Who Was Hermann Weyl?
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Weyl's Spinor and Dirac's Equation
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A Note on Weyl Conformal Gravity
Weyl's 1918 Theory
Weyl's 1918 Theory Revisited
Weyl v. Schroedinger
Why Did Weyl's Theory Fail?
Weyl and the Aharonov-Bohm Effect
Weyl's Scale Factor
Weyl's Spin Connection
Weyl and Higgs Theory
Lorentz Transformation of Weyl Spinors
Riemannian Vectors in Weyl Space
Introduction to Quantum Field Theory
Electron Spin
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Bell's Inequality
The Four-Frequency of Light
There Must Be a Magnetic Field!
Kaluza-Klein Theory
A Brief Look at Gaussian Integrals

Kets, Bras and All That
General Relativity
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God and Light

Particle Chart (Courtesy CPEP)

Uncommon Valor

Sophie did not forget Jesus
Long live freedom!

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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.

 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. 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 $$S = \frac{1}{c}\frac{\partial \Phi}{\partial t} + \vec{\nabla}\cdot\vec{A}$$. Just for the hell of it, let's set $$S = 0$$. A gauge transformation then gives $$\frac{1}{c}\frac{\partial \Phi}{\partial t} + \vec{\nabla}\cdot\vec{A} \rightarrow \frac{1}{c}\frac{\partial \Phi}{\partial t} + \vec{\nabla}\cdot\vec{A} + \frac{1}{c^2} \frac{\partial ^2 \lambda}{\partial t^2} - \nabla^2 \lambda = 0$$ Thus, $$S = 0$$ can be satisfied only 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) therefore implies that $$\frac{1}{c}\frac{\partial \Phi}{\partial t} + \vec{\nabla}\cdot\vec{A} = \partial_\mu A^\mu = 0,$$ 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, $$\Box^2 A^\mu = 4\pi j^\mu$$ 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 notable 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 theoretical 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