) book is probably not worth purchasing, but it's definitely worth reading (my copy was a loaner from the California Lutheran University, of all places), as it sheds a little more light on why the otherwise brilliant Einstein was so reluctant to give up the unification idea, while others (notably Weyl) saw the writing on the wall and moved on to more productive endeavors.
 Kaluza-Klein at 100 — Posted Thursday, June 26 2014 Finnish relativist Gunnar Nordström (1881-1923). In 1919 the German physicist Theodor Kaluza proposed a novel generalization of Einstein's 1915 gravity theory that posited the existence of a fifth dimension, in addition to time and the usual 3-space we all know and love. More importantly, Kaluza believed that all of Maxwell's electrodynamics lived in this fifth dimension. He communicated a paper he wrote on the idea to Einstein, who was enthralled. But when Kaluza asked Einstein to recommend the paper for publication, for some reason Einstein sat on it for two years. It didn't see the light of day until 1921, but it caused quite a stir. In 1926 the Swedish mathematician Oskar Klein made important improvements to Kaluza's theory, which then became known as Kaluza-Klein theory (in my opinion, it should have been called Klein-Kaluza theory). The beauty of KK theory is that it automatically undergoes dimensional reduction to give both Einstein's gravitational field equations and Maxwell's equations. All that by simply adding another space dimension! Unfortunately, that dimension is unimaginably small, on the order of the Planck length, which is why you don't back over it when you leave for work in the morning. The basic idea is extended by superstring theory, which posits no fewer than 10 space dimensions, of which 6 are, like the KK fifth dimension, unimaginably tiny. But the idea of a fifth dimension seems to have originated with the Finnish physicist Gunnar Nordström in 1914 (I knew a German kid named Gunnar in high school). But, according to the Wikipedia article, the idea was either ignored or forgotten. However, Nordström was not forgotten — he went on to discover (with Hans Reissner) the metric for an electrically charged black hole. But that's another story. In his wonderful book Einstein Gravity in a Nutshell, Physicist Anthony Zee beautifully expresses his hope that a modern version of Kaluza-Klein theory will someday answer all our physics problems. I hope he gets his wish.
 More History of Unified Field Theories — Posted Tuesday, June 24 2014 The dream of unifying all fundamental interactions in a single theory by one common represention still captures the mind of many a theoretical physicist. — Hubert Goenner Hubert F.M. Goenner, Professor of Physics Emeritus at Germany's Göttingen University, has written extensively on general relativity and on early attempts to unify it with electromagnetism. Years ago I posted a link to a 2004 article he wrote (it's more like an online book) entitled On the History of Early Unified Field Theories, a concise and very readable introduction into the work of Hermann Weyl, Theodor Kaluza, Oskar Klein, Arthur Eddington, Oswald Veblen and others, who all tried their hand at generalizing Einstein's 1915 gravity theory in the hope that other forces of Nature were somehow also describable in terms of pure geometry. (Although a German, Goenner's English is superb, far better than mine, and he thankfully provides translations for many of the original German scientists' quotes into English.) Yesterday Goenner posted a much longer article on the same subject, this time covering the period 1930 to 1965. I've only just started reading it, but it includes an extensive discussion of Einstein's last unification efforts and attempts made by both physicists and mathematicians to connect quantum theory with general relativity. Fascinating stuff, and the best part is that the interested physics or mathematics undergraduate should have no trouble following most of the material.
 Free Relativity Book — Posted Saturday, May 31 2014 Not much is known about science author Kevin S. Brown. He doesn't show up on the Internet, and no one I've talked to has any idea who he is. But he has a fantastic (and at over 700 pages, very comprehensive) book that you can read online for free over at Mathpages. It's an enormous collection of articles that Brown has written and also collected into book form called Reflections on Relativity. While the online book is free, the math typsetting — though clean — is a tad awkward. I'd recommend buying the much better version at Amazon, but at $40 I'll stick with the online version. Furthermore, while Brown makes numerous references to Hermann Weyl and his theories, the online book is not indexed, making it somewhat difficult to find specific topics. The book's many quotes and stories remind me of Anthony Zee's Einstein Gravity in a Nutshell, which is probably the best book on general relativity ever written. However, Brown has a number of calculations you won't find in Zee's text, and it's just as much fun to read. Enjoy.  Einstein and Eddington — Posted Thursday, May 29 2014 Do not Bodies act upon Light at a distance, and by their action bend its Rays, and is not this action strongest at the least distance? — Isaac Newton I just finished watching Einstein and Eddington, a 2008 film dramatizing the mutual personal, political, religious and scientific struggles that Germany's Einstein and England's Arthur Stanley Eddington endured while the two countries fought one another in World War I. Einstein was desperately trying to finalize his general theory of relativity, which held little interest for the proponents of the German war machine, while the devoutly religious Eddington, a Quaker, was troubled over the potential religious aspects of the theory's possible overturning of Newton's law of gravitation and its perfect "clockwork" description of God's universe. The film is decent as a dramatization, but its historical accuracy leaves much to be desired. For one, Einstein wasn't quite the doting father or the brash, wisecracking know-it-all he's portrayed in the movie, while the film's treatment of Newton's law of gravitation would lead one to believe that starlight is gravitationally bent only in Einstein's theory. (In truth, Newtonian physics predicts one-half the relativistic deflection; by a neat coincidence, the Newtonian deflection in radians is the Sun's Schwarzschild radius divided by its actual radius.) The film also glosses over the significant technical problems Eddington experienced when trying to quantify the deflection of starlight by the Sun. He had to wait for a solar eclipse in May 1919 to do the experiment, which he personally supervised on a months-long journey to Principe, West Africa. Almost miraculously, the cloud cover broke just minutes before totality, but damaged photographic equipment allowed him to get only two decent plates of the eclipsed Sun and the star field surrounding it. And even then, the few stars visible on the plates were significantly displaced from the solar disk, decreasing the amount of observable relativistic deflection (the path a light ray takes skimming past the Sun is an exceeding flat hyperbola). Nevertheless, Eddington was able to prove that deflection had occurred in accordance with Einstein's prediction. Here is one of the plates, which is overlaid with the same star field in the absence of the Sun, which allowed Eddington to measure the deflection: The results of Eddington's Principe expedition literally made Einstein a scientific superstar, and within just days of the news ordinary people around the world were talking about the warping of spacetime. The movie portrays Einstein's achievement in light of the political differences between Germany and England at the time — England hoping that his theory was wrong, in deference to Newtonian (and British) science, and the Germans hoping that Einstein was right, as a means of exemplifying German scientific genius, but in fact it didn't play out that way. Germany lost the war, thus nullifying a lot of its scientific pride, while Einstein became a reviled Jew in his own country. German right-wing attacks began against Einstein and "Jewish physics" almost immediately, intensifying with the rise of the Nazis in the early 1930s. Indeed, right-wing hatred of legitimate science continues to this day in our own country. As we approach the 100th anniversary of Einstein's 1915 gravity theory, I see a world becoming more and more like it was in the days of Einstein and Eddington. Militarism, nationalistic pride, racial bigotry, greed, anti-gay hatred and willful scientific ignorance abound, as evidenced by the recent disturbing resurgence of extreme right-wing politics in Europe. Something seems to be brewing, perhaps the unconscious acknowledgment of catastrophic global climate change, the coming end of the age of cheap oil and food, or perhaps it's just a kind of growing mental vacuity or spiritual nihilism on the part of the planet's increasingly restless 7.2 billion inhabitants. It would be a shame if the centennial of the theory of general relativity is met with indifference or, if the Republicans win big in November, outright rejection of science.  Where is that Damned Dark Matter? — Posted Thursday, May 29 2014 The stars at the outer portions of most spiral galaxies are moving too fast, and still nobody knows why. The observed matter density of these galaxies is too low to hold the outer stars in place, even when only Newtonian physics is assumed. Most astrophysicists believe there is an unseen type of exotic matter called dark matter responsible for this anomaly, but what it's composed of no one knows, as all detection efforts so far have failed. First noticed in the 1930s, it's now an 80-year-old mystery. Lisa Grossman at New Scientist writes that it's becoming do-or-die time with respect to WIMPs, the hypothetical weakly interacting massive particles many believe constitute dark matter. Numerous highly sensitive detection experiments have been conducted over the past decade, but to date nothing has been seen, not counting a few candidate events, but these outliers have been dismissed as spurious. By comparison, the masses of the three known flavors of neutrino are too small to account for the anomaly, spurring ideas that there might be fourth kind of neutrino (the sterile neutrino), but this is just another kind of hypothetical matter. (Time is also running out on supersymmetry, the theory whose particles were deemed certain to be spotted at the Large Hadron Collider, but were not.) An increasing number of astrophysicists believe that the missing dark matter isn't there at all, and that stellar velocities can be explained by assuming that Newton's law of gravity needs to be modified. Modified Newtonian Dynamics (MOND) is the theory that, in addition to the usual Newtonian gravitational acceleration $$GM/r^2$$ acting on a massive body, there is a much smaller fixed acceleration term $$a_0$$ that has to be added to the calculations. This has been done, and the results agree with observation, but exactly what is the justification for assuming $$a_0$$, other than that it makes theory fit observation? I've mentioned more than once on this site the theory known as conformal gravitation, which is based on Hermann Weyl's conformal tensor $$C_{\mu\nu\alpha\beta}$$. This quantity replaces the Ricci scalar $$g^{\mu\nu}R_{\mu\nu}$$ in the Einstein-Hilbert Lagrangian with $$C_{\mu\nu\alpha\beta}C^{\mu\nu\alpha\beta}$$. As numerous investigators have discovered (notably Mannheim and Kazanas), the equations of motion associated with such a Lagrangian explains the apparent hypervelocity of outer galactic stars quite well (alas, the inclusion of the mass-energy tensor in the theory makes it quite complicated, but at first glance it can be ignored). A more recent paper by Mannheim and O'Brien can be read here; it shows the velocity-distance curves for dozens of galaxies, including the one for NGC 3198 depicted above. This is Mannheim's stellar velocity-distance curve for the galaxy NGC 3198; the abcissa is the distance from the galactic core in kiloparsecs, and the ordinate is the star velocity in km/sec. Dots are observations, and the heavy dashed line is the expected data based on Newtonian gravitation. The solid line is calculated from conformal gravitational theory, which matches observation fairly well. The other two curves are lower-order contributions to the full theory and can be ignored here. Will dark matter turn out to be just another erroneous aether theory, phlogiston, or a Cheshire Cat grin? Time will tell, hopefully.  Two New Quantum Interpretations — Posted Monday, May 26 2014 Two new papers are out by physics Nobelists Steven Weinberg (1979) and Gerardus 't Hooft (1999) dealing with quantum measurement, wave function collapse and entanglement, problems which have defied solution since quantum theory's earliest days. Weinberg's paper, which proposes dumping the state vector in favor of the density matrix, is 28 pages long but is relatively readable. 't Hooft's paper is 202 pages long and appears less accessible, but I haven't read the whole thing yet so maybe I'm wrong. You can read an elementary introduction to the papers here, which includes links to the papers on arXiv.org that you can download for later reading. Richard Feynman once famously remarked that nobody really understands quantum mechanics. But we can still hope.  Are We Being Had? — Posted Monday, May 26 2014 It is hard to say to someone face to face: Your paper is rubbish! — Max Planck If you see fraud and don't shout fraud, you are a fraud. — Nassim Taleb I have several books by City College of New York professor of physics Michio Kaku, a highly respected theoretical physicist with a solid background in string and quantum field theory. But his primary occupation in the last decade seems to be publishing a lot of tripe about hyperuniverses and such, while making appearances on speculative whoosh-bang science programs of the purely infotainment variety ("If you see a black hole coming, watch out!") Nowadays, when he shows up on one of these programs I just turn the station. I read two recently-published books over the weekend, each dealing with what many have called "fairy-tale physics." No, one of them is not the ridiculous The Physics of Star Trek (although I'm an admirer of its author, Arizona State University physicist Lawrence Krauss). Instead, the books I read involve very speculative and untested physics, the kind that is being peddled today as mostly silly CGI entertainment on cable television. The books also talk about how over-attention to these theories is corrupting the field of physics itself. Science writer Jim Baggott's Farewell to Reality: How Modern Physics Has Betrayed the Search for Scientific Truth is arguably the better of the two. It challenges many of modern physics' latest concepts, such as brane worlds, supersymmetry, string theory and inflationary cosmology, primarily from a set of six principles that the author asserts are necessary to establish the validity of any physical concept: the expectation of reality, not fantasy; the acquisition of facts; the creation of a sensible theory; the testability of the theory; the theory's veracity; and something that Baggott calls the Copernican principle, which is basically the requirement that a theory not rely on any kind of anthropogenic authority. Taken together, these principles essentially constitute the scientific method, albeit in a rather longwinded manner. Then there's Alexander Unzicher, a German theoretical physicist who, with co-writer Sheilla Jones (acting mainly as his German-to-English translator), is the author of Bankrupting Physics: How Today's Top Scientists Are Gambling Away Their Credibility. The entire book is basically one long rant against the silliness of fairy-tale physics, but the book's far more pithy, interesting and entertaining than Baggotts' rather dry critique of modern trends. Filled with personal experiences (as a high school physics teacher, yet), spot-on quotes by famous theorists and a wonderfully sardonic sense of humor, I couldn't put the book down (even though he appears to have a caustic attitude toward black holes, one of my favorite subjects). Unzicher has just released another book called The Higgs Fake: How Particle Physicists Fooled the Nobel Committee. Based on initial reviews of the book it would appear that Unzicher may have gone too far with his criticisms, but when it shows up at my local library I'll give it a look. Regardless of the reviews, these books made me rethink my own attitudes towards theories involving parallel universes, string theory, extra dimensions, supersymmetry and brane worlds. Both authors bemoan the fact that, for the past 30 years or so, string theory has dominated every other field of physics despite having provided no evidence whatsoever to show that it mght be true. Indeed, if it is necessary that we be able to probe Nature at the Planck scale ($$10^{-35}$$ meter) to test the theory (which will almost certainly be forever impossible), then string theory and its ilk might end up being a kind of religious faith among physicists who simply refuse to let go of it.  Memorial Day — Posted Monday, May 26 2014  Mister Feynman's Neighborhood — Posted Wednesday, May 21 2014 This will sound presumptuous, but intellectually I have one thing in common with the late Caltech physicist and 1965 Nobel Prize winner Richard Feynman: a need to understand things for myself. I can't just read about physics, I have to actually do the calculations myself before I can believe that something's true. That probably explains why popularized "physics for the layperson" books don't do much for me. At best they're just entertainment, and at worst they're just meaningless handwaving. I was thinking about Feynman this morning for a reason I will get to in a moment. I recently came across some old 8mm home movies in the garage that my father had shot in Duarte, California back in 1955-56, and I can't believe what a different world that was (I transferred them to DVD, and will send my sister a copy, as it includes views of the Wayfarer's Chapel in Palos Verdes, where she got married in 1955). We moved to Duarte in 1949 just after I was born in Altadena, California, and for some reason I decided to drive by the old house in Altadena today. It's still there, but it hasn't changed much since 1946: (That's my mother and sisters, but the boy isn't me, as I hadn't arrived yet. Also, in the original photo I can see my Dad taking the photo in the car's reflection. Like me, he didn't like having his picture taken.) Anyway, the house is a stone's throw from the Mountain View Cemetery, where Feynman and his third wife Gweneth were buried in 1988 and 1989, respectively. Here's the shot I took this morning: I don't get reverential very often, but I couldn't help but reflect on the fact that the guy six feet below my shoes had changed the world with his ideas and discoveries. He was a friend, colleague and contemporary of the likes of Einstein, Weyl, Dirac, Pauli, von Neumann, Bethe and so many others, yet here he lies with his wife in a grave as modest as those of all the ordinary people laid out around him. Dying of stomach cancer in 1988, Feynman's last words were "I'd hate to die twice — it's so boring." He got his wish. Adieu, Professor Feynman.  A Miracle — Posted Wednesday, May 21 2014 I've been reading Anthony Zee's Einstein Gravity in a Nutshell again, whose nearly 900 pages of dense text will likely take me the rest of my life to work through. Published in 2013, it's already a classic on general relativity, easily eclipsing all the other texts out there. Zee's presentation of just about everything involving Einstein's 1915 theory (including its most recent applications) is also replete with the author's many clever insights, jokes, puzzles, witticisms and personal stories. The last chapter of the book describes Zee's take on Kaluza-Klein theory, along with his sincere hope that some modern form of the theory will eventually lead to a complete description of reality. In my own previous take on the theory I expressed the opinion that the original 5-dimensional theory's dimensional reduction to the usual 4-dimensional Einstein-Hilbert-Maxwell lagrangian was only a fortuitious coincidence of the mathematics. The calculation, which both Kaluza and Klein had to do by hand in the 1920s, is complicated to say the least, and some years back I did the calculation with the help of some computer software. Maybe that's why I was not so impressed, since by taking a computer shortcut I missed out on all the (gory) details. Tonight I could not sleep, so I decided to do it by hand myself. It took three hours, making me question my sanity in my old age. Those of you familiar with tensor calculus will know that such calculations produce many terms, but reduction (and much relabeling of dummy indices) invariably eliminates nearly everything through cancellation, leaving a nice, simple result. The Kaluza-Klein problem is that kind of calculation, but on steroids. The initial expansion of the 5-D Ricci scalar $$\tilde{g}^{AB} \tilde{R}_{AB}$$ results in about two hundred terms, which then slowly collapse down to just two, the famous $$R + 1/4\, F_{\alpha\beta}F^{\alpha\beta}$$, $$F$$ being the electromagnetic tensor. I now stand chastened by the experience — it's a magical result, and Zee's remark that he would be sorely disappointed if Nature didn't somehow involve higher dimensions truly hits home. Like Hermann Weyl, Theodor Kaluza was also a German mathematician who happened to be born on the same date as Weyl, 9 November 1885. I'm not aware of any correspondence beteen the two (although they must have written each other), but it's inevitable that Weyl would have seriously considered the possibility that the world was, as Kaluza was presumably the first to suggest, five dimensional. As others have noted, Weyl was a lifelong prolific mathematical physicist, while Kaluza was your typical one-hit wonder (though he is said to have been fluent in more than a dozen languages). Even then, Kaluza's theory would have probably been stillborn without the added insight of the Swedish mathematical physicist Oskar Klein, who considerably improved the theory while showing that it might also have application to quantum physics. Zee's book expresses the Kaluza-Klein 5-D metric tensor in a slightly more compact form than the one I used years earlier, but it still has the superfluous constant $$k$$ (which I suppose could simply be absorbed into the identification of the electromagnetic four-potential $$A_\mu$$). Zee uses $$\tilde{g}_{AB}= \begin{bmatrix} g_{00}+k^2A_{0}A_{0} & g_{01}+k^2A_{0}A_{1} & g_{02}+k^2A_{0}A_{2} & g_{03}+k^2A_{0}A_{3} & kA_{0} \\ g_{01}+k^2A_{0}A_{1} & g_{11}+k^2A_{1}A_{1} & g_{12}+k^2A_{1}A_{2} & g_{13}+k^2A_{1}A_{3} & kA_{1} \\ g_{02}+k^2A_{0}A_{2} & g_{12}+k^2A_{1}A_{2} & g_{22}+k^2A_{2}A_{2} & g_{23}+k^2A_{2}A_{3} & kA_{2} \\ g_{03}+k^2A_{0}A_{3} & g_{13}+k^2A_{1}A_{3} & g_{23}+k^2A_{2}A_{3} & g_{33}+k^2A_{3}A_{3} & kA_{3} \\ kA_{0} & kA_{1} & kA_{2} & kA_{3} & 1 \end{bmatrix}$$ or, in $$2\times2$$ shorthand notation, $$\tilde{g}_{AB} = \begin{bmatrix} g_{\mu\nu}+k^2A_\mu A_\nu & kA_{\mu} \\ kA_\nu & 1 \end{bmatrix}$$ from which we see that the metric determinant in 5-D is the same as it is in 4-D, or $$\sqrt{-\tilde{g}} = \sqrt{-g}$$ (I had to use Mathematica to prove this, as I wasn't about to do the $$5\times5$$ determinant by hand). It should be easy to see that the inverse 5-D metric is just $$\tilde{g}^{AB} = \begin{bmatrix} g^{\mu\nu} & -kA^{\mu} \\ -kA^\nu & 1+k^2 A_\mu A^\mu \end{bmatrix}$$ The Ricci scalar $$\tilde{R} = \tilde{g}^{AB} \tilde{R}_{AB}$$ is then $$\tilde{R} = g^{\mu\nu} \tilde{R}_{\mu\nu} + 2 g^{\mu5} \tilde{R}_{\mu5} + g^{55} R_{55}$$ (Computationally, the last two terms are a snap. It's the first term that's the troublemaker.) The Kaluza-Klein action then reduces to $$\int d^5x \sqrt{-\tilde{g}} \tilde{R} = \int dx^5 \int d^4x \sqrt{-g} \left( R + \frac{1}{4} k^2 F_{\mu \nu}F^{\mu\nu} \right)$$ Remarkably, in Zee's earlier text Quantum Field Theory in a Nutshell he states that the entire calculation can be avoided using basic symmetry arguments. If I only had a brain like that!  The Money Pit Syndrome, or It's Time to Grow Up — Posted Tuesday, May 20 2014 Oak Island off the southern coast of Nova Scotia is the location of the infamous Money Pit, which supposedly contains a treasure that was buried there over two centuries ago. Over the years, treasure hunters have discovered tantalizing traces of jewelry and other objects buried at increasingly deeper levels in the pit, which is naturally flooded at its deepest level by seawater. Numerous searchers have lost their lives looking for valuables, while vastly more treasure has been sunk into the hole by disappointed searchers than has been taken out. Still, there seems to be no end of treasure hunters and wealthy investors who are convinced that a large treasure awaits anyone who can successfully overcome the pit's constant cave-ins, floodings and accidents. Somewhat more inaccessible are carbon-rich planets believed to harbor fantastic quantities of pure diamond, and even further out may be entire cold, dead stars (carbonized white dwarfs) consisting of single crystals of pure diamond (but I think de Beers has already staked a claim on them). Closer to home are the asteroids that orbit the Sun between Mars and Jupiter, many (if not most) of which are made primarily of primordial iron and nickel, though there's no reason not to think they may also have huge amounts of gold, platinum and other precious metals. All of the gold ever produced on Earth would fit into a cube 50 feet on each side, and a single gold asteroid could easily surpass that amount. We can forget about diamond planets, as the logistics of finding and exploiting them are surely insurmountable. But while the Money Pit is probably only a myth or elaborate hoax, it is nevertheless enticing. Also enticing is the thought of somehow capturing a precious-metal asteroid. But why? I believe it has to do with the fact that we have the technologies to get at them, and so we tend to think of them as free resources waiting to be plundered. That is certainly true of asteroid mining, a subject that is becoming more and more of interest as precious metal resources here on Earth are diminished. The idea of space mining is not new, but a recent article in New Republic addresses the recent heightened interest in the subject, particularly by wealthy entrepreneurs who believe that the not-inconsiderable logistical problems can be overcome with existing technologies. And what are those technologies? Current mining practices and rocket propulsion, both of which are ancient, inefficent and environmentally destructive. That's about it. And that's what's driving these entrepreneurs to think about space mining in the first place — the hope of using these rudimentary technologies to make even greater fortunes for themselves. Even someone as normally level-headed as astrophysicist Neil deGrasse Tyson is excited about space mining, as the article notes. Tyson asserts that an asteroid (or meteorite) the size of a house might contain more platinum than has ever been mined on Earth. And even if that asteroid contained little platinum, the iron and nickel metal alone would still be worth billions of dollars. But getting either an asteroid or the metal it contains down to Earth is horrifically problematic, and that's where I see these optimistic entrepreneurs getting the cart before the horse. Take the case of Tyson's house-sized asteroid. Space miners wouldn't be interested in rocky or stony asteroids; they'd certainly focus on metallic asteroids, which have an average density of around 7 or 8 grams per cubic centimeter. A house-sized object would therefore have a mass of roughly $$10^{7}$$ kilograms, or about 25 million pounds. But you're not likely to find one of these just hovering out in space, available for the plucking; they typically travel at about 25,000 meters/second, giving our house-sized object a kinetic energy of about $$10^{16}$$ joules. That's the energy equivalent of some 250 million gallons of gasoline, which is what you'd have to burn with 100% efficiency to bring the asteroid to a halt. And assuming you could stop it in its tracks 10,000 miles from Earth, you'd have to expend the equivalent of millions more gallons to provide the associated gravitational braking energy to bring it to Earth's surface safely. (It has been suggested that refined materials could be delivered back to Earth using space parachutes to avoid the problems of braking. But if that's such a good idea, why aren't parachutes used today?) Space-mining apologists counter such arguments with tragically broken logic. You don't have to stop an asteroid, they claim, just hop onto it with your mining equipment, mine it at your leisure, and ship the refined resources down to Earth via space shuttles. But think about it — the energy requirements I mentioned above would be exactly the same. Even more problematic would be getting the mining equipment up to the asteroid in the first place. The current cost to transport one pound of equipment into space via space shuttle is roughly$10,000. Apologists have suggested that we shoot the equipment up instead with giant Earth-based space cannons, but cost issues associated with propulsion energy and air friction are essentially the same as with shuttle transport, while simply capturing a payload at the mining site would also require energy. Furthermore, since gasoline doesn't burn in space, the miners would have to use storage batteries or nuclear power to run their operations. I can promise you that those heavy materials won't be sent into space for a long time. Space-mining entrepreneurs talk a lot about how mankind's future home is in the stars, while the need for metals will only accelerate in the here and now, so we might as well get started. Unfortunately, they're only interested in making themselves richer, not advancing the condition or welfare of mankind. To date, you can hop on a Russian rocket for a trip to the International Space Station if you have the requisite $25 million, while Richard Branson and his ilk are promising more local trips via hypersonic aircraft to the same wealthy travelers. But these technologies are all still based on rocket propulsion, a low-tech and wasteful technique invented by the Chinese a thousand years ago, and they will never be made available to the average person. Well then, they say, how about antigravity propulsion, powered by matter-antimatter annihilation? Or maybe dilithium crystals? Yeah, how about that? As long as we're driving, flying and riding around in fossil-fuel powered cars, planes and trains, the prospects for such futuristic technologies will likely remain with the magic elixir that turns a tankful of water into gasoline.  Weyl (Early 1940s?) — Posted Tuesday, April 29 2014 My computer crashed recently, and while recovering some files I came across this photo. It's undated, but I had labeled it as "Fuld Hall," so it must have been taken at the Institute for Advanced Study (IAS) in Princeton. My guess is that it was taken around 1945 or a little earlier. I can recognize four individuals: the first person on the left is mathematician James Alexander (a topologist); don't know the next guy; then Einstein; don't know the next guy; then Hermann Weyl (mathematician); and the lanky guy on the far right is Oswald Veblen (mathematician). Weyl, Einstein, Veblen and John von Neumann were the very first to sign on with the IAS when it opened in 1933. (Did Einstein ever wear a suit and tie?)  Annemarie Schrödinger, 1886-1965 — Posted Tuesday, April 29 2014 Erwin and Annemarie Schrödinger wedding photo, March 1920. In April 1963 the noted American physicist and philosopher of science Thomas Kuhn traveled to Vienna to interview Annemarie (Anny) Schrödinger, the widow of 1933 Nobel laureate Erwin Schrödinger (1887-1961). The American Institute of Physics recently posted a transcript of the interview (in English), which you can read here. In the interview the then 67-year-old Anny shared many memories of her husband and his colleagues, which included Hermann Weyl. She doesn't go into any details, but she and her late husband shared a notoriously open marriage. Erwin had many girlfriends, and he took one with him on a Christmas tryst to the Alps in December 1925, where the seemingly indefatigable Austrian physicist somehow also found the energy to discover wave mechanics. A close friend, Weyl would later characterize Schrödinger's discovery as the intellectual result of "a late erotic outburst" in the Nobelist's life (he was only 38 at the time, but I guess that was considered past one's prime in those days, and not just in physics). I was surprised to learn some years ago that Anny had a fling of her own, and that her lover was none other than Weyl himself. Schrödinger knew about the affair, but the free-swinging physicist didn't mind, and he and Weyl remained close lifelong friends. I was disappointed that Kuhn's interview didn't reveal any details of the Annemarie-Weyl affair, but then 1963 was a different time as well. As Mark Twain once remarked in The Adventures of Tom Sawyer, "Let us draw the curtain of charity over the rest of this scene."  The Unified Field Theory of Inequality* — Posted Friday, April 25 2014 Inexplicably, French economist Thomas Piketty's book Capital in the Twenty-First Century, at nearly 700 pages of dense economic prose, is a runaway best seller. The Los Angeles Times reports that Piketty has hit an ideological nerve in America, which partly accounts for its popularity. Liberals love it because it brings attention to the growing problem of wealth inequality, while conservatives hate it for the same reason. Amazon's bi-modal book reviews seem to verify the love-hate aspect of Piketty's work. Stranger still is the fact that Piketty's thick French accent, exhibited in the numerous television interviews I've seen, would ordinarily stanch any interest in his book. My e-book copy clocks in at only 642 pages, though I'm only halfway through it. I'm not particularly adept at economics, but I can understand the data and his analyses well enough to say that Piketty's assertions are spot-on. But we're in big trouble, because not even Piketty can see a way out of the mess we're in. His basic conclusion is this: if $$r$$ is the long-term private rate of return on capital and $$g$$ is the rate of income and output growth, then The inequality r > g implies that wealth accumulated in the past grows more rapidly than output and wages. This inequality expresses a fundamental logical contradiction. The entrepreneur inevitably tends to become a rentier, more and more dominant over those who own nothing but their labor. Once constituted, capital reproduces itself faster than output increases. The past devours the future.Piketty goes on to note that capital growth averaging 4 or 5 percent per year will always outstrip income growth, leading to ever greater inequality. Historically, wealth inequality was countered somewhat by two world wars, whose emphasis on production far exceeded private wealth accumulation. But those wars were primarily the result of geopolitical instability, not proletarian worker discontent over income. Today, both of those conditions exist. (By the way, if you can't or won't read the book, at least read Nobel economist Robert Solow's review, which will bring you up to speed on the book's main points.) I have not finished the book, but so far I feel that Piketty has missed an important aspect of wealth inequality, at least as far as this country is concerned. That is the idea that people today have become so overly arrogant when issues of financial success and prosperity are raised that they're willing to tolerate any level of inequality provided either that they are wealthy or are able to maintain the expectation that they will become wealthy. In addition, many of the wealthy today are not only greedy but feel entitled to their wealth, and are not ashamed to express their pride and haughtiness about it. Worse are the sanctimonious conservatives who believe their prosperity is God's gift to them for being so good, and whose revilement of the poor stems from a core belief that poverty is God's punishment of the poor's unworthiness. You may recall around the time of the 2012 presidential election that the issue of "Who built my business" became a conservative rallying cry. Obama rightly if naively asserted that no entrepreneur becomes successful on his or her own — the country and its people as a whole contibute to their success through public education, infrastructure and other "village" amenities. Mitt Romney and Paul Ryan stoked conservative hubris by claiming that succesful people have earned their prosperity all by themselves. It worked, but it didn't win them the election. A few days ago on my way to the gym I pulled up next to a large motorhome pulling an SUV. The large, white stenciled letters on the back window read "The government didn't build my business. I BUILT IT!" I rolled down my window and said something like "Did you really get no help from anyone else?" to the driver. Instantly annoyed, he responded to my query with the bird. I suppose I deserved that. * What Nobel economist Paul Krugman calls Piketty's book  Quantum Mechanics, Susskind/Friedman — Posted Thursday, April 24 2014 I friend loaned me this book, wanting to know what I thought about it. I have the first of Lenny's "Theoretical Minimum" books (classical physics) and wasn't too impressed with it, mainly because of the book's awful typeset equations (although the contents are pretty good). This new one's on quantum mechanics, and overall it's a distinct improvement. If you managed to get through all of Susskind's YouTube video lectures on the subject you'll find this a great reference. The sections on quantum entanglement are especially clear and informative, requiring only a little familiarity with complex numbers and linear algebra. It even manages to get into what's known as the density matrix which, besides being a fascinating subject all its own, is arguably the best way to approach the topic of quantum entanglement, which a number of notable physicists have called "the only mystery." Like the lecture series, Susskind intended this book for those who majored in science or engineering in school and managed to maintain an active curiosity of the physical world, even if they didn't go into physics as a career. His video lectures were videotaped before an informal Stanford University audience comprised of mostly aging computer scientists, software/hardware engineers and other geek types who kind of remember the math and want to satisfy their intellectual cravings concerning the Higgs field and all the new stuff that's going on. BTW, co-author Art Friedman is the owner of School Year Data (whatever that is), which is located in the San Francisco Bay Area. He's a software engineer who's also taught high school math and science. His CV includes the above book, which he considers suitable "for mathematically literate nonphysicists." If that describes you, invest the$17, read the book, and become an enlightened human being.
 "Duck Soup, My Ass" — Posted Monday, April 21 2014 Speaking of lead poisoning, the latest episode of Cosmos tells the story of how Caltech scientist C.C. Patterson discovered the true age of the Earth (4.55 billion years) in 1956 by analyzing the relative concentrations of lead and uranium in zircon crystals. Patterson was also the guy who discovered the dangers of lead in the environment caused by the anti-knock gasoline additive tetraethyl lead. The discovery almost destroyed his career, since the chemical industry (primarily the Ethyl Corporation) was making billions with the additive. They tried hard to shut him up, but Patterson persisted, and by analyzing Arctic snow and ocean water samples he finally conviced the EPA (in 1973) to ban lead compounds in gasoline. (Modern radiometric dating methods give the age of the Earth as 4.54 $$\pm$$ .05 billion years, nearly identical to Patterson's finding.) Besides showing some nice shots of Pasadena and Caltech, the Cosmos episode also relates how, for hundreds of years, the "official" age of the Earth was the one determined by Bishop James Ussher in the 1600s. By carefully studying all the "begats" in the genealogies of the Old and New Testaments, Ussher calculated that the Earth was created on Saturday night on 22 October, 4004 BC (and at 9 pm, by golly). Incredibly, there are still many Americans today, perhaps 20% of the population, who still believe the Earth is only about 6,000 years old. And that percentage is much higher on the Republican side of the House of Representatives. An interesting story, and one showing that lead, which featured prominently in the fall of the ancient Roman Empire and the demise of the Franklin Expedition, can't be blamed for the insanity running amok in our country today.
 Some Things Never Change — Posted Tuesday, March 25 2014 Hermann Weyl and Paul Dirac both considered the possibility that the Newtonian gravitational constant G might be changing with time, as this would support theories they developed to explain why a certain large number ratio seems to appear again and again in Nature (for example, the ratio of the electromagnetic to gravitational force is about $$10^{40}$$, as is the ratio of the radius of the observable universe to that of the electron). This idea became known as the Large Number Hypothesis, which assumed that G is getting slightly smaller over cosmic time periods. The notion of a non-constant G has also been used by young-Earth creationists to argue that radioactive decay rates are getting smaller, which would (not really) explain why radiometric dating techniques appear to indicate a much older Earth than that indicated in the Bible. But now new research has shown that, at least for the last 9 billion of the universe's 13.8-billion-year history, the gravitational constant has not varied more than (at most) one part in a billion. After an exhaustive study of 580 observed supernovae events, professor Jeremy Mould and his PhD student Syed Uddin at the Swinburne Centre for Astrophysics and Supercomputing and the ARC Centre of Excellence for All-Sky Astrophysics showed that the Newtonian constant G has not changed appreciably over cosmic time. Their research, which focused on Type 1a supernovae, demonstrated a constant G within an upper bound of $$\dot{G}/G$$ of $$10^{-10}$$. The legitimacy of Type 1a supernovae studies was demonstrated in 1998 when three astrophysicists, Saul Perlmutter, Adam Riess and fellow Australian astronomer Brian Schmidt, used Type 1a supernovae to show that the universe is expanding at an accelerated rate, thus proving the existence of dark energy (see my post dated November 11, 2013 for more information). In 2011, the three scientists were awarded the Nobel Prize in phyics in recognition of their work. A Type 1a supernova occurs when a white dwarf accretes sufficient matter from a companion star to attain critical mass. It then explodes, and the light given off by the explosion is used as a standard candle to measure the supernova's distance from the Earth. This critical mass is the same for all Type 1a events, and it depends on the value of G in the relativistic calculations used in its determination. Observations of billion-year-old Type 1a events demonstrated the constancy of G to high precision. A less technical overview of the Swinburne study can be read here. An absolutely fascinating discovery.
 Black Friday — Posted Friday, March 21 2014 While I don't expect newscasters to be astrophysicists, I don't expect them to be complete idiots, either. CNN anchor Don Lemon, a nice enough guy, unfortunately represented the bulk of clueless humanity when he seriously asked a guest if the ill-fated Malaysian Flight 370 might have been the victim of a black hole. Also unfortunate was the fact that his guest 1) did not break out in hysterical laughter, and 2) reminded Mr. Lemon that a black hole would "suck up the entire universe." I guess she didn't know that the universe's countless black holes have somehow failed to do that just yet. And while I understand that reality shows like Life After People and Strip the City might be entertaining in a fun if impossible sort of way, I'm waiting for someone at CNN to seriously suggest that if we have the technology to Drain the Ocean, then we should use that technology to find Flight 370 (and Captain Kidd's treasure). But then if Animal Planet and even National Geographic can seriously suggest that mermaids exist, perhaps humankind has already reached the end of its rope.
 Primal Fear — Posted Friday, March 21 2014 Religion is the great salve that protects believers from death and the great unknown, right? If you believe that, then watch this short (2:42), award-winning video, Lights Out. And watch it ALONE, AND IN THE DARK. (Duct tape helped get George W. Bush reelected. But it ain't gonna save you, lady.) Sweet dreams.C'est une croix qui de l'enfer nous garde. — Gounod, Faust
 Improbable v. Impossible — Posted Tuesday, March 18 2014 Here's a highly improbable story, fictional but one that I absolutely guarantee will play out some day, and exactly as I've written it. In the not too-distant future, computers will be able to calculate the transcendental number $$\pi$$ out to previously unimaginable accuracy — to be precise, let's say $$10^{1,000,000}$$ decimal places — in a matter of only a few weeks or so. Mathematicians, being a curious sort, will concurrently develop sophisticated programs to determine if there are any unusual or weird numerical patterns in $$\pi$$, like the not too-unusual $$999999$$ sequence around the 762nd decimal you see here: Then, using the trivial code $$A=1, B=2, C=3$$ etc., at some unbelievably-distant decimal place in $$\pi$$ they discover the phrase I AM THE LORD YOUR GOD YOU HAVE AT LAST DISCOVERED PROOF OF MY EXISTENCE AND MY POWER TO MANIPULATE ALL TRANSCENDENTAL THINGS INCLUDING PI BEHOLD MY WORKS YE MIGHTY AND DESPAIRUpon publication of this discovery there is a momentous worldwide resurgence in religious belief, based on the apparent statistical impossibility of finding such a detailed, prescient message in the seemingly random infinite number $$\pi$$. But upon further investigations at an even more distant decimal place, mathematicians find the message JUST KIDDING THE BIBLE AND THE TORAH AND ALL THAT ARE JUST NONSENSICAL GIBBERISH CREATED BY FEARFUL HUMANS I DONT EVEN EXIST YOU ARE TRULY ALONEUpon publication of this numerical discovery, the human race goes into a long period of self-reflection and doubt, not knowing what to think or believe anymore. In 1985 I traveled to New Zealand on a two-week business trip. I caught a taxi at Auckland Airport and directed the driver to my hotel. It was a long drive, and along the way the cab driver and I struck up a conversation about various things, and at one point I asked him how long he'd been driving a taxi. He told me he only moonlighted as a cab driver to make ends meet, as the taxes in New Zealand, a socialist country, were pretty steep. He then told me that he worked as a water resources engineer at his day job. I informed him that I also worked in water resources, and was in New Zealand to inspect some water treatment equipment for possible purchase for my company. He replied that he had an appointment the next day to meet with a guy from Los Angeles regarding water quality issues and a potential sales contract. Upon exchanging names, we realized that he and I had been talking to each other by phone for the previous six months about water quality issues, the drought in nearby Melbourne, Australia, and life in general. We had a good laugh over this coincidence, which seemed pretty amazing at the time. But the next day in his office he introduced me to a consulting engineer for his firm, and I experienced another improbability: the consultant and I recognized each other immediately, as we had both majored in chemistry at California State University at Long Beach, graduating in 1971. He too had left chemistry and had gone into engineering. I've experienced a number of coincidences like these, but they're insignificant compared to some of the stories that Imperial College mathematics professor David J. Hand relates in his new book The Improbability Principle: Why Coincidences, Miracles, and Rare Events Happen Every Day. He opens the book with an unbelievable incident that actor Anthony Hopkins experienced in London while preparing for a role. But Hand notes that, given the billions of human beings that have ever lived and the countless experiences they have shared, unbelievable incidents are simply par for the course, mathematically speaking. In previous posts I've talked about science writer Michael Shermer and his thoughts on patternicity and agenticity. Humans are hardwired to see patterns in nature, Shermer notes, probably as a survival mechanism, even when there are no patterns. The tendency for humans to see unusual or improbable patterns and associate them with an agent (like God) is also probably hardwired into our brains. But, as Hand points out, given an infinity of alternatives and enormous spans of time for which these alternatives to play out in, and there is absolutely nothing unusual about highly improbable events. On the most recent episode of Cosmos, astrophysicist and host Neil deGrasse Tyson talked about evolution and intelligent design. He raised a favorite topic of the IDers, namely the human eye and the notion that it's too complex to have come into existence via evolution — it simply had to have an Intelligent Designer behind its construction. But Tyson notes that innumerable numbers of ocean-borne molecules (simple and complex), coupled with billions of years of random interactions, made the eye and even life itself not only probable but a certainty. We do not know if there's a Great Designer behind all things, and we'll likely not know until we die. In the meantime, I would suggest that we put our biases aside, along with our fears of death and the unknown, and try to put this universe and all its inconceivable wonders into a context that leaves us simply in awe, and not afraid. PS: I can't remember who said "It's difficult to convince someone of a factual truth when their faith or their salary depends on their not believing it." That's what we mean by bias. Here's a cute cartoon demonstrating the concept:
 Gravity Waves Detected? — Posted Monday, March 17 2014 Years ago it was noted that the observed cosmic microwave background (CMB) temperature of the universe appears to be far too uniform. If the Big Bang was literally the beginning of everything (including time and space), the expansion of the universe, however rapid it might have been, would be expected to produce some non-uniformity in the observed CMB temperature pattern (it averages only about 2.7 Kelvin, close to absolute zero). To explain this uniformity, Alan Guth proposed the theory of inflation, which conjectured that within the first $$10^{-35}$$ second or so after the Big Bang the universe experienced a brief but extremely rapid expansion, so rapid that any non-uniformities would have been effectively smoothed out. Inflation has since become the leading theory of how the universe got to be so uniform, although direct experimental evidence has been lacking. If inflation (and general relativity) is correct, then gravitational waves resulting from the Big Bang would have been produced copiously and with unimaginably small (and immeasurable) wavelengths, but within a short time inflation would have stretched the waves out to more reasonable size, making them (at least indirectly) observable. The results of the latest research appears to show that this is correct. The preliminary papers are out on arXiv.org (links here). I haven't read them yet, but notables like Lawrence Krauss, Alan Guth and Andrei Linde are saying that if the data holds up this will be the biggest thing in cosmology in the last 30 years, and certain to earn at least one Nobel Prize.
 Weyl as Art — Posted Thursday, March 13 2014 This is interesting — a beautiful interpretation of Hermann Weyl by artist Sarah Kaiser, which she created for the cover of the June 3, 2010 issue of Nature magazine.
 Multiverse — Posted Saturday, March 8 2014 After reading an interview with Wesleyan University professor of religion Mary-Jane Rubenstein on the Religion Dispatches website, I picked up her new book Worlds Without End: The Many Lives of the Multiverse, in which she tries to come to terms with the commonalities and differences between science and religion as envisioned by her in the multiverse theory, which is currently enjoying a vogue with the lay public right now. Imagine flipping a coin five times and getting heads with each flip. No big deal, you say, it's common enough. If you flipped it ten times and got heads on each flip, you'd probably think it was kind of remarkable, but still no big deal. However, if you flipped it twenty-five times and got heads each time, you'd likely think the coin was flawed, or rigged in some way, perhaps heavily weighted on the tails side. But you could still entertain the possibility that it was still only a statistical fluke, albeit a very unusual one. The odds of getting heads 25 times in a row on an honest coin are only about 0.000003%, but that's still far from zero. You say to yourself, hey, it could happen. You now flip the coin $$10^{100}$$ times (you couldn't live long enough to do it for real, but a computer might simulate the flips in a few million years or so). If the results were all heads, and you were absolutely positive that the coin (or the computer) was not flawed or rigged in any way, and you had absolutely eliminated the possibility of an outside physical force or other influence acting on the coin (electromagnetic induction, gusts of wind, losing your mind, etc.), then you would almost certainly ascribe the result to some supernatural cause or entity. In a nutshell, that's the situation between religion and the multiverse theory today. A religious person would say that it's absolutely impossible for the universe to exhibit the multitude of precise and apparently inviolable physical laws and biological processes we observe by chance alone. But a physicist or mathematician would say that the odds, while inconceivably small, are not zero. And if you have a theory in which the number of possible universes is infinite, or if the amount of time available is essentially unlimited, then the universe we live in is not just a fluke, they say, but a statistical certainty. We just happen to be living in it. If we weren't, we wouldn't be wondering about all this. That is what Stephen Hawking meant when he claimed that the universe does not need God (which isn't the same as saying that God doesn't exist). Noted science writer and author Michael Shermer (who lives a few blocks from me in nearby Altadena) has asserted the idea that humans are endowed with two attributes that together conspire to create and enforce religious or superstitious belief. The first is a survival instinct that he calls patternicity. Humans tend to see patterns in things, even when there are no actual patterns. To one person, a random collection of data points is just a meaningless scatter plot, while another person might see a definite or meaningful pattern or trend in the data. A more basic form of patternicity that humans developed long ago involves imagining dangerous animals (or rival tribe members) lurking in bushes and forests. If you see a nearby shrub shaking unexpectedly, most likely it's just the wind blowing the leaves around. But why take a chance? You run like hell! This instinct gave rise to the concept of false positives and false negatives — better to run from both if you want to survive. If it's just the wind, then all you've lost is a little energy. If it's a hungry Smilodon or cave bear, then you've improved your survival chances by hightailing it out of there. And it worked great for thousands, maybe even millions of years of human existence. The second attribute is what Shermer calls agenticity, which is the tendency to ascribe supernatural intent or force to things we do not understand or are afraid of. This also provided a sense of comfort and security and control over the natural world to early humans, which for much of our existence was totally beyond control. Agenticity also provided humans with a sense of purpose and meaning; belief in supernatural forces gave rise to acknowledgment over their power and influence, which then led to ritualized worship, obedience and even sacrifice, actions that provided a measure of purpose in human affairs. Ritual practices also made supernatural belief more "believable," even sensible. That is why atheists, agnostics and "nones" even today are shunned, if not persecuted outright by believers — non-belief on anyone's part serves to induce doubt in the believers, and they don't like that. In addition, just the act of tolerance on the part of believers induces the fear that their god will be angry with them for not getting rid of the non-believers. The Old Testament books of Exodus, Numbers and Deuteronomy are classic examples of extreme religious intolerance, while today American fundamentalist Christians fear God's wrath if they do not expunge the country from the evil-doers in their midst. But what the multiverse scientists cannot adequately explain is why physical and mathematical laws are so beautiful. And "beautiful" here is not simply a subjective description, but an undeniable attribute of our universe that chance alone would seem to have no business in creating. I suppose it's fair to say that if there is truly an infinite number of possible universes, then there is also an infinite number (or at least a large number) of worlds in which non-subjective "beautiful" physical laws and their underlying mathematical symmetries exist. But to date, I haven't been able to accept that. If there's anything interesting in Rubenstein's book worth sharing, I'll come back to it.
 Hidden Variables — Posted Thursday, March 6 2014 I'm laying hardwood flooring and, having removed all the carpeting, I carefully covered all the tack strips so I wouldn't step on them (those little nails are sharp). Well, I didn't step on any, at least until I took my shoes off for bed. I then immediately stepped on one I somehow forgot to cover. Now aching from a tetanus shot and a pierced foot (not to mention a sore back, because at 65 I'm too damned old to do this anymore), I have little to do but lay here and talk about gauge theory, one of my favorite topics. The Weyl gauge in electrodynamics is an exceptionally simple prescription that gives the electric potential $$\Phi$$ rather directly; in fact, it's just $$\Phi = 0$$. Why this prescription is attributed to Weyl escapes me, but in certain situations it's obviously very handy. But even then, the electric field $$E$$ itself remains largely undetermined. It has always amazed me that electric and magnetic fields are so easily detected and measured (an EMF meter, which is actually a kind of antenna that detects and quantifies deflections, like the jump of the needle in a voltmeter, is one such device), while the underlying four-potential $$A^\mu = (\Phi, \vec{A})$$ that $$E$$ and $$B$$ are made from is essentially undetectable and immeasurable. There is no device that can tell us in a straightforward manner that a "bare" electric potential $$\Phi$$ is nearby, or that a non-zero vector potential $$\vec{A}$$ is lurking about. And, for that matter, no device can measure their intensity. The reason for this has to do with the gauge freedom of the four-potential. As is well-known, Maxwell's equation are unchanged under the pair of gauge transformations $$\vec{A} \rightarrow \vec{A} - \vec{\nabla} \lambda, \quad \Phi \rightarrow \Phi + \frac{1}{c} \frac{\partial \lambda}{\partial t},$$ where $$\lambda(x,t)$$ is a completely arbitrary scalar function of space and time. Consequently, there is no such thing as the four-potential $$A^\mu$$ because $$\lambda$$ can have any value. Nevertheless, the gauge parameter $$\lambda$$ can be specified in such a way that makes Maxwell''s equations easier to solve. (In fact, the potentials were originally viewed as just a mathematical convenience with no physical validity, while gauge invariance was seen as a mere happenstance of the formalism.) Recall that the electric and magnetic fields can be expressed in closed form by considering the homogeneous set of Maxwell's equations $$\vec{\nabla} \times \vec{E} + \frac{1}{c} \frac{\partial \vec{B}}{\partial t} = 0, \quad \vec{\nabla} \cdot \vec{B} = 0,$$ which can be solved using simple vector identities to give $$\vec{E} = -\vec{\nabla} \Phi - \frac{1}{c} \frac{\partial \vec{B}}{\partial t}, \quad \vec{B} = \vec{\nabla} \times \vec{A}$$ As is easily shown, these physical quantities do not change under the transformations given above. But what about the inhomogeneous Maxwell's equations, which specify the sources? They are $$\vec{\nabla} \cdot \vec{E} = 4\pi \rho, \quad \vec{\nabla} \times \vec{B} - \frac{1}{c} \frac{\partial \vec{E}}{\partial t} = 4\pi \vec{j}$$ Using the identities for $$E$$ and $$B$$ above, these go over to $$\nabla^2 \Phi + \frac{1}{c} \frac{\partial (\vec{\nabla}\cdot\vec{A})}{\partial t} = -4\pi \rho$$ and $$\nabla^2 \vec{A} - \frac{1}{c^2} \frac{\partial^2 \vec{A}}{\partial t^2} - \vec{\nabla} \left( \frac{1}{c}\frac{\partial \Phi}{\partial t} + \vec{\nabla}\cdot\vec{A} \right) = - 4\pi \vec{j}$$ As many have noted, these last two expressions are ugly as hell, not to mention the fact that they're inextricably coupled in $$\Phi$$ and $$\vec{A}$$. But they are both invariant with regard to a gauge transformation, and we can use that fact to simplify them. Consider the scalar quantity $$S = \frac{1}{c}\frac{\partial \Phi}{\partial t} + \vec{\nabla}\cdot\vec{A}$$ A gauge transformation then gives $$S \rightarrow S + \frac{1}{c^2} \frac{\partial ^2 \lambda}{\partial t^2} - \nabla^2 \lambda$$ Thus, $$S$$ can be made gauge invariant if the gauge parameter satisfies the wave equation of light, which is $$\Box^2 \lambda = \frac{1}{c^2} \frac{\partial ^2 \lambda}{\partial t^2} - \nabla^2 \lambda = 0$$ Selecting a gauge parameter $$\lambda$$ whose d'Alembertian $$\Box^2 \lambda$$ vanishes (which we can always do, since it's arbitrary) thus turns $$S$$ into an arbitrary gauge invariant scalar, and for simplicity we may as well set it to zero: $$\frac{1}{c}\frac{\partial \Phi}{\partial t} + \vec{\nabla}\cdot\vec{A} = \partial_\mu A^\mu = 0$$ This is called the Lorenz gauge, and its primary value is that it uncouples the above equations in $$\Phi$$ and $$\vec{A}$$ to give the beautifully symmetric expressions $$\frac{1}{c^2} \frac{\partial^2 \Phi}{\partial t^2} - \nabla^2 \Phi = 4\pi\rho, \quad \frac{1}{c^2} \frac{\partial^2 \vec{A}}{\partial t^2} - \nabla^2 \vec{A} = 4\pi \vec{j}$$ or, in lovely covariant language, $$\Box^2 A^\mu = 4\pi j^\mu$$ where $$j^\mu = (\rho, \vec{j})$$. In principle, if you can solve one of these equations, then you can solve the other using the same approach. I've always had problems with the Lorenz gauge. For one thing, I always confused Ludwig Lorenz with the far more famous Hendrik Lorentz of Lorentz contraction fame. They're not the same guy. For another, setting $$S = 0$$ requires that the gauge parameter $$\lambda$$ satisfy the wave equation, but the opposite is not necessarily true. The only sure way of justifying the Lorenz gauge is by appealing to a rather nasty theorem in vector calculus called Helmholtz's theorem, which states that any well-behaved vector function can always be expressed as the sum of a divergence and a curl. For the vector $$\vec{A}$$, the curl is already specified in terms of the magnetic field via $$\vec{B} = \vec{\nabla} \times \vec{A}$$. But the divergence of $$\vec{A}$$ is undetermined, so we can select any arbitrary value for it. That's what I was taught, but I still don't get it. I know it should have something to do with gauge transformations, but I'll be darned if I know how. At any rate, to me the four-potential is something like God — it never makes its existence known, and is a total and profound mystery, yet it's somehow there, and can be deduced mathematically and by physical reasoning. I cover some of these thoughts in my elementary write-up on the Aharonov-Bohm effect, which explains how the physical existence of the four-potential was finally demonstrated by a very clever (and beautiful) quantum-mechanical experiment. By the way, noted UC Berkeley physics professor J.D. Jackson has written a lengthy paper detailing many useful types of gauge transformations. But be warned — Jackson is also the author of many a grad student's greatest nightmare, the seemingly impenetrable textbook Classical Electrodynamics.
 Quote of the Week — Posted Monday, March 3 2014 US Secretary of State John Kerry, on Russian warmongering in Ukraine:"You just don't invade another country on phony pretext in order to assert your interests." And that's from 2004 Democratic presidential candidate John Kerry. I can only wonder if he was intentionally including a "dog whistle" message in that statement, or if he's even aware of how monstrously hypocritical it sounds. And speaking of that erstwhile "Empire of Evil" or "Axis of Evil" country, I can remember George W. Bush (easily the stupidest and most insanely corrupt President we ever spawned) talking about Vladimir Putin not that long ago, when Bush "looked into his eyes and saw his soul." Do you remember that? Well, it seems the country's conservatives don't. But they're all over President Obama for being a total wuss for not getting tough with Putin. Fratboy George W. Bush in happier, cockier days. He's still an asshole. Yeah, let's shoot off those nuclear-tipped ICBMs and get it the fuck over with.
 Schrödinger Again — Posted Saturday, February 22 2014 Most people know Erwin Schrödinger as the father of wave mechanics and the co-recipient (with Dirac) of the 1933 Nobel Prize in physics. But he was also interested in numerous other scientific areas, including biology, genetics, general relativity and color measurement (he was also a noted womanizer, but that ain't scientific). In the 1940s his interests turned to a fundamental topic in differential geometry, that of affine connections. I just posted an online paper (suitable for undergraduates) concerning one particularly simple connection that Schrödinger presented in his short but illuminating 1950 book Space-Time Structure. I bought the book back around 1978 and still turn to it on occasion. One warning — I kind of dump on Hermann Weyl in this paper, as I believe Schrödinger's connection makes more sense than Weyl's. But whatever.
 Einstein's Cake — Posted Monday, February 17 2014 I'm getting a renewed interest in (and even appreciation of) the latter work of Erwin Schrödinger in what he referred somewhat extravagantly to as "the final laws" of gravity and electromagnetism, which he developed in the years immediately following the end of World War II. This work in many ways paralleled that of Einstein, whose interest in a unified theory of gravitation and electromagnetism continued unabated from around 1925 until his death in 1955. By 1939 Schrödinger had moved to Dublin, Ireland from his native Austria, following a long bout of political persecution from Nazi Germany, which had annexed Austria two years prior to the war. (He was not a Jew, but his progressive ideas were nevertheless annoying to the Germans. But as a co-recipient of the 1933 Nobel Prize in physics his fame fortunately outweighed his infamy in Nazi eyes, so his life was never in danger.) Schrödinger helped establish the Institute for Advanced Study in Dublin and became a naturalized citizen there in 1948. Following his retirement in 1955 he moved back to Austria, where he died in 1961. During his years in Ireland he wrote numerous papers on unified field theory (while simultaneously siring several illegitimate children with two Irish women), one series of which was titled The Final Affine Field Laws. Like Einstein, Schrödinger had decided that the symmetry of the affine connection $$\Gamma_{\mu\nu}^\lambda$$ in the lower two indices should be abandoned in order to derive a workable theory. This idea had been considered by many physicists, even as far back as 1918, but it introduces many problems. Any asymmetry in the connection of course goes unnoticed in the equations of the geodesics $$\frac{d^2x^\lambda}{ds^2} + \Gamma^\lambda_{\mu\nu} \frac{dx^\mu}{ds} \frac{dx^\nu}{ds} = 0,$$ but quantities like the Riemann-Christoffel and Ricci tensors become unwieldy, and the field equations associated with these more general tensors don't seem to produce any useful physics. Furthermore, the mathematical notation itself is messy, as one must continually work to keep the symmetric and asymmetric pieces distinct from one another throughout the derivations. In my humble opinion, it's all an exercise in futility, and I suspect Einstein and Schrödinger both feared this was indeed the case. But there are many physicists today who remain undaunted by these difficulties. Notable is physics professor Nikodem Poplawksi of the University of New Haven in Connecticut (previously with Indiana University), who has authored many papers involving general affine and asymmetric connections. His work (some of which has been featured on television programs) ranges from conventional to truly interesting, even profound to crackpot. But I don't think there are many who have investigated the connection and its possible relation to gravitation and electromagnetism as much as he has. His many papers, most of which are available on arXiv.org, have the benefit of being accessible to the motivated undergraduate, and I encourage the interested student to look into the ideas of this young New Haven professor. Near the end of his life, when he had completed his failed work on unified field theory, Einstein's personal secretary Helen Dukas had a cake baked for the great scientist in honor of his questionable "achievement." The cake was decorated with the field equations themselves, in red icing. I have a neat photo of the cake laying around here somewhere on my hard drive, and if I can locate the damned thing I'll post it here. In the meantime, you can pop over to the jstor.org academic publishing website where you can read Schrödinger's own papers (in English) on the final affine laws yourself (requires a free subscription).
 And the Survey SAYS — Posted Sunday, February 16 2014 In Arthur Conan Doyle's A Study in Scarlet, Sherlock Holmes admits to an astonished John Watson that he is not aware that the Earth orbits the Sun, nor does he care: "What the deuce is it to me?” he interrupted impatiently; “you say that we go round the Sun. If we went round the Moon it would not make a pennyworth of difference to me or to my work."By way of explanation, Holmes reveals that he is careful not to commit such "useless" information to his brain, lest it interfere with more important things, like the science of deduction. Doyle's novel was written in England in 1886, yet even then it would have been impossible to find someone who thought that the Sun revolves around the Earth. But here in 2014 America such a finding would not be unusual at all. A recent survey conducted by the National Science Foundation showed that 26% of Americans actually believe just that. Similarly, only 39% of Americans believe in the Big Bang, and only 48% express a belief in evolution. And half of Americans believe that antibiotics are effective against viruses. Not surprisingly, Europeans and Asians fared much better in the survey. Any astrophysicist will tell you that the Earth and Sun actually revolve about a common center of mass located very close to the Sun's core. I'll bet nearly 100% of Americans polled would not know that fact, but that is of course quite excusable. But to have 26% of Americans think that the Sun goes round the Earth confirms my theory that a quarter of us are certifiably insane. And they're called Republicans.
 Making the Necessary Adjustments — Posted Saturday, February 15 2014 The first-ever winner of the Hermann Weyl Mathematics Prize (2002), Edward Frenkel is a Russian-born professor of mathematics at UC Berkeley whose short NY Times article this Sunday touches on the subjectivity and objectivity of mathematics. Like Einstein's colleague Kurt Gödel, Frenkel asks whether mathematics simply "is," and is therefore subject only to discovery and analysis by humans, or if it's purely an invention, in which case it is subjective to some extent. Frenkel also addresses a favorite topic of mine, which is the question of what the ultimate reality might be. His article references a recent paper by physicists Silas Beane, Zohreh Davoudi and Martin Savage, which considers the possibility that our universe is actually a computer simulation (I posted the URL for this article two years ago, but you can also link to it from Frenkel's article). Frenkel suggests that if we are indeed living in a simulation then our mathematics might not be inherently absolute but simply a kind of artificial version handed down to us from our simulators, who decided early on that $$1+1 = 2$$ and not $$3$$, for example. The Beane et al. paper actually addresses the possibility that the proposed computer simulators' mathematics is not an invention at all, but a fixed logic like ours from which their simulation is based. What makes the paper interesting is the authors' supposition that, either due to oversight or the limitations of their technology, the simulation is "flawed" at some level, making it possible for us otherwise unwitting humans to discover that we've been had. I once suggested that far more powerful Large Hadron Collider-like machines might someday reveal such flaws — for example, we may not discover a wealth of new physics, particles and forces at all, but a barren desert representing the "pixel" limits of the simulators' impressive but ultimately constrained technology. For some reason, the article reminded me of an Amazing Stories episode from long ago (or something like it*), which featured a man who suddenly realizes that nearly everything he knows is wrong. In the end (if I remember it correctly) he has to be re-educated by his 4-year-old daughter, who reads to him from a child's reading primer. She shows him a picture of a cake, which is labeled "dinosaur" in the book. Needless to say, the man realizes he has a lot to learn (or relearn). If we are ever granted access to the true reality behind our existence, I wonder if it will be like that. But perhaps it will be like this: From The Thirteenth Floor (1999). Simulant (Vincent D'Onofrio) meets simulator (Craig Bierko) with unpleasant results. * A friend has since informed me that the episode was "Wordplay", from the newer (1985) Twilight Zone series.
 Schrödinger on Weyl — Posted Friday, February 7 2014 In 1922, Schrödinger submitted a paper to Zeitschrift für Physik that apparently represents the first attempt to tie Hermann Weyl's 1918 gauge theory to quantum mechanics. Schrödinger's On a Remarkable Property of the Quantum-Orbits of a Single Electron (Zeit. f. Phys. 12 1922, 13) notes that Weyl's proposed metric $$\hat{g}_{\mu\nu} = e^{-k\int \phi_\mu dx^\mu} g_{\mu\nu}$$ explains the energy spectrum of an electron in the hydrogen atom if the term in the exponential is an integral multiple of $$ie/\hbar c$$. Schrödinger adds that It is difficult to believe that this result is merely an accidental mathematical consequence of the quantum conditions, and has no deeper physical meaning.At the same time, he is hesistant (or unable) to expound on the role that Weyl's theory might actually play in quantum theory, which was then still in its infancy. Perhaps Schrödinger's observation that the factor was pure imaginary bothered him, since Einstein's and Weyl's theories were, after all, classical theories. Two pages from Schrödinger's notebook (mid-1925) — the genesis of his wave function concept Note that Schrödinger wrote this paper three years before his own seminal announcement of the wave equation, which for the first time fully explained the strange quantum behavior of hydrogenic electrons that Bohr originally reported on in 1913. Indeed, the then still-emerging quantum theory had not advanced appreciably beyond Bohr's work, and the paper demonstrates the kind of brilliant thinking that was to characterize Schrödinger's contributions to physics, which were finally rewarded when he shared the 1933 Nobel Prize with Dirac. One must also remember that Weyl was essentially trying to eliminate the subjective concept of scale in his theory. This has since led to conformal (scale- or length-independent) cosmological theories, which may or may not have anything to do with the problems of dark matter and dark energy. Certainly, when the Universe was born out of the Big Bang, the concept of scale or length had little if any physical meaning, since spacetime "outside" the Big Bang did not even exist! I have been unable to locate an English translation of Schrödinger's paper and, loath as I am to translate the entire (rather lengthy) article from my original German copy, am presenting here the abbreviated version reproduced in the late Lochlainn O'Raifeartaigh's indispensable 1997 book The Dawning of Gauge Theory. The Alpbach, Austria graves of Erwin and Anny Schrödinger, with perhaps the most profound physics equation of all time (and yes, it beats $$E=mc^2$$)
 Mathematical Justice? — Posted Wednesday, January 29 2014 Here's an interesting paper from last year by Alexander Afriat of the Université de Bretagne Occidentale entitled How Weyl Stumbled Across Electricity While Pursuing Mathematical Justice (see also this paper). The "justice" that Afriat talks about has to do with the equality of direction and distance in Riemannian geometry — vectors can have any direction they want, but their lengths are required to be fixed. This is in direct conflict with quantum mechanics, where the direction of a state vector $$|\psi\rangle$$ can be anything, likewise its length; multiplying the state vector by any real or complex number doesn't change the vector at all — the length of the vector is in fact essentially meaningless. But I think Afriat has confused mathematical justice with mathematical symmetry, which is what I believe Weyl was actually interested in. All of our physics appears to arise from mathematical symmetry, which is essentially the invariance of our theories with respect to coordinate change, linear and rotational translation, time translation and quantum-mechanical gauge or phase translation. These symmetries also give us the conservation laws, like those for energy, linear and angular momentum and electrical charge. To me, these symmetries are the most sublime and beautiful evidence we have that there is a Great Intelligence behind everything. And what is that Great Intelligence, you ask? I haven't the faintest idea. At any rate, Afriat's paper provides lots of neat quotes from Weyl that appear to support the contention that Weyl was indeed on a kind of philosophical or spiritual quest for the truth. I just wouldn't call it "justice."
 Entangled Systems — Posted Wednesday, January 1 2014 Although Hermann Weyl's 1918 theory of conformal invariance failed as a model for the unification of gravity and electromagnetism, it was a phenomenal success when it was applied ten years later to quantum physics. Even today, it seems remarkable that a theory that is invariant with respect to the simple phase transformation $$|\psi\rangle \rightarrow e^{i \theta} |\psi\rangle$$, where $$\theta(x)$$ is an arbitary function of the spacetime coordinates, could explain the conservation of electric charge. Renamed gauge invariance, Weyl's idea is a cornerstone of modern quantum theory. But there is another way of dealing with quantum state vectors that does not involve phase arbitrariness, and that is the density operator approach. If the state vector $$|\psi\rangle$$ contains everything we are allowed to know about the quantum state $$\Psi$$, then surely the slightly more complicated dyad operator $$|\psi\rangle\langle\psi|$$ contains the exact same information. Indeed, it not only contains the same information and is phase invariant, but it also provides a means for understanding random collections of quantum states (or mixed states), a topic usually skipped in undergraduate courses. The density operator formalism also opens the door to the branch of quantum physics known as quantum information theory, in which the mystery of quantum entanglement is explained. Although most of his texts are highly technical and written in German, University of Konstanz professor of physics Jürgen Audretsch's 2007 book Entangled Systems: New Directions in Quantum Physics is written in English at an undergraduate level that is accessible to students of physics, mathematics, chemistry and even computer science. While the book presupposes a beginner's understanding of basic quantum mechanics, it's the most accessible introduction to quantum entanglement, information and entropy I've ever seen (even better than Leonard Susskind's video lectures). The book is a great self-teaching tool, and includes many exercises. (Amazon wants seventy bucks for the book, but you can read a good portion of it for free over at Google Books.) I'm particularly impressed because for years now I've ruminated on the idea that physical reality is fundamentally rooted in the creation, propagation and annihilation of information and that, for whatever reason, the universe itself is somehow tied to the notion of "interestingness." For many years, physicists have asked the question "Why is there something rather than nothing in the universe?" The answer might simply be "Because something is more interesting than nothing." [On the downside (at least for me), Audretsch's book addresses some topics that I wanted to use in a book of my own. How can I write anything when other people keep beating me to it?] Anyway, Happy New Year!
 Is We Evolving? — Posted Tuesday, December 31 2013 To me, the most beautiful and profound aspect of physical law is that Nature invariably strives to be as efficient as possible. This is succinctly demonstrated by the fact that the mathematical quantity known as the action, which Nature is somehow intimately familiar with, is invariably extremalized (and usually minimized) in all physical interactions, from the very small (quantum physics) to the very large (gravitation). Since action is always expressed in units of momentum-displacement ($$p \cdot x$$) or energy-time ($$E \cdot t$$), Nature evidently likes to do things using the the least momentum along the shortest path, or the least energy in the least time. Indeed, the principle of least action as first developed in the 18th century was viewed as the best scientific evidence for the existence of God. When biologist Charles Darwin visited the Galápagos Islands in the 1830s he noted that there was a diverse variety of birds (notably finches) whose food preferences depended to a great extent on the size and shape of their beaks. Those who fed primarily on small seeds had small beaks, while large-beaked birds ate larger, harder seeds. Some birds, which fed mostly on hard-to-reach seeds (like those in cactus), had long, narrow beaks. While not a physicist or mathematician, Darwin saw this diversity as evidence of an evolutionary tendency that trends toward efficiency — for example, birds who feed on small seeds do not carry around heavy beaks, as this would be a waste of energy. I doubt very much if Darwin was ever aware of the principle of least action, but if he was he would probably have viewed evolution as a good example of it. True, Darwin likely saw evolution as a slow process, taking many generations of animals over huge time periods, but given the fact that environmental stressors such as climate change, disease, inter-species competition and predator population change all vary slowly with time, Darwin probably considered evolution to be a very slow but efficient process overall. [Side story — in 1971 I took an undergraduate class in biochemistry, and the professor calculated the energy efficiency of the electron-transport mechanism at 67%. He compared this with the efficiency of a car engine, which at best is only around 25%. Nature wins this contest, hands down.] But for many people at the time (and even today), Darwin's On the Origin of Species, written in 1859, went too far. It implied that humans themselves had physically evolved over time, in seeming contradiction to the Bible. Most people in Darwin's time believed that God had created man roughly 6,000 years earlier, and that man and all the other creatures were created by God in the exact same form that we see them today. Worse, Darwin hypothesized that humans had evolved from earlier primate forms related to monkeys and apes. Human ego rejected the notion that people were descended from monkeys — "If monkeys had evolved into people", the saying went, "then why are there still monkeys walking around?" Although we now know from sound genetic evidence that the genetic lines of apes/monkeys and man diverged some 5 million years ago (which is why monkeys are still around) and that physical evolution can take place startlingly fast, it should not be surprising that the theory of evolution is still being questioned today. After all, Darwin's work is rarely actually read by anyone (much less studied), and biblical genealogy from Adam and Eve on down to Jesus supports the notion that only about 6,000 years have elapsed since time began. To a person of strong Christian, Jewish or Muslim faith, evolution is not only puzzling but contradictory to religious belief as well. But what is surprising, however, is that in spite of new, ongoing and profound fossil and genetic discoveries, evolution is being increasingly rejected by people of faith in America today. Some 1,983 American adults from all 50 states were polled in a new study conducted by the Pew Research Center regarding their views on human evolution. Although 60% of Americans indicated they believe human evolution has occurred, fully 33% believed that humans have remained physically the same since they were first supposedly created by God. While marginally more Americans believe in evolution today, the percentage of white, evangelical Protestants who reject evolution has increased. Indeed, nearly two-thirds of white evangelicals polled do not believe in evolution: I don't know about you, but this scares the crap out of me. When I was teaching I used to tell my students that there are no laws in science, only theories. Electromagnetism, quantum mechanics, gravitation, chemistry, aerodynamics and germ theory are and will always be just theories, no matter how much evidence is compiled to support them — that's what the scientific method is all about. People universally believe these theories, but 33% hold that evolution is "just a theory." And I suspect that a sizable subset of that 33% is also wondering why monkeys are still walking around if there's anything at all to Darwin's ideas. Study after study now confirm that the country is becoming ever more politically and culturally polarized. I keep asking myself how otherwise intelligent people can reject facts, reason and empirical evidence in favor of dogmatic allegiance to illogical, self-contradictory religious nonsense — people who can drive a car, hold a job, program a DVR, balance a checkbook and even teach at university, but choose willful ignorance over a life of rational thought. I keep asking myself what they're afraid of, what is it that they find so frightening that they would abandon the thinking, reasoning brain that God gave them. I don't have any answers, and I have no idea what's going on around here. I can only hope that 2014 turns out better than 2013. Those who can make you believe in absurdities can make you commit atrocities. — Voltaire