2013 Archives

In this video, noted biologist David Suzuki explains the dangers of exponential growth in a way that anyone, even an American, can understand:

The most fundamental quantity in differential geometry is arguably the Riemann-Christoffel tensor \(R_{\mu \nu \lambda}^{\,\beta}\), which describes how spacetime is curved or warped. In an \(n\)-dimensional space each index in the tensor can take on \(n\) values but, because of the various symmetry properties of the tensor, there aren't \(n^4\) independent components but only \(\frac{1}{8} n(n^2 -n+2)(n-1)\) components. For ordinary 4-dimensional spacetime, this computes to 21 components. For students trying to sort out Einstein gravity, that's a lot of components to keep track of.

The supergravity theory known as M-theory assumes the existence of 11 spacetime dimensions, so anyone having the courage to tackle the Riemann-Christoffel tensor in that theory has to contend with 1,540 independent components. If you're thinking that it might be time to bring out a computer to do the work, you would be right.

Huffington Post reports a recent paper by some bright Japanese physicists who've done just that. Their calculations support the idea that the universe we all know and love is really an 8-dimensional hypersphere, which in somewhat simpler language means that we might be living in a hologram.

The researchers' 30-page paper is mostly a description of long calculations done with the help of the computer algebra system Mathematica, which can manipulate tensor indices. I really can't follow what they're doing, but one of the main researchers in the field of quantum gravity, Juan Maldacena of Princeton's Institute for Advanced Study, says that the calculations are correct. I'll take his word for it.

I believe Stanford physicist Leonard Susskind was the first to suggest that all the information that falls into a black hole is stored on the hole's event horizon. So when a book falls into a black hole, its information is not destroyed forever but stored (albeit in a very scrambled form) on the hole's horizon. Susskind has also posited the idea that all the information in the universe (including us) is stored on a far-distant surface, which for all intents and purposes is a hologram.

I stand aloof, and will no reconcilement. — Laertes, Hamlet, Act 5, Scene 2

Although Hamlet apologizes to Laertes for accidentally killing his father, Laertes has other issues with Hamlet that he is unwilling to dismiss. Rather than go their separate ways, they end up killing each other in the play's final duel.

Like Hamlet and his intended brother-in-law, America's liberals and conservatives seem destined for violence, with America having finally devolved into a deep political-religious über-polarization that seemingly cannot be reconciled. Books like What's the Matter with Kansas? - How Conservatives Won the Heart of America, Idiot America: How Stupidity Became a Virtue in the Land of the Free and Better Off Without 'Em: A Northern Manifesto for Southern Secession (an excellent book) describe the problem in detail, and even propose efforts that both sides could make to alleviate the country's cultural and ideological schizophrenia, but it ain't gonna happen, folks. In Better Off, author Chuck Thompson seriously suggests that we just admit that a divorce is probably the best solution and recommends secession of the offending Southern states. I agree.

To date the Southern states have not actively promoted secession (despite Texas governor Rick Perry's lip service to his intended presidential base), it being contentious if not outright unconstitutional, so they prefer more subtle tactics such as nullification of federal laws, rules and regulations based on very narrow interpretations of states' rights (translation: President Obama is a Marxist, America-hating liberal Kenyan nigger who deserves lynching, but they can't say that outright). So I think it's high time to reconsider secession and, if the South is still somewhat tentative about the issue, then the rest of the country should at least start talking obout it.

I cannot even begin to summarize the lies that conservatives in this country have fallen victim to, thanks to the pandering efforts of folks like Rush Limbaugh and Sean Hannity, nor can I count the times that liberals have shaken their fists in frustrated disbelief over how those lies continue to gain traction with those same conservatives. But it's obvious that both sides are now firmly entrenched in their positions, with the media serving only to further that entrenchment, so let's stop jabbering about it.

I gave up on the whole thing a long time ago, mainly because I see conservative ideology as religiously motivated, and you cannot argue with someone about their religious beliefs. For example, anyone who believes in the Old Testament story of Balaam's donkey (it's a talking donkey, but that's not the point — the story makes absolutely no sense whatsoever) is not going to be a reasonable person, period, so let's move on and get that divorce.

You've no doubt heard of Blaise Pascal, the 17th-century Christian French philosopher and mathematician. You've probably also heard about Pascal's Wager, in which he argued that being religious, even if it's not sincere, is more advantageous in the betting sense than being an atheist or agnostic.

There are several variants, but this is basically how it goes: There either is a God or there is not. If there is a God, then you have two choices — to obey Him or not to obey Him. If you obey, then you win eternal life. If you do not obey, then you suffer eternal torment and damnation. On the other hand, if God does not exist, then by obeying you gain nothing, but neither do you suffer eternal torment. However, if God exists and you obey Him, your reward is commensurately far greater than if he does not exist and you obey anyway, because going to church and kneeling and praying and giving to the poor and all that is at most a minor inconvenience by comparison. Pascal therefore argues that it is better to believe in God because the potential return or "payoff" far outweighs the investment or "ante."

Of course, God would have to be pretty stupid to reward anyone who just goes through the motions out of fear of damnation rather than a true love of goodness. In my opinion this completely wrecks Pascal's argument. The only way to salvage the argument would be to have God demand that you absolutely believe — no pretense allowed. But you may see things differently.

[You may recall in George Orwell's dystopian novel 1984 that the Ministry of Truth didn't want you to just say that \(2+2=5\); the Ministry wanted you to actually believe it. Consequently, to get on God's good side you can go ahead and play Pascal's Wager, but first you have to allow yourself to be totally brainwashed into believing the innumerable contradictions, inconsistencies and magical silliness in the Bible. But that's another story.]

An interesting modern take on Pascal's wager is this short scenario written by Oxford professor of philosophy Nick Bostrom, in which Pascal is accosted by a mugger. (Bostrom is not a mathematician, so we can forgive him for his erroneous statement that doubling one's money is equivalent to a 200% return on investment.) Bostrom's scenario very cleverly mimics the same fear and investment issues that Pascal raised, but includes an interesting philosophical twist that you may find amusing.

Bostrom is more famously known for his work on the computer simulation argument, which posits the possibility (though I don't think Bostrom actually believes in it) that the entire universe is merely a gigantic digital simulation run by future post-human computer programmers. The argument not only retains the idea of an omnipresent, all-powerful God (though He's now a nerdy, Professor Frink-type entity), but it also neatly answers the age-old problem of theodicy (that is, if God is beneficent and loving, why does He also allow suffering). Think of it this way: when you play one of those violent video games (Grand Theft Auto, for example) in which you cause the maiming and deaths of innocent characters, you feel no sense of guilt because they're only computer simulations. Bostrom's simulation argument provides a similar "out" for God: it's not that He doesn't care, it's just that His creations are essentially non-existent characters in a very sophisticated video game. (Okay, so maybe He doesn't care.)

But could we really be living in a computer simulation? I think it is actually very plausible, perhaps even probable. But read some of Bostrom's papers on the idea (you'll find them on his website) and give it a think.

One episode of The Simpsons has Homer apparently disproving Fermat's Last Theorem, which states that there are no positive, non-zero integers \(x,y,z\) and \(n\) that satisfy \(x^n + y^n = z^n\) (\(n \ne 1\)) other than \(n=2\). A rigorous proof of the theorem took almost 400 years, but in the episode Homer simply writes down $$ 3987^{12} + 4365^{12} = 4472^{12} $$ on the blackboard, in contradiction to Fermat and a 150-page proof given by British mathematician Andrew Wiles (shown) in 1995. What gives?

In truth, there are an infinite number of such "disproofs." The two numbers above don't sum to \(4472^{12}\), but to \(4472.0000000070592...^{12}\) [Thank you, Mathematica.]

Fermat's Last Theorem, and Wiles' proof, are safe from The Simpsons.

In the days before Galileo, everyone in the world instinctively knew that heavier objects fell faster than lighter objects. Sure, you could climb up a leaning tower like the one in Pisa, Italy and test the idea by dropping two dissimilar rocks and timing their fall, but why bother? It was common sense that the heavier rock would reach the ground first. But then Galileo actually tried it out (he was too old to climb the tower, or maybe just afraid of heights, so he used inclined planes with rolling cylinders in his workshop instead) and easily showed that everyone's common sense was wrong.

And back in 1887 everyone in the world knew instinctively that light, being a wave, needed something to "wave against" in order to move, just like ripples on a pond need water to move. So common sense told people that outer space must be filled with an "ether" — a thin, invisible, undetectable air-like medium — that allowed light from the distant stars to reach Earth. But a series of simple experiments by Michaelson and Morley proved that the ether did not exist — light could travel through a vacuum without having to wave against anything.

These examples show how simple-mindedness and conventional common sense can be overcome with a little experimentation. But the situation is not so simple anymore. Today we have two other seemingly undetectable ethereal media, dark energy and dark matter, and they're giving physicists apoplexy. But I wonder if we're experiencing a similar problem with common sense.

Consider the dark matter problem. Decades ago it was noticed that the outer stars of galaxies were moving far too fast to be held by the gravity of their galactic cores. The velocities of these stars, which were easily measured by their luminal Doppler shifts, were so great that the stars could simply not be held by gravitational force. After eliminating potential sources of "unseen" gravity such as galactic gas, dust and neutrinos, common sense told astrophysicists that there must be a type of matter embedded in the galaxies that provided the missing mass. Unfortunately, this missing mass could not be detected by ordinary means, so common sense again said that this matter must be a brand new kind of matter that they dubbed "weakly interacting massive particles" (or WIMPS), not generally seen in ordinary matter. Reasoning that such particles must interact at least occasionally with ordinary matter, scientists constructed detectors deep underground (to eliminate the effects of cosmic rays) to find the WIMPs. The latest of these efforts is the Large Underground Xenon experiment (LUX), in which nearly half a ton of liquid xenon cooled to ultra-low temperatures was lowered more than a mile into an abandoned mine in the Black Hills of South Dakota. Although the experiment would seem to have sufficient sensitivity to detect any WIMPs passing by, to date nothing has been found. This seems a trifle odd, given the fact that dark matter is estimated to make up about 25% of all the mass in the observable universe.

Consider also the similar phenomenon known as dark energy. In the late 1990s astrophysicists noted that the observed expansion of the universe appeared to be speeding up, not slowing down, in a manner exactly analogous to throwing a rock into the air and watching it speed up past escape velocity and leave the Earth, eventually achieving the speed of light. Again, common sense told scientists that there must be a hidden force in seemingly empty spacetime that exerted a kind of "anti-gravity effect" on ordinary matter. In 2011, the Nobel Prize in physics was awarded to three researchers for their verification of the accelerated expansion of the cosmos, all on the basis of supposedly tight physics work. Today, conventional cosmology tells us that about 70% of all the mass-energy in the universe is in the form of dark energy. If these theories of dark matter and dark energy are correct, then the Earth and solar system and all the observable stars and galaxies in the universe account for only 5% of the mass-energy that's out there. In other words, we seem to be missing 95% of the universe.

Of course, there are alternatives to these theories, and the simplest of these seem far more plausible to me.

The theories known as modified Newtonian dynamics (MOND) and modified gravity (MOG) are especially attractive (no pun intended) because in principle they do not rely on the existence of strange new undetectable sources of matter and energy. Instead, they tell us that our 100-year-old Einteinian gravity theory is simply not quite correct. Recall that Einstein, Hermann Weyl, Arthur Eddington, Wolfgang Pauli and many others tried unsuccessfully to unify gravity with the other fundamental forces of Nature by "tweaking" Einstein's gravitational field equations. MOND and MOG are different — being strictly gravity theories they do not try to unify anything, but instead propose that gravity is a bit more complicated than we originally thought. The work of researchers like the Perimeter Institute's John W. Moffat and Philip Mannheim of the University of Connecticut are particularly interesting, though they are not nearly as beautiful as Einstein's original theory (Mannheim's conformal gravity theory is very "clean" in that it's based solely on Weyl's conformal tensor, while Moffat's is complicated by the presence of various scalar, vector and tensor quantities). Both theories appear to provide evidence that the effects of what we call today "dark matter" and "dark energy" are simply spacetime consequences of a more complete gravity theory.

It is interesting to note that Einstein himself had proposed a more accurate theory based on a very slight tweaking of his own field equations. Recall that the field equations for empty space go like \(R^{\mu\nu} - \frac{1}{2} R g^{\mu\nu} = 0\). The divergence of these (ten) equations is identically zero, a fact that can be viewed as merely a consequence of the tensor mathematics or the assumption that mass-energy is conserved in a fully covariant manner. Since the metric tensor \(g^{\mu\nu}\) is also divergenceless (actually, it acts as a constant under covariant differentiation), Einstein simply tacked on an additional term in his equations proportional to the metric tensor (the factor of proportionality was of course his famous cosmological constant \(\lambda\)). Since the bare field equations necessarily predict a non-static universe that is either expanding or contracting in time, the cosmological constant can be used to either halt the expansion, reverse it or accelerate it. When the noted Pasadena astronomer Edwin Hubble proved in 1929 that the universe is in fact expanding, Einstein's hopes for a static, unchanging universe were dashed. He subsequently called the cosmological constant his "greatest blunder," but today it provides a handy (and perhaps correct) explanation for dark energy.

I can't help but think that when astrophysicists announced one day that 95% of the universe is missing, perhaps they were all missing something themselves. Perhaps the universe is not that complicated, but that our mathematics just needs some adjusting (but I'm probably wrong).

Let me wrap this up by giving you two graphs that were instrumental in getting Saul Permutter and two co-workers the 2011 Nobel Prize for their validation of the dark energy idea. The graphs rely on the accuracy of distant Type 1a supernovae as "standard candles," that is, means for accurately measuring extragalactic distances. Though I readily admit that Perlmutter et al. are far above my pay grade in terms of brilliance, the data summarized on the graphs seem open to debate in terms of exactly what they're predicting. Consider the error bars, and judge for yourself:

The top graph depicts the redshifts (\(z\)) for observed Type 1a supernovae. The distances of those supernovae with very high redshifts are greater than they would be expected for "conventional" expansion of the universe (heavy dotted line), indicating that the universe is expanding at an accelerated rate. The bottom graph summarizes the same data, but with an ordinate scale designed to enhance the apparent variation from the dotted line.

Article from The Daily News-Post, dated 15 August 1968.

I was at a Rite Aid this morning to pick up some dental floss when a middle-aged woman happened into the same aisle. I wouldn't have noticed her at all if not for the fact that she was wearing a perfume that I instantly recognized as Fabergé's Tigress, a truly ancient scent from the late 1960s. While I'm generally immune to such things, whether they're discrete, delicate aromas or the overpowering stenches that some older women seem intent on dousing themselves with, this one instantly transported me back to August 1968, because my very first girl friend wore that particular perfume.

While my aging memory banks are also sensitive to certain songs from the past (the Beatles' annoying Hey Jude also takes me back to her), my brain appears to be especially sensitive to smells. The scent of pine oil sends me reeling, as it reminds me of long hikes in pine forests and a truly awful experience I had as a child, but certain industrial chemical odors whip up intense past memories as well. I still recall as an undergraduate chemistry student taking a whiff from a reagent bottle of acetic anhydride, which actually locked up my breathing apparatus for a terrifying ten seconds or so. But there's one chemical odor my brain will never be free of.

The week after graduation from high school in 1967, I started working for a chemical production company that specialized in liquid and solid organic peroxides, which were (and still are) used as catalysts for epoxy resins. For the most part we made benzoyl peroxide (later to be recognized as an effective ingredient in topical acne medication) and methyl ethyl ketone peroxide (MEKP), but we also made lots of tert-butyl hydroperoxide. I routinely cooked up TBHP in huge, glass-lined reactors by mixing tert-butyl alcohol and hydrogen peroxide under acidic conditions, both the reactant (TBA) and the product having wonderful, sweet, almost intoxicating aromas (I didn't learn until later that TBHP is carcinogenic and hazardous to inhale. *Cough*).

But on the morning of August 15, 1968, a 250-pound batch of dry benzoyl peroxide ignited in the plant's bunkerized vacuum oven, instantly killing my friend, a 19-year-old co-worker who was removing the dried product for packaging. I was only about 50 feet from the explosion, which shook the entire complex, but thankfully no one else was hurt. The vacuum oven and surrounding railroad-tie bunker were destroyed, and upon arriving at the scene we found my friend, Julio, his head missing, his broken body flattened against the wood ties. While we all fanned out to make sure nothing had caught fire, the police and fire department arrived at the site. I distinctly remember one of the fire department personnel asking us to help locate the victim's missing head, presumed blown off. Only later did we discover that the explosion had actually driven his head into his chest cavity, from which we ascertained that he was probably bending forward toward the oven when the explosion occurred. We later found pieces of bone embedded in some of the ties. A local mortuary was notified, and they came and dug them out of the wood.

That was on a Thursday, and the following Saturday I had my first date with Karen (my first date with any girl, for that matter). But the first thing I noticed about her was her perfume, Tigress. It was not unpleasant, but it had the same acrid, cloying, sweetish smell of burned benzoyl peroxide, a smell that permeated the air around the chemical plant for weeks following the explosion.

We dated for only eight weeks that summer and early fall, and each time I saw her she wore that perfume. I never said anything to her about it, but the events of that August had an enormous impact on me. I lost weight and just wasn't myself, and by mid-October I had lost her as well.

I quit my job right after breaking up with her. Eight years later most of the chemical plant was destroyed when hundreds of pounds of MEKP exploded in the reactors, killing four workers.

I still love chemistry, but I will never get over the smell of that summer, that odor of death.

I've become fascinated with the history of the so-called "Schwarzschild solution" of general relativity, which as every student learns represents the first cosmologically important solution to Einstein's gravitational field equations (see also this paper, which includes a discussion of the early rejection of the notion of black holes).

But as Anthony Zee notes in his book Einstein Gravity in a Nutshell, Karl Schwarschild was apparently not the first person to derive the static, spherically symmetric solution to the field equations, which he published in January 1916, just two months after the publication of Einstein's gravity theory. Zee's book mentions the Dutch physicist Johannes Droste as being the first one to actually arrive at the solution, but there is evidence that it was derived independently by the German mathematician Hermann Weyl and his PhD thesis advisor and colleague, the famous Göttingen mathematician David Hilbert.

Recall that Einstein's gravity theory was published only ten years after his special theory of relativity of 1905, which dealt strictly with flat space. Recall also that as late as the early 1930s (and even today, if you count the disgustingly conservative website Conservapedia), Einstein's special theory was not universally understood or considered correct. Furthermore, Einstein's November 1915 gravity theory was based on the then relatively new tensor calculus, a field of mathematics that the 1905 theory did not utilize. Consequently, the 1915 theory must have appeared as a complete shock to most physicists of the day, who would have had to drop whatever they were doing and spend a lot of time just getting acquainted with the mathematics of tensors. So it should come as a surprise that Weyl, Hilbert, Droste and Schwarzschild were each able to come up with a solution to Einstein's field equations within a month or so of the theory's publication.

[Conservapedia's relativity article starts out claiming that the theory has been contradicted by experiment. Why do Phyllis Schlafly and her son Andy (the site's developers) disagree so strongly with Einstein? Because Einstein's relativity disproves the notion of action at a distance, which runs counter to God's supposed I-snap-my-fingers-and-boom-it-happens capability. Conservapedia, not to dismiss a theory without offering a viable alternative, prefers to promote the concept of magic and wishful thinking as a more accurate description of how Nature God works.]

I remember the first time (around 1975) I was able to derive the Schwarzschild solution from the condition \(R_{\mu\nu} = 0\). I immediately proceeded to rederive Weyl's solution for a massive, charged point field. At the same time I was infatuated with the problem of unifying gravitation and quantum mechanics, so I derived the field equations associated with the hermitian metric \(\hat{g}_{\mu\nu} = g_{\mu\nu} + i F_{\mu\nu}\), where \(F_{\mu\nu}\) is the electromagnetic stress tensor ("Hey, the Christoffel symbols are still symmetric because they automatically incorporate Maxwell's equations!") Of course it was a dead end, but I learned an important lesson — I'm not only a very mediocre physicist but a complete idiot as well, and I should leave such problems to the experts.

Lastly, I would like to note that, taken together, Einstein's special and general theories can be viewed as the most accurate theory ever conceived by mankind. The famous binary pulsar system known as PSR 1913+16 lies in the constellation Aquila, roughly 30,000 light years from earth. The system consists of a collapsed dead star and a companion pulsar, each about 1.4 times the mass of the Sun, rotating about one another with a period of 7 hours, 45 minutes and 6.9816132 seconds. Like the general-relativistic advance of the planet Mercury's perihelion, the orbit of the stars in PSR 1913+16 can be both computed and observed. In addition, the pulsar is emitting radio energy at a very precise rate, allowing accurate measurement of the system's energy loss due to gravitational radiation with time. The most recent calculations performed by astrophysicist Joseph Taylor and his colleagues confirmed the computed and observed changes in the stars' orbit to an error of less than 10\(^{-14}\), unquestionably the most precise calculated/observed agreement known to man:

(Observations in red; black line is calculated)

Taylor and colleague Russell Hulse received the 1993 Nobel Prize in physics for their work. Not too shabby.

So help me, if I hear another conservative moron claim that "it's just a theory" I'll scream. (And maybe stamp my foot. Then they'll be sorry.)

Several people emailed me to ask about a mathematical trick that Hermann Weyl used in 1918 to derive the Schwarzschild solution to Einstein's gravitational field equations, as described in Anthony Zee's amazing book Einstein Gravity in a Nutshell (p. 374, "Weyl's shortcut to Schwarzschild"). Although Zee committed a slight typo in his book (\(R_{\mu\nu}\) is not a metric), the intent is clear — Weyl in effect assumed a solution before he actually carried out the variational analysis of the Einstein-Hilbert action. But it worked anyway.

Weyl committed essentially the same faux pas when he derived his own field equations for the Weyl action \( \int \, \sqrt{-g}\, R^2\, d^4x\) (see the Adler-Bazin-Schiffer book Introduction to General Relativity for details), in which he set, from the start, the Riemann scalar \(R\) equal to a constant, \(\lambda\). But oddly enough, Weyl's trick actually works in both cases. As Zee mentions in his book, it took almost 100 years for researchers to verify that Weyl's trick was in fact valid (see R.S. Palais, Comm. Math. Physics 69, 1979 p. 19). To any student familiar with variational calculus Weyl's trick was singularly improper, but the results were correct nevertheless. Weyl was a mathematician of the very first order, yet he somehow knew he was right in spite of the liberties he took in achieving a solution.

We are now just two years away from the 100th anniversary of Einstein's theory of general relativity. It is indisputably one of the most profound and beautiful achievements of the human mind, yet it pains me that this milestone will likely go largely unappreciated by the general public in November 2015. Although Einstein's theory is rather mathematical and perhaps even somewhat nonintuitive, it is not terribly difficult to understand. I wish to God that more people would read Zee's book, which is by far the most comprehensive and understandable text of its kind. I cannnot say enough positive things about the book — by all means, BUY IT AND READ IT, if only for the good of your soul.

PS: Page 375 of Zee's book recounts how the Dutch physicist Johannes Droste actually arrived at Schwarzschild's metric before the German Schwarzschild did, but his work was ignored by the physics community. Perhaps Karl Schwarzschild's early death in 1916 (brought on by an automimmune disease he acquired while fighting on the front in World War I) provoked a sympathetic Einstein to champion Schwarzschild's efforts over those of Droste, who today remains forgotten to the world. C'est la guerre.

Isaac Newton's personal journal, which I once held in my gloved hands. It includes a description of how Newton probed his own eye socket with a blunt needle to observe its effects on color perception. Yow.

A few years back the Huntington Library in Pasadena exhibited a number of Newton memorabilia, which included his death mask. I thought it remarkable, given the small size of the man's head and the numerous scars (smallpox?) on the face. The exhibit did not delve into Newton's many eccentricities, which you can read about here.

Chico (Ravelli): Well, let's see how we stand.

Groucho (Capt. Spaulding): Flat footed.


Peter Higgs has won the 2013 Nobel Prize in physics, and the Higgs boson is now more or less an official member of the Standard Model (though it sticks out like a family's bastard cousin):

The quarks and leptons (fermions) account for all the observable matter in the universe, while the other particles (bosons) represent the forces. All told, this accounts for roughly 5% of all the mass-energy in the universe, with dark matter and dark energy representing the remaining 95%.

It has taken civilization about 10,000 years to get to this point. I wouldn't exactly say that we're only 5% of the way to true understanding, but it sure looks that way.

More good news: at the same time, the theory known as supersymmetry (SUSY) seems to be in dire straits, with some physicists expressing the view that supersymmetry has gotten screwed (almost literally!) by the Higgs theory. That's fine with me, as I could never understand SUSY anyway.

A friend sent me this photo, taken at Princeton Cemetery. Since I know Hermann Weyl is buried in Zürich, my guess is that he also has a memorial marker in Princeton, where he was an active faculty member at the Institute for Advanced Study for nearly twenty years (1933-1952). I've never been to Princeton, so I have no idea was the hell it is.

Meanwhile, I found an obituary notice for Weyl's younger son Michael, who died in 2011 at the age of 93. The notice indicates that he entered Princeton University (which adjoins the IAS) at the age of 16 despite the fact that he was completely nonconversant in English (his father no doubt helped him get into the school). While his older brother Fritz became a mathematican like his father, Michael majored in German literature.

Everything should be made as simple as possible, but not any simpler.
— Einstein

It is futile to do with more things that which can be done with fewer.
— "Occam's Razor"

I have discovered a truly marvelous proof, which the margin of this page is too narrow to contain. — Pierre de Fermat

A physicist friend of mine sent me a remarkably simple proof of Fermat's famous last theorem, which asserts that there is no set of positive integers \(x,y,z,n\) that satisfies the simple relation \(x^n + y^ n = z^n\) for \(n \gt 2\). Proof of the theorem has bedeviled mathematicians since Fermat wrote about discovering it in 1637, but in 1995 a mathematically consistent proof finally emerged (unfortunately, the proof was more than 100 pages long and incomprehensible even to most mathematicans). I studied my friend's proof, thinking that he must have divided by zero somewhere or done something equally lame, but he hadn't. He has since sent his paper to a prestigious mathematics journal where, as I have warned him, they'll almost certainly spot the fatal flaw (if there is one).

The idea of making things as simple as possible is a central tenet of scientific thinking, though on the surface things may appear to be quite complicated to the casual observer. The variational principle is perhaps the most notable of this idea. It's profoundly simple and beautiful in concept, yet its implementation can appear mathematically daunting. That probably explains why it isn't taught in high school.

In my little write-up on Gaussian integrals I echoed Anthony Zee's lament about Feynman's infinite-dimensional path integral, which essentially contains all of quantum theory but can't be evaluated in closed form: Ah, if only we could do the integral! So for 60 years physicists have resorted to breaking the integral down and doing the calculations piecemeal, each iteration being computationally more laborious but giving ever more accurate results. But now a handful of theorists have discovered what they hope might be a simple alternative to the path integral based purely on geometry. The geometric object in question, the amplituhedron, has an ugly name but a beautiful purpose — to do in a few lines of calculation everything that the Feynmann path integral can do in theoretical closed form.

As reported in the journal Quanta, the amplituhedron supposedly yields probability amplitudes for particle interactions without doing any heavy quantum lifting. One of the theory's discoverers is Nima Arkani-Hamed, who describes the work in a one-hour video that you can watch here.

However fundamentally simple or beautiful it might be, the theory reportedly violates two of conventional quantum theory's most cherished assumptions, that of locality (particles exist and interact at a point in space) and unitarity (deterministic time evolution of quantum states and unit probability). Consequently, the theory effectively does away with the notions of space and time, a consideration that many theorists believe to be desirable, yet mathematically the geometric thing in question is itself a multi-dimensional object.

String theory also has a geometric quantity that is responsible for the theory's 10- or 11-dimensional form. It's the Calabi-Yau manifold, a six-dimensional geometric object that you can read about (if not understand, like me) in Brian Greene's 2000 book The Elegant Universe. The book includes a two-dimensional picture of the manifold, which in its full form is impossible for humans to visualize. If this is simplification, then I'm missing something.

The Arkani-Hamed video is entertaining, if not particularly simple. But to me, any scientist who has the nerve to make a formal presentation dressed in shorts and dress shoes without socks can't be wrong!

UC Riverside mathematical physicist John Baez reexamines the Elitzur-Vaidman bomb testing method in his recent short but interesting article, in which he posits the possibility of performing a mathematical calculation without actually turning on the computer! This kind of seemingly nonsensical thinking is made reality by merely considering quantum possibilities, which the Elitzur-Vaidman procedure amply demonstrates (the procedure has actually been tested experimentally). I tried my hand at explaining it in my post of June 21, 2011 (I use a little more math), but Baez explains it better.

While not strictly an example of quantum entanglement, the E-V problem does exemplify the crazy thinking that pervades quantum physics. The high school version of entanglement is fairly simple to explain, but the phenomenon goes far deeper into the nature of reality (and the reality of nature) than most people realize. Alas, in my own search for an accessible explanation of entanglement (see my own effort posted May 4) I've been consistently disappointed. Amir Aczel's otherwise excellent 2001 book Entanglement: The Greatest Mystery in Physics is really too elementary, so a mystery it is likely to remain if you limit yourself to that book. At a more advanced level, Dagmar Bruss' arXiv article Characterizing Entanglement makes a more heroic effort, but it's more for a mathematician than a physicist. Consequently, I still feel that Leonard Susskind's YouTube video series on quantum entanglement (you can find the lecture notes here) is the best available for the informed non-expert.

Caltech physicist Richard Feynman once called the quantum version of Young's double-slit experiment the "only mystery" in quantum physics, but entanglement takes us far beyond (the double-slit experiment is almost quaint by comparison). Entangled particles somehow influence one another instantaneously over vast distances (a fact that has been experimentally demonstrated many times over the past 30 years), and while the mathematics of the "connection" is now fairly well understood (not by me, of course), just why this should be so remains a mystery.

Given the presumed symmetry of space and time, the entanglement phenomenon makes me wonder whether entire systems of particles can be entangled in time as well. It was the late physicist David Bohm who developed an alternative approach to quantum mechanics along these lines, in which the universe is characterized by a kind of quantum potential that permeates all spacetime and inexorably connects all particles and fields to one another. This may sound more like metaphysics, but Bohm's thoughts were actually considered dangerous by none other than Senator Joe McCarthy, whose 1950s witch hunt got the politically liberal Bohm fired from his professor position at Princeton (I'm sure that McCarthy did not consider the quantum potential itself a threat, though most ultraconservatives today would consider all higher-level thinking subversive, if not outright dangerous).

The great Austrian physicist Erwin Schrödinger coined the word "entanglement," which in German is Verschränkung (I see a slightly erotic meaning in the word, apropos of Schrödinger, but I could be wrong). The word must have arisen following publication of the famous Einstein-Podolsky-Rosen paper of 1935, which first addressed the ambiguities of the entanglement issue. (In what may have been the last gasp of Einstein's noted "battle" with Niels Bohr over the validity of quantum mechanics, the anti-quantum Einstein appeared to have the pro-quantum Bohr by the short hairs. But Bohr prevailed, in what must surely have been as much a surprise to him as it was to Einstein.)

I think entanglement is the only mystery. When we've sorted it out, and we will, we'll understand more about who we are as a species and how we fit into this crazy universe. Everything else is just details.

In his 1918 book The Continuum: A Critical Examination of the Foundation of Analysis, Hermann Weyl considers an arbitrary function \(f(x)\) of some real number \(x\). If two numbers satisfy the inequality \(|x-y| \lt \delta\), where \(\delta\) is an arbitrarily small quantity, then Weyl shows that we can always find a small quantity \(\epsilon\) such that the inequality \(|f(x)-f(y)| \lt \epsilon\) holds. Weyl asserts that, because \(\delta\) can be made as small as one wants (but not zero), then the function \(f(x)\) can be considered uniformly continuous. Weyl's argument thus parallels the familiar \(\delta,\epsilon\) limit formalism of high school calculus.

But the flip side of an arbitrarily small number is an arbitrarily large number, and here we run into a problem. For while small numbers all share a very clearly defined limit (zero), there is no corresponding limit to large numbers. While we can get closer and closer to zero without actually hitting it, we always know we are in its proximity. For large numbers this is simply not the case. True, we have the concept of infinity, but it acts more like a placeholder term representing our total ignorance of what mathematical infinity really is. Infinity plus \(\delta\) is still infinity, but so is infinity plus infinity.

Perhaps the greatest problem quantum physicists have with infinity lies in the process known as renormalization, in which infinity (as Richard Feynmann once said) is simply "swept under the rug." Renormalization typically means taking a God-awful integral that diverges and redefining the integrand so that the integral splits into two terms — one finite, which provides the sought-after information (often to great accuracy), and the other infinite, which is ignored (the infinite part typically includes things like infinite electron masses and charges, which actually are hard to ignore, but what the hell).

Science writer Amanda Gefter reports on efforts to tame the infinity concept (Amanda's article is behind a subscription wall, but you can read the entire article here), and maybe do away with it altogether. Perhaps the first to attempt this was the 19th-century German mathematician Georg Cantor, whose invention of the concept of sets in effect made infinity a member of a mathematical set. A more recent idea, as Gefter reports, comes from Doron Zeilberger of Rutgers University, who likens infinity to a kind of computer overflow error — if you exceed a computer's integer math limit, it either flashes OVERFLOW or resets the offending number to zero. (This reminds me of Roger Penrose's idea of ultimate heat death in the far future, when all the black holes have evaporated and the universe is composed of nothing but stray, high-entropy radiation. At that point the universe "forgets" what it is, resets itself, and a new low-entropy universe is born from another Big Bang.)

Another way to deal with infinity, Gefter suggests, is to assume that it doesn't exist in the first place. She notes that the so-called "holographic universe" (an idea first proposed by Leonard Susskind) may consist of nothing but information that is stored on the surface of the universe's horizon, which is roughly the 3-dimensional shell that encloses the cosmos. The area of this shell, based on estimates of the current size of the universe, would hold something like \(10^{122}\) quantum bits of information. While a truly enormous number, it doesn't even begin to come close to infinity, leading researchers like Zeilberger to believe that infinity is a non-physical concept that exists only in our minds. Indeed, Weyl's Continuum would appear to reduce many of our problematic mathematical concepts to symantics, a purely human concept devoid of objective reality. Perhaps we should relabel infinity then as just "an unimaginably large number," and be done with it.

There's a research paper making big news right now purporting to show that agnostics and atheists are generally smarter than people of strong religious faith. The 30-page August 2013 paper, The Relation Between Intelligence and Religiosity: A Meta-Analysis and Some Proposed Explanations by social scientists Miron Zuckerman, Jordan Silberman and Judith A. Hall of the University of Rochester and Northwestern University is sadly behind a paywall, but I managed to find a copy (which I regrettably cannot share here out of copyright concerns). However, Ars Technica has an excellent online review of the paper that will serve you just as well.

In grade school I was somehow declared "gifted," but subsequent tests I took as a freshman entering high school showed that I was mentally deficient (to this day I don't know how I screwed up so bad). I recall that in October 1963 I spent two uncomfortable sessions with my school's vice principal, Mr. Scheel, going over the pathetic results of several intelligence tests I had taken earlier in the term. He initially wanted to place me in a "special" program under Mr. Boaz, but recanted when I begged him to give me a chance with the "normal" students. As a consequence of this unpleasant experience, I decided that intelligence tests at best don't always tell the whole story and at worst are useless. Today, I consider myself to be of average intelligence, though Mr. Scheel, if still alive, might contest that claim.

This sad story notwithstanding, the ZSH paper does appear to show a statistically meaningful inverse correlation between intelligence and religiosity. The statistical parameter of interest in the study is Pearson's \(r\) coefficient, which ranged from -0.02 to -0.75 for 53 investigations, while 10 studies actually showed small positive \(r\) values. In the hard sciences and engineering, the square of the coefficient (\(r^2\)) is normally used, as the data are generally easier to obtain and of greater reliability; a value of \(r^2 = 0.7\) (\(r = 0.84\)) is usually considered to be of marginal statistical significance, but in the social sciences much lower values are acceptable given the greater inherent difficulties in data gathering and interpretation (in my doctoral dissertation, the \(r\)-squared's for my experiments ranged from 0.6 to 0.999).

The Ars Technica article includes the following neat graph that plots IQ against the extent of atheism for some 137 countries (I haven't read the associated 2009 study from the University of Ulster, but the IQs seem unnaturally low, unless some unfamiliar metric for IQ was used):

The bottom line of the ZSH study is that nonconformance with authoritarian belief systems and a tendency toward independent thinking is what gives agnostics and atheists the edge in intelligence — that is, if the results are truly statistically meaningful. The study is already being attacked by American conservatives.

It's frustrating to read about an interesting new idea in which the source paper is published behind an obnoxious paywall. Such was recently the case for the idea of Lawrence Krauss (Arizona State University) and James Dent (University of Louisiana), which tries to explain the problem of dark energy by tying it to the Higgs boson. Often, however, a quick search over at arXiv.org shows that the paper is available for free.

The paper, A Higgs–Saw Mechanism as a Source for Dark Energy, is very brief (four pages) and readable. Undergraduates will recognize the idea as basically a variant of \(\phi^4\) scalar field theory with a proposed new field \(\sigma_H \) (associated with the Higgs) that mixes and "see-saws" with the Standard Model field to produce a vacuum energy consistent with the observed dark energy level. I'm still scratching my head over it, since the idea of extra scalar fields isn't new, but maybe there's something to it. Take a look at it for yourself.

An old friend sent this photo of Einstein to me. Believe it or not, I'd never seen it before, and now that I have I will probably try to forget it.

One way Einstein liked to relax was to go sailing, and I think this picture was taken during one of those outings. After he'd docked his little sail boat he most likely succumbed to a local photographer's request for a candid picture, and candid it was. Einstein was an extremely informal person, and I would not be surprised if he'd borrowed his wife Elsa's shoes for the photo. Ugh.

"Blind belief in authority is the greatest enemy of truth."

"As punishment for my contempt for authority, God made me an authority myself." — Einstein

Last year psychologist Bruce E. Levine wrote an article for Alternet.org entitled Would We Have Drugged Up Einstein? in which Levine questioned whether today's anti-authoritarians, like Julian Assange, Edward Snowden and Bradley Manning, are being treated (or at least viewed) as mentally ill. Levine noted that Einstein, who famously detested authoritarianism, was, in addition to being seen as a genius, also considered dangerous by numerous authorities, including the Gestapo and the FBI (the FBI, responding to Einstein's pacifist sentiments and statements, compiled an enormous dossier on him). Levine also notes that today Einstein would likely be diagnosed with attention deficit hyperactivity disorder (ADHD) or opposition defiant disorder (ODD).

Levine has now written another article for Alternet that asks if the American lifestyle is literally driving its citizens insane. In particular, he looks at the rise of the pharmaceutical industry, which in recent decades has come up with a plethora of antidepressant, anti-ADHA and other enormously profitable psychotropic drugs, not to mention questionable medications for shyness, social anxiety and restless leg syndrome (the CDC reported in 2011 that the use of antidepressants has increased 400% (that's 5 times, Republicans) in the last two decades, a trend that is disturbingly being played out with America's youth). And let's not get started on the line of erectile dysfunction drugs like Viagra and Cialis (the very idea of two aging, obese, wrinkled spouses climbing into a bathtub down at the beach would seem to have the opposite effect). God, George, we're in our seventies now — give it a rest, fer chrissake!

Levine also points out that a 2013 Gallup poll showed that 70% of American workers hate their jobs and/or have mentally "checked out" of them, and are simply going through the motions to get a paycheck. While this is not in itself indicative of mental illness, years of workplace dread, helplessness, hopelessness, boredom and fear undoubtedly carry a price tag on the body's immune system, and a similar degrading effect on the mind is also likely.

But the real issue I want to raise here has to do with a nation that may be going incrementally insane at a pace that the people, the politicians, the sociologists, the psychiatrists and the scientists cannot measure. I find it amazing that, in spite of the medical, behavioral and psychological data and metrics we have at out disposal today, the vast majority of us are just shrugging our shoulders and dismissing everything as the price of progress or the inevitable effects of the modern world.

Given the current scenario in which the world's food and water supplies are gradually but incessantly being contaminated with synthetic chemicals, I wonder if it's even possible that we can ever come to the realization that something's just not right and that it's affecting our mental faculties to the extent that, like the proverbial unthinking lemmings leaping over a cliff or the frog blithely being boiled alive, we no longer even question what's going on around us.

In my neighborhood is a wonderful charitable organization that cares for the adaptably insane — profoundly mentally impaired but non-violent adults who can carry out basic tasks under supervision. Once a week they are brought to the local public library, where they are read to and given very basic social and educational training. On occasion I've participated in this training, and what always stood out to me is the fact that these people are unaware that they are insane or that the others around them are insane. I've then asked myself what people would be like if they were insane but somehow capable of safely driving a car, holding a job, having a family, balancing a check book, programming a DVR, and running a government. That kind of insanity would have to be the result of a slow, incremental descent into madness whose effects would be imperceptible to themselves, but possibly quite obvious to a minority of others for whom the world truly has gone insane.

I completely agree with Levine when he raises the possibility that for many people insanity has become a refuge from the world or a kind of self-induced rebellion against a society in the throes of national suicide. He writes

Once it was routine for many respected social critics ... to express concern about the impact of modern civilization on our mental health. But today the idea that the mental illness epidemic is also being caused by a peculiar rebellion against a dehumanizing society has been, for the most part, removed from the mainstream map. When a societal problem grows to become all encompassing, we often no longer even notice it.

To me, it is entirely possible that our willful ignorance of major problems such as global pollution, global climate change, overpopulation and the widespread incidence of endocrine disruptors in our water and food is not due to any lack of data or evidence, or even a healthy difference of opinion based on available contrary facts, but a widespread mental disability that renders us incapable of responding rationally to problems we inwardly fear (or know) are insoluble.

No odder a couple than Albert Einstein and actor Lloyd Bridges once argued for a benevolent world government whose sheer expanse and power would resolve all political problems. I was reminded of this today by an article in Religion Dispatches, a liberal Christian website (and a favorite of mine) that involves, of all people, New York Times writer David Brooks, perpetually a favorite hate of mine.

The article, which I advise you to read, talks about the rise and current dominance of secularism. It takes as its base the lengthy book by Charles Taylor (A Secular Age), which I have not read, along with notable references to Thomas Hobbes' Leviathan, which I have read (the books would appear to be dissimilar, except perhaps for the accidental commonality of man's ability to reason and his right to self-governance).

The article's author, the appropriately-named John Modern, writes

To be clear, such Leviathanism does not merely signal the presence of power, hierarchy, injustice, and bad things impinging but — and perhaps more importantly — that the world, at base, does not give a flying fuck about you or your fate, family, and friends, but may actively be working to ensure your obliteration. (This is what I have gathered from watching the news at night, from reading history, etc.).

Modern is referencing recent blatherings by Brooks regarding how it could be that widespread secularism, which would have been all but impossible in 1500, is today not only commonplace but gaining predominance as an alternative to religious dogma. Brooks, which I need not remind you, always tries to have it both ways:

Orthodox believers now live with a different tension: how to combine the masterpieces of humanism with the central mysteries of their own faiths. This pluralism can produce fragmentations and shallow options, and Taylor can eviscerate them, but, over all, this secular age beats the conformity and stultification of the age of fundamentalism, and it allows for magnificent spiritual achievement.

Thus, according to Brooks, humanistic secularism (perhaps even atheism!) and religious dogma can play nice together ("Masterpieces"?! "Central mysteries?!" Jesus Christ!) I suspect that if Brooks had ever studied mathematics, his response to the liberals' "\(1+1=2\)" and the fundamentalists' "\(1+1=3\)" would be to assert that "\(1+1=2.5\)". A happy compromise!

Modern credits scientific discovery as the real reason why we don't think the same way people did in 1500. But even today you hear the fundies screaming that science is just "theory," as if facts and evidence were no different than opinion. Not too long ago, right winger Bill O'Reilly monumentally embarrassed himself on camera by claiming that "Nobody knows what causes the tides" as proof of the existence of God. Given the predominace of blinkard, fundamentalist radio talk shows, I strongly suspect that the shows' adherents don't know what causes the tides, either. In my opinion, this kind of ignorance makes these people dangerous in the extreme.

I used to believe that science and religion could not only coexist but be mutually supportive, even self-validating. I no longer believe that. This is bloody, f**king war.

Those who can make you believe in absurdities can make you commit atrocities. — Voltaire

The August 2013 issue of Scientific American has a thought provoking article on the wave-particle duality problem, with a take that most people (myself included)ver have considered. Written by philosophy professor Meinhard Kuhlmann of Bielefeld University in Germany (who is a "dual" himself in physics and philosophy), the article explores the problems that both the particle and field concepts exhibit under close examination.

When is a particle not a particle? When it's a field. What is a field? We don't really know. A classical field, like temperature, electromagnetism or gravity, is characterized by a number (or numbers) assigned at every point in spacetime. But a quantum field is described by a state vector \(|\Psi\rangle\) at every point, and the state vector can have any number of properties assigned to it, most of which are stochastic in nature. So if the concept of a quantum field is problematic, that of a particle (which is simply a highly-localized quantum field), is also problematic.

Kuhlmann points out that there are also problems associated with the observer. Given a box containing no particles or fields (a "vacuum"), an observer at rest looking at the box would rightly conclude that it's empty. But another observer in an accelerated reference frame would observe the box to be full of a thermal gas of radiating particles:

Kuhlmann concludes that this simultaneous "nothing" and "something" can only make sense if the wave-particle picture is replaced by a reality that is based on abstract "relationships", and it is relationships like shape and geometry that create the structure behind a reality that humans tend to define by things like particles and fields. For example, Kuhlmann explains quantum entanglement as simply a relationship that exists among two or more particles. This relationship is not affected by spacial separation, and so it should not be surprising that a measurement performed on one particle (its spin, say) somehow instantaneously affects its partner's spin, even if the particles are separated by many light years. In precisely this same sense, two bananas share the property of being yellow, a property that does not depend on things like distance.

This kind of relational thinking also appears in the theory known as loop quantum gravity, which consists mainly of interconnected, numbered lines and nodes devoid of any spacetime background. The relationships between these lines and nodes have a precise mathematical description; indeed, loop quantum gravity is one of the very few theories that to date has successfully predicted the correct black hole entropy formula.

Eddington (and Haldane) once remarked that the universe is not only stranger than we imagine, but stranger than we can imagine. It is my great hope that the reality behind this world will one day be revealed to us all.

"Measurement always disturbs, yet that didn’t stop classical physicists from in principle knowing position and velocity simultaneously." This gem comes from the pen of Craig Callender, professor of philosophy at UC San Diego, in his recent NY Times piece Uncertainty about the uncertainty principle. Perhaps I've misread Prof. Callender's article, which tries to explain what Heisenberg's uncertainty principle actually says. But the classical physicists that he refers to were simply wrong — there's no sense trying to avoid that fact, or to allude to some divine point in time in which physics made more sense. Callender also refers to the larger theory of quantum mechanics as an "oddsmaker" that provides nothing but the statistical chances of getting a particular outcome from a measurement, although I personally think it goes far beyond that.

The uncertainty principle (you can see a simple derivation from my May 3, 2011 posting) in fact relates the standard deviations in the measurements of two complementary quantities, like position and momentum or energy and time. The product of these standard deviations provides an absolute minimum that cannot be violated. Given the standard deviation of a position measurement \(\Delta x\) for some particle, the complementary "fuzziness" \(\Delta p\) of the particle's momentum is such that the inequality $$ \Delta x \Delta p \ge \hbar/2 $$ always holds. Thus, the strictly classical statement "Given a particle fixed at \(x\) with the exact momentum \(p\)" is nonsensical — there can be no such exactitude in Nature.

String theory modifies the uncertainty principle, but it only makes matters worse. In addition to the \(\hbar/2\) term, there is now a positive-valued term proportional to the tension in the string (if the term were negative, then \(\Delta x \Delta p\) could be made equal to zero). Even if string theory is proved wrong, the uncertainty principle would still hold. But what if quantum mechanics is proved wrong? We'd then be in a real fix, because to date quantum theory has never made a wrong prediction. Classical reasoning also tells us that the uncertainty principle is in effect a consequence of the second law of thermodynamics, so if quantum theory is wrong, then so is classical theory. And if classical theory is wrong, then there'd be no sense to anything — a human being would have been better off with the mind of a cow. (Although I tend to believe that some of us — notably Republicans — fit that description.)

The phenomenon known as the butterfly effect has an interesting if tangential relationship with the uncertainty principle. The initial conditions of some physical process of course determine the future evolution and final state of the process, but in chaos theory the final state of a process can be profoundly sensitive to the exactitude of the initial conditions. Thus, a change in the 1000th decimal place of some initial condition may completely change the character of the final state. The seemingly marginal effects of a butterfly flapping its wings in the Amazon can therefore in principle determine whether or not a tornado forms over some Kansas corn field. Similarly, one might naively think that the preposterous smallness of Planck's constant \(\hbar\) (roughly \(10^{-34}\) joule-sec) would not be much different from zero (indeed, we recover much of classical physics from such an approximation), but the fact that it is not zero explains to a great extent the difference between the universe we have and probable non-existence.

Callender's article references a much better article by Slate Magazine's Jim Holt, which to me is of both philosophical and historical interest. There you'll discover what an interesting (and enigmatic) scientist the German physicist Werner Heisenberg really was, in particular with regard to his ambiguous attitudes toward the Nazi regime.

My younger son is a scientist at the Centers for Disease Control in Atlanta, Georgia, and visiting him is a much looked-forward-to event for me as I enter aged decrepitude. Atlanta has always presented surprises to me, not just because it happens to be a very liberal city in the Deep South, but because of its almost lackadaisical attitude toward the Civil War, which Georgia played a key role in. Many great Civil War battles (including the Battle of Atlanta), were staged there, but you'd hardly notice it today. Nearly everything about that conflict is gone now, and it isn't just a consequence of Atlanta's awful problem with urban sprawl, which seems to be burying everything. It's like the Southerners decided that the best way to expunge William Tecumseh Sherman from their minds was to simply burn and blow everything up, build over the land, and get on with life. Today's Atlantans appear to be far more concerned with how the local teams are playing. But reminders of the past are there, if you just bother to look. And there's lots of other neat stuff in the city as well.

This summer's visit included the Archibald Smith Plantation, one of the few intact reminders of Georgia's antebellum slave days. The house (constructed in 1845) and external buildings are still there, having been faithfully preserved by the original family and its heirs, but the 300 acres of slave-worked cotton and wheat are all gone. The plantation's three-story house contrasts sharply with the shabby slave quarters nearby, also preserved, which comprise perhaps 150 square feet of living space encompassing a few windows, a crude stove and dirt floor. Life for the slaves there consisted mostly of rising at dawn, working in the fields until dark, and going to bed in preparation for another day of the same. Records show that Archibald Smith and his wife were kindly slave owners who encouraged their flesh and blood property to follow Christian ethics, in keeping with all those inerrant biblical principles that required slaves to be obedient to their masters. (I'm being sarcastic here, folks.)

Just five minutes away is Vickery Creek, a stunningly beautiful area adjacent to the nearby Chattahoochee National Forest, with abundant greenery, hiking trails, a covered bridge, a fast-running creek (I'd call it a river), waterfalls and friendly locals. But the area's wildlife, at least what I saw of it, consisted mostly of insatiable mosquitoes, chiggers, biting flies, gnats and copperhead snakes (which, unlike the mosquitoes, are actually quite tame — I easily caught one, and let it go).

The area is also home to the still-standing but non-operational Ivy Mill, constructed in 1832. The mill utilized hydropower from Vickery Creek to cut nearby timber for the area's burgeoning population, made possible by the federal Indian Removal Act of 1830, which resulted in the forced extraction of the area's Cherokees, Choctaws and other indigenous tribes, who were marched off at gunpoint to what was then considered worthless land in Oklahoma and Arkansas. Many thousands of Indians perished along the way. (There's a biblical precedent for that, too. It's the entire Old Testament.)

But my favorite part of the visit was Stone Mountain, located about 30 miles northeast of Atlanta. With a peak elevation of about 1,680 feet, it's a short (1.3 mile) but grueling slog to the top over solid granite. The view from the top was worth it, in spite of my sputtering cardiovascular system. There's a neat aerial tram ride down the sheer eastern face (one poor girl tossed her cookies on the way), and the view from the visitor's village below is equally impressive: a huge bas relief sculpture cut into the mountain side depicting Confederate heroes Robert E. Lee, Jefferson Davis and Stonewall Jackson astride their horses (pointed toward eternity, no doubt). Being no fan of the Confederate South, I shrugged it off, thinking it was carved in simpler times. But then a guide informed me that the sculpture had been completed only 40 years ago, and until fairly recently had been a traditional site for Ku Klux Klan rallies. This kind of spoiled things for me, and the winding path around the park below the carving, extolling the virtues of the Southern States that seceded from the Union, was not exactly uplifiting, either, but perhaps I'm just being overly sensitive. (My ever-positive son had a blast, but his ever-dour father was less impressed.)

On my last day we visited the Michael C. Carlos Museum at Emory University, which adjoins the CDC. I have not seen so many ancient Egyptian relics since my visit to the Cairo Antiquities Museum several years ago (the museum guide told me that noted Egyptologist Dr. Zahi Hawass considers the Carlos Museum his home away from home). While the bulk of the material ranges in age from 1,700 to 3,500 years old, many of the pieces are over 4,500 years old.

Here my half-Egyptian son enjoys a one-sided conversation with a distant relative

The museum's holdings include an extensive collection of ancient Roman statuary, jewelry, coins and related artifacts, along with many ancient Greek, Persian and Assyrian relics (statuary, pottery, cuneiform tablets, jewelry and armor). But what impressed me most was its collection of Middle Bronze Age pottery from Levantine Jericho circa 1700-1900 BCE. (Sorry, fundamentalists, but when the post-Exodus Israelite commander Joshua supposedly leveled the city around 1200 BCE, there was no city to destroy and no population to slaughter. There were also no walls; the fabled Walls of Jericho were just that — a fable.)

I can't say for the same for all of Georgia but, in spite of its past and current Southern attitudes, I'm giving Atlanta a break. The people are cultured, educated, courteous and nice, there's a lot of racial and ethnic diversity, the houses are relatively inexpensive ($500,000 will buy just about anything there), and the environs are gorgeous — lush greenery and trees are everywhere (but it all comes with hoardes of mosquitoes — who seem to prefer dining on out-of-state blood — and humidity, which is almost always above 90%).

Kid, you picked a great place to work and live. God bless you always!

With many in-laws living in Egypt, I am understandably thankful that the latest uprising there has apparently come off with hardly any violence. It was stunning to see literally millions of people lining the streets of Cairo, people who have no jobs, no savings, no food, no health care and little future, peacefully protesting yet another corrupt government that until now enjoyed the backing of the United States in its usual role of cynical Realpolitik string-puller. The Muslim Brotherhood, a strange bedfellow even for the United States, appears to be no more, at least in Egypt.

America is now backtracking, of course, threatening to cut off its annual $1.6 billion support to Egypt (almost all of it in military weapons systems) unless Egypt's pro tem leader, constitutional judge Adli Mahmood Mansour, toes the line. Well, we'll just have to see how it plays out this time around.

The noted pre-Civil war abolitionist William Lloyd Garrison once said

"Breaking down the narrow boundaries of a selfish patriotism, I have inscribed upon my banner this motto — My country is the world. My countrymen are all mankind."

Garrison thus preempted a similar sentiment made 100 years later by Albert Einstein who, revulsed by the world's rampant patriotic militarism (exacerbated as always by religious fervor), also declared himself a citizen of the world. Like organized religion, which enslaves the mind, patriotic nationalism enslaves entire countries. Egypt seems to have escaped for now. Will we?

As successful as the Standard Model of physics is, to date there has been no evidence that it can accommodate a massive, spin-1/2 fermion that is its own antiparticle. This seems odd, given the fact that force-carrying bosons (with integral spin) commonly serve as their own antiparticles (the photon being the most notable example).

But if such a particle is found to exist, then we already have a name for it: the Majorana fermion. It was dreamed up by the brilliant and enigmatic Italian physicist Ettore Majorana (pronounced eh-TOR-ay my-or-AHN-ah) in the 1930s. The Majorana fermion is one of a triad of (to me) weird spin-1/2 spinor particles, the others being Weyl's zero-mass neutrino and the Dirac electron.

On the basis of radioactivity studies performed by Irène Curie, in 1931 Majorana (1906-1938) deduced the existence of a neutral particle having about the same mass as that of a proton. The great Italian physicist Enrico Fermi urged him to publish his work, but Majorana didn't think much about the idea and so dropped it. The neutron was subsequently discovered a year later by the British scientist James Chadwick, who earned the Nobel Prize for the discovery in 1935. Perhaps still smarting for this oversight, in 1938 Majorana boarded a boat in Palermo bound for Naples, but he never arrived at his destination. Whether he was kidnapped, murdered, committed suicide by drowning or simply took a powder from life to become a hermit, the only thing we know for sure is that he was never seen again.

Being its own antiparticle, the Majorana fermion must by necessity be electrically neutral, but that does not strictly prevent its interaction with certain types of electromagnetic fields. Indeed, there is much current interest in identifying the Majorana fermion as the intermediary between dark matter (the mysterious quantity known to make up about 27% of all the mass-energy in the universe) and ordinary matter interaction through what is known as an anapole (or toroidal) electromagnetic field. The most recent paper to appear on this topic, from two researchers at Vanderbilt University, is short, fairly readable, and provides theoretical support for recent XENON100 dark matter detection studies.

Did you ever Google "naked girls" on your browser? Have you ever downloaded anything from Pirate Bay? Have you uploaded any "special" files on the Cloud, you know, those sick graphics you wouldn't want your children to ever see or know about? How about those Facebook conversations you've been having, you know, those innocent chats with old high school girlfriends that you'd just as soon not have your wife know about? Did you ever utter the words "assassinate," "kill" or "nitrogen triiodide" in a cell phone discussion? Got your passwords all nice and stored away, hidden from prying eyes?

Seven years ago, when we found out the Bush administration was tracking international calls to our country to foil those nasty, sophisticated box-cutter terrorists, we were horrified to discover that the government might not be all that trustworthy when it came to our supposed privacy. But then president-to-be Obama promised that he would put an end to all that, and we breathed a sigh of relief.

Smarter people than you and me saw this as just the tip of the iceberg. Not only has the government been tracking your phone calls, but now we discover that our Orwellian overlords have been tracking us on the Internet as well. Did you think that just because they now have the technology to record every click and keystroke on your computer that they would never use it against you? Too intrusive, you say?

Well, now they have it. Your passwords, your files, your dirty pictures, your confidential conversations, your innermost secrets. It's on those millions of terabyte solid-state drives at the National Security Agency, CIA and Pentagon that have the amazing capacity to track and record just about everything we do in real time, and they did it with our tax dollars, bless 'em. Of course, they assure us that they won't actually look at or listen to the data unless they get a federal court order to do it, but each day now we learn about some new revelation that informs us just how low the stinking human race has sunk.

"Yer Honor, we need a joint search warrant to data-mine these here 1.3 trillion recorded phone conversations and decode these 210 billion iCloud, GoogleDrive and SkyDrive files."

"Granted! We gotta keep our country safe!"

"And some of these here Facebook records indicate that Senator Sneak might be fooling around on his wife. Could be useful to us in the next election."

"Go ahead and get 'em. I always hated that son of a bitch!"

I have a better idea, one that's more cost effective. Just have every human being on Earth implanted with a miniaturized audio-video recorder/transmitter that will track everything they say and do, with the data RF'd to a worldwide network of receivers linked to existing cell phone towers and satellites. And we won't need drones anymore! Sound paranoid? Then just remember that later this year Google will start selling its eyeglass-mounted Google Glass computer that can do essentially the same thing. People will love them because the glasses will make them look just like Patrick Stewart's Borg character from Star Trek ("But I wanna be a Borg, Mommy!") Best of all, Google's already on board with the NSA — no coaxing needed!

Take a good look around you, Amerika. This is what we annihilated millions of indigenous people for. This is what we built from the stolen labor of four million black slaves. This is what we are destroying the natural world for. This is what we have evolved into.

Go to, I’ll no more on 't. It hath made me mad. I say, f**k this stupid country.

So there's this guy named Eric Ross Weinstein, see (who has a Harvard PhD in mathematical physics but is a hedge fund manager at some outfit called Natron) who's come up with a Theory of Everything™ based on Einstein-Cartan geometry in 14-dimensional spacetime. It seems everybody is talking about it, including Peter Woit at Not Even Wrong, despite the fact that not a single paper has been published or issued by Dr. Weinstein, nor does a publication appear imminent.

A few years back the non-academic physicist-surfer-adventurer Garrett Lisi published an influential paper called An exceptionally simple theory of everything, which is based on the \(E_8\) Lie group (the paper is not too difficult to follow, but requires some knowledge of quantum field theory). Lisi's \(E_8\) root system consists of some 240 nodes, each of which is associated with an elementary particle:

Why, a four-year-old child could understand this report! Run out and find
me a four-year-old child -- I can't make head or tail of it! — Groucho Marx

(Weinstein's theory similarly has many new particles gushing out of it, like Lisi's, but at least Lisi surfs, rock-climbs and prefaces his papers with "mathematics should be beautiful" while including Hermann Weyl in the analysis to boot. And he has papers to show for it.)

Well, like Groucho's character Rufus T. Firefly in the immortal Duck Soup (1933), I can't make head or tails out of any of this either, and Lisi's choice of the phrase "exceptionally simple" just makes me feel exceptionally stupid (which I am).

Now, way the hell over on Page 775 of Anthony Zee's Einstein Gravity in a Nutshell (yes, another damned plug for the book!), the author expresses his love for higher-dimensional Kaluza-Klein theory, and how disappointed he would be if Nature didn't utilize it. To me, it seems that Einstein-Weyl-Cartan theory (non-symmetric connections) and Kaluza-Klein theories, with \(n>4\) spacetime dimensions (including strings) still hold promise, while overly complicated theories like supersymmetry that predict scads of new particles (which should have been detected at the LHC, but were not) aren't gonna make it (at least I hope not). Meanwhile, if Eric Weinstein ever decides to publish his theory, we can at least hope that there's a new approach that will finally tell us what this crazy universe is all about.

It is a great pity that most people, needing to put food on the table to feed themselves and their families, cannot do what they really dream about doing with their lives. — Einstein

Einstein really did say something like that, but I can't find the exact quote or its source. Perhaps some of it just came from my aging brain, reflecting a lament I've carried around all my life.

For many years I worked as a chemist and then engineer for a water utility in Los Angeles. It had state-of-the-art water quality and biological laboratories, hydrological and snowpack survey facilities in the Sierras (skiing!), high-end computers, a top-notch science and engineering library, a gymnasium and even a medical center for its employees. For a while the utility even employed one lone physicist, who performed x-ray crystallography studies. Although times have changed, with budgetary considerations eliminating most of these things, it seems that the utility today provides one job I would have died for — archaeologist.

Today's Los Angeles Times features a discovery made by Los Angeles Department of Water and Power archaeologists in the long-dry bed of Owens Lake. (Once a thriving body of water that hosted boats and ferries carrying workers to and from mines and ore-processing facilities, the lake dried up when the DWP began exporting local water supplies to the then-young but burgeoning city of Los Angeles in 1913. But that's another story.) Archaeologists found evidence in the exposed lake bed of a famous massacre that occurred on the lake shore on the night of March 19, 1863, when dozens of US soldiers and local vigilantes slaughtered some 35 Paiute Indians over a dispute involving cattle, sheep and land rights. The story of the massacre has been handed down for generations of local Native Americans, but the exact location was unknown until researchers found bullets, musket balls, cavalry buttons and Indian artifacts recently exposed by wind and water erosion.

Local Native Americans, representing the Lone Pine Paiute-Shoshone Reservation, assisted in the conservation of the artifacts. (I have fond memories of Lone Pine, where I stumbled upon Misner/Wheeler/Thorne's Gravitation text, but that too is another story.)

Coincidentally, I recently finished reading 38 Nooses: Lincoln, Little Crow, and the Beginning of the Frontier's End by Scott Berg, Professor of English Literature at George Mason University. The book describes the Dakota War of 1862, which culminated, on December 26, 1862, just three months prior to the Owens Lake incident, in the hanging of 38 Dakota warriors for insurrection against the United States. The executions, ordered by a reluctant but resolute President Lincoln, also represented the beginning of the end for the indigenous American Indian nations.

Admittedly, I'm now a long way off from the topic of archaeology (I do have a tendency to wander mentally at my age), but I find stories like this to be more than a little compelling, particularly in view of my country's ongoing issues involving race and minority rights. But consider this: The United States committed genocide against the American Indians while concurrently carrying out the barbaric practice of enslavement of over four million Africans. Today, after 150 years of so-called Black Emancipation and Indian treaties (all of which were broken by the US government), America's remaining Indians and (to a lesser extent) their black cousins represent decimated races of people. Conservatives today complain bitterly about why these minorities, having been supposedly appeased with civil rights legislation, can't simply shrug off 400 years of inhuman treatment and get on with their lives. My own feeling is that it will probably take another four hundred years just to undo the damage that has been done, a damage that might well now be genetically ingrained in these persecuted minorities.

And please also consider this: Despite all their earnest and poetic prayers, soaring appeals to God and good intentions, America's Christians did nothing to stop the genocide or the mistreatment. Indeed, even today many Christian evangelicals persist in the belief that Indians and blacks are inherently inferior to whites, a belief that is still predominant in the South.

They made us many promises, more than I can remember, but they never kept but one: they promised to take our land, and they took it. — Red Cloud, Oglala Lakota (Sioux) Warrior and Chief

The moving finger writes and, having writ, moves on; and neither all thy piety nor all thy wit shall lure it back to cancel half a line, nor all thy tears wash away a word of it. — The Rubayaat of Omar Kayyam

As the Nobel laureate economist Paul Krugman would say, what follows is rather wonkish. But it seems I have nothing better to do today.

I still get the occasional inquiry from my microscopic coterie of past students, interested laypersons and legitimate physicists regarding the need for local conformal symmetry in the world (and by conformal symmetry I mean the physics does not change for the metric gauge transformation \(g_{\mu\nu} \rightarrow \lambda(x) g_{\mu\nu}\). Hermann Weyl believed in such a symmetry, or at least he did until Einstein showed him that the measure of the re-gauged line element \(ds^2\ = \lambda \, g_{\mu\nu} dx^\mu dx^\nu\), which can be made equivalent to a meter stick or a ticking clock, would depend on an object's history. For example, a locally re-gauged metric would mean that the spacings of the atomic spectral lines of an excited sodium atom would vary as the atom was carried about from one place to another (and they would also vary with time if the atom was simply left alone). Obviously this has never been observed, and Weyl's otherwise brilliant 1918 theory was all but dead by 1921.

Although Weyl successfuly resurrected the idea as quantum-mechanical phase invariance in 1929, interest in conformal invariance lagged for many years until fairly recently, when it was discovered that it may help explain the dark energy/dark matter problem.

Here I will once again try to show that Weyl's 1918 theory, which was originally developed as a unified theory of gravity and electromagnetism, has nothing whatsoever to do with electrodynamics—it's probably just empty formalism. The importance of conformal symmetry in cosmology is an issue I will leave to the experts.

Although some who have written me think that conformal invariance can be established via a local phase factor attached to objects like vectors and spinors, the magnitudes (local or global) of these quantities are determined solely by the metric tensor \(g_{\mu\nu}\). Weyl himself realized that attaching a phase factor to the metric could render the line element \(ds^2\) conformally invariant, so let's start with that.

We assume that there is a quantity \(\varphi(x)\) that makes the re-gauged metric $$ \bar{g}_{\mu\nu} = e^{-2 \varphi} g_{\mu\nu} $$ invariant under the infinitesimal gauge (or conformal) transformation \(g_{\mu\nu} \rightarrow (1+\epsilon \pi) g_{\mu\nu}\), where \(\pi(x)\) is an arbitrary scalar quantity. We then have \(\delta g_{\mu\nu} = \epsilon \pi g_{\mu\nu}\) and \(\delta \bar{g}_{\mu\nu} = 0\), where the operator \(\delta\) means gauge variation. It is easy to see that this requires that the parameter \(\varphi\) obey the condition \(\delta \varphi = 1/2 \epsilon \pi\), but it is otherwise arbitrary. Armed with the new metric \(\bar{g}_{\mu\nu}\), we see that any quantity that involves the metric directly, including the line element \(d\bar{s}^2 = \bar{g}_{\mu\nu} dx^\mu dx^\nu\) and the metric determinant \(\sqrt{-\bar{g}}\), is automatically conformally invariant.

If Weyl's theory has anything to do with electromagnetism, we should be able to derive the geodesic equations of motion for a charged particle directly from a coordinate variation of \(d\bar{s}\). This is a standard, elementary exercise in general relativity, the only complication here being our use of a re-gauged metric. Beginning with the relativistic action \(\bar{S} = - m c \int d\bar{s} \,\) or $$ \bar{S} = -m c \int \bar{g}_{\mu\nu} \frac{dx^\mu}{d\bar{s}} \frac{dx^\nu}{d\bar{s}} \,d\bar{s} $$ we want to compute its variation with respect to \(x^\beta\), or $$ \frac{\delta \bar{S}}{\delta x^\beta} = - m c \frac{\delta}{\delta x^\beta} \int \bar{g}_{\mu\nu} \frac{dx^\mu}{d\bar{s}} \frac{dx^\nu}{d\bar{s}} \,d\bar{s} $$ which we will then set to zero. The calculation is straightforward but tedious (as well as boring), so I will just write down the result: $$ \frac{d^2 x^\alpha}{ds^2} + \Gamma_{\mu\nu}^\alpha \frac{dx^\mu}{ds} \frac{dx^\nu}{ds} = \frac{\partial \varphi}{\partial x^\mu} \frac{dx^\mu}{ds} \frac{dx^\alpha}{ds} - g^{\alpha\beta} \frac{\delta \varphi}{\delta x^\beta} $$ where I have expressed everything back in terms of the ungauged line element and metric (\(\Gamma_{\mu\nu}^\alpha\) is the ordinary Christoffel symbol). Note that the lhs of this equation is just the familiar geodesic term, which vanishes for a free particle. If electromagnetism has anything to do with all this, we'd expect the rhs to be the Lorentz force term, which is proportional to $$ g^{\alpha\beta} F_{\beta\lambda} \frac{dx^\lambda}{ds} $$ where \(F_{\beta\lambda}\) is the electromagnetic tensor $$ F_{\beta\lambda} = \frac{\partial A_\beta}{\partial x^\lambda} - \frac{\partial A_\lambda}{\partial x^\beta} $$ and \(A_\beta\) is the four-potential. Up to this point we have allowed the gauge parameter \(\varphi\) to be arbitrary. Can it be defined in such a way as to reproduce the Lorentz force term? Weyl assumed that \(\varphi = \int A_\mu dx^\mu\), which makes the above equations of motion equal to $$ \frac{d^2x^\alpha}{ds^2} + \Gamma_{\mu\nu}^\alpha \frac{dx^\mu}{ds} \frac{dx^\nu}{ds} = A_\mu \frac{dx^\mu}{ds} \frac{dx^\alpha}{ds} - g^{\alpha\beta} \! \int \! F_{\beta\lambda} dx^\lambda $$ The rhs of this expression does not resemble the Lorentz force at all, so we are forced to assume that Weyl's \(\varphi\) has nothing to do with electromagnetism.

Weyl's 1918 theory failed, and for good reasons, perhaps the primary reason being that it was just too simple. It wasn't until quantum mechanics came along that Weyl redeemed himself through the principle of phase symmetry. Today, this principle, which is still called gauge invariance, underlies all modern quantum field theories. While we can be grateful that Weyl discovered it, it's so important that its discovery was inevitable — sooner or later, somebody would have found it! Weyl was justified in hoping that it might also be applicable to gravitation but, as we've seen for almost 100 years now, gravity, perhaps the most subtle and beautiful of Nature's forces, is a bitch.

Google is teaming up with NASA to implement what may be the first commercial application of a quantum computer. The D-Wave computer reportedly uses entangled electrons to represent "fuzzy" on-off logic as a superposition of 0 and 1, making numerical computing up to 3,600 times faster than the fastest current conventional computer. To paraphrase Adult Swim, I don't know how it works, and I don't ask.

Quantum computing uses entangled computing states to essentially seek all possible solutions to a mathematical problem, then picks the best answer. The linked article talks about application to the famous traveling salesman problem, but the first thing I thought about was Feynman's path integral, which represents all possible paths that a quantum particle or state can take for a given problem.

Conservatives will continue to complain that "it's just a theory," but in a few years they'll all be using the technology without giving it a single thought (indeed, they try to avoid thinking at all costs). I would add that it must be comforting to believe that an electron goes from Point A to Point B by the hand of God or fate or whatever, and not in accordance with the probability amplitude \(\langle B|A\rangle\), but then I would sound like an elitist. (Elitist? Moi?)

I came across this photo while cleaning out a closet today (dang, escrow's closing later this week and I'm still cleaning). I was only 13 at the time I graduated from the 7th grade at Northview Junior High School in Duarte, California but I remember this photo being taken as if it was yesterday (I'm the dopey-looking one between the tall redhead and the sultry brunette). As far as I know, only three of the kids became scientists or engineers. The tall kid near center top row (Daniel M.) was my best friend, and he actually got me interested in physics and chemistry (and a lot of dangerous chemistry at that). He was probably the only truly brilliant member of Mr. Weaver's class, having previously skipped two grades (he's only 11 in the picture, despite being nearly 6 feet tall). We were very competitive academically, but he invariably got the top class scores, while I was invariably second, always unable to match him. I seemed destined to be second in everything I did in life (assistant division manager, assistant executive director, etc.). Oh well.

Several of the kids in this photo are dead now (as is the school itself, which was torn down many years ago). While visiting the grave of a beloved high school French teacher some while back (see my post of March 1), I inadvertently stepped on the grave of one of them while returning to my car. I hope that gave you a little smile, Gary, wherever you are.

One final memory: Around Christmas time one of the girls in the photo (Cindy F.) learned that Santa Claus was not real, and she cried like a baby in class. That's the way kids were in those days. (Cindy was also a stone fox, but I was too young to appreciate it at the time.)

Wolfgang Pauli at 20 months, with mum Bertha. Pauli was about to enter the Terrible Twos, which seemingly lasted the rest of his life.

I finished reading Charles P. Enz' 2002 book No Time To Be Brief: A Scientific Biography of Wolfgang Pauli to the end, mainly to see what it had to say about Pauli's relationship with Hermann Weyl. The correspondence between the two men, spanning from 1918 until Weyl's death in 1955, is fascinating in view of how much it reveals about the changing character of Pauli.

Born in 1900, Pauli was a true prodigy who entered high school at the age of ten, mastered calculus at 14, and by the age of 18 was nearly as conversant in general relativity as Einstein or Weyl. In May 1919 Weyl wrote to him, saying

I am extremely pleased to be able to welcome you as a collaborator. However, it is almost inconceivable to me how you could possibly have succeeded at so young an age to get hold of all the means of knowledge and to acquire the liberty of thought that is needed to assimilate the theory of relativity.

Pauli was "uncharacteristically respectful" towards Weyl around this time, but that would quickly change as Pauli gained greater and greater confidence in his abilities. Later in 1919 Pauli politely pointed out a "small oversight" in Weyl's own relativity work, and by December of that year Pauli was already beginning to demonstrate the biting criticism that eventually made him the enfant terrible of physics, mostly feared by less talented physicists but also sharply felt by many renowned scientists as well, Einstein included. Three of my favorites: Das ist falsch (that's wrong); Das ist ganz falsch (that's completely wrong); and Das ist nicht einmal falsch (that's so bad it's not even wrong).

After obtaining his PhD from Munich's Ludwig Maximilian University in July 1921, Pauli eventually found his way in 1928 to the ETH in Zürich, where Weyl had taught since 1913. Initially Pauli expressed some regret for his earlier criticisms of Weyl:

Without falling into elegiac considerations I wish to express the hope that in the meantime you may find me to have become yet somewhat more grown up.

Within a year, however, Pauli was back to his characteristic sniping. Noting that as a mathematician Weyl had made much progress in the field of theoretical physics, in 1929 Pauli wrote

The conclusion appears irrefutable that, at least for some time, you wish to be judged not for your successes in the domain of pure mathematics but for your loyal but unhappy love for physics.

Enz goes on to note that Pauli seemed to be especially harsh toward Weyl because of the latter's tendency to express himself in gentlemanly prose when writing, and always displayed very polished manners.

Weyl often did not appreciate Pauli's sarcastic criticisms, but the two men managed to remains friends for life, no doubt due to their shared love of physics. Whenever Pauli's criticisms were justified, Weyl acknowledged them graciously. When speaking at a dinner given in honor of Pauli's 1945 Nobel Prize, Weyl admitted

Perhaps I am among the first with whom [Pauli] established scientific contacts, for the first papers he published dealt with a unified theory of gravitation and electromagnetism which I had propounded in 1918. He dealt with it in a truly Paulinean fashion — namely, he dealt it a pernicious blow.

The physicist Enz, Pauli's assistant at the time of Pauli's death in 1958, has written a highly erudite and comprehensive book (it's nearly 600 pages) that covers all of the Nobelist's work from early relativity to quantum field theory and beyond. My only crit icism is Enz' terse description of Pauli's very last day, which ended due to a malignant abdominal tumor on Dec ember 15, 1958. Enz writes "Ms. Pauli called me to say that Pauli had died." That's about it. I would like to have read more about Pauli's reflections on his life at this stage. I might also add some quibbling about Enz' spelling and his occasional destruction of an English word or expression, but overall the book is a very worthwhile biographical contribution to the life of a particularly fascinating and brilliant, if somewhat grumpy, physicist.

Last night I was reading an arXiv paper posted early last year claiming that the quantum violation of Bell's inequality could be understood using Hermann Weyl's "conformal geometrodynamics." I know what the term means, though I'd never seen it in print before, but just how Bell's inequality has anything to do with Weyl conformal geometry beats me. And after reading the paper, it still beats me.

Anyway, last night I also watched Leonard Susskind's excellent video on Bell's inequality, and I got fascinated again by what Sakurai has called "one of the most astonishing consequences of quantum mechanics." It is indeed astonishing, although Susskind is undecided about whether the quantum violation of Bell's famous inequality is truly profound or just one of those ho-hum (if crazy) predictions of quantum theory, which Feynman once asserted that nobody really understands, anyway.

Susskind's analysis is straightforward but the details are messy, and he goes through the exercise for only a single set of angles in the inequality (all in the same plane). In fact, he uses the same set that Sakurai uses in his book, and the same set I used in my own Kindergarten analysis that I posted some years ago on this site. As is my wont, I wanted to derive the violation for an arbitrary set of angles, in arbitrary directions, and do it in as elegant a fashion as possible. It took me an hour and a half, but since it was way past my bedtime when I finished I can't be sure if my aging brain got it completely right (though it reproduces Sakurai's general formula). At any rate, it's quite suitable for a third-year undergraduate, and it also serves to demonstrate the important fact that quantum mechanics actually upholds Bell's inequality for certain angle sets (if there's any profundity going on, it's due to the fact that the inequality is always maintained classically, whereas quantum mechanics can violate it, at least on occasion).

I'll post it here when I have time and energy to write it up (I'm busy trying to sell some property right now). And if I can figure out what Weyl has to do with any of it, I'll pass that along, too.

Update: Okay, so I don't have the time or energy to write it up! Here's a shortened version, but it will only be comprehensible if you watch Susskind's lecture first.

To derive Bell's inequality for any set of directions, we first express the Pauli matrices in terms of a sum of matrix operators that will act on an arbitrary normalized eigenvector: $$ \vec{\sigma} \cdot \vec{n} = n_x \left[ \begin{array}{cc} 0 & 1\\ 1 & 0 \end{array} \right] + n_y \left[ \begin{array}{cc} 0 & -i \\ i & 0 \end{array} \right] + n_z \left[ \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right] $$ where \(\vec{n}\) is a unit vector (\(n_{x}^2 + n_{y}^2 + n_{z}^2 = 1 \)). This comes out to be $$ \vec{\sigma} \cdot \vec{n} = \left[ \begin{array}{cc} n_{z} & n_- \\ n_{+} & -n_{z} \end{array} \right] $$ where \(n_{\pm} = n_{x} \pm i n_{y} \). This matrix has eigenvalues \(\pm 1\), so there are two eigenvectors. We'll only need one, and a suitable normalized eigenvector (for eigenvalue +1) is $$ |n+\rangle = \frac{1}{\sqrt{2(1 - n_{z})}} \left[ \begin{array}{c} n_{-} \\ 1 - n_{z} \end{array} \right] $$ The projection operator (or density matrix) for this eigenstate comes out to be the simple quantity $$ |n+\rangle \langle n\!+\!| = \frac{1 + \vec{\sigma} \cdot \vec{n}}{2} $$ We want to use this operator on the two-electron singlet eigenstate $$ |\psi\rangle = \frac{1}{\sqrt{2}} \left( |ud\rangle \pm |du\rangle \right) $$ where \(|ud\rangle\) means Electron 1 is up and Electron 2 is down, etc. ("singlet" just means a state with zero total spin). We'll also leave the sign between the two kets undecided for the time being. Since there are two electrons, we need two projection operators. It is conventional to let the matrix $$ |n+\rangle \langle n\!+\!| = \frac{1 + \vec{\sigma} \cdot \vec{n}}{2} $$ operate only on Electron 1, while $$ |n+\rangle \langle n\!+\!| = \frac{1 + \vec{\tau} \cdot \vec{n}}{2} $$ operates only on Electron 2. The matrix \(\tau\) is identical to the Pauli matrix \(\sigma\), but it's renamed as a mnemonic device to tell us which electron it should operate on.

We now remind ourselves how the Pauli matrices operate on ket states: \begin{eqnarray*} \sigma_{x} |u\rangle & = & |d\rangle \\ \sigma_{y} |u\rangle & = & i|d\rangle \\ \sigma_{z} |u\rangle & = & |u\rangle \\ \sigma_{x} |d\rangle & = & |u\rangle \\ \sigma_{y} |d\rangle & = & -i|u\rangle \\ \sigma_{z} |d\rangle & = & -|d\rangle \end{eqnarray*} Assume now that we're handed a 2-electron singlet state ket whose spin directions are completely arbitrary. What we want to do is essentially "filter out" those portions of the state that are not parallel to a direction that we specify for Electron 1. Let us describe that direction by the unit vector \(a\). T he projection operator for this direction is then $$ |a+\rangle\langle a\!+\!|= \frac{1 + \vec{\sigma} \cdot \vec{a}}{2} $$ The same electron may also exhibit some tendency to spin in a different direction, say direction \(b\). The projection operator for that direction would be $$ |b+\rangle\langle b\!+\!| = \frac{1 + \vec{\sigma} \cdot \vec{b}}{2} $$ If Electron 1 were to be found not spinning in direction \(b\), then it would have to spin in the direction \(-b\). But if that were the case, entanglement would require that Electron 2 spin in the direction opposite to \(-b\), which is just direction \(b\)! Consequently, this strange (and perhaps confusing) logic implies that if the spin directions of Electron 1 and Electron 2 are considered for two different values \(a\) and \(b\), then the projection operators for the system "a not b" have to be $$ |a+\rangle \langle a\!+\!| = \frac{1 + \vec{\sigma} \cdot \vec{a}}{2}, \quad \quad |b+\rangle \langle b\!+\!| = \frac{1 + \vec{\tau} \cdot \vec{b}}{2} $$ The probability that the arbitrary ket state spins will be aligned in accordance with "a not b" (or, in set notation, \(a\backslash b\)) is given by the expectation value $$ P(a\backslash b) = \frac{1}{\sqrt{2}} \left( \langle ud| \pm \langle du| \right) |a+\rangle \langle a\!+\!|b+\rangle \langle b\!+\!| \left( |ud \rangle \pm |du \rangle \right) \frac{1}{\sqrt{2}} $$ (It is easy to show that \(P(a\backslash a)\), which is the probability that the spins of two entangled electrons will be found pointing in the same direction, is identically zero, as it must be.) From here on, I'll just refer to the bra and ket as \(\langle\psi|\) and \(|\psi\rangle\), respectively.

To set up the quantum version of Bell's theorem, we consider three such situations involving three spin combinations for the electrons: \(a\backslash b\), \(b\backslash c\), and \(a\backslash c\): \begin{eqnarray*} P(a\backslash b) & = & \langle \psi|a+\rangle\langle a\!+\!|b+\rangle\langle b\!+\!|\psi\rangle \\ P(b\backslash c) & = & \langle \psi|b+\rangle\langle b\!+\!|c+\rangle\langle c\! +\!|\psi\rangle \\ P(a\backslash c) & = & \langle \psi|a+\rangle\langle a\!+\!|c+\rangle\langle c\!+\!|\psi\rangle \end{eqnarray*} Using the orthogonality of the various bras and kets, you should be able to show that the \(a\backslash b\) case works out to be simply $$ P(a\backslash b) = \frac{1}{4} \left( 1 \pm a_1 b_1 \pm a_2 b_2 - a_3 b_3 \right) $$ We now see that this expression can be meaningful only if we assume a minus sign (\(-\)) between the original two kets, so that $$ P(a\backslash b) = \frac{1}{4} \left( 1 - \vec{a} \cdot \vec{b} \right) $$ But the dot product \(\vec{a} \cdot \vec{b}\) is just \(\cos\theta_{ab}\), where \(\theta_{ab}\) is the angle between the two spin directions (remember that \(\vec{a}\) and \(\vec{b}\) are unit vectors). We now use a simple trigonometric identity to write $$ 1 - \cos\theta_{ab} = 2 \sin^{2}\frac{1}{2}\theta_{ab} $$ (The appearance of half angles in all this almost screams "spin one-half"!) To complete Bell's inequality, we then have, for the three spin directions, the probabilities \begin{eqnarray*} P(a\backslash b) & = & \frac{1}{2} \sin^{2}\frac{1}{2}\theta_{ab} \\ P(b\backslash c) & = & \frac{1}{2} \sin^{2}\frac{1}{2}\theta_{bc} \\ P(a\backslash c) & = & \frac{1}{2} \sin^{2}\frac{1}{2}\theta_{ac} \end{eqnarray*} Bell's theorem in quantum mechanics is then given by the inequality $$ \sin^{2}\frac{1}{2}\theta_{ab} + \sin^{2}\frac{1}{2}\theta_{bc} \ge \sin^{2}\frac{1}{2}\theta_{ac} $$ and we're done.

It's actually not that easy to find a set of angles where the inequality is violated. Almost every textbook will use the combination \(\theta_{ab} = \theta_{bc} = \pi/4, \theta_{ac} = \pi/2\), which does violate the inequality. But the main point is that the inequality is always upheld in classical theory, while it can be violated in quantum mechanics.

If you have Mathematica installed on your computer, you can see which angles in the above inequality lead to violation of Bell's theorem.

The great Austrian physicist Wolfgang Pauli (1900-1958) has been mentioned on this site numerous times, but outside of a few comments he made regarding Hermann Weyl's 1918 and 1929 gauge theories he's been pretty much ignored here. But there's a wonderful 20 04 biography of Pauli (revised in 2010) that details the man's life while recounting many of his dealings with Weyl.

Charles Enz' No Time To Be Brief: A Scientific Biography of Wolfgang Pauli not only provides a comprehensive overview of the scientist's personal and professional life, but goes into some of the mathematical details of his theories as well. It is a refreshing step away from the typical biographies of scientists like Einstein, where \(E=mc^2\) is the only equation the reader is likely to encounter.

For many years I've owned and read a tattered copy of Pauli's seminal 1921 book on relativity theory. Written at the tender age of 21, just a few months after receiving his PhD at the Ludwig Maximilian University in Munich, it was later expanded by the author to include a detailed analysis of Weyl's 1918 metric gauge theory. However, I was unaware that in 1919 Pauli himself wrote several papers on the theory in which he examined the various terms allowed into the Weyl action Lagrangian. In his second paper Pauli considered

$$ S = \int d^4x\, \sqrt{-g}\, \left(c_1 R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta} + c_2 R_{\mu\nu}R^{\mu\nu} + c_3 R^2 + \frac{1}{4} F_{\mu\nu}F^{\mu\nu} \right) $$ to be the most general Weyl action possible (the various \(c\)'s are constants). It amazes me that here we see a mere 19-year-old university undergraduate going head-to-head with the already famous and highly accomplished 33-year-old Weyl.

Although both Pauli and Einstein would subsequently attack Weyl's 1918 theory as unphysical, Pauli was so impressed with the electric charge-conservation aspect of the theory that he relied on it later to defend the idea of energy conservation itself. As recounted in Enz' book, \(\beta\)-decay experiments conducted by Lise Meitner in 1929 appeared to conclusively demonstrate that energy was not a conserved quantity after all. This in fact was the opinion of the great Niels Bohr himself, who saw no way to avoid the non-conservation conclusion. However, Pauli saved the notion of energy conservation by proposing that \(\beta\)-decay was accompanied by the emission of unseen massless particles, which he called neutrons. When the actual (massive) neutron was discovered by Chadwick in 1932, Pauli's particle was renamed the neutrino. Thus, largely on the basis of Weyl's purely mathematical charge-conservation discovery, Pauli proposed a particle whose later experimental confirmation upheld what is arguably the most important conservation law ever devised by man.

The year 1929 was also groundbreaking for Weyl, who resurrected his failed 1918 theory in the guise of quantum-mechanical phase invariance, a concept that today underlies all fundamental quantum field theories. Even Pauli, who by that time had become something of a feared scientific curmudgeon, was impressed with Weyl's achievement. In 1929 he wrote to Weyl, saying

In contrast to the nasty things I said ... Here I admit your ability in physics. Your earlier theory with \(g_{\mu\nu} \rightarrow \lambda(x)g_{\mu\nu}\) was pure mathematics and unphysical. Einstein was justified in criticizing and scolding you. Now your hour of revenge has arrived!

Pauli went on to win the Nobel Prize in physics in 1945 for his famous exclusion principle (which every high school chemistry student knows well), by which time he was primarily involved in quantum mechanics. Pauli is also well known for the Pauli effect, a dubious honor normally ascribed to laboratory klutzes. In Pauli's case, his mere presence in a lab seemed to bring on disaster, and many experimentalists, rational though they were, would not permit Pauli to enter their labs.

Ever fascinated (even obsessed) with the dimensionless fine structure constant \(\alpha\) (which is roughly 1/137) and its importance in quantum theory, Pauli later contracted cancer and was taken to the hospital, where he passed away in Room 137 at the relatively young age of 58.