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AfterMath

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Zurich Bildarchiv

Who Was Hermann Weyl?

Wheeler's Tribute to Weyl (PDF)

Old Stuff
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Math Tools
Weyl's Spinor and Dirac's Equation
Weyl's Conformal Tensor
Weyl Conformal Gravity
Weyl's 1918 Theory
Weyl's 1918 Theory Revisited
Weyl v. Schrodinger
Why Did Weyl's Theory Fail?
Did Weyl Screw Up?
Weyl and the Aharonov-Bohm Effect
The Bianchi Identities in Weyl Space
Conformal, Parameter-Free Riemannian Gravity
Gravity Wave Tutorial
Conformal Kerr-de Sitter Gravity
A Child's Guide to Spinors
Levi-Civita Rhymes with Lolita
Weyl's Scale Factor
Weyl's Spin Connection
Weyl and Higgs Theory
Weyl & Schrodinger - Two Geometries
Lorentz Transformation of Weyl Spinors
Riemannian Vectors in Weyl Space
Introduction to Quantum Field Theory
A Children's Primer on Quantum Entanglement
Veblen and Weyl
Electron Spin
Clebsch-Gordan Calculator
Bell's Inequality
The Four-Frequency of Light
There Must Be a Magnetic Field!
Non-Metricity and the RC Tensor
Curvature Tensor Components
Kaluza-Klein Theory
The Divergence Myth in Gauss-Bonnet Gravity
Schrodinger Geometry
A Brief Look at Gaussian Integrals
Differential Forms for Physics Students
Particle Chart

Uncommon Valor

She did not forget Jesus!
"Long live freedom!"

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2011 Archives

 Weyl and Higgs Theory -- Posted on Friday, December 9 2011 Our hope of finding a gauge theory of weak interactions with massive gauge bosons looks forlorn. It appears that we shall also have unwanted (unobserved) massless scalar particles to worry about. Nevertheless, let us proceed from a global to a local gauge theory. A miracle is about to happen. — Halzen and MartinToday's Los Angeles Times has an article on the elusive Higgs boson, the "God particle" that the Large Hadron Collider (LHC) is (hopefully) zeroing in on. The Higgs particle represents the final missing piece of the Standard Model of physics, and its existence (or rather the associated Higgs field) would explain why elementary particles such as electrons and muons have the masses they have. The LHC has been running for nearly two years now, and its scientists have accumulated enough data to make a fairly accurate assessment of whether they have found the Higgs among the trillions of collision events stored in the collider's massive computer system. A public announcement is expected to be made early next week, possibly on 13 December. It should be remembered that the Higgs theory is critically dependent on Hermann Weyl's gauge invariance principle which, prior to Peter Higgs' theoretical work of 1964, was believed to be strictly applicable only to theories involving particles of zero mass. However, unlike the zero-mass carriers of the strong, electromagnetic and gravitational forces (gluons, photons and the as-yet undiscovered graviton), the weak force is mediated by the $$W^{\pm}$$ and $$Z^{0}$$ vector bosons, which are anything but massless. The work of Higgs (and a few others) showed how gauge invariance could be preserved in the Lagrangian describing spin-zero particles. I've given an elementary mathematical treatment of how this is done, although there are much better explanations available: Griffiths' Introduction to Elementary Particles is one, while Halzen and Martin give a somewhat older treatment in their Quarks and Leptons: An Introductory Course in Modern Particle Physics. At any rate, the Higgs mechanism is profoundly ingenious if subtle, and if the LHC finds the Higgs (which Leon Ledermann has also called the "goddamned particle") Higgs should surely be considered for the Nobel prize. By the way, the journal New Scientist has an article that posits the question "What if there is no Higgs particle?" It answers the question by saying that the Standard Model will surely survive through other theoretical avenues. But I liken confirmation of a non-existent Higgs boson with faster-than-light neutrinos: the world will go on spinning, but it will be a much more c onfusing place. Lastly, and by way of apology, those of you who have waded through my dreary write-ups will have probably noticed that I base my belief in the existence of God on the fact that mathematical symmetry governs the laws of nature, yet that symmetry cannot be directly observed. Whatever your own personal preference is regarding this matter, I wish you all a joyous and safe holiday season! UPDATE, 12-13-2011: Gotta wait some more. "Tantalizing" data, but no definitive proof that the Higgs particle exists. Peter Higgs in 2008.
 Weyl at the IAS -- Posted on Sunday, September 18 2011 One more. The Institute for Advanced Study, where Hermann Weyl went after fleeing Germany in 1933, has three pages on Weyl's life and legacy. I don't know how long these pages have been up, but they provide a very nice summary of the man and his contributions to mathematics and physics. One page has a photo of a bust of Weyl that his widow, Ellen Bär, sculpted in 1957, two years after his death. The IAS site includes this beautiful (but undated) photo of Weyl, which I'd like to believe was taken early in his tenure at the school. If true, then he's either working on a math problem or trying to understand the Internal Revenue Service's tax tables of which, as a newly naturalized American citizen, he later wrote about! Co-founded by the noted American educator Abraham Flexner in 1930, the IAS has been home to many of the world's greatest scientists and mathematicians. The Institute imposes no teaching or administrative requirements on its faculty, and they are generally free to conduct whatever research interests them. Photo from the Shelby White and Leon Levy Archives Center, Institute for Advanced Study, Princeton, NJ, USA, with special thanks to Ms. Erica Mosner, Assistant Archivist.
 Weyl's Princeton Home -- Posted on Saturday, September 17 2011 Here's a very recent photo of Hermann Weyl's home in Princeton, New Jersey, close to the Institute for Advanced Study (where Weyl taught from 1933 to 1951). Located at 284 Mercer Street, it's a few blocks from Einstein's home at 112 Mercer Street. Fellow mathematician Kurt Gödel also lived nearby, and he and Einstein often walked to and from the IAS together. Much thanks to Dr. Gualtiero Badin of Princeton University for sending me the picture and Prof. C.N. Yang for sending me the address. Yang bought Weyl's house in 1957 (the year Yang won the Nobel prize in physics) and lived there until 1966.
 Weyl Gravity Again -- Posted on Thursday, August 18 2011 I don't post to this site anymore, but I wanted to toss in this last item. There are a zillion papers on arXiv.org dealing with Weyl conformal gravity theory, and I really can't contribute anything to the literature. But I did derive a much simpler alternative to Cornelius Lanczos' treatment of the mathematics, and here it is. Two of the papers I reference can be found here and here. A snapshot of the DeWitt reference can be found here (specifically equations 16.34 and 16.35). Enjoy. Whether or not Weyl gravity is a valid theory with regard to the dark matter/dark energy problem is anyone's guess. I suppose it's as good as any right now. Cornelius Lanczos was a Hungarian-American mathematical physicist (1893-1974). After receiving his PhD in physics in 1921, he discovered an exact solution to Einstein's field equations representing a cylindrically symmetric distribution of matter. Lanczos was Einstein's assistant in Berlin in 1928-29 before settling at Purdue University, where he spent the bulk of his academic career.
 Classes of Infinities -- Posted on Monday, August 1 2011 Somewhere in the infinite sequence of digits comprising the transcendental number pi (π) is the string 9142085257914481477154This apparently random sequence of numbers is guaranteed to be there, as π has an infinite number of digits. But perhaps the string "In the beginning God" is too easy. It doesn't take much imagination to realize that the entire Old Testament is transcripted in π as well, along with the New Testament (in koine Greek), and all of Shakespeare's plays and sonnets (in Kurdish, yet). Somewhere, many, many trillions of decimal places into π, those books are there. So is My Pet Goat. It was the German mathematician Georg Cantor (1845-1918), the inventor of set theory, who was the first to rigorously study infinite sets of numbers. He apparently was also the first to recognize that there are infinities that are fundamentally different from other infinities, such as the difference between the infinite set of cardinal numbers (0, 1, 2, 3) and real numbers (pretty much everything else). There are infinitely more real numbers (13.002123 and π, for example) than integers and infinitely more integers than prime numbers. But they're all infinite, so big deal, right? Today's New Scientist has an article that examines this question in light of recent and current mathematical research. You can read the article for yourself, but to me it implies that infinity might have a kind of mathematical structure all its own, a structure in which infinite sets display individual, distinct properties that are only now being examined. One aspect that the structure idea touches on has to do with the continuum hypothesis, which basically posits that there are two kinds of infinity—discrete infinities (e.g., integers) and continuous infinities (real numbers), with nothing in between. They're both without limit, but each member of the discrete set is entire to itself (the number 13, for example), while $$&pi$$ has an infinite number of digits and cannot be written down save by the purely human invention of designating a symbol to represent the number. So why is this a big deal? Infinity is infinity, right? Here's my take. In quantum physics we have a number Z that represents Feynman's path-integral transi tion amplitude for some initial and final state. The integral must be integrated over an infinite number of space-time points and so cannot be done in closed form. But the integration can be done in piecewise fashion, i.e., by perturbation. Each step in the integration gets exponentially more difficult, though, so the calculations must be taken only as far as absolutely necessary. With each step, more and more state/particle paths (or virtual particles) enter into the calculation, but 6-digit accuracy is usually achievable after only a few steps. Assume now that some genius comes along who figures out a way to do the integration exactly, without any approximation at all. That is, when given the path integral for some specific problem, she goes to the blackboard and simply writes down the complete answer, presumably a complex number that contains combinations of Planck's constant $$h$$, the speed of light $$c$$, the electronic charge $$q$$, $$\pi$$ and maybe even the gravitational constant $$G$$. Question: if we could dispense with the perturbation approach, what form would the answer have? What would it look like? How would particles be represented? And what would it mean? It seems to me that here we would be crossing over from an infinite number of discrete quantities to a continuous set that could be written down at once. Although the two approaches might yield the same result (to some specified number of digits), the mathematical, physical and philosophical consequences would be enormous. I think we would, in a very real sense, finally behold infinity itself.
 Causality Holds -- Posted on Wednesday, July 27 2011 A group of physicists at the Hong Kong University of Science and Technology has reported that the velocity of a photon in a vacuum is indeed limited to the usual 3×10^8 meters/second speed limit. However, this finding, which appears in Physical Review Letters, is being picked up by the news media as proof that time travel is impossible. Their reasoning is that, according to relativity, time slows down as velocity is increased. At the speed of light, time freezes; ergo, if something goes faster than light, time must go backward. This is a logic I don't understand at all. Republican "science"?
 Nothing v Something -- Posted on Tuesday, July 26 2011 New Scientist's Amanda Gefter's article Existence: Why is There a Universe? touches on the problem of entropy and nothingness: although a zillion moles of a randomly-dispersed gas has a very high entropy (the gas has almost no order), a state of nothingness must also have a very high entropy because "you can shuffle it around all you want and it still looks like nothing." By the same logic, according to physics nobelist Frank Wilczek of MIT, nothingness is a state of perfect symmetry. But in physics, broken symmetries are the rule, so nothingness would have to be an unstable state. Arizona State astrophysicist Lawrence Krauss has a new book out (I can't recall the title just now, as I haven't read it) that deals with this kind of phenomena. A flat universe (one in which there is zero overall curvature) would have a total energy content of exactly zero: the positive energy of matter and fields would be canceled precisely by the negative binding energy of gravitation. Thus, asserts Krauss, a state of nothingness (zero energy) could give rise to a state of somethingness, which still has zero energy. This is surely the most extreme example of mass-energy conservation one could think of. One version of the Heisenberg uncertainty principle states that the energy content of a vacuum can be very high over very small intervals of time: $$\Delta E \Delta t > \frac{1}{2}\hbar$$ When the universe was created time did not exist, so the energy uncertainty must have been very great, indeed. But, as Gefter rightly points out, Heisenberg's principle is a law of physics, and physical laws must have existed prior to the creation of the universe. How did this come about? This question takes us back to Gefter's original inquiry, which is: Why is there something rather than nothing?
 Weyl Gravity Theories -- Posted on Saturday, July 16 2011 Weyl gravity appears to be a very peculiar theory. — Sophie PireauxSome rambling thoughts tonight. Research involving Hermann Weyl's conformal gravity theory continues unabated—many papers are appearing on arXiv that deal with it in one form or another. I can't keep up with the ones I can actually understand, while the ones I can't understand I ignore. Here's a small glimpse of what I see going on. Weyl's 1918 theory was an attempt to explain electrodynamics as a purely geometrical phenomenon. In spite of its elegance the theory failed, although the manifold that Weyl had invented (Weyl geometry) has remained a topic of great interest to this day. Why? In my opinion it is because (apart from the purely conformal aspect of the theory, which is itself aesthetically beautiful) it provides a tantalizing new approach to current problems, including the vacuum energy and dark matter problems (galactic rotation rates, etc.) and quantum gravity. This view is supported by many recent papers, of which the following are just a few: Juan Maldecena, Einstein gravity from conformal gravity, June 2011 Philip D. Mannheim, Making the case for conformal gravity, January 2011 Philip D. Mannheim, Conformal gravity challenges string theory, July 2007 Sophie Pireaux, Light deflection in Weyl gravity: critical distances for photon paths, February 2004 A. Edery and M.B. Paranjape, Classical tests for Weyl gravity: deflection of light and time delay, August 1997 These papers are all very readable and interesting, but I have a few beefs regarding these applications of Weyl geometry. The primary problem involves the Weyl vector $$\phi_\mu$$ (which Weyl tried so valiantly to identify with the electromagnetic potential in his theory). In almost all current theories this vector has been scrubbed out (though I doubt that anyone seriously thinks that Weyl geometry has anything to do with electrodynamics nowadays). But it is this vector that makes the Weyl geometry work; in particular, it makes the Weyl connection conformally invariant. When $$\phi_\mu$$ is zero this is no longer the case; Weyl's geometry reverts back to ordinary Riemannian geometry, so Weyl's gravity theory no longer makes any sense. In addition, the Weyl Lagrangian provides a theory of gravitation at the expense of making the associated Lagrangian of fourth order wi th respect to the metric tensor $$g_{\mu\nu}$$ and its derivatives. Such higher-order theories involve "ghost fields" that are non-physical (of negative norm). Furthermore, none of these approaches deal with the "divergence problem." That is, mass-energy conservation still relies on covariant derivatives rather than ordinary derivatives. Also, no one knows what the mass-energy tensor should look like for 4th-order theories. Lastly, with respect to quantum gravity theories, we see unappealing (at least to me) combinations of classical geometrical quantities (like the Ricci scalar $$R$$ and similar scalars) mixed up with scalar and spinor quantum fields (like $$\Psi$$ ). But what the hell. Let's move on. Weyl proposed using the square of the Ricci scalar ( $$R^2$$ ) as the basis for his gravity theory solely because the density of the Einstein-Hilbert term $$R$$ alone is not conformally invariant; that is, it's not invariant with respect to some arbitrary change of scale $$g_{\mu\nu} \rightarrow \lambda(x) g_{\mu\nu}$$. Weyl's $$R^2$$ is conformally invariant, and is the simplest one available (Weyl was aware that some combination of $$R_{\mu\nu} R^{\mu\nu}$$ and $$R_{\alpha\beta\mu\nu} R^{\alpha\beta\mu\nu}$$ would also work, but I suspect he considered these scalars to be less "pretty.") The gravitational equations of motion for empty space are easy to derive for the Weyl Lagrangian (I've worked it out in at least one of the write-ups available on this site, so I won't do it here). The answer is $$R(R_{\mu\nu} - \frac{1}{4} g_{\mu\nu} R) + R_{|\mu||\nu} - g_{\mu\nu}g^{\alpha\beta}R_{|\alpha||\beta} = 0$$ Contraction with $$g^{\mu\nu}$$ shows that $$g^{\mu\nu}R_{|\mu||\nu} = 0$$, leaving $$R(R_{\mu\nu} - \frac{1}{4} g_{\mu\nu} R) + R_{|\mu||\nu} = 0$$ The trivial solution is $$R = R_{\mu\nu} = 0$$, which is Einstein's equation. But for a constant, non-zero $$R$$ we also have the second-order expression $$R_{\mu\nu} - \frac{1}{4} g_{\mu\nu} R = 0$$ For a Schwarzschild-like line element of the type $$ds^2 = A c^2 dt^2 - B dr^2 - r^2 d\theta^2 - r^2 \sin^2 \theta d\phi^2$$ where $$A$$ and $$B$$ are functions only of the radius parameter $$r$$, the equations of motion give the identifications $$A = B^{-1} = 1 - \beta/r + k r^2$$ with $$\beta$$ and $$k$$ non-zero constants. This is just the Schwarzschild line element with an additional term with one very interesting property—it can be used to explain the observed cosmological acceleration effect, in which the universe seems to be expanding at an ever-increasing rate (the constant parameter $$k$$ is in fact proportional to the cosmological constant $$\Lambda$$). But there's a better candidate for Weyl gravity, one that even Weyl himself seems to have overlooked. As I mentioned, the quantities $$\sqrt{-g}R^2$$, $$\sqrt{-g}R_{\mu\nu} R^{\mu\nu}$$ and $$\sqrt{-g}R_{\alpha\beta\mu\nu} R^{\alpha\beta\mu\nu}$$ are all available, and it would appear that some linear combination of these scalar densities might serve as a good Lagrangian candidate for gravity. As luck would have it, there exists a scalar density in which all these terms appear together automatically. It is, appropriately, the Weyl conformal density $$\sqrt{-g}C_{\alpha\beta\mu\nu} C^{\alpha\beta\mu\nu}$$ itself. Best of all, this density is conformally invariant in ordinary Riemannian geometry; you don't need Weyl's $$\phi$$-field. I've talked about the Weyl conformal $$C$$ tensor several times here before; it is responsible for the tidal, deforming (but volume-preserving) effects of gravitational fields in the absence of matter, and in some respects is more fundamental than the Riemann-Christoffel tensor $$R_{\alpha\beta\mu\nu}$$, which vanishes in the absence of matter. But it never dawned on me to actually compute the $$C^2$$ scalar itself, which turns out to be $$C_{\alpha\beta\mu\nu} C^{\alpha\beta\mu\nu} = R_{\alpha\beta\mu\nu} R^{\alpha\beta\mu\nu} - 2 R_{\mu\nu} R^{\mu\nu} + R^2 /3$$ and so has everything built into it. Two notable researchers who worked out the equations of motion associated with this quantity are Mannheim and his associate D. Kazanas (the pdf file is here). By varying the Weyl Lagrangian with respect to the metric $$g^{\mu\nu}$$ they got the Bach equation $$B_{\mu\nu} = 0$$, where $$B_{\mu\nu} = 12 R_{\mu\lambda} R_{\beta\nu} g^{\beta\lambda} - 3 g_{\mu\nu} R_{\alpha\beta} R^{\alpha\beta} -6 g_{\mu\nu} R_{\:\:||\alpha||\beta}^{\alpha\beta} + 12 g_{\alpha\nu} R_{\:\:||\mu||\beta}^{\alpha\beta} - 6 g^{\alpha\beta} R_{\mu\nu||\alpha||\beta} + 1/2 g_{\mu\nu} R^2 - 2 R R_{\mu\nu} + 2 g_{\mu\nu} g^{\alpha\beta} R_{|\alpha||\beta} - 2 R_{|\mu||\nu}$$ Note that this tensor is traceless ($$B = B_{\mu\nu} g^{\mu\nu} = 0$$) and admits the identification $$R_{\mu\nu} = \Lambda g_{\mu\nu}$$ as a solution ($$\Lambda$$ is usually identified with the cosmological constant). This (along with $$R = 0$$) is called the trivial solution. ArXiv has many papers that detail searches for other solutions. Mannheim and Kazanas solve the Bach equation (not a fun job!) and get the surprisingly simple result $$A = B^-1 = 1 - \beta/r + k r^2 + \gamma r$$ along with a constant term. This is the Schwarzschild/Weyl result with an additional parameter $$\gamma$$ that might also have something to do with dark matter (note that the $$k$$ and $$\gamma$$ coefficients would have to be very small to have gone unnoticed, particularly on galactic scales). Researchers now seem to have plenty of parameters to play with to explain their cosmological and quantum gravity theories. None of them, to me, is very attractive, particularly in light of Dirac's old admonition that "mathematical equations must be beautiful" if they are also to describe the truth. Please give these papers a look. They're at least interesting, if not mathematically beautiful.
 Bored Tonight -- Posted on Wednesday, July 13 2011 Several people have asked where the imaginary "i" comes from in Penrose's treatment of the Elitzur-Vaidman bomb problem that I talked about earlier. I didn't know myself (hell, I just pass this stuff along) but suspected it could be derived by considering a 90^o momentum change in the photon wave function. Assuming an overall arbitrary phase factor of $$\exp(i\theta) = \cos \theta + i \sin \theta$$ would seem to do it (let $$\theta$$ go from zero to $$\pi/2$$ ), but that's not very satisfactory (and probably wrong). But then I asked what the wave function of a photon might actually look like, and realized that I'd never given it any thought. Neither, it seems, has anyone else (just try looking for a photon ket state vector on the Internet). The fundamental quantity that describes a photon is the four vector $$A_\mu$$, which is described by the source-free Lorentz gauge condition as ◻^2 A_mu = 0 The solution to this partial differential equation is A_\mu = exp (-ip⋅x/ℏ) \epsilon_\mu (p) where \epsilon_\mu is the polarization vector (which actually has only two independent terms, not four). Numerous writers call A_\mu the photon wave function, but it would seem that you can't actually do anything with it, much less pull down a factor of i from it under a reflection. Penrose's treatment references the book Optics by Miles V. Klein and Thomas E. Furtak. I borrowed the text from Caltech, but it still doesn't explain where the damned i comes from in the Mach-Zehnder beam splitter experiment. But more searching l ed me to a very interesting paper that presents the photon wave function in a form I've never seen before. It's a 2006 paper by M.G. Raymer and Brian J. Smith of the University of Oregon at Eugene called The Maxwell wave function of the photon. This little five-page paper (suitable for undergraduates) includes a nifty flow chart that takes Einstein's E^2 = m^2 c^4 + c^2 p^2 mass-energy relation in two directions, one for massive particles (Dirac's equation) and one for massless photons (Maxwell's equations). Assuming a three-component vector wave function \Psi(x, t) for the photon having the simple form \Psi(x, t) = E(x, t) + iB(x, t) (where E and B are the electric and magnetic field vectors), we easily see that i {\partial\Psi}/{\partial t} = c∇ × \Psi reproduces all four Maxwell equations for free space. In addition, it is easy to see that the integral ∫ |\Psi|^2 d^3 x normalizes \Psi with respect to the total energy of the electromagnetic field. By the way, this version of the photon wave function \Psi isn't new; Iwo Bialynicki-Birula of the Polish Academy of Sciences has published extensively on it and its role in quantum field theory (but to me this is no wave function; it's just an interesting combination of vectors with some interesting properties). Still, I haven't been able to use any of this to get the i factor from the Mach-Zehnder experiment. Let me know if you find the solution.
 Bombs Away -- Posted on Tuesday, June 21 2011 While re-reading Shadows of the Mind by Roger Penrose (1994) I came across his description of a fascinating quantum problem that I had completely forgotten about. It's called the Elitzur-Vaidman bomb problem, and it neatly demonstrates how a real problem that cannot be solved classically can be solved using quantum mechanics. Here we'll walk through it using Pen rose's diagrams but with a few changes to shorten the solution (Penrose's treatment is many pages long). You're given a quantity of bombs, many of which are duds. The live ones are equipped with a highly sensitive trigger device—the trigger is so sensitive that it will respond to the action of a single photon and set off the bomb. The duds are also equipped with triggers, but they are defective—the triggers are jammed and an impinging photon has no effect. Your task is to sort out the duds and to secure a quantity of live bombs. Obviously, there is no classical way you can test any of the weapons without destroying the live ones; the best you can do is find the duds. However, the principle of superposition in quantum mechanics can be used to find the live ones. Here's how. First a preliminary situation to acquaint you with the math and experimental setup (which is known as a Mach-Zehnder interferometer): A single photon emerges from a light source. It will have some state vector associated with it; call this vector |A〉. The photon encounters a half-silvered mirror, so that it has an equal chance of being reflected or transmitted. If it is transmitted, it will enter the state designated as |B〉; if reflected, it will be in the state |C〉. The photon thus finds itself in a superposition of the two states, |A〉 = |B〉/√2 + i|C〉/√2 (The square root is there just to make things look better at the end, and the imaginary number i is added to the C state to account for the fact that a reflection changes the photon's phase by π. Also, I'm using equal signs, but they should be read here as "goes like.") If transmitted, the photon encounters a fully-silvered mirror and is reflected upward. We thus have a new state given by |B〉 = - i|D〉/√2 (The minus sign is purely arbitrary.) If it is reflected by the original half-silvered mirror, the photon proceeds to another fully-silvered mirror, and its state is described by |C〉 = - i|E〉/√2 The total superposition thus far is then described by |A〉 = |B〉/√2 + i|C〉/√2 = - i|D〉/√2 + i(-i|E〉/√2) = |E〉/√2 - i|D〉/√2 Now the superposed photon encounters another half-silvered mirror, where two photometers are located. The photon's state goes either like |D〉 = |G〉/√2 + i|F〉/√2 or |E〉 = |F〉/√2 + i|G〉/√2 depending on which path you're considering. Of course, being superposed it goes both ways, so that the final state becomes |A〉 = (|F〉/√2 + i|G〉/√2)/√2 - i(|G〉 + i|F〉)/√2 = |F〉 The photon thus collapses back to its original state (|F〉 = |A〉), which was inevitable—the photon's initial energy and momentum must be conserved. The photometer at F always registers the photon, while the photometer at G always remains silent. Now consider the revised set-up above, where we have replaced the fully-silvered mirror on the bottom right with a similar mirror attached to the trigger of a bomb. Again, we send one photon out from the source. If the bomb is a dud, the jammed trigger will not budge. The mirror remains in place; we have the same situation as before, and the photometer at G will not register. All dud bombs will activate the photometer at F and leave the photometer at G alone. However, if the bomb is live the situation changes drastically. Consider the scenario where the photon emerges from the source and is actually transmitted through the original half-silvered mirror. The photon encounters the bomb mirror and triggers the bomb, which explodes. In this case, the photon's state has collapsed in accordance with |A〉 = |B〉 The bomb thus acts as a detector (and in this case a rather noisy one). However, if the photon is reflected the photon's state still collapses, this time in accordance with |A〉 = i|C〉 The photon is now reflected by the fully-silvered mirror at the upper left, giving it the new state |C〉 = i|E〉 so that |A〉 = i(i|E〉) = - |E〉 Now, when the photon (still collapsed) encounters the half-silvered mirror at the photometers, it goes back into a superposition of states in accordance with |E〉 = |F〉/√2 + i|G〉/√2 so that overall |A〉 = -|E〉 = -|F〉/√2 - i|G〉/√2 The photon's state will now collapse either at the photometer at F or at G, with a 50% probability for each. Thus, overall a live bomb has only a 25% chance of triggering the photometer at G (the photon has a 50% chance of being reflected at the first half-silvered mirror, then another 50% chance of being reflected to G). But the only ones that can possibly set off the photometer at G are definitely live bombs (or bad, depending on how they will be used). Summary: if you test a bomb, and the photometer at G registers a photon, then the bomb is definitely live. Convince yourself that the live-bomb case makes the state |D〉 unavailable to the photon, and this makes all the difference. Israeli physicist Avshalom Elitzur earned a PhD in physics without any formal education, but his accomplishments are notable and he has wor ked with some great physicists, including Yakir Aharonov. You can find the 1993 Elitzur-Vaidman paper here. In 1994, the bomb-testing problem was verified experimentally by Anton Zeilinger and his colleagues. [Of course, things are more complicated than what's described here. Real beam splitters aren't made from infinitely thin reflecting surfaces; they consist of slabs of glass or other dielectric material of finite thickness. For a simple explanation of why this is important, see How does a Mach-Zehnder interferometer work?]
 Lots of Black Holes -- Posted on Saturday, June 18 2011 Scientists, using a combination of data obtained from the Hubble telescope and the Chandra X-ray orbital observatory, have discovered a large group of supermassive black holes at the edge of the universe, hiding within a larger group of galaxies. The finding adds to growing evidence that black holes and galaxies are somehow intimately related, and that this relationship stretches back to the time of the Big Bang. Artist's conception of an early-universe black hole The presence of the black holes was deduced using "image stacking." The Chandra observatory, which was trained continuously on the study region for 46 days, detected glimmers of X-ray radiation but discerned no individual candidates for black holes. But when combined with ultra-deep field data from the Hubble, the black holes were revealed. Of the 250 or so galaxies studied, a large percentage (perhaps 100%) appeared to contain black holes. (The Chandra observatory was specially designed to detect and measure X-rays given off by cosmologically-distant celestial objects.) X-ray radiation is a high-energy form of light that can be emitted by stars and matter as they accrete closer and closer to a black hole. As matter piles up in the accretion disk (or is compressed as it falls into the hole), friction heats up the matter to the point of iridescence. For this reason some black holes, while actually invisible to the naked eye, are considered "bright" objects. (The Los Angeles Times erroneously reported that X-rays, unlike visible or infrared light, can escape the powerful gravitational pull of a black hole.No, light is light, and X-rays are no better at escaping the pull of a black hole than any other light.) The red shift of light from the galaxies studied had an average Z of about 6, meaning that the galaxies are receding from us at 96% of the speed of light. The Hubble equation can be used to show that the light from these galaxies was emitted some 12.8 billion years ago, or less than a billion years after the Big Bang. The existence of black holes so early in the universe's history seems to imply that they are common and fundamental celestial objects. Numerous scientists familiar with the study have expressed their belief that intelligent life might not have arisen without them.
 The Weyl Curvature Tensor and the End of Everything -- Posted on Thursday, June 9 2011 Roger Penrose's fanciful view of God setting up the universe. In the phase space representing the entire universe, the Creator would have had to set the initial conditions to within one part in 1010123 (depicted here with a very fine placement needle!) to produce the universe that we see today (this is not, however, proof of the existence of God). From Road to Reality. I've touched on the Weyl curvature hypothesis once or twice (an odd term, considering the curvature tensor is Weyl's but the hypothesis is due to Penrose), but here I'll mention two very readable papers that discuss the hypothesis in greater detail. First, some background. Roger Penrose published his idea in 1977 in an attempt to explain a second law of thermodynamics that is compliant with gravitation. By way of example, Penrose began by noting that the second law (which basically states that entropy can only increase with time) requires that a high-pressure volume of gas will invariably expand into a larger volume of lower pressure, the idea being that the entropy S of the gas molecules will increase via Boltzmann's equation S = k log V, where k is Boltzmann's constant (a variation of this equation is inscribed on Ludwig Boltzmann's head stone; see below). However, a large volume of diffuse gas molecules will also tend to collapse under the influence of gravity, resulting in just the opposite effect. The question is: how can this contradiction be resolved, given the assumption that entropy always increases? Penrose decided that there must be a form of entropy contained in gravitational fields, such that a large volume of gas would have a smaller gravitational entropy than, say, a coalesced object like a star or a black hole. This line of thought took Penrose into deep considerations of the second law at the time of the Big Bang singularity and its continuing role in the evolution of the universe (including its possible collapse back to the singularity under the Big Crunch, if that should be the case). When the universe was born, Penrose reasoned, the expanding ball of mass-energy would likely have had a minimal Weyl curvature, while the Ricci curvature would have been large (for a brief description of the two, see my write-up on the Weyl conformal tensor). As the universe evolved, gravitational coalescence occurred with a corresponding increase in Weyl curvature. Penrose therefore reasoned that Weyl curvature might have something to do with entropy, given the fact that they both increase as the universe gets older. Penrose maintained that as the universe ages, more and more black holes will accumulate, so much so that at some point nearly all the mass of the universe will reside in black holes, with a corresponding large Weyl curvature. This will not quite mark the end, however, as the black holes themselves will all slowly evaporate via Hawking radiation. In the end, the universe will consist only of stray photons, and entropy will have reached its maximum extent (perhaps it will be infinite). In this sense, Penrose believes, Weyl curvature and entropy are related. Indeed, the Bekenstein-Hawking black hole entropy-area formula $$S = \frac{1}{4}\frac{kc^3 A}{G \hbar}$$ (which can be derived a number of different ways) demonstrates the intimate relationship between entropy and the surface area A of the ultimate gravitational object, the black hole. Most interestingly, the entropy and all the information it contains exists on the surface of the event horizon! In 2006, O. Rudjord and O. Gron of the University of Oslo investigated the Weyl curvature hypothesis and its possible relationship to gravitational entropy in a beautiful paper, The Weyl curvature conjecture and black hole entropy. The authors assumed that, if Penrose is right, the Weyl conformal tensor Cαβγδ would have to be proportional to gravitational entropy. However, since the contracted trace of this tensor is zero, the only scalar they could form is CαβγδCαβγδ, which they call P2. Then begins an amusing search for a radial vector field whose divergence is proportional to the entropy. After laboriously trying out three alternatives for the proportionality term, they finally settle on the Kretschmann scalar, RαβγδRαβγδ. They then explore this choice in both Schwarzschild and de Sitter spacetimes. Conclusion: at the cosmological horizon (end of time), entropy must be of non-geometrical origin. Oh well. Maybe something else will work. Penrose has written numerous papers himself on the subject, of course, but the most readable I've seen to date is the one he presented to a conference in Edinburgh in 2006. It also introduces the idea that at the universe's end, only photons will exist, so that the geometry of the universe must be conformal (again, see my write-up for a explanation of this point). Weyl himself explored a strictly conformal space-time in his 1918 theory, but it failed, mostly because the line element ds of matter cannot be made scale-invariant. Penrose's latest book is Cycles of Time. I have not read it yet but the reviews, which have been very positive, indicate that Penrose now believes that at the end of the universe all information will have been destroyed via black hole evaporation. The universe, consisting of nothing but radiation, will then have no concept of time or history, and so will "reset" itself by forgetting about its large entropy content. At that point there will be nothing to distinguish the universe from its pre-Big Bang state. Thus, a new Big Bang will occur, possibly with a different set of fundamental physical constants; then another, and another after that, for all eternity. This is pretty heavy stuff, and I have not nearly done it justice in this overly-long post. Penrose's earlier books, The Emperor's New Mind and The Road to Reality, along with the papers cited above, will be of much greater use to those of you want more information. Personally, the subject matter just blows me away.
 Uncertainty Principle Not So Uncertain? -- Posted on Tuesday, May 3 2011 Werner Heisenberg's (1901-1976) famous Uncertainty Principle states that there is a fundamental limit to how accurately we can measure certain quantities simultaneously. The best known example is position and momentum; in summary, the principle says that if you measure a particle's position and its momentum at the same time, there will be a certain finite amount of "fuzziness" in the values you get. This fuzziness is a limit imposed by Nature; it cannot be eliminated or even reduced. Or can it? The journal New Scientist is reporting the recent work of several researchers who believe that quantum entanglement might provide a way to improve upon Heisenberg's famous principle. If it's true, one of the foundational tenets of quantum mechanics is going to be seriously shaken up. You can find the two most recent papers here and here. For a very elementary overview of the quantum entanglement problem, see my write-up on Bell's inequality here. While a rigorous derivation of the precise uncertainty principle is usually given in beginning graduate school, I can give you a very simplified derivation here that any undergraduate can follow. Start with any two arbitrary hermitian operators A and B in a space spanned by some arbitrary state vector |ψ〉. Now define the two associated operators ΔA = A −〈A〉ΔB = B − 〈B〉 where 〈A〉 = 〈ψ|A|ψ〉 is the expectation value of the operator A, with a similar definition for B. The new operators thus define the "distance" of A and B from their "average" values. If we now take the expectation values of the squares of ΔA and ΔB themselves, we get 〈(ΔA)2〉 = 〈A2〉 − 〈A〉2〈(ΔB)2〉 = 〈B2〉 − 〈B〉2 Let us call these quantities σ2A and σ2B, respectively. The significance of this notation will be apparent shortly. Now let's construct a new arbitrary state vector |β〉 from ΔA and ΔB and some real scalar λ via |β〉 = (ΔA + i λΔB)|ψ〉 Its associated bra vector is |β〉*, or 〈β| = 〈ψ|(ΔA − i λΔB) The inner product is then 〈β|β〉 = 〈σ2A + λ2 σ2B + i λ [A, B]〉         (1) where [A, B] is the anticommutor AB - BA and we have used the fact that [ΔA, ΔB] = [A,B]. Now, 〈β|β〉 ≥ 0 but is otherwise just a number. Its extremal value is found by taking the derivative of (1) with respect to λ and setting the resulting expression to zero. This gives λ = − ½ i [A, B]/σ2B Plugging this into (1), we have the condition σ2A σ2B ≥ − ¼ [A, B]2 or σA σB ≥ − ½ i [A, B]      (we take the negative root for a good reason) Convince yourself that whether λ maximizes or minimizes 〈β|β〉, the inequality still holds. For the case where A is the position operator x and B is the momentum operator p we then have, from the basic quantum identity [x, p] = iℏ, the precise Heisenberg uncertainty principle, σA σB ≥ ½ ℏ For classical systems, the anticommutator [A, B] is usually zero, so there is no uncertainty at all in the measurements of classical parameters, such as the position and velocity of an automobile. For quantum systems, the situation can be drastically different! So what are σA and σB? In elementary statistics, these quantities are nothing more than the standard deviations of the position and momentum measurements, while their squares σ2A and σ2B are called variances. (Why the statistical nature of these identities is never made clear in most textbooks escapes me.)
 Weyl and Ellen -- Posted by on Wednesday, April 27 2011 I often get emails from people who want to know more about Hermann Weyl's personal life. Here's a clue: I actually know very little about Weyl's life outside of his physics and mathematics. I've been to Switzerland, but I haven't even been able to locate the guy's grave. (Photo: Weyl in Jena, Germany, sometime in 1930.) But here's something I was able to share recently. Weyl's first wife Hella died in 1948, and two years later he married Ellen Lohnstein Bär (1902-1988), the daughter of noted physicist Richard Bär. She was the member of a very prominent and wealthy Swiss family with ties to banking and finance. Ellen was herself a noted sculptor (Bildhauerin), artist and violinist, and her works can still be found on online auction sites. Here is her son's online family album (warning: 4.7 MB). Ellen and her family moved in high circles: Wolfgang Pauli, Isaac Stern, Georg Solti and Chaim Weizmann were all family friends. It is said (see Wheeler's tribute to Weyl elsewhere on my website) that Weyl's last uttered word on this Earth was "Ellen," when he died of a heart attack while posting some letters on 8 December 1955. Update: I couldn't find any copyright information on the Bär album, so here is a photo taken of Weyl and wife Ellen, presumably around the time they were married in 1950. He was around 65 at the time, while Ellen (still beautiful) was 48. They had five years together before Weyl passed on in 1955. As Ellen's previous husband had been a physicist (she became a widow in 1940), I suspect this had something to do with her attraction to the rather frumpy-looking Weyl.
 Weyl at 35 -- Posted on Tuesday, April 26 2011 Yesterday I found a couple of new photos of Hermann Weyl on the ETH-Zürich website. This is a portion of a picture taken during a visit Weyl and his wife made in 1921 to the home of Fritz Medicus, a Swiss-German philosopher (go to the website to see the copyrighted photo). Weyl married his first wife Fredericke Berta Hellene Joseph ("Hella," 1893-1948) in 1913. She came to the University of Göttingen in 1911 to study under the great German phenomenological philosopher Edmund Husserl, who Weyl had known there since his student days. A translator of Spanish novels to German and English, Hella died in 1948, with Weyl remarrying two years later. Weyl and Hella had two sons who both became noted mathematicians. Speaking of philosophy (not my best subject), some years ago I recommended Thomas Ryckman's excellent book The Reign of Relativity: Philosophy in Physics, 1915-1925. It's an invaluable source of information on Weyl's physics and philosophy at a time when Einstein's general theory of relativity had changed the scientific and philosophical world. I still recommend it. (Copyrighted photo courtesy of the Eidgenössische Technische Hochschule.)
 Krauss on Nothingness -- Posted on Sunday, April 24 2011 Noted Arizona State University astrophysicist Lawrence Krauss recently gave an interesting talk called A Universe from Nothing, which you can watch on YouTube. In his talk Krauss addresses the cosmological constant, which Einstein fudged into his 1915 gravitational field equations in order to "freeze" the universe, which he believed was static, neither expanding nor contracting. In a few years, observational evidence appeared that seemed to show that the universe was not static but expanding. H ermann Weyl was one of the first physicists to suggest that the universe was indeed expanding, and even proposed an early form of Hubble's Law (which states that cosmological objects are receding from one another at a rate proportional to their distance). The famed astronomer Edwin Hubble observationally proved the expanding universe concept in 1929 (right here in Pasadena!) Einstein initially thought the idea " abominable," but by 1923 he admitted the possibility that it was true. His 1923 postcard to Weyl (reproduced above) includes the fascinating remark "Wenn schon keine quasi-statische Welt, dann fort mit dem kosmologischen Glied" (If there is no quasi-static world, then away with the cosmological term). The noted Swiss physicist (and fellow Weyl enthusiast) Norbert Straumann writes about Einstein's struggles with the cosmological issue in this short paper from 2002 (which includes the translation of Einstein's postcard remark to Weyl). By the way, Krauss is also a noted proponent of evolution (you can watch many of his talks on this subject online), and is a dedicated (nay, virulent) atheist. In the linked video he states that all of creation could simply be the result of a vacuum fluctuation some 13.7 billion years ago. Interesting, but this does not account for the universe's apparent need to produce complex structures, such as human beings (although such "islands" of extremely low entropy would be permissible). The "driving force" of the universe (which doesn't necessarily have to be God) remains a mystery.
 Where is Everything? -- Posted on Friday, April 15 2011 For decades, astronomers have realized that there is simply not enough observed matter in most galaxies to keep them from flying apart. The measured orbital velocity of galactic stars, particularly those on the rims of their galaxies, is too great for gravity to maintain them in orbit. This puzzle has led to the dark matter conjecture, which posits that much of the matter in galaxies in unseen or "dark." The most likely candidates for dark matter, which doesn't interact with ordinary matter except through gravity, are weakly interacting massive particles, or WIMPS. Neutrinos (which hardly interact with matter at all) are now considered to have mass, but the upper bound of neutrino mass is to o small for them to be considered WIMPS. Two days ago, scientists in Italy studying WIMPS released a paper summarizing the results of the most ambitious WIMP detection project to date. The heart of the detector (buried deep in a mine to minimize cosmic ray interference) held 161 kilograms of liquid xenon, which the scientists hoped would interact with any passing WIMPS. The results: The graph depicts all the detection events that occurred over the project's 101-day observation period. All but three events (red dots) were the result of background noise or other extraneous events considered statistically meaningless. However, the scientists also expected to see two or three hits as a result of stray cosmic rays penetrating the detector. Their conclusion: the red dots were cosmic rays, and so no WIMPS were detected. All matter (or mass-energy) in the universe is now considered to consist of dark energy (73%), dark matter (23%), and baryonic/fermionic matter, the ordinary stuff we all know and love (4%). Incredibly, 96% of the universe is missing.
 Neutrinos Again -- Posted on Thursday, April 14 2011 Yesterday I posted a blog about neutrinos, then quickly took it down because (knowing little about the subject) I had no right to say what I did. But I did want to reaffirm my recommendation for the second edition (2008) of David Griffiths' Introduction to Elementary Particles, as the revised edition now includes a good overview chapter on neutrinos. (Most of the revised edition is available on Google Books.) In 1929, Hermann Weyl took Dirac's relativistic electron equation (which actually holds for any fermion) and found that the four coupled, simultaneous differential equations decouple into just two equations when the particle mass is set to zero. This gave Weyl two equations which have since been traditionally viewed as the mathematical foundation for neutrinos , particles of zero mass and spin ½ that travel at the speed of light. Alas, recent observational evidence now pretty clearly demonstrates that the three neutrino flavors can all oscillate into one another, meaning that an electron neutrino νe can be spontaneously converted to a muon neutrino νμ or a tau neutrino ντ, etc. Neutrino oscillation solves the so-called neutrino problem, which describes the discrepancy between the observed flux of solar electron neutrinos and that predicted by the Standard Model. You can read about the problem here. Anyway, Griffiths presents the derivation for a simple 2-neutrino oscillation situation which can be followed by any undergraduate (last night I repeated the calculation for the three neutrino situation, which is far messier but yields the same results). For a more detailed discussion, Griffiths refers us to the work of Fermilab's Boris Kayser, who seems to be a leading researcher on the topic; a good, accessible reference (and with one of the wildest graphics I've ever seen) can be found here at arXiv.org. Back in 1933 Weyl got sucker-punched by Wolfgang Pauli for his (Weyl's) neutrino equations because they did not preserve parity symmetry. The experimental verification of broken parity was not made until 1956; regretfully, Weyl's victory over Pauli was unknown to him, as he passed away the year before. Pauli died in 1958. For an excruciatingly detailed treatment of neutrino oscillation at a relatively accessible level, see Giuti and Kim's 2007 book Fundamentals of Neutrino Physics and Astrophysics (chapter six has no fewer than 438 numbered equations!)
 Eddington, Just Once More -- Posted on Tuesday, April 5 2011 The British astronomer and mathematical physicist Arthur Stanley Eddington died in November 1944. His last book, Fundamental Theory, was published posthumously. I pieced it together from portions available on the Internet. It's a large pdf file (18.8 MB), but here it is if you want it. The book includes a concise summary of Weyl's 1918 gauge theory, a fascination for which I guess Eddington took with him to his grave. His understanding of the replacement φμ → i φμ is most interesting. The book's appendix also supplies the basis for Eddington's calculation of the number of protons and electrons in the universe, which I posted a few days ago in full. For brevity, I used Mathematica to calculate it to only 20 decimal places: (Notice the 136 figure in the input. It seems Eddington never could get over that integer, which he earlier surmised was the exact inverse of the fine-structure constant.) Oddly, the last line of Fundamental Theory reports the number to be 3/2 times the above figure. It looks as if Eddington inserted it in light of the thermodynamic formula for the molar kinetic energy of an ideal gas, E = 3/2 nRT. (In fact, the last chapter of the book is called "The Molar Electromagnetic Field".) Huh? I really don't get it. Maybe you can figure it out.
 Nonsymmetric -- Posted on Tuesday, April 5 2011 You probably already know that the affine connection of differential geometry is usually taken to be symmetric in its lower indices (Γλμν = Γλνμ). But scores of researchers over the decades have sought to generalize Einstein's gravity theory by abandoning this condition. Notably, Einstein himself tried it (this page of handwritten connection calculations was found near his death bed), and Schrödinger also gave it a shot in a series of papers he wrote in 1947 (google "The Final Affine Field Laws," the full text of which I have been unable to find in pdf format). You can take my word that this generalization led nowhere, but it's interesting. I haven't seen any evidence that Hermann Weyl ever investigated a nonsymmetric connection. He probably saw it as a waste of time. A very readable review of what a nonsymmetric affine connection entails can be found in the 2007 paper "On the Nonsymmetric Purely Affine Gravity" by Indiana physics professor Nikodem Poplawski, whose math can be followed by any undergraduate. "Purely affine" simply means that the metric tensor gμν is not taken into account. By the way, I scanned the above page of Einstein's calculations from the book Albert Einstein: Creator and Rebel by Banesh Hoffman (Einstein's friend and colleague) and Helen Dukas, who was Einstein's personal secretary for many years. It's a great book that any Einstein admirer should have in her library.
 Eddington -- Posted on Sunday, April 3 2011 "I believe there are 15 747 724 136 275 002 577 605 653 961 181 555 468 044 717 914 527 116 709 366 231 425 076 185 631 031 296 protons in the universe, and the same number of electrons." — Sir Arthur Stanley Eddington, 1938I've written about Eddington (1882-1944) before, and whatever you want to say about him, imprecise would definitely not describe the man. The above number is known as the Eddington Number (quite a coincidence, yes?), usually given as 1.57×1079. Clearer minds have since truncated that figure to simply 1080, which is the basis of Hermann Weyl's large number hypothesis (and subsequently extended by Paul Dirac). It was Eddington who, back in the days when the fine structure constant α was thought to be very close to 1/136, stepped up with a detailed proof that the inverse was exactly 136. Later, when experiment showed it was closer to 137, Eddington provided a revised proof showing the inverse was exactly 137 (he blamed an earlier algebraic error on the discrepancy). Like Weyl, Eddington was also by Einstein's general theory of relativity, and again like Weyl he attempted a generalization of Einstein's tensor mathematics in the development of a unified field theory. Weyl gave up his own 1918 theory within a few years, but Eddington was more dogged. He in fact loved Weyl's theory, which he described as "unquestionably the greatest advance in the relativity theory after Einstein's work." When Einstein showed that Weyl's theory was invalid, Eddington resolutely set about generalizing Weyl's work. Alas, Weyl subsequently referred to Eddington's revision as "not fit for discussion" (undiskutierbar). Eddington with friend. Eddington wrote two books in the period 1921-22 on Einstein's gravity theory, both of which included a detailed analysis of Weyl's theory. I have a raggedy copy of Eddington's The Mathematical Theory of Relativity (1922), which is interesting if something of a quaint mathematical antique, along with a reprint of his Space, Time and Gravitation (1921), which covers much the same material. Eddington also wrote a curious book called Relativity Theory of Protons and Electrons (1936), which appears to be more philosophical (though I haven't read it). If you have nothing better to do, you can download it legally here. I recently discovered a book on Eddington by King's College professor of mathematics C.W. Kilmister called Eddington's Search for a Fundamental Theory: A Key to the Universe (1994). I've only just started reading it, and it looks to be very interesting. Also interesting is the author's unabashed fascination with Eddington's theories, which he reveals in statements such as"It is half a century since I succumbed to the Eddington magic — I paraphrase Thomas Mann's phrase to try to do justice to my youthful if uncritical absorption in Relativity Theory of Protons and Electrons, which Eddington had published five years or so earlier, in 1936."Imagine that—a grown man fascinated by the work of an obscure scientist! :-) But Eddington deserves praise if only for the fact that he led the 1919 solar eclipse expedition whose results famously proved Einstein's theory predicting the bending of light by gravity (it also made Einstein a scientific superstar).
 Newton's Belated Triumph -- Posted on Monday, March 28 2011 While I'm on the subject of newspapers, here's an amusing story about an editorial that ran in The New York Times on January 13, 1920, ridiculing American rocket pioneer Robert H. Goddard about his proposal to shoot rockets into the vacuum of outer space. It read, in part:That Professor Goddard, with his 'chair' in Clark College and the countenancing of the Smithsonian Institution, does not know the relation of action to reaction, and of the need to have something better than a vacuum against which to react—to say that would be absurd. Of course he only seems to lack the knowledge ladled out daily in high schools.It wasn't until July 17, 1969, with the Apollo moon landing, that the newspaper printed an apology:A Correction. On Jan. 13, 1920, "Topics of the Times," an editorial-page feature of the The New York Times, dismissed the notion that a rocket could function in vacuum and commented on the ideas of Robert H. Goddard, the rocket pioneer. Further investigation and experimentation have confirmed the findings of Isaac Newton in the 17th Century and it is now definitely established that a rocket can function in a vacuum as well as in an atmosphere. The Times regrets the error.The Times' apology took nearly 50 years. "Further investigation and experimentation," indeed.
 Scientific Honesty -- Posted on Monday, March 28 2011 Just a note to say that the April 2011 issue of Scientific American has a nice article by Princeton physics professor Paul Steinhardt on the cosmological theory known as inflation, the idea that the very early universe experienced an extremely brief (∼ 10-30 second) period of hyper-expansion. Inflation was predicted by Alan Guth some 30 years ago and has become a cornerstone of conventional modern cosmology. It not only accounts for many observations, it is also highly predictive. Scientifically speaking, it has passed all the tests. But Steinhardt, a 30-year adherent of inflation theory, isn't so sure, and he brings up some good arguments as to whether inflation is correct or even necessary. In short, Steinhardt puts inflation on trial in his article, and finds that there are numerous arguments that could easily overturn it or make it redundant. But what's particularly interesting is that Steinhardt's article is a good example of how scientists—even when they have a perfectly valid, testable theory on their hands—continue to objectively test and re-evaluate their ideas to the point where, even when their theories are shown to be right but unnecessary, are willing to throw them out and start over. How many politicians, theologians and neoconservative ideologues are willing to do the same? The article cites a recent paper by G. W. Gibbons and Neil Turok (The Measure Problem in Cosmology), which you may find enlightening. Another paper that I have found interesting is Inflation with a Weyl Term, or Ghosts at Work by Nathalie Deruelle et al., in which Hermann Weyl's conformal tensor is tossed into a model Lagrangian for the universe. (The authors refer to it as constantly popping up "on the market of gravity theories." Ha.) Just the stuff for the beginning of the week!
 Weyl and Dark Matter -- Posted on Saturday, March 26 2011 A little looking around on the online physics journal arXiv.org shows that there continues to be much interest in identifying Weyl's vector field φμ with dark matter. A particularly simple example was published recently by MIT's H. Cheng, who also reserves a few choice comments regarding the basis for Einstein's rejection of Weyl's original 1918 conformal theory. Cheng believes that fermions would not interact with an all-permeating bosonic Weyl field, negating Einstein's assertion that the presence of the field would alter the history of electrons passing through it (making distinct atomic spectral lines impossible). Cheng does not explain why he says that "Weyl made the mistake of identifying [the φ-field] with the photon," but he is actually referring to the fact that the line element ds2 in Weyl's conformal theory is invariant only for photons (i.e., ds = 0). In reality, Weyl struggled valiantly to find a general invariant form for ds2. He never did, and it was primarily for this reason that he abandoned the theory. I would prefer to think that dark energy, not dark matter, would have more in common with Weyl's conformal geometry, but I have little to justify that point of view other than a layperson's sense of aesthetics. In my naïve write-up on Weyl's theory, I used Weyl's lagrangian along with φμ = 0 to indicate that just about any alteration of the Einstein-Hilbert action will result in something that looks interesting, but isn't. Still, the holy grail of the Large Hadron Collider is the Higgs field, and the Higgs is predicted to be a boson. The Higgs field is presumed to permeate all space, and it also presumes to explain the basis of particle mass, as particles of varying kinds slog their way through the field. Could Weyl's field have anything to do with the Higgs? Probably not, but it's an interesting thought, and one that seems to keep popping up in the literature. I think Weyl would be amused were he around today.
 Emmy Noether -- Posted on Thursday, March 17 2011 Physicist and writer Ransom Stephens gave a lecture last year on Emmy Noether, the female German mathematician (1882-1935) and friend/colleague of Hermann Weyl that I occasionally talk about on this site. Below is Stephens' lecture on YouTube. Stephens' talk is not perfect, and it's an hour long, and there's a pitch thrown in at the end for his book, but I recommend that you watch it because he provides a pretty decent overview of the relationship between mathematical symmetries and conservation laws (which Noether formalized in her famous 1918 theorem). The humanity of the woman is also much in evidence in the lecture. Plus, Hermann Weyl's 1935 eulogy of Noether is described. I was heartened to hear Stephens admit that, as a physicist, he doesn't understand Noether's pure mathematics at all (ring theory, non-commutative algebra, that sort of thing). Hell, I don't, either.
 "It's got to be there, damn it!" -- Posted on Wednesday, March 16 2011 After straining innumerable brain cells trying to understand supersymmetry (SUSY, see my post of 25 January), I now learn that it may be all wrong, after all. In her article "What if Supersymmetry is Wrong?", science writer Amanda (and New Scientist book editor) posits the very real possibility that decades of forlorn searching for the tell-tale signs of SUSY (including preliminary results from the Large Hadron Collider) might be telling us that one of the leading contenders for a "theory of everything" might simply be a wild goose chase. String theorists, rejoice. For me, this leaves loop quantum gravity, as I have given up trying to figure out string theory (it's just too damned difficult). Hopefully, by the time Gefter writes "What if LQG is Wrong?" I'll be comfortably ensconced in a nursing home.
 Why You Want to Believe -- Posted on Wednesday, March 16 2011 In our hunter-gatherer days, a rustling sound in the bushes could mean either the wind or a predator, so the fledgling human mind had to make a decision—stay or run. Running could mean saving your life (predator) or just wasting your time (the wind). Not running meant being eaten (predator), or being relieved that it was only the wind. Even a dim-witted Australopithecus could figure that one out: she ran, because it was the best survival option. ←Here's a young Australopithecus I snapped at the Museum für Naturkunde in Berlin. She evidently did not run. New Scientist has an interesting article called "Why You Want to Believe in the Paranormal" by British psychologist Richard Wiseman (you have to log in to read the New Scientist article, but reader membership is free). We all have eccentric friends and relatives who believe in psychic abilities, paranormal phenomena, dowsing and magnet therapy. I think it's all nonsense, and science has shown it to be nonsense, but people will believe what they want to believe, and they will never be convinced they are wrong. Wiseman's article tells us why, and it seems to go back to our hominid days. Here's a snippet:Every moment of our waking lives we are bombarded with huge amounts of sensory information which we struggle to make sense of. Our visual system tries to detect objects and faces, and our auditory system works hard to identify sounds and understand conversation. These pattern-detecting processes are so important to our survival that we have evolved to err towards false positives, preferring to "see" nonexistent patterns than miss a genuine one. As a result, our brains have a tendency to perceive meaning in random input. This is why some people believe astrological predictions, see faces on Mars, mistake lenticular clouds for UFOs and hear the voices of the dead in static noise.George W. Bush, who in my humble opinion should have been hanged, used Americans' fear of "rustling sounds" to scare us into invading Iraq. He lied, and thousands of our military people died for nothing. But better safe than sorry*, right? America still hears those "rustling sounds," which is why we spend a trillion dollars a year on weapons systems, covert surveillance and secret prisons. That we do this and still call ourselves Christians is another neat psychological trick, but I'll talk about that some other time. * A chemistry student of mine brought up this excuse after class. I replied that it's easy to say when you're sending other people off to die.
 Weyl and Gravity Waves -- Posted on Monday, March 14 2011 I was flipping through an old book this afternoon when this clipping fell out. I'd forgotten all about it — it's an obituary notice (dated 10 October 2000) I had saved for physicist Joseph Weber, who was a UC Irvine laser expert. He was also known as the father of gravitational radiation detection. Shortly after publishing his theory of general relativity in November 1915, Einstein discovered that gravity waves, traveling at the speed of light, should exist. Yet despite ninety years of dedicated effort, these waves have not been conclusively detected. Weber (who used 1.4-ton cylinders of aluminum outfitted with highly sensitive strain gauges to detect the anticipated slight warping effect of passing gravitational waves from outer space), obtained inconclusive (and most likely null) results. Einstein's gravitational field equations are highly non-linear and difficult to solve for all but the most simple and symmetric applications (Schwarzschild solved them for the static, spherically symmetric case in 1916, and it took almost another 50 years to do it for a simple rotating mass). Researchers today use high-speed computers to solve the equations with a variety of numerical approximation techniques, but in Einstein's day there was only one known approximate approach — linearizing the field equations. This entails assuming that the metric tensor gμν can be expressed as the constant Lorentz metric coupled with a new metric that represents a slight departure from flat space-time: gμν = ημν + εγμν where ε is some small constant such that all Einstein field term s proportional to order ε2 and higher can be ignored. It is interesting that Hermann Weyl, in his investigations of the linearized field equations in 1918, discovered a metric γμν that satisfied the wave equation, thus showing that gravitational effects must propagate at the speed of light. It was typical of Weyl to employ a clever mathematical trick* to do this: he decomposed the metric according to γμν = ∂μφν + ∂νφμ where φμ(x) is an arbitrary, twice-differentiable quantity (but not the vector 4-potential of his unified theory of gravitation and electromagnetism). This allowed Weyl to write the linearized field equations as ◻2γμν = (∂0∂0 - ∇2) γμν = 0 which is the wave equation (I've avoided a subtlety here for simplicity). Thus, gravity propagates through space at the speed of light. For the static case (time independence) the μ, ν = 0 equation reduces to ∇2 γ00 = 0 which is Newton's law of gravitation. Surprisingly, Weyl's metric itself is not a gravitationally significant quantity! It is amazing how much information can be extracted from the linearized field equations, despite the fact that they are only approximations to reality. By further decomposing the γμν metric it can be shown that gravity, like electromagnetism, propagates at the speed of light but, unlike electromagnetism, has far different polarization properties. These properties can be utilized in the design of gravity wave detectors, the most modern of which is the LIGO detector. Although gravitational radiation has not been directly detected, its existence has been confirmed by observing the rotational characteristics of binary neutron star systems. Neutron stars rotate at precise rates, making them excellent clocks. However, binary systems can radiate gravity waves as they rotate about one another, with the resulting loss of energy; this in turn results in the slow orbital collapse of the system. Detailed observations by Russell Hulse and Joseph Taylor of Princeton University precisely (±0.2%) confirmed the theoretical energy loss of the PSR B1913+16 binary system (which they discovered in 1974), an effort which garnered them the Nobel prize in physics in 1993. * Adler, Bazin & Schiffer, Introduction to general relativity, 2nd ed., 1975.
 Weyl's Postulate -- Posted on Thursday, March 10 2011 First, a definition and a quote:The world lines of galaxies on average form a family of non-interacting geodesics converging toward the past in accordance with the existence of a global Gaussian time coordinate. — Weyl's Postulate, 1923 "It appears that the velocities between distant celestial objects on average increase with their mutual separations." — Hermann Weyl, 1923In 1923, Hermann Weyl proposed what is now called Weyl's postulate, which has to do with the large-scale behavior of the universe. I've experienced some difficulty following the reasoning behind Weyl's postulate, but there are those who do understand it, and I wanted to pass along the thoughts behind one short and very readable article that explains Weyl's reasoning about as well as anyone. It's interesting because Weyl's postulate, according to the authors, predates Edwin Hubble's 1929 discovery that our universe is expanding by six years. The paper is entitled Weyl's Principle, Cosmic Time and Quantum Fundamentalism by S.E. Rugh and H. Zinkernagel, dated 30 June 2010. The following is not a review of the paper, but rather my rough take on Weyl's postulate according to the authors. The cosmological principle basically states that the universe on average is both isotropic (meaning that it looks the same in every direction) and homogeneous (meaning that the universe looks the same to any two observers, no matter how far apart they might be). Although stars, galaxies and and their interactions are very complicated, at some large enough scale galaxies and other megastructures can be considered as point objects uniformly distributed throughout the universe. These points undoubtedly interact to some extent (galactic collisions, for example, are known to occur), but on a truly grand scale all interaction can be ignored. If the universe is isotropic and homogeneous, it might resemble a uniform ball of matter points, with each particle of matter following a geodesic world-line (assuming no interaction). Spherical symmetry would require that each particle have a counterpart on the other side of the ball, and the movement of each "shell" of particles would define a smooth 3-hyperspace in which the particles are "co-moving" or fixed with respect to one another at every moment in time. Weyl reasoned that with such a model, a "cosmic time" could be imagined in which the particle shells converge at earlier times and diverge as cosmic time increases. At every moment, particle movement is orthogonal to the shell. Two views of the universe: On the left, galaxies move chaotically, their world lines are allowed to cross, and a consistent mathematical description of the universe is impossible. On the right, world lines follow geodesics that are orthogonal to smooth hyperplanes of constant cosmic time. From J. Narlikar, Introduction to Relativity, Cambridge Univ. Press, 2010. (Weyl did not really assume any spherical symmetry, but I've used it to help me visualize the idea.) More conventionally, in the above graph we imagine hyperplanes of 3-space at every point in a universal, cosmic time. If these hyperplanes are all orthogonal to the world lines, then (with the unphysical exception of "parallel" matter flow) convergence and divergence of the world lines becomes unavoidable. Cosmic time thus has a "direction" consistent with the concept of universal expansion or contraction. Weyl's cosmic time thus becomes a global, standard clock time that applies to every observer in the universe, making possible simultaneity of events. Unfortunately, this kind of cosmic time flies in the face of relativity, where time is always relative, depending on things like particle velocity and gravitational effects. Consequently, Weyl's postulate appears to prevent a completely covariant treatment of the simple cosmological models that utilize his postulate (however, realistic cosmological models have been developed that are remarkably similar to the universe we actually observe). One way of seeing this is to recall the Schwarzschild metric of general relativity, which exhibits radial-dependent terms in the time and space coefficients. With the global time marker assumed in Weyl's postulate we can set the time coefficient equal to unity, so that the line element becomes ds^2 = c^2 dt^2 - g_{ij} dx^i dx^j (i, j = 1,2,3), where the g_{ij}  are functions of space and the global time t. This line element resembles that of ordinary flat space, but it's actually much more complicated than the Schwarzschild case. The most important contemporary model of the inverse is the Friedmann-Lemaître-Robertson-Walker (FLRW) model, which utilizes a line element of this type. The big pay-off of these global-time models is the fact that they provide a solution to Einstein's fields equations that includes an unambiguous formula for the Hubble equation, which relates the velocity of a galaxy or galactic cluster to its distance from the observer. They also also yield the usual formula for the red shift of the object as a function of its velocity. Weyl's postulate would appear to be essential to global-time cosmological models like the FLRW model, but Rugh and Zinkernagel note that Weyl has gotten scant attention or credit for his idea. They point out that many recent texts on general relativity don't even mention Weyl in the derivation of the models, preferring to focus instead on the cosmological principle. For example, Adler, Bazin and Schiffer (Introduction to General Relativity, 1975) ignore Weyl's cosmic time idea altogether and state only that a global time coordinate "is the price one has to pay to simplify the cosmological models and to describe physical reality in convenient mathematical terms." Rugh and Zinkernagel's paper is only 14 pages, but provides much more information on the history of Weyl's postulate and its role in cosmology. The final section of the paper considers the fate of the postulate as global time is reversed back to the poin t of the Big Bang, when quantum effects dominated. A good undergraduate-level derivation of the FLRW cosmological model, Hubble's law and the redshift formula can be found here (courtesy Prof. Michael Kachelrieß of the Norway Institute of Physics).
 CMB and Graviton Handedness -- Posted on Saturday, March 5 2011 In his excellent 2004 book The Road to Reality, Sir Roger Penrose (a hero of mine) discusses the similarities and differences between the two most famous of Nature's massless particles of integer spin—the photon, a particle of spin 1 that is the carrier of the electromagnetic force, and the graviton, a hypothetical spin 2 particle that is the carrier of the gravitational force. Penrose notes that, like photons, gravitons may come in chiral (right-handed and left-handed) forms, adding that This may seem strange, from the physical point of view, because there is no evidence of any left/right asymmetry in the gravitational field, and there is certainly none in the standard Einstein theory of general relativity.Still, he reasons, there is definitely something asymmetrical about Nature ("left" being preferred by neutrinos in the weak interaction), so gravity might have a chiral asymmetry at the quantum level. This is indeed the point of view of loop quantum gravity, and it is also the subject of some new research as reported by an article in the latest New Scientist journal. In an upcoming paper by João Magueijo and Dionigi Benincasa of Imperial College London, it is suggested that any neutrino-like behavior of the graviton might be discovered by a more detailed study of the cosmic microwave background (CMB), the relic radiation from the Big Bang that is responsible for the observed residual 2.7 Kelvin temperature of our universe. The researchers posit that powerful gravitational waves at the time of the Big Bang might have polarized the CMB's photons in some measurable way. For those of you who have fast Internet connections, here is the most highly-detailed map available of the latest Wilkinson Microwave Anisotropy Probe (WMAP) data . The discreteness is due to micro-level temperature gradients in today's universe, which resulted from the extremely fine anisotropy of the Big Bang (if the Big Bang had been perfectly uniform, you would not seen any temperature differences across the universe). Astrophysicists who have at least some concept of a Creator have called the WMAP the closest thing yet to seeing the face of God. If this is true, then God has a distinct preference for imperfection, slight though it may be. Indeed, if not for the observed spontaneous symmetry breaking of quantum field theory, there would arguably be no quantum theory at all.