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Why Did Weyl's Theory Fail?
Did Weyl Screw Up?
Weyl and the Aharonov-Bohm Effect
The Bianchi Identities in Weyl Space
Conformal, Parameter-Free Riemannian Gravity
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Riemannian Vectors in Weyl Space
Introduction to Quantum Field Theory
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Veblen and Weyl
Graphing the Lorentz Transformation
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Is There a Flaw in the FLRW Metric?
On the Mannheim-Kazanas Spacetime
Electron Spin
Clebsch-Gordan Calculator
Bell's Inequality
The Four-Frequency of Light
There Must Be a Magnetic Field!
Non-Metricity and the RC Tensor
Curvature Tensor Components
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The Divergence Myth in Gauss-Bonnet Gravity
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Differential Forms for Physics Students
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2013 Archives

Turing Is Forgiven, or There's One in Every Southern Family — Posted Tuesday, December 24 2013
The highly underrated 2001 movie Enigma (featuring a very nerdy Kate Winslet) is a somewhat fictionalized version of the Bletchley Park effort to crack the German U-boat codes Enigma and Ultra. It stars Dougray Scott as the group's principle code breaker Tom Jericho, although his character is actually a straight version of the gay mathematical genius Alan Turing, who was responsible for breaking the codes. In the film Jericho is tortured by his longing for blond seductress and spy Claire Romilly (played by Saffron Burrows), while in real life Turing was tortured by the British government for being a homosexual. Despite his acknowledged contributions to the war effort (which saved countless lives), Turing was forced to undergo chemical castration for his tendencies, which the British government considered criminal at the time. As a direct result of this treatment, Turing committed suicide in 1954 at the age of 41. What a tragedy, and what a waste.

As a singularly heterosexual male, and a product of a much earlier generation, I really don't understand the gay thing at all. While I have to admit a certain degree of revulsion (which I oddly don't feel regarding lesbianism), I have accepted the biological fact that gays are simply born differently than the majority of the population, and should be treated like everyone else. (I really need to watch what I say about this subject. I corresponded for awhile with the gay son of a famous British mathematician until I said something he took as offensive, and I never heard from him again.)

Turing's mathematical genius was on par with that of John von Neumann, who also worked on computation theory. Anyone who doubts this should consider the fact that with little to go on, Turing was able to decipher the workings of the German Enigma machine, a portable, battery-operated device consisting of a bewildering combination of multiple lamps, rotors and mechanical encryption algorithms. Several thousand of these machines were constructed by the Germans, and at least one found its way into the hands of the cryptanalysts at Bletchley Park. Months of dedicated, non-stop trial and error allowed Turing and his colleagues to understand how the device functioned.

[On very rare occasion an original Enigma machine goes up for auction, but the prices are well beyond most of us. I'm still hoping to land a Type II Curta machine in decent condition, but that looks doubtful as well. I recall well a fellow undergraduate having one back around 1970, but at \$125 my student poverty prevented my acquiring one. They now run in the thousands of dollars. Sigh.]

But I'm digressing again. No doubt in view of the upcoming 60th anniversary of Turing's death, the British government has seen fit to issue a royal pardon for Turing. I'm not sure, but they may have already issued a similar pardon for noted gay author Oscar Wilde (The Importance of Being Earnest, etc.), who actually spent time in prison in the 1890s for his then-illegal indiscretions.

Lastly, I find it amusing and more than a little maddening that fundamentalist Christian pastors like Ted Haggard can be flamboyant homosexuals and convicted drug felons while continuing to be forgiven by their stupid flocks (and yes, Haggard is back, bigger than ever, raking in more millions from his idiotic followers). The hypocrisy of the Bible-thumping South never fails to amaze me.
Exponential Growth — Posted Sunday, December 22 2013
In this video, noted biologist David Suzuki explains the dangers of exponential growth in a way that anyone, even an American, can understand:

The RC Tensor in 11 Dimensions and Other Stuff — Posted Wednesday, December 11 2013
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.
No Reconcilement — Posted Friday, December 6 2013
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.
Pascal, Bostrom and Frink — Posted Monday, December 2 2013
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.
Homer Beats Fermat? — Posted Saturday, November 23 2013
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.
Common Sense Theories — Posted Monday, November 11 2013
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.
What's That You're Wearing? — Posted Tuesday, November 5 2013
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.
Hilbert-Droste-Weyl — Posted Friday, November 1 2013
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.)
Weyl's Trick — Posted Tuesday, October 29 2013
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.
Ouch — Posted Thursday, October 10 2013
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.
Here's How We Stand — Posted Tuesday, October 8 2013
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.
What's This? — Posted Saturday, September 28 2013
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.
Keeping It Simple — Posted Friday, September 20 2013
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!
More on Entanglement — Posted Thursday, August 29 2013
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.
Sweeping Infinity Under the Rug — Posted Wednesday, August 21 2013
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.
I Read It For You — Posted Thursday, August 15 2013
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.
Krauss/Dent Theory — Posted Wednesday, August 14 2013
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.
Not Just a Pretty Face — Posted Friday, August 2 2013
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.

The Mental Illness Epidemic — Posted Wednesday, July 31 2013
"Blind belief in authority is the greatest enemy of truth."

"As punishment for my contempt for authority, God made me an authority myself." —
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.
Bloody War — Posted Sunday, July 28 2013
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
Particle or Field? — Posted Sumday, July 28 2013
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.
Uncertainty — Posted Tuesday, July 22 2013
"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.

Dear Diary — Posted Tuesday, July 22 2013
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!
Happy Fourth of July for Egypt — Posted Thursday, July 4 2013
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?

Hardly Neutral — Posted Monday, June 17 2013
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.
If You Haven't Done Anything Wrong, You Have Nothing to Fear — Posted Thursday, June 6 2013
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.
The Latest Theory of Everything — Posted Wednesday, June 5 2013
"We exist in a universe described by mathematics. But which math? Although it is interesting to consider that the universe may be the physical instantiation of all mathematics, there is a classic principle for restricting the possibilities: The mathematics of the universe should be beautiful. A successful description of Nature should be a concise, elegant, unified mathematical structure consistent with experience." — Garrett Lisi
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.
My Next Life, and Other Things — Posted Monday, June 3 2013
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
Gravity's a Bitch — Posted Thursday, May 23 2013
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.
A Quantum Traveling Salesman I — Posted Thursday, May 16 2013
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?)
Reminiscing — Posted Sunday, May 12 2013
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.)
Pauli Again — Posted Saturday, May 10 2013
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.
Weyl and the Violation of Bell's Inequality — Posted Saturday, May 4 2013
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.
Pauli and Weyl — Posted Sunday, April 28 2013
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.
WMAP Upstaged — Posted Thursday, March 21 2013

The Wilkinson Microwave Anisotropy Probe (WMAP), launched in 2001, has given scientists unprecented insight into the age, nature and makeup of the universe. But today scientists at the European Space Agency released data from its Planck Space Telescope, launched in 2009, that is fundamentally remaking our views of the cosmos. In short, WMAP's ass has been kicked, but good.

Most notably, the universe is slightly older (13.8 billion years) than previously believed (about 13.7 billion years), and its complement of energy and matter (dark energy, dark matter and ordinary, or baryonic, matter) will also have to be revised (see above graphic). In addition, the universe appears to violate the cosmological principle, which basically states that the distribution of matter and energy is the same no matter where an observer is standing or in which direction he looks.

The Planck project, which includes important contributions from NASA, has also provided the best view to date of the cosmic microwave background (CMB):

This graphic, which cosmologists have likened to seeing the face of God or the birth of everything, reveals tiny fluctuations in the temperature of the CMB relating to the distribution of matter and energy in the universe about 370,000 years after the Big Bang. Without these minute differences, which formed at the very beginning of the universe, gravitational clustering of hydrogen gas into galaxies would not have been possible (and neither would we). I see it as another example of how a slight discrepancy (broken symmetry?) in the birth of the universe made all the difference. Perfection is not always a good thing!

Better and better

Higgs Announcement — Posted Thursday, March 14 2013

With 2.5 times more data than when the discovery was announced at the Large Hadron Collider (LHC) on July 4, 2012, CERN scientists now say that the Higgs boson has been confirmed to a statistical reliability of 7.4 sigma, meaning that they are about 99.99999999998637% sure that it is a Higgs. The particle is thought to decay almost instantly into two photons, and the above graph shows a particle having a mass of about 126 GeV doing just that.

The Higgs predicted by the Standard Model of physics is a massive spin-zero particle, exactly what the data indicate, but there could be more than one variant of the Higgs showing up at the LHC. Resolution of this issue will have to wait until the collider (which has been shut down for a year of upgrades) is brought back on line with its full (14 TeV) energy capability.

Another outcome of the Higgs confirmation is the apparent nullification of the supersymmetry idea. This would be good news to me, because supersymmetry is the most convoluted, notation-drenched theory there is, and since I could never understand it, much less do the math, I'll be happy when the theory is abandoned for good.
Pi Day — Posted Thursday, March 14 2013
Today is Pi Day, so named because of the way Americans mark their calendars — March 14 is 3.14, which is a rough approximation of the transcendental number \(\pi\) (Europeans would write it as 14.3, but that's an even worse approximation). A better one is

but it's still an approximation.

The University of Oxford is conducting an interactive Pi Day event in which people around the world are invited to perform ancient experiments designed to approximate the number. For example, in the 1700s the Frenchman Georges-Louis Leclerc, the Comte de Buffon (sounds like a pretty fun guy) calculated that if one drops \(N\) needles, each having a length of \(L\) units, on a hardwood floor whose planks are \(T\) units apart, then \(\pi\) can be approximated using $$ \pi \approx \frac{2LN}{TH} $$ where \(H\) is the number of needles that cross or touch a join line between the planks (assuming \(L \lt T\)). About 30 years ago I actually programmed a PC to numerically simulate this experiment (I'm a fun guy, too, but for some reason girls always avoided me in school). Since the integer limit in BASIC in those day was about 32,000 my result wasn't very accurate. But the Oxford Pi Day handlers figure that if many millions of people participate in the event and share their data (Oxford's got a lot of nerdy optimists in its math department), they may be able to determine \(\pi\) to at least three or four decimal places. Inefficient, perhaps, but fun (if that's your idea of fun).

I still recall when I got my first non-programmable handheld calculator in January 1972 (it cost me about 100 dollars), I had to approximate \(\pi\) using the ratio 355/113 = 3.14159292 ..., a method that was known to the ancients. I'll bet you anything that Oxford doesn't even come close.

Update: With a total of less than 300 responses (as of March 18), the University of Oxford announced a new experimental value for \(\pi\). It's

Not great, but it's better than the Old Testament's estimate (1 Kings 7:23-26). Maybe Oxford will have a better turnout next year. (Hey, Oxford — what the hell is a mean average?)
The Theoretical Minimum — Posted Tuesday, March 12 2013
The great Russian physicist Lev Landau famously devised an entry exam to test incoming students. His "theoretical minimum" contained everything he considered elementary but mandatory for a young aspiring physicist. It was notoriously difficult and comprehensive, and few of his students passed it.

I'm a huge fan of Stanford physicist Leonard Susskind, but I debated buying his latest book The Theoretical Minimum: What You Need to Know to Start Doing Physics for fear that it would be too simplistic. But it was only about 15 bucks, so I picked it up, hoping it would be at least entertaining, if not informative.

Well, Prof. Landau would be disappointed. I was too, but not simply because Susskind's book, which deals only with classical physics, covers ground that any third-year undergraduate physics student should know intimately. I don't like the book because the equations have been typeset using what appears to be 1950s technology. In short, it's awful and it's distracting, and there's no excuse for it. Susskind himself could have typeset it using LaTeX in just a few hours.

That said, the book reads pretty much like one of Susskind's YouTube lectures, and that's good. Susskind explains in the book what motivates him at the age of 73 to teach the way he does, and that alone is worth the price of the book. But if you can follow his online lectures, you can safely skip this text.
I'm Getting the Band Back Together — Posted Tuesday, March 12 2013

I barely knew my father, but I remember on several occasions he talked about the bands and orchestras he played in as a young man in Quincy, Illinois. One story had him coming out to Long Beach, California in the summer of 1921 for an out-of-state play date. I never gave this story much credence, considering the fact that he was only 16 at the time and a mere sophomore in high school.

Many years later, not long before Dad died in 1981, I took him and Mom on a sentimental journey to Long Beach. They showed me the place (the Breakers Hotel) where they had stayed when they first came out to California in 1943 (my father and mother had secured lucrative book-keeping and riveting jobs, respectively, at Lockheed). Dad also showed me the place where he had stayed that summer in 1921, which looked to be by then a very rundown apartment building (probably torn down now). I recall wondering at the time how his brain, then severely impaired by years of heavy drinking and smoking, could have retained the memory of something that almost surely never happened, but I didn't say anything.

Yesterday I came across the above photo from the Illinois Digital Achives website. It was taken in 1921 in Quincy, and I'd never seen it before. Although Dad never mentioned the "Melody Boys Orchestra" to me, I remember him getting a letter one day in the late 1970s. When he'd read it, he simply said "Waldo died." I assumed Waldo was just another distant relative that I'd never heard about, and ignored it. So I suppose Dad's Long Beach story could be true after all, though I wouldn't know how to confirm it.

I've ordered a clean print of the photo from the Archives, along with another I discovered at the same source:

(Dad's third from the right; he and saxophonist Parker Gates are flanking the pianist, Mim Broderick). I remember this photo of the "Tivoli Orchestra" well from my earliest kid days in Duarte, California because my parents had kept the original in our old family scrapbook (which my sister has now). Dad had written the first names of the other orchestra members on the photo, but without the last names (or the orchestra's name) I was never able to track any of them down. But the Archives copy includes that information, so I can start looking again. One of the sax players — Parker S. Gates — is notable because he went on to found a highly successful radio and sound recording business that by the 1930s had achieved worldwide eminence. He took over the fledgling Quincy company from his father in 1926, so when the photo was taken (1928) he was already well on his way. Why Dad didn't follow him into the business is anyone's guess (my father never made more than 2.75 an hour).

I took Dad's trumpet (or maybe it's a cornet) out of the garage today to look at again. It's the same one in the Tivoli photo, and it has that same strange smell I remember as a kid.

Update — The Social Security Death Index tells me that trombonist Waldo Specht was born May 20, 1903 in Illinois and died August 1, 1977 in Barstow, California. (Of all the places I would choose not to die in, Barstow (or maybe Needles, CA) would be right at the top of the list.)
Ménage à Trois — Posted Monday, March 11 2013
Jackson Pollock? No, just three bodies locked in perpetual, repetitive orbit.

Everyone learns in school that the gravitational two-body problem in orbital mechanics can be solved exactly using the reduced mass for one of the bodies, which essentially fixes one body (usually the more massive one) at the origin. Similarly, the hydrogen atom (proton and electron) can be solved exactly by the same approach in the non-relativistic Schrödinger equation. But the three-body problem is a completely different animal. Newton tried it for three identical gravitating masses and failed, while the atomic equivalent (helium, nucleus plus two electrons) cannot be solved exactly, either. Instead, some kind of numerical, perturbative approach must be used in both cases. And the \(n\)-body problem is far and away more difficult.

Until recently, mathematicians could find only three stable, cyclical cases for three co-orbiting Newtonian masses. But amazingly a group of Serbian mathematicians has now found 13 new classes of repeating orbits for three bodies. Some of these orbits are fairly simple, while others resemble a complicated looping mesh of seemingly chaotic motion. But they're all stable nevertheless, and many mathematicians are dumbfounded by the discovery.

You can investigate these 13 orbital classes for yourself here.
Spooky Einstein — Posted Thursday, March 7 2013
In a famous March 1935 paper, Einstein, Podolsky and Rosen questioned the completeness and validity of the theory of quantum mechanics by posing what is now known as the EPR paradox. Basically a thought experiment, the scientists reasoned that quantum mechanics as it was then (and still is) understood violated the principle of locality, that is, the belief that no signal can travel faster than the speed of light. Einstein famously remarked that if locality can be violated then there must be a kind of spooky action at a distance (his actual words, albeit in German) involved in which the speed of light is exceeded.

You undoubtedly already know that experiments have irrefutably proven that quantum mechanics does indeed violate locality (and hence objective reality as well) through quantum entanglement. But this does not necessarily mean that locality is violated instantaneously. Experiments to date indicate that the effects of quantum entanglement probably do occur instantly, but setting precise lower bounds on the minimum "velocity" of spooky action at a distance has been elusive.

Now a group of Chinese physicists has done just that, showing that spooky action at a distance occurs at least 1.38\(\times\)10\(^4\) times the speed of light. (The linked article poses the possibility that faster-than-light quantum computing and other applications might someday be developed, but Einstein's special relativity still places a firm limit on how fast actual signals can be transmitted.) This is indeed a great velocity, and the results can be viewed as more evidence that Einstein's "spooky" explanation does not hold up, representing yet another victory for entanglement.

The scientists' experimental set-up and results can be read in the original paper, available here.
Black Beaut y — Posted Friday, March 1 2013
Singular surfaces surrounding a rotating black hole. Frame-dragging, non-inertiality, points of no return—and something else.

Today's Azimuth has a wonderful article about black holes and the Golden Ratio, which is defined as the value of \(x\) satisfying $$ x = -1 + \frac{1}{x} \rightarrow x = \frac{\sqrt{5} - 1}{2} $$ or \(x = \) 0.6180339... . A rectangle having sides with the ratio \(x\):1 is called golden due to its presumed beauty, and was known to (and used by) the ancients. It turns out that certain types of spinning black holes have the Golden Ratio built into their angular momenta, a surprising (if not astounding) finding that was apparently first deduced by noted New Zealand physicist Paul Davies. I have several books by Davies (who's also a Christian of some fame), but I was never aware of this little discovery.

Black holes are just remnants of exploded or dead stars whose gravitational fields are so strong that the stars have collapsed in on themselves, with whatever matter remaining in the star crushed down to infinite density and zero volume. The hole's event horizon marks the distance from the hole's center at which point nothing (not even light) can escape the inward gravitational pull. Though nothing that has fallen in can escape a black hole, in the early 1970s Stephen Hawking proved that black holes lose mass via particle/antiparticle creation/annihilation just outside the event horizon. Large, non-rotating black holes may take eons to "evaporate" in this manner, but as they get smaller they radiate much faster. As the Azimuth article explains, black holes get "hotter" as they shrink, contrary to common sense, and they consequently possess negative specific heat. But a rotating black hole is different. If it rotates sufficiently fast, it can undergo a kind of "phase transition" to positive specific heat. Davies has shown that this transition point occurs when the hole's angular momentum reaches a very specific figure.

Most if not all stars rotate to some extent, and their angular momentum is conserved after collapse; consequently, almost all black holes rotate, and the spin produces weird physical effects. Equatorially (\(\theta=\pi/2\)), a massive, spherically symmetric spinning object is described by the Kerr metric $$ ds^2 = \left(1 - \frac{2m}{r}\right) c^2dt^2 - \frac{4mac}{r}\, d\phi dt - \frac{dr^2}{1 - \frac{2m}{r} + \frac{a^2}{r^2}} - \left(1 +\frac{a^2}{r^2} + \frac{2ma^2}{r^3}\right) r^2 d\phi^2 $$ Here \(m = GM/c^2\) is the metricized mass of an object of mass \(M\) kg and \(a = Jc^2/GM\), where \(J\) is the angular momentum. Like the Schwarzschild metric (\(a=0\)), the Kerr metric exhibits problems when the denominator in the third term on the right goes to zero. This happens when \(r^2 -2mr +a^2 =0\), or $$ r = m + \sqrt{m^2 - a^2} $$ where we have taken the positive root. Obviously, the square root will exhibit its own problems when \( a \gt m\), and for this reason a spinning object with \( a = m\) is called an extremal Kerr black hole. Using the above definitions for \( m\) and \(a\), this corresponds to a black hole having an angular momentum of $$ J = \frac{G^2M^2}{c^4} $$ Davies showed that the transition point of a rotating black hole from negative to positive specific heat occurs at the point $$ J = \sqrt{\frac{\sqrt{5} -1}{2}} \frac{G^2M^2}{c^4} $$ which is about 78.6% of the hole's maximum allowable angular momentum.

What is so special about the golden term in the above equation? I suspect that it's just a coincidence, but it's also possible that black holes possess another kind of beauty we never expected.

Update — Maybe and maybe not. Azimuth now reports some corrections.
Not Forgotten — Posted Friday, March 1 2013
Having been blessed (or cursed) with a near-photographic long term memory (though I can't remember what I did last week), I recall in vivid detail my high school's announcement of the assassination of President Kennedy. It was my freshman year at Duarte High School, just around 11:30 am, Friday 22 November 1963, and I was on my way to the cafeteria when my friend Greg Schubert came up to me to say that the president had been shot in Texas. Arriving at the cafeteria, I noted several televisions were set up there, reporting the unfolding events in Dallas. Being a complete idiot then (though, as Stan Laurel might say, "I'm better now"), I got my usual carton of milk and 25-cent hamburger (which I can still taste) and ignored the whole thing. My first class after lunch was French I, taught by Mrs. Eleanor Farrell. She was tall, gray, imperially slim (as Edwin Arlington Robinson would put it) and she would eventually tolerate me for three more years of French instruction. (Mrs. Farrell did eventually teach me the language, although today I can only read it.)

Shortly after the class started, a messenger (girl student) entered the class and handed Mrs. Farrell a note. She had been writing something on the blackboard, but she stopped to read the note and then intoned, without any emotion whatsoever, "The President is dead. He died at 1:00 pm, Dallas time." She then went back to the blackboard and continued her lesson without saying another word about it.

Anyway, I was reading today that our newly appointed Secretary of State John Kerry was in France, meeting French president François Hollande to discuss Syria, Mali and some transatlantic business matters. Kerry later addressed the press corps there, speaking in their language, which he is quite fluent in. I was then disturbed to learn that my country's conservative morons were upset to discover that Kerry chose to speak entirely in French, rather than use gutteral English to denounce France for having not joined us in our glorious 2003 Iraq War. Planning to post something snarky about this (jeez, Mitt Romney also speaks French, having gone to that country to escape the draft), I suddenly remembered Mrs. Farrell, and wondered what had become of her.

I'd have sworn that if still alive she'd be in her 100s today, but a Google obituary search informed me that she had died in December 2010 at the age of only 89. I also learned that she was a high school valedictorian, a graduate of the Claremont college system here in Southern California (very elite, expensive and tough schools to get into today) with B.A. and M.A. degrees in language and music performance, a WAVES ensign in the US Navy in World War II, and a 40-year organist and choraler with her church. Needless to say, I was quite impressed, but I wish I had known what an accomplished teacher she was back in 1963-67.

Today, Mrs. Farrell's remains lie in Live Oak Memorial Park in Monrovia, California, not too far from my parents' graves. Next time I visit the place, I'll say hello and a very belated Merci!
Not the First — Posted Tuesday, February 26 2013
German physicist Gustav Mie (1868-1957), frien d and colleague of Einstein and Hermann Weyl. Mie's theories are now largely forgotten.

The history of rational efforts to discover a unified basis of Nature's interactions has a long history. Hermann Weyl, who arguably understood general relativity as well as Einstein, may have been the first scientist to apply Einstein's 1915 discovery to the problem of unification (which at the time involved electromagnetism and gravitation, the only forces known), but he was not the first to attempt unification by other means. In 1864, near the end of his groundbreaking A Dynamical Theory of the Electromagnetic Field, the great Scottish physicist James Clerk Maxwell turned his attention to gravity (Pages 35-36), writing
"After tracing to the action of the surrounding medium both the magnetic and the electric attractions and repulsions, and f inding them to depend on the inverse square of the distance, we are naturally led to inquire whether the attraction of gravitation, which follows the same law of the distance, is not also traceable to the action of a surrounding medium ... [but] as I am unable to understand in what way a medium can possess such properties [of both attraction and repulsion], I cannot go any further in this direction in searching for the cause of gravitation."
That is, since Maxwell could not explain why gravitation and electromagnetism could be so similar and yet so different, he simply gave up.

I believe that the great German mathematician Bernard Riemann, having access to the then-emerging tensor formalism of fellow German mathematician Elwin Christoffel, probably considered gravitation as a field expressible in tensor language, but his failure to consider time as a necessary fourth component to the formalism prevented gravitation's full exposition until Einstein came along 50 years later. (Riemann, probably the greatest mathematician who ever lived, might have eventually discovered general relativity had he not died of tuberculosis in 1866 at the age of 39). Riemann was also well aware of Maxwell's work, and I'm tempted to think that he must have seriously considered how the forces of electromagnetism and gravitation might be connected.

Weyl too was well aware of the work of all these earlier researchers, but he was also influenced by another German physicist by the impossible-to-remember name of Gustav Adolf Feodor Wilhelm Ludwig Mie (1868-1957). In 1912 Mie developed a variant of Maxwell's electrodynamics in which he tried to explain the origin and structure of electrically charged particles. While he did not specifically address gravitation, Mie's theory touched on the nature of matter, a subject that much interested Weyl at the time. Weyl believed that Mie's theory itself might be used to explain the asymmetry of electrical charge, and he proposed an action lagrangian designed to do just that. But none of this went anywhere (Mie's theory was not gauge invariant, and did not agree with observation), and despite some excellent research that survives to this day (the chromatics of gold colloids), Mie's work is little more than a footnote in the archaeological dustbin of physics.

Still, Mie had a long and interesting life. Pedro Lilienfeld has written a brief biography of this now-obscure physicist that can be downloaded here.
Time to Worry — Posted Thursday, February 21 2013
It seems distinctly possible in simple unified models, that our present vacuum is only metastable, and that nevertheless, the Universe would have chosen to get 'hung up' in it. If this is the case, then without warning, a bubble of true vacuum could nucleate somewhere in the Universe and move outwards at the speed of light, and before we realized what swept by us our protons would decay away.

— Michael S. Turner (Univ. Chicago) and Frank Wilczek (UC Santa Barbara) in
      Is Our Vacuum Metastable?, Letters to Nature, Vol. 298, 12 August 1982
Thirty years ago physicists like Wilczek (Nobel prize, 2004) and Turner pondered the possibility that our universe might have been caught in a quantum vacuum state that was only a local minimum, that is, that the true (lowest) vacuum state lay somewhere below it. Since Nature always seeks a state of minimum energy, these scientists believed that at some point in the future the universe would undergo a spontaneous collapse to the true minimum, the result being the destruction of everything in the universe we have come to know and love, including ourselves. However, the so-called "metastable vacuum" that Wilczek and Turner fretted about would only be a problem if the mass of the Higgs boson (which was completely unknown in 1982) were smaller than about 130 giga electron-volts.

When a particle believed to be the Higgs was discovered at the Large Hadron Collider last July, the data indicated a mass of 125 to 126 GeV. In view of this, Joseph Lykken, a theoretical physicist with the Fermi National Accelerator Laboratory in Batavia, Illinois, raised the issue of cosmological self-destruction again. He noted that if the true mass of the Higgs is within one percent of that found by the LHC, then we could be in for big trouble, or "fireballs of doom," according to Lykken.

However, Wilczek and others have calculated the decay rate of the metastable vacuum, giving a lifetime on the order of 10\(^{10}\) to 10\(^{30}\) years. Long before either of these time periods elapse, our Sun will become a red giant engulfing the inner planets, including Earth. Roughly three billion years from now, in the words of the late Carl Sagan, the Earth will experience "one last perfect day." One or two billion years later, the Earth will either be vaporized or reduced to a molten cinder within the outer envelop of the Sun. Bottom line: "Don't worry about the metastable vacuum," says Lykken.

Yeah, that's what he says. Like the smart folks in Doomsday Preppers, I'm gonna be ready for it.
The Scholls — Posted Thursday, February 21 2013
Friday 22 February marks the 70th anniversary of the executions of Hans Scholl, his sister Sophie and fellow student Christoph Probst by the Nazis. Their crime was to print and disseminate anti-Nazi leaflets and posters in their hometown of Munich and neighboring cities in southeastern Germany. Their tragic but ultimately uplifting story can be found elsewhere on this website.

My son and I visited Ludwig Maximilian University in Munich several years ago, and we stood at the spot where students Hans and Sophie were arrested by an overly-zealous school janitor, who had spotted Sophie tossing the last of her load of leaflets into the university's atrium. Briefly imprisoned at Munich's Stadelheim Prison, they and fellow doomed student Probst were tried by Roland Freisler, the notorious Nazi judge who sentenced the majority of his victims to death by hanging or guillotine. The Scholls and Probst got the latter, on the same day of their trial.

I wonder how many Americans today would risk their lives protesting what they knew to be a great evil. And I wonder—could I?

Long live the White Rose.
How Old — Posted Wednesday, February 20 2013
This book was recommended to me by an old student of mine (no—she's still young, I'm the one who's old). Though I lost interest in astronomy years ago I still find some aspects that are fascinating, such as the evolution and fate of the universe. The book, by Vanderbilt University astronomer David A. Weintraub (author of Is Pluto a Planet?), focuses mainly on the various methods that astrophysicists and others have used to determine the age of the universe, but its evolution and fate are inextricably tied to the findings of these scientists.

Of particulat interest is the author's description of how the observed expansion of the cosmos can be explained. One theory posits that matter created by the Big Bang is simply expanding into an open and available empty space that has always existed. This explains the observed red shift of light coming to us from extragalactic stars and is in agreement with Einstein's theory of general relativity. But there is another explanation (also in agreement with red shift data and Einstein) in which matter created by the Big Bang is essentially fixed in position, with the space between galaxies being stretched or created as the universe evolves. Weintraub notes that the latter explanation is the more plausible one in view of the cosmological principle, which states that the universe should look essentially the same in all directions to all observers, regardless of where they are and where they may happen to be looking.

The concept of stretched or created space makes even more sense in view of the discovery made by UC Berkeley physicist Saul Perlmutter (and concurrently by Adam Wiess and Brian Schmidt), who determined that light from distant supernovae is fainter than it should be. Their observations told them that these exploding stars are themselves farther away than they should be, implying that the expansion of the universe is actually accelerating. This discovery, which earned the three scientists the 2011 Nobel Prize in Physics, only makes sense if space is actually being created between galaxies as the cosmos expands. In his book Weintraub suggests that the additional space provides more room for dark energy, a hypothetical force that serves to push galaxies apart (kind of like gravity's evil twin). Gravity would have been dominant in a younger universe, when the volume of space was relatively small and the effect of dark energy would have been comparatively weaker. Now the roles have been apparently reversed, and gravity's the weaker of the two.

Taken from Perlmutter's Nobel lecture, this historic graph conclusively shows the deviation of observed universal expansion from the expected decelerating or coasting expansion rate (bottom line, as extrapolated from smaller red shifts). The yellow data points from previous (1996) studies were limited to relatively small stellar red shifts and no deviation could be reliably discerned. The more recent red data points from Perlmutter, Wiess and Schmidt unambigously demonstrate that for large red shifts (more distant stars) the stars are farther away than expected (that is, their effective stellar magnitudes \(m_B\) are larger than normal). The universe is therefore expanding at an accelerated rate.

To me, the creation or stretching of space says that things like length, area and volume are really of little importance in the universe; if this is the case, then scale invariance should definitely be incorporated into our physical theories. As successful as it has been, Einstein's 1915 gravity theory does not include this type of symmetry, though it can accommodate universal acceleration via his cosmological constant. But the cosmological constant exists merely as a consequence of the fact that the metric tensor \(g^{\mu\nu}\) has zero divergence (in fact, it acts like a constant under covariant differentiation), so it's hardly a dynamical quantity.

(The only plausible quantity that can be used to build a scale invariant theory is Hermann Weyl's conformal tensor \(C_{\mu\nu\alpha\beta}\), which I've discussed before. Numerous theories based on this tensor have been developed in recent years, and they do indeed provide some indication that the dark energy/dark matter problem might yield to a fully conformal approach.)

Weintraub's book summarizes how recent and/or refined observations of supernovae, the cosmic microwave background and radiometric dating have come together to produce a precise and consistent measure of the age of the universe, which is 13.75\(\pm\)0.11 billion years. The combined scientific efforts that led to this result can easily be considered one of the most profound achievements of the human mind.

But no discussion of the universe's age would be complete without some mention of the famous (or infamous) estimate made by James Ussher (1581-1656), an archbishop with the Church of Ireland. Weintraub relates in loving detail how earlier estimates of the age of the universe culminated in Ussher's stunning 1654 announcement that, based on a careful consideration of Old Testament genealogies, God created the universe on Saturday, October 22, 4004 BC. This estimate, which stood as gospel until the early 20th century, was further refined by Cambridge academic John Lightfoot, who proclaimed that Creation took place at 9:00 am (undoubtedly Universal Time, UT). It is interesting to note that in 2011, a Gallup poll showed that 30% of Americans believe in a literal interpretation of the Bible, consistent with the poll's finding that 46% of Americans believe God created man in his present form within a time frame comparable to the Ussher estimate.

Idiot nation, indeed.
This Huge Shed — Posted Sunday, February 17 2013
I recently visited St. Peter's Basilica in Rome for the first time and, like most people, was enormously impressed. Upon entering the cathedral you see the large marker on the floor commemorating the spot where Charlemagne was supposedly coronated in 800 AD, while over to the far right is Michelangelo's incomparable Pietà, sculpted by the artist in 1498-99. Straight ahead lies the interior of the huge central dome, soaring more than 400 feet above, with breathtaking views of carved marble statues and majestic columns, stained glass, paintings, tapestries, and inlaid tiled marble floors. It's literally almost more than a person can take in at first sight. While not considered the official church of the Roman Catholic faith, St. Peter's is widely held to be a holy place. Indeed, the apostle Peter, the Rock of Christ's church on earth, was crucified just outside the Basilica's doors, in what is today the sprawling expanse of St. Peter's Square.

Groundbreaking on the Basilica took place in 1506, but construction was not completed until 1626. The enormity of the undertaking overwhelmed the church's ability to pay for the ongoing construction, and in 1517 it initiated the selling of indulgences, whereby the faithful could pay the church to have their deceased relatives (and later themselves) released from Purgatory, giving rise to the jingle

When a coin in the coffer rings, a craven soul from Purgatory springs.

This and other such practices resulted in Martin Luther's revolt and subsequent establishment of Protestantism after his excommunication from the church in 1520.

While it is difficult to chronicle the 600-year record of Roman Catholic offenses against humanity and the God the church professes to serve, the more recent widespread (not to mention horrendous) sexual abuse of young boys by priests is truly hard to comprehend. I suspect that this ongoing scandal is largely behind the almost unprecedented resignation of Pope Benedict XVI (the last papal resignation took place in 1415), though the official reason is the Pope's declining health (he could hardly claim that he wants to spend more time with his family). And I also suspect that after he has left, the true extent of his involvement in covering up the abuse and protecting his wayward priests will go public. No wonder the Pope wants to "disappear from the world" after he's left. (Me, I'd go scrag myself, but then I'm not the Pope.)

Today, the Washington Post has a lengthy article on the Pope's resignation and its tie-in with last year's so-called "Vati-leaks" scandal, in which leaked documents revealed the vast extent of corruption in the Vatican. At least sometimes the truth wins.

But here's the kicker. In the world today there are more than 1.2 billion Roman Catholics, considerably more than the estimated 900 million Protestants. As the largest Christian church on earth today, I have to ask myself how so many people could be so terribly misled if God is truly guiding their hearts and minds. St. Peter's Basilica may be lavish and beautiful, but in reality it's little more than a monument to fear and ignorance, just as the nearby Colosseum is little more than a defunct, open-air torture chamber.

But the Pope will leave, Easter will come, the sheep will quickly forget, and the shearing will go on as if nothing has happened.
I, too, misjudge the real purpose of this huge shed I'm herded in: not for my love or
lovely wool am I here, but to make some world a meal.
See, how on the unsubstantial air I kick, bleating my private woe, as upside down
my rolling sight somersaults, and frantically I try to set my world upright; too late
learning why I'm hung here, whose nostrils bleed, whose life runs out from eye and ear.

— Alfred Hayes
Update, 23-02-2013 Could the erupting scandal about gay Vatican priests paying male prostitutes to keep their mouths shut (no pun intended) have anything to do with Benedict XVI's abdication? Oh, Heavenly shades of Pope Innocent VIII!
Throwing in the Towel — Posted Saturday, February 16 2013
In the summer of 1990 I attended a graduate class on quantum physics given by Fernando Morinigo, now retired (in fact it was the last class he taught before he retired). What I remember most was his occasional digressions about physicist Richard Feynman, who Morinigo worked with as a post doc at Caltech in 1962-63. Years later this led to a book that Morinigo and fellow post doc William Wagner wrote based on Feynman's lec tures on general relativity (I pulled the book down from my shelf to look at again today, and even though the book came out in 1995 the material itself is quite dated).

One interesting anecdote in the book is given by Caltech's Kip Thorne, who co-wrote the book's 33-page (!) foreward, where he writes that in the 1950s Feynman turned his attention to the problem of quantum gravity, which he believed at the time would be a "piece of cake" to solve! This led to a brief romance between Feynman and the theory of general relativity (in those days called geometrodynamics), culminating in the early 60s with a series of lectures Feynman gave at the school on the subject (which in turn became the basis of Morinigo's book). According to the foreward, Feynman abruptly called off the relationship when quantum gravity proved too elusive, even for the great Feynman.

I have a much shorter book on general relativity by Paul Dirac, in my humble opinion the greatest physicist who ever lived. He too had a run with the quantum gravity problem, and likewise got nowhere. Many others, notably Einstein, Weyl, Eddington, Schrödinger and Pauli, also had their turn at quantum gravity (perhaps more commonly known as unified field theory), all to no avail. The most recent example to my knowledge is UC Riverside's John Baez, an expert in loop quantum gravity (and just about everything else), who threw in the towel a few years ago to focus on environmental problems (and God bless him for that).

Readers of this website will know that my interest in Hermann Weyl was based primarily on his gauge invariance idea, its early application to his failed 1918 gravity theory and its resounding success when Weyl applied it to quantum physics. But his gravity theory is still an active subject of current research, though today it appears tied to a bewildering array of theories involving various scalar, spinor and tensor fields.

While this ongoing quest continues to fascinate me, I am also frustrated by the fact that Einstein's original theory, which by comparison seems so simple and straightforward, refuses to be successfully generalized to provide a description of anything other than basic gravity. This in effect is a restatement of the fact that while gravity was the first force known to man, we haven't really been able to get very far with it in comparison with the other forces of Nature (Newton's constant \(G\) is known accurately only to four decimal places, whereas the fine structure constant is known to 10 decimals and the electron gyromagnetic ratio is known to an astounding 12 decimals).

I suppose what I am saying here is that Nature seems to be conspiring against us, having frustrated the efforts of many brilliant scientists for nearly 100 years now, while at the same time motivating them to keep at it. Just when will the breakthrough arrive?
Largest Prime Number to Date — Posted Thursday, February 7 2013
In the following I've tried to adapt an old high school lesson into something more contempor ary.

Mathematician Curtis Cooper of the University of Central Missouri has found the largest prime number to date. A Mersenne prime, it's 2\(^{57885161}\) - 1, a number that is 17,425,170 digits in length, much longer than the previous record holder from just a few years ago (about 13 million digits).

The French monk Marin Mersenne discovered a method for predicting prime numbers over 350 years ago. The method simply takes 2 to some positive integer power \(n\) and subtracts 1 from the result. It works better for large exponents (for example, 2\(^4\) - 1 = 15 is not prime, nor is 2\(^6\) - 1 = 63), and it was not until the advent of powerful computers that it became practical to use the method for large \(n\). Today, the Great Internet Mersenne Prime Search uses some 360,000 computers harnessed together to produce a computational capability of about 150 trillion calculations per second, and it's still a struggle.

I may be missing something, but as the Mersenne method has produced a total of only 48 prime numbers to date, I would hardly call it a method. But as test numbers get truly huge, the density of primes thins out enormously, making them progressively harder to find.

I think I am safe in saying that most physicists couldn't care less about prime numbers, as they're mostly of interest to pure mathematicians. Still, most people are aware that large prime numbers are used as the basis for credit card encryption, of which the RSA method is a prime example (no pun intended). It relies on the fact that while it is easy to multiply two prime numbers together, factoring a large product of primes is difficult. For a product that is thousands of digits long, the factoring process is essentially impos sible, and that explains why credit card numbers are stolen, not hacked.

To get some appreciation for this, let's try a simple example of what's involved. Let's say I want encode my password for some important application using prime numbers. Since I'm a very unimaginative person, my password is "HI" which, using my equally unimaginative alphabetical code H = 8 and I = 9, makes HI = 89. This is the password I want to use.

First, I pick two arbitrary prime numbers P and Q. Keeping things small, I set P = 7 and Q = 13. The product of these numbers is N = P*Q, or N = 91. The reduced product M is next calculated via M = (P-1)*(Q-1), or M = 72. Now I want to pick another prime number E that's less than M but not a factor of M. For that reason, I can't choose E = 3, because that's a factor of M = 72. So for the heck of it I'll pick E = 23, which works just fine. The set of numbers {N,E} = {91,23} defines what is known as the public key set. That is, you can use them, your friends can use them, and the world at large can know what they are. However, the set {P,Q} should be discarded without anyone else knowing what they are, though the numbers are not considered particularly secret.

Here's the only hard part. I now want to find a number D that will serve as my secret key. For the RSA method, it must satisfy the seemingly arbitrary condition

(E*D - 1)/M = Z

where Z is any integer (think about that for a second!) There are fancy numerical methods for finding D when E and M are truly enormous numbers (the extended Euclidean algorithm is one example), but for small (very small!) problems you can use a simple BASIC program such as

FOR D = 2 TO M

For the variables given above, I get D = 47. I store this number in my head, or equally inaccessible space. I also give this secret number to my bank, which currently holds the entire 14.88 in my account. They put it in their computer so it will know who I am and grant me the authorization I need to plunder my money at will.

I now want to access my account, so I have to send the bank's computer a code number that will get me in. This means I have to encrypt my "89" password into something that the bank can then decrypt. Setting this code to K = 89, I calculate an encrypted code S using

S = K^E - N*INT((K^E)/N)

Plugging in the numbers given previously, I get S = 45. Note that this calculation does not involve the secret key D.

When the bank gets my withdrawal request, it verifies the demand on my account using essentially a reversed version of the above calculation. The bank's computer calculates the quantity

W = S^D - N*INT((S^D)/N)

which, again plugging in the variables, gives W = 89, which is my secret code (even the bank security guard can convert this to "HI"). Satisfied that I am who I say I am, the bank then sends along the message Sorry, insufficient funds. Damn—overdrawn again!

Sadly, even for this simple example your home computer will probably not be able to handle the huge numbers involved in the calculations, so you'll have to resort to something more powerful. (But not to worry, your computer's browser does it all automatically.) I recommend the free online program Wolfram Alpha, which is kind of like having a free version of Mathematica on your computer. (In that case, you'll have to replace the BASIC integer operator INT with the operator FLOOR, which works the same way.) The following graphic shows the calculation of S:

In practice, the above variables are all huge numbers, and the messages to be encoded are also enormous. You can see from my little Kindergarten example that very powerful computational techniques are required to handle practical encryption/decryption problems, especially considering the exponentiation calculations involved (89\(^{23}\) is a truly huge number).

In real life, you would have access to the public keys of just about any encryption problem. Some day, you or some other genius may discover a way to factor large numbers into their constituent prime numbers, as this is the primary roadblock to figuring out secret keys. If you do, you should realize the great potential danger of this knowledge, given the fact that the world's financial centers could be destroyed overnight by it (not to mention the CIA and NSA—a great pity). It would be far better not to publish your discovery, but to take it to the appropriate authorities, who would make you either very dead or rich beyond your wildest dreams. You make the call.
Enemies, Again — Posted Wednesday, February 6 2013
Not enough guns.

I posted an exposé on the NRA's enemies list last week, and only now is the story going viral. Looks like my puerile mixture of science, math, politics and religion doesn't resonate with very many people.

But I'm mentioning the enemies list again mainly because of what I'm currently reading (or re-reading, in the case of Chuck Thompson's wickedly funny Better Off Without 'Em: A Northern Manifesto for Southern Secession).

I recently watched author and civil rights activist Randall Robinson on C-SPAN talking in depth about his 2004 book Quitting America: The Departure of a Black Man from his Native Land. His observations on the state of race relations in America were deeply moving, so I got the book to read for myself. In it he writes
Perhaps Americans, or more specifically white Americans, have behaved such because they only really value or respect what they either crave or fear: money or might, all else to be belittled, distained, dismissed, even desecrated. Could it be that in America the unexcelled bigness of all things material has resulted in the concomitant relative smallness of all values nonmaterial? Moribund ethics. The death of the spirit. An unexamined and withering national soul. The commercialization of everything from school to pew.
One of the issues Robinson tries to answer is white America's unceasing demand that blacks, having been officially unshackled in 1865 and given civil rights in 1964, get on with their lives and put the past behind them. His response is something along the lines that four hundred years of slavery and stolen labor; broken lives, families, traditions and culture; and Jim Crow cannot be legislated away. Indeed, Robinson compares America's nuclear annihilation of Hiroshima and Nagasaki with its treatment of black people— as disastrous as the bombings were, they did not destroy Japan's culture or traditions, while a thousand of years of black heritage were systematically wiped out by slavery and mistreatment. Robinson reminds us of the old African proverb

The axe forgets, but the tree remembers

For me, all this brought back memories of Thompson's more recent (2012) book (which has almost replaced Charles Pierce's Idiot America: How Stupidity Became a Virtue in the Land of the Free as my favorite text on what's wrong with the country). Reading both books together, and considering the insanity that the National Rifle As sociation is perpetrating on us once again, makes it quite clear that the problem is actually rooted in Christian fundamentalism (or, as Thompson calls it, KKKristianity), specifically that brand of unthinking religious belief that is practiced primarily in the American South.

Do not be misled by the apparently tongue-in-cheek title of Thompson's book. While his recommendation that we do indeed allow the eleven American states he identifies as southern (I would add Missouri and Oklahoma to the list) to leave the union, his analysis of what's wrong with those miscreants and how the rest of the country allows the tail of southern stupidity to wag the national dog is penetrating and unassailable.

Where these three books really come together is what I see as the inability of the southern mentality to truly empathize with others they see as different. In spite of the "Love others as I have loved you" mandate of their favorite religious icon, the South's inability to feel for others translates inevitably into the triad of fear, hatred and authoritarianism, traits that so characterize the South. And like Thompson, I don't believe that this mindset can be changed—it's essentially a genetic defect that was probably handed down from the English (you know, the guys that came here to escape religious persecution from the Anglicans, only to impose their own Puritan brand of persecution on the Indians and everybody else).

Lastly (and getting back to the issue of empathy), have a look at this video from RSA Animate, which explains it much better than I can:

Christ taught us to understand, empathize with and have compassion for our enemies. What does the NRA teach?
Rudely Stamp'd — Posted Monday, February 4 2013
I don't know the reason, but Abraham Lincoln was so impressed with Shakespeare's tragedy Richard III that he committed the play's opening soliloquy to heart.

The play's subject has become a mixture of fact and fiction. He definitely suffered from scoliosis (curvature of the spine), was of slight build, and died in battle in 1485 from a pole ax blow to the back of the head (ouch), but Shakespeare's less-than-flattering depiction of King Richard as the murderer of his little nephews and other dastardly acts was probably just theatrical surmise.

Much of all this will undoubtedly change now that Richard's body, believed found in an unmarked grave in Leicester five months ago, has been positively identified through DNA and osteoarchaeological analysis. In addition, radiocarbon dating of two ribs bones found in the grave indicate a date between 1455 and 1540, consistent with the known August 1485 death of the king. Confirmation was made when maternal DNA matched that of a 55-year-old, 17th-generation nephew living in Canada. I think Shakespeare would have found it fitting that Richard's remains were found unceremoniously lodged beneath an automobile parking lot.

Much more to come, I'm sure of it. National Geographic and the Science Channel are already at work on documentaries of the discovery.

Last year I communicated with John G. Sotos, MD, whose intriguing book T he Physical Lincoln (besides being an exhaustively complete overview of everything we know about Abraham Lincoln's physical self) posits the very real possibility that our 16th president was suffering from a rare, Marfan-like condition called MEN-2. Sotos (who's also a fan of Hermann Weyl!) was also the principal investigator on the National Geographic's excellent documentary Lincoln's Secret Killer. I told Sotos that I found it ironic that while we can analyze the genetic material of ancient pharaohs to establish their lineage (performed on King Tutanhkamun, who lived 3,300 years ago), we cannot similarly test Lincoln's DNA because surviving blood and tissue samples have become too contaminated for analysis. We can now add another king, Richard, to that list of ancient rulers whose DNA can tell us so much more than that of the much more recent Lincoln, arguably our greatest president.
Gödel's Universe — Posted Wednesday, January 30 2013
On the occasion of Einstein's 70th birthday in 1949, the Austrian mathematical physicist Kurt Gödel (pronounced like girdle) presented the famed aging scientist with an unusual birthday gift—a novel solution to the Einstein field equations of gravitation that allowed for time travel to the past. To derive the result, Gödel abandoned the idea of a universal time coordinate that was seen by many as a defect in the Friedmann?Lemaître?Robertson?Walker cosmological model (which even today represents the standard model for cosmological behavior and evolution). However, Gödel had to assume that the entire universe was rotating. You might ask, as I have, "Rotating with respect to what?" Indeed, there's the rub, and so Gödel's universe is not physically realizable. But it did give rise to the concept known as the closed timelike curve of science fiction lore, and many stories have been written involving CTCs as a time-travel plot device. (And it's possible that CTCs actually exist.)

The latest online edition of New Scientist includes a video showing what time travel along a Gödel CTC would look like, based on the recent work of University of Ulm physicist Wolfgang Schleich and his colleagues. They have developed optical ray-tracing techniques based on Gödel's gravitational metric, and the results are weird, to day the least. Their paper "Visualization of the Gödel Universe" can be downloaded here. It's long (32 pages), but suitable for anyone with some familiarity with Einstein's field equations. It's best to wade through the paper first to get some feel for what's going on, but if you want you can watch the video right now:

Gödel was a close colleague of both Einstein and Hermann Weyl at the Institute for Advanced Study in Princeton, and he often walked to and from his office with Einstein. Gödel is most famous for his incompleteness theorems, with which he proved that no axiomatically simple but consistent mathematical system is complete. Basically, Gödel proved that not every mathematical problem can be solved, even for apparently simple and logical problems. The famous "THIS STATEMENT IS FALSE" problem is a somewhat oblique example, but it has mathematical underpinnings that cannot be resolved. Gödel was brilliant, but like many brilliant people he exhibited mental instability, especially in later life. Convinced people were trying to poison him, he would only eat food prepared by his wife. When she was hospitalized in 1978 and unable to cook, Gödel simply stopped eating, dying that same year of self-starvation.
Ticking Mass — Posted Monday, January 21 2013
Two items, somewhat related.

The January 19 issue of New Scientist has a nice article about the equivalence principle that is worth reading. Galileo is generally credited for demonstrating the principle, which just says that the inertial mass of \(F = ma\) and the gravitational mass of \(F = G m M/r^2\) are the same. It is said that he dropped two balls of different weight off the Leaning Tower of Pisa (a myth, he actually used inclined planes in his workshop) to show that they would fall at the same rate. Apparently no one had ever tried doing this, as most people at the time would have said that it goes without saying that the heavier weight would fall faster (thus proving that Republicans existed even in Galileo's day). But the weights landed at the same time, and a bit later Newton proved why. It is now a standard lecture in high school physics that, if inertial and gravitational mass is the same, we must have \(ma = - mg\). Dividing out the \(m\), we see that the acceleration of falling bodies in a gravitational field is independent of mass, as Galileo showed. Newton also believed that inertial mass and gravitational mass are one and the same, but it is by no means obvious why this should be the case. Since Newton's time, the question of mass equivalence has been addressed again and again by physicists great and small and, having gotten nowhere, the issue has simply been conveniently ignored. And no wonder—the most elaborate experiments to date have shown the equivalence of inertial and gravitational mass to 13 decimal places. So maybe they are the same after all.

Meanwhile, the January 10 issue of Science has an article on the possibility of using particle mass to measure time. Using the basic energy-frequency relation \(E = mc^2 = h\nu\), Holger Müller and his colleagues at UC Berkeley noted that a massive particle, having a de Broglie-Compton int rinsic frequency equivalent to \(\nu = mc^2/h\), might be used to measure time. But since even an electron has an enormous intrinsic frequency (with a corresponding period of roughly 10\(^{-19}\) sec), measuring such a frequency would not be easy. You can read Müller's approach for yourself (it involves cesium atoms and an appeal to special relativity), but in principle it offers the possibility that measuring mass would be tantamount to measuring time (and vice versa).

While 10\(^{-19}\) sec is really not as small as it looks (many particle decay processes are even smaller), it is infinitesimal when compared to human concepts of time.

Although particle masses might indeed be used as clocks someday, we positively must hold to the concept of invariance of particle mass with time. But like the possible difference between inertial and gravitational mass, this may be difficult to prove when dealing with differences involving many decimal places. The concept of mass-energy conservation was almost overturned in 1930, when particle decay processes seemed to indicate it was being violated, but Pauli bailed out the conservation principle when he postulated that unseen neutrinos were carrying away some of the energy. But mass-energy conservation is in fact routinely violated in fundamental particle interactions due to internal, off-mass shell quantum processes and the Heisenberg uncertainty principle (though the energy deficits have to be "paid back" in the end).

Hermann Weyl's 1918 theory of the combined gravitational-electromagnetic field was in fact doomed by the fact that particle mass appears to be rigidly conserved. One unintended consequence of the theory was that the magnitudes of vectors were required to change, even though there are vectors associated with particle mass that must be invariant. One is the momentum four-vector, whose relativistic definition is given by \(p^\mu = mc \frac{dx^\mu}{ds}\). Its magnitude is therefore \(\eta_{\mu\nu} p^\mu p^\nu = m^2c^2\), and this quantity must absolutely not change with time or location (to paraphrase David Griffiths, "They don't make 'em any more invariant than that!"). This argument against Weyl's theory was originally given by Einstein in a different context, and though Weyl tried desperately to save the theory he couldn't overcome the fact that quantities such as the spacing of atomic spectral lines (which can be associated with vector quantities) are invariant.

But what if the spacings are not truly fixed, but vary infinitesimally over spans of time that humans have yet to experience? After all, Dirac himself believed that the gravitational constant \(G\) might be getting smaller with time but unnoticeable over periods of many millions of years. It's like the old story of the hypothetical sentient ephemeral fly, whose lifespan of several hours would force it to assume that the trees, grass and flowers it witnessed as it lived out its life were eternal objects.

Perhaps experiments like that being undertaken by Müller will shed some light on these questions. Perhaps inertial and gravitational masses are not the same, and maybe nothing we observe is truly eternal. We'll just have to keep measuring things to see how invariant some of our concepts really are.

The Very End — Posted Sunday, January 6 2013
When informed by doctors that he had inoperable pancreatic cancer, the brilliant Hungarian-American mathematical physicist John von Neumann (1903-1957) was terrified. Only 51 years of age, he had already achieved many lifetimes' worth of accomplishment in mathematics, physics, computer logic, linguistics and quantum mechanics, and was undoubtedly hoping to achieve much more. Reputed to be a secular Catholic, most of his close associates knew him to be a confirmed atheist, and so were rather disappointed when the mortally-ill von Neumann embraced rigid Catholicism in his last months on Earth. I forget who said it first, but a life-threatening illness or accident "has a way of focusing one's mind on the eternal." Von Neumann's reconfirmation of the Catholic faith was apparently sincere but it did not give him the solace he sought, and he remained terrified of death to the very end. (Conversely, the great Caltech physicist Richard Feynman, who died of stomach cancer in 1988, found his own very last moments "boring.")

Today's news of a close family member's heart attack came while I was reading writer Susan Jacoby's latest contribution to the New York Times on the "blessings of atheism." I've read several of her books, notably the one about the age of American unreason (I've forgotten the exact title), and I find myself generally in agreement with what the noted atheist has to say about American culture. But in her article she asserts that an atheist's disbelief in God and the hereafter is also able to convey a sincere sense of condolence to those who are suffering (she mentions the assault-rifle tragedy in Newtown, Connecticut as an example) as well as a sense of comfort to the suffering atheist himself. But here I have to disagree, though not for the standard reasons.

Although I try to live in accordance with the teachings and philosophy of Jesus Christ, I long ago gave up believing the metaphysical nonsense that makes up the vast bulk of the Old and New Testaments. Having never witnessed a miracle myself (I did see a UFO once, though it was probably an atmospheric disturbance), my rational mind simply cannot accept what I read there, nor can it untangle the uncountable inconsistencies, contradictions and errors I find in the Bible's pages (Mark Twain once claimed that it contained "a thousand lies"). My faith instead rests on an acute awareness of a Great Intelligence that sits behind the workings of an ordered universe whose physical laws could not have simply come into being without purpose or reason (in Melville's Moby-Dick, Ahab ascribes it somewhat less nicely as the "malevolent and inscrutable thing" that hides behind the pasteboard mask of life). I call that Great Intelligence God, and while it is possible that He/She/It may have subtly whispered into the minds of certain historical people, God appears to remain generally aloof from human affairs. As for the reason for God's allowance of suffering (and in many cases horrible, abject, unbearable, inhuman suffering), I have no answer for it, but then neither do all the theologians, priests, pastors and ministers who nobly feign to possess an explanation. (My car's license plate holder proclaims that "God is love," but if He is then He has a peculiar way of showing it.) "We don't know" is simply as good as it gets. And as for an afterlife, it there is one we'll know it, and if there isn't we won't.

That doesn't, however, address the very important teleological issues of purpose and meaning. It is said that toward the end von Neumann embraced the concept of Pascal's wager, which briefly states that belief in God is essentially a non-zero-sum game in which it's best to believe and do what the Good Book says. But a god who would grant everlasting life to an inwardly insincere, feigning "believer" whose primary motivation is fear would be a pretty pathetic god. And besides, Pascal's approach doesn't say much about meaning and purpose, if all we're given is poorly-documented commands to do as we're told. So here again we find no answers for why we're here. It may be that God is deficient in musical ability, and simply enjoys our music. Or perhaps He's a poor writer, and likes Shakespeare and Conrad. But as these are all purely human concepts, God certainly has His own reasons for having created us and everything else.

Faced with these seemingly unanswerable questions, and the futility and meaninglessness of living a strictly hedonistic life (eat, drink and be merry), I suppose that it's best to just resolve to be helpful, loving and caring toward one another, as Christ suggested, and to try to set aside enough time in our busy lives to study and be in awe of this wonderful place called the universe.
The Other Way Round — Posted Sunday, January 6 2013
Conformal symmetry is a fundamental ingredient for modern theoretical physics, according to a December 2012 paper by Sofiane Faci of the Institute of Cosmology, Relativity and Astrophysics in Rio de Janeiro. In the paper the author presents an interesting application of Hermann Weyl's conformal geometry in the construction of conformally invariant Riemannian expressions. Whereas Weyl started with Riemannian geomet ry and modified it in the development of his 1918 unified field theory, Faci goes the other way, starting with Weyl geometry and going back to the Riemannian case. The author's "Weyl-to-Riemann" algorithm shows how conformally invariant Riemannian expressions can be rather easily derived by simply setting the Weyl vector \(\phi_\mu\) equal to a gradient field. Although the electrodynamic aspects of the original Weyl theory are ignored, the Weyl vector, which was originally thought to represent the electromagnetic four-vector, cannot be a gradient field unless the space is truly Riemannian.

I haven't tried it yet, but Faci's procedure should be able to reproduce Lanczos' identity, which I've detailed in my short write-up regarding his notable 1938 paper.

For those of you who are new to Weyl's geometry (and this site), you may want to know that Riemannian space is characterized by a symmetric metric tensor \(g_{\mu\nu}\) and the invariance of vector magnitude under parallel transport. That is, you can move a vector quantity around any way you like, but its length will remain constant (rather like the invariance of a vector's length under an arbitrary rotation). Given that the length \(L\) of a vector \(\xi^{\alpha}(x)\) is defined as \(L^2 = g_{\mu\nu}\xi^\mu \xi^\nu \), its variation under transport can be written as $$ 2L\,\delta L = g_{\mu\nu||\alpha} \xi^\mu \xi^\nu dx^\alpha $$ where the double subscript stands for covariant differentiation. (Here we have used the Cartan-Weyl formula \(\delta \xi^\alpha = \Gamma_{\mu\nu}^\alpha \xi^\mu dx^\nu \), where \(\Gamma_{\mu\nu}^\alpha \) is the connection.) Thus, in Riemannian space the non-metricity tensor \(g_{\mu\nu||\alpha}\) vanishes. This imposes the requirement that \(\Gamma_{\mu\nu}^\alpha\) reduce to the Christoffel symbol.

Hermann Weyl saw this invariance as an unnecessary restriction of Riemannian space. He chose to remove it by assuming that vector length could vary in a way analogous to that of the Cartan-Weyl expression; that is, \(\delta L = L \phi_{\alpha} dx^\alpha\), where the vector field \(\phi_\alpha\) is a kind of proportionality factor. This then imposes the identity \(g_{\mu\nu||\alpha} = 2 g_{\mu\nu} \phi_\alpha\); in other words, a non-vanishing non-metricity tensor is the sole quantity that allows for vectors to change both in direction and magnitude under parallel transport.

The primary advantage of Weyl's geometry is that the Weyl connection is itself invariant with regard to a conformal transformation of the metric tensor \(g_{\mu\nu} \rightarrow \Omega^2 g_{\mu\nu}\), where the scale factor \(\Omega(x)\) is completely arbitrary at every spacetime point. Thus, all quantities built from the connection (like the Riemann-Christoffel tensor \(R_{\mu\nu\alpha}^\lambda\) and the Ricci tensor \(R_{\mu\nu}\)) are automatically invariant. The Ricci tensor and its scalar cousin \(R = g^{\mu\nu}\,R_{\mu\nu}\) appear in Einstein's gravitational field equations, although the equations themselves are not invariant. While aesthetics alone demands that spacetime be conformally invariant, the spirit of relativity itself seems to demand it as well; how else would a vector's magnitude be absolutely maintained from point to point unless that information were somehow given simultaneously to all of spacetime?

There is indeed much interest today in gravitational theories involving strict conformal invariance. A few of these theories, which I have written about earlier on this site, show real promise in elucidating the nature of dark matter and dark energy. Only time will tell if these efforts pay off.