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

Einstein's 1931 Pasadena Home Today

Uncommon Valor

Sophie did not forget Jesus
Long live freedom!

2949689 visits since 11/1/2004.

My work always tried to unite the
Truth with the Beautiful, but when I
had to choose one or the other, I usually chose the Beautiful.

Hermann Weyl

I died for Beauty, but was scarce
Adjusted in the tomb,
When one who died for Truth was lain
In an adjoining room

Emily Dickinson

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

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

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

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

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

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

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

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

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

In Search of Elizabeth — Posted Wednesday, 4 March 2015
The film is set in the life of the poet in the turbulent times in which he lived. In Episode One, we are introduced to the dark side of Queen Elizabeth I's police state — in a time of surveillance, militarism and foreign wars.
This week PBS has been airing the much-lauded four-part series In Search of Shakespeare, filmed in 2004 and hosted by British historian Michael Wood, and I finally sat down and watched the whole thing. I'm kind of a history buff, and a sometime Shakespeare fan, though my son Kurt is the real expert, having read all his works. I tend to quote Hamlet a lot on this website, as if I know anything, but it's one work of Shakespeare I actually have read — many times — and turn to whenever I need an understanding of this stupid species called humankind.

At the same time I am quite illiterate when it comes to Elizabethan history, and I think I learned far more from the PBS series concerning 'Good' Queen Bess than I did about Shakespeare. It was turbulent times, indeed, when the otherwise Catholic Henry VIII, infatuated with Anne Boleyn and seeking separation from first wife Catherine of Aragon, abandoned Roman Catholicism when the Pope denied Henry's request for a divorce and turned the country toward Protestantism. Sometime after his death in 1547, 'Bloody' Queen Mary took the country back to Catholicism, and bloody she was, imprisoning, torturing and murdering all those who stood in her way, particularly if they were Protestants. But after she died Elizabeth took the throne, and the country went back to Protestantism. She carried on her predecessor's bloody ways, only this time against Catholics, and it was in these times of religious and political tumult that Shakespeare was born in 1564.

In those days England literally had an official state religion, though one might be hard pressed to know whether being Catholic or Protestant was a good thing. At any rate, the official state religion was just that, whatever it might be at the time, and religious observation was not only encouraged but mandated — you had to attend and donate to the church whether you were Catholic, Protestant, agnostic or atheist, although being one of the latter two would have seen you being fined or getting thrown into prison.

But why am I really telling you all this? It's because the State of Mississippi now wants to establish Christianity as a state religion. Yes, I know, that would violate the 1st Amendment to the Constitution, but the Magnolia State never really got over the 13th Amendment abolishing slavery anyway, and the right-wing notion of state's rights is as powerful there today as it was in 1863 (and, presumably, in Queen Elizabeth's time). Mississippi also wants to establish English as the official language (Liz would be okay with that too), adopt "Dixie" as the official state song, and make April the official "Confederate Heritage Month" (April 26th is already recognized in most Southern states as "Confederate Day"). Yee haw!

Surveillance, militarism and foreign wars. Dear God in Heaven, it's been over 400 years since the time of Elizabeth. Do we never fucking learn?

You know, for some reason plane turbulence is always at its worst when I fly over Mississippi. It never fails — either leaving Arkansas heading east or leaving Alabama heading west, the plane always acts up. I think it's because Mississippi knows how I feel about the state, and it's her way of getting back at me.
Several Roads — Posted Wednesday, 4 March 2015
The close similarity of the inverse-square laws of gravity and electromagnetism and that of the electromagnetic and gravitational potentials themselves has been known for hundreds of years. All school children are aware of this similarity, and most wonder if electricity and gravity are somehow the same thing.

I recently returned from a long visit to the east coast, where my younger son just bought a house. As I've never experienced weather like the severe cold snap that coincided with my visit, my outings consisted only of numerous trips to Home Depot for house repair materials and a single two-mile hike in 13-degree weather with snow flurries. Outside of that I did a lot of reading, which included an interesting paper that Hermann Weyl wrote back in 1944. You can read it yourself here, if you have the energy (and that's a pun).

In Weyl's 13-page "How far can one get with a linear field theory of gravitation in flat spacetime?", Weyl looks at the linearized form of Einstein's field equations, their relationship with Einstein's notorious 'pseudo-tensor' (which is used to get a mass-energy conservation law out of the field equations), and the connection between Maxwell's electrodynamics and gravitational waves. I found it amusing to note that, in getting essentially nowhere in the paper, Weyl (primarily a mathematician) defers to "the attention of physicists" regarding some of the paper's more theoretical remarks. He also uses the French term faute de mieux, implying that crazy theories often result from a lack of any better alternative.

Nevertheless, gravitational waves almost certainly exist, and as such they are a form of radiation, and radiation is described by Maxwell's equations. Consequently, one could naively say that gravitation and electromagnetism must be unified at some level. Currently, there seem to be three approaches: non-Riemannian geometry, which somehow embeds the electromagnetic 4-potential \(A_\mu\) into spacetime, à la Weyl's 1918 theory; gravitoelectromagnetism, which uses Weyl's conformal tensor \(C_{\mu\nu\alpha\beta}\) to derive variants of the electric and magnetic fields, albeit in a rather goofy way; and quantum theory, which seeks a much more formal, non-classical unification of gravity and EM. (There is yet another description that Nordström, Reissner and Weyl derived in 1916-18, which is to use the electromagnetic energy tensor \(T_{\mu\nu}\) in the field equations themselves to derive a Schwarzschild-type metric, but this is not really a unification.)

Anyway, what Weyl was getting at in his paper is that unlike classical electromagnetism, in which Maxwell's equations are linear and well-defined in terms of conservation of electromagnetic energy, the gravitational field equations are neither linear (because a gravitational field, possessing energy, can act on itself) nor well-defined in terms of energy conservation. To fix this, Einstein had to linearize his equations (which required a weak gravitational field via \(g_{\mu\nu} \rightarrow \eta_{\mu\nu} + \epsilon h_{\mu\nu}\)) while proposing a tensor \(t_{\,\mu}^\nu\) that has a vanishing ordinary divergence (thus making the field conserved). But Einstein's tensor is only a tensor in special coordinate systems, and so it is not really a valid gravitational quantity. His first task was successful — he was able to show that the quantity \(h_{\mu\nu}\) satisfies the wave equation, proving that in the weak-field limit gravitational waves exist and that gravity propagates at the speed of light. But Einstein's pseudo-tensor \(t_{\,\mu}^\nu\) was a total bust, of some limited theoretical interest today but wholly unsuitable as a description of gravitational energy conservation.

Einstein and Weyl both went to their graves in 1955, having never solved the problems of unifying gravity and electromagnetism or gravitational energy conservation. These issues still have not been resolved and now, 100 years after Einstein's unveiling of the general theory of relativity, exactly what gravity is, how it fits into quantum theory and what its relationship to electromagnetism might be continues to represent the most pressing and exasperating problem of physics today.
Peak Morality v. Jerkdom — Posted Monday, 2 March 2015

Get thee to a nunnery. Why wouldst thou be a breeder of sinners? I am myself indifferent honest, but yet I could accuse me of such things that it were better my mother had not borne me.
— Hamlet, Act 3, Scene 1

In today's Salon magazine, writer Dan Falk talks about the moral failures of our heroes, people such as Einstein, Churchill and even John Lennon, whose achievements in some cases barely outweighed their shortcomings. By comparison, how often have I reflected on my own failures as a father, husband, son, brother, friend, erstwhile Christian and self-described scientific moralist, failures that have far outweighed whatever meager success I've achieved in life. Yet, writes Falk, science and reason can be credited for gradually producing a greater awareness of our hypocritical moral failures, thus echoing the thoughts of Harvard psychologist Steven Pinker and writer/psychologist Michael Shermer, whose recent books argue that science and reason are slowly bringing about the better angels of our nature. Indeed, the "moral arc" that Pinker talks about appears capable of overcoming the moral shortsightedness of past generations:
Though they were surely decent people with perfectly functioning brains, the collective moral sophistication of the culture in which they lived was as primitive by modern standards as their mineral spas and patent medicines are by the medical standards of today. Many of their beliefs can be considered not just monstrous but, in a very real sense, stupid. They would stand up to intellectual scrutiny as being consistent with other values they claimed to hold, and they persisted only because the narrower intellectual spotlight of the day was not routinely shone on them.
Although one can argue to what extent science and reasoning are producing better humans — strict Christian dogma would instantly denounce the notion as "rags fit for burning" (Isaiah 64:6) — there is little doubt that the snake oil mentality that Pinker mentions has largely disappeared due to worldwide education and the instantaneous nature of mass communication.

While I don't disagree with Falk's essay, he nevertheless omits the fact that while scientific reasoning is progressive and morally beneficial (we'll ignore things like nuclear/biological weapons and drones for the time being), most conventional religious philosophies are static and, anti-intuitively, actually counterproductive to human moral progress. Falk's single comment on the influence of religion on human betterment (the Golden Rule) does nothing to sweep away the fact that the inhuman barbarity and genocides of the Old Testament, for example, are immutably and permanently fixed in time and cannot be altered by any progressive notions of modern morality.

Case in point: When an advisor asked Israeli prime minister Benjamin Netanyahu in 2009 what his view of Iran's nuclear program was, Netanyahu simply remarked "Think Amalek," referring to the ancient Amalekites who hindered the Israelites in their sojourn from Egypt into the Promised Land. Netanyahu was also referring to God's command to Saul in 1 Samuel 15:3:
Now go, attack the Amalekites and totally destroy all that belongs to them. Do not spare them; put to death men and women, children, infants and sucklings, cattle and sheep, camels and donkeys.
Netanyahu thus sees Iran as an existential threat to Israel, no less worthy of utter destruction than was Amalek. And, lest you think that Netanyahu sees Iran only as a contemporary threat, remember that God's command to Saul was recorded some 600 years after the Israelite exodus from Egypt — God never forgot Amalek, and neither will Netanyahu dismiss Iran.

On Tuesday night Netanyahu (whom some have referred to as the Republican senator from Israel) will address the US Senate on President Obama's negotiations with Iran, which Netanyahu views as an existential error. The prime minister's goal will be to convince Congress to instead join with Israel and attack Iran militarily. Keep in mind that Netanyahu does not want to end up like King Saul, whom God killed because he did not carry out the destruction of Amalek to the letter. And remember also that while Saul had only swords and bows and arrows at his disposal, Netanyahu's Israel has some 200 nuclear weapons.

Science and reason are progressive, while religion, being set in stone, sadly is not. Falk's article should have addressed this as well.
The Moving Finger writes; and, having writ,
Moves on: and neither all thy Piety nor Wit
Shall lure it back to cancel half a Line,
Nor all thy Tears wash away a Word of it. — Rubáiyát of Omar Khayyám
Galileo Meets Tom the Dancing Bug — Posted Thursday, 26 February 2015
It is difficult to get a man to understand something when his salary depends upon his not understanding it. — Upton Sinclair

For every PhD there's an equal and opposite PhD, especially when a person's livelihood is involved. — Kenneth S. Deffeyes
Ruben Bolling's comic today tells of the birth and death of scientific logic, and the fifth panel says it all.

By the way, there is little historical evidence that Galileo ever tested the equivalence of inertial and gravitational mass by laboriously trudging up the Pisa Tower, which at his age at the time would have made the trip problematic. He instead did his tests with polished blocks sliding down polished inclined planes. But it is interesting to note that before Galileo, the notion that heavier objects fell faster than less massive ones was simply taken for granted — there was no need to actually test the notion, since common sense told people what the outcome should be.

I fervently believe that such common-sense notions are inherent in the human mind, and are the primary basis for religious belief. But, unlike Galileo's experiments, religious belief cannot be tested or empirically confirmed. This is both the power and the weakness of religion, since anything that cannot be experimentally confirmed must be taken on faith alone — and blind faith sadly becomes dogma. The last panel of Bolling's comic anticipates Galileo's battle with organized religion, which very nearly resulted in his being tortured and then burned at the stake.
If English Was Good Enough for Jesus \(\ldots\) — Posted Wednesday, 4 February 2015
The gods have imposed upon my writing the yoke of a foreign language that was not sung at my cradle. —Hermann Weyl, 1939

If English was good enough for Jesus Christ, it ought to be good enough for the children of Texas.
— Attributed to Miriam Ferguson, first female governor of Texas
No one knows what the first formal spoken or written language was, although it was probably some variant of cuneiform or ancient Egyptian, but there is little doubt that Greek was the first language used to express higher scientific and mathematical thought. Much later was Latin, used first by the Romans and then much later by Isaac Newton and his crowd to express their technical ideas; Latin was the lingua franca of Newton's day. By the turn of the 20th century, scientists and mathematicians alike routinely published their work in French, German, Russian, Italian and English, and the likes of Einstein, de Broglie, Schrödinger and many others were fluent in several languages (even Hitler knew French). But today it's mostly English that's used. Why is that?

As noted by Princeton University science historian Michael Gordin, English is used today in about 99% of published technical papers. While it wasn't always that way, polyglotism has given way to monoglotism. Gordin traces that change back to World War I, although he also notes that Hitler's summary expulsion of Jewish professors and intellectuals in 1933, which resulted in their emigration to America or other English-speaking countries, also contributed to monoglotism. But Gordin infers that the refusal or inability of Americans to learn other languages probably was the greatest factor in the world's adoption of English as the lingua franca of science and math.

I can read French and German, but I'm not truly fluent in those languages. My excuse is that I don't use them frequently enough to be good at them. But I have family members who speak Arabic around me all the time, yet in nearly 40 years I've learned very little Arabic. So I suppose it's an American thing — we're monoglotic simply because we're basically lazy and expect others to accommodate us.
"In France, oeuf means 'egg' and chapeau means 'hat.' It's as if those French have a different word for everything!" — Steve Martin
Moving On — Posted Wednesday, 4 February 2015
As I've noted repeatedly on this website, Hermann Weyl's 1918 theory continues to show up regularly in physics journals and papers, though I believe it's quite wrong in its present form. That's not to say that the notion of conformal invariance is wrong (it's beautiful, and may actually be a valid symmetry, at least cosmologically), but Weyl's idea that the electromagnetic field is derivable from geometry simply doesn't appear to work, Einstein's arguments against it notwithstanding.

Here I will take an approach different from Einstein's and show that any non-Riemannian geometry is unworkable — and by that I mean any tensor formalism in which the non-metricity tensor \(g_{\mu\nu||\lambda}\) (that is, the covariant derivative of the metric tensor) does not vanish. It's a real problem, and I wish researchers would look at ways around it.

Consider any symmetric affine connection \(\Gamma_{\mu\nu}^\lambda\) that is not identically the Christoffel symbol of Riemannian geometry. It is easy to show that the familiar cyclic symmetry condition
$$ R_{\alpha\mu\nu\lambda} + R_{\alpha\lambda\mu\nu} + R_{\alpha\nu\lambda\mu} = 0 $$
still holds, where \(R_{\alpha\mu\nu\lambda}\) is the lower-index form of the Riemann-Christoffel tensor. It is also easy to show that the Bianchi identity
$$ R_{\,\,\mu\nu\lambda||\beta}^{\,\alpha} + R_{\,\,\mu\beta\nu||\lambda}^{\,\alpha} + R_{\,\,\mu\lambda\beta||\nu}^{\,\alpha} = 0 $$
is also valid. The only other symmetry property is the antisymmetry of the last two indices of the RC tensor itself.

When space is Riemannian we have two additional symmetry properties:
$$ R_{\alpha\mu\nu\lambda} = - R_{\mu\alpha\nu\lambda} $$
along with the index-pair exchange property
$$ R_{\alpha\mu\nu\lambda} = R_{\nu\lambda\alpha\mu} $$
Using these expressions, it is a standard undergraduate exercise to show that the Bianchi identity leads to the covariant conservation property of the Einstein tensor:
$$ \left( R^{\mu\nu} - \frac{1}{2} g^{\mu\nu} R \right)_{||\nu} = G^{\mu\nu}_{\,\,\,||\nu} = 0 $$
Ever since Einstein came out with his theory of general relativity 100 years ago, this expression has been viewed as the fully-covariant version of mass-energy conservation. But it's not really a true divergence in the traditional mathematical sense, and the question of mass-energy conservation in general relativity has long been (and still is) plagued by confusion. However, for brevity we'll pass over this issue for the time being.

For a general non-Riemannian space (like the one Weyl came up with), the non-metricity tensor \(g_{\mu\nu||\lambda}\) no longer vanishes. But we still have the definition
$$ g_{\mu\nu||\alpha||\beta} - g_{\mu\nu||\beta||\alpha} = -g_{\mu\lambda}R_{\,\,\nu\alpha\beta}^{\,\lambda} - g_{\lambda\nu}R_{\,\,\mu\alpha\beta}^{\,\lambda} $$
$$ g_{\mu\nu||\alpha||\beta} - g_{\mu\nu||\beta||\alpha} = - \left( R_{\mu\nu\alpha\beta} + R_{\nu\mu\alpha\beta} \right) $$
Consequently, the antisymmetry property of the first two indices in the RC tensor no longer holds. Furthermore, it is also easy to show that
$$ g_{\mu\nu||\alpha||\beta} - g_{\mu\nu||\beta||\alpha} = \left( R_{\nu\beta\mu\alpha} - R_{\mu\alpha\nu\beta} \right) + \left( R_{\mu\beta\nu\alpha} - R_{\nu\alpha\mu\beta} \right) $$
and so now the index-pair exchange property of the RC tensor is gone as well. But this is a disaster: without these properties, we can no longer derive a divergence/conservation expression for the Einstein tensor \(G^{\mu\nu}\). Indeed, without a specific definition for \(g_{\mu\nu||\lambda}\), it is impossible even to have a formal theory of gravity anymore. In short, the non-vanishing of the non-metricity tensor ruins everything.

But this did not stop Weyl from trying (although he likely never derived any of these expressions). In fact, he was never able to derive a mass-energy expression for his theory. While it did result in a geometrical form for the Maxwell electromagnetic source vector density
$$ \sqrt{-g} \,s^\mu = \sqrt{-g}\, g^{\mu\nu} \left( R \varphi_\nu + \frac{1}{2} R_{||\nu} \right) $$
(where \(\varphi_\nu\) is the Weyl vector), it is not divergenceless. Consequently, the Weyl vector cannot be the electromagnetic four-potential \(A_\nu\) and Weyl's \(s^\mu\) cannot be the Maxwell source vector.

While Weyl's 1918 theory doesn't work in its present form, ten years later it led him directly to the notion of gauge invariance in quantum theory, where it was a profoundly successful idea. In fact, today Weyl's idea is arguably the major cornerstone of quantum physics, and without it modern physics would likely be completely lost.

More Fun — Although I love Roger Penrose's books and writing style, he has a rather nasty habit of setting complicated mathematics into unintelligible graphical format. Here is his "explanation" of the Riemann-Christoffel tensor and its symmetry properties. taken from Chapter 14 of his otherwise excellent book The Road to Reality:

I finally worked out what this thing means schematically, but I can't figure out what it's good for. It certainly doesn't teach anyone what the Riemann-Christoffel tensor is or what it does. Maybe you'll have better luck with it.
Exaggerating Eddington — Posted Thursday, 29 January 2015

On 29 May 2014 I posted an article on the 2009 BBC film Einstein and Eddington, which dramatized Eddington's 29 May 1919 solar eclipse expedition to Principe, an island on the west coast of Africa. I included a screen capture of the "evidence" of starlight deflection around the Sun (right photo, above), thinking it was an enlargement of one of Eddington's few good photographic plates (shown on the left).

This morning I got an email from Delft University's Leo Vuyk informing me that the photo I posted is probably a fabrication created by the filmmakers to exaggerate the deflection effect. I've actually seen the real photographic plates, which show only a few stars in the Sun's field along with tiny deflections, and I have a several good reproductions of the photos, but the one from the film is so patently false that I'm ashamed I didn't suspect "artistic license" from the filmmakers from the beginning.

The starlight deflections that Eddington actually measured were indeed tiny, and his results just barely proved out Einstein's general theory of relativity (later, more careful photographic examination showed that Einstein's gravity theory was indeed upheld). As this article on the film notes, Einstein was a relative unknown prior to 1919, and Eddington's Principe expedition made Einstein a worldwide scientific superstar.

It's still a good film, and noted Shakespearean actor David Tennant (Hamlet) is excellent as Eddington. But I promise to be more careful in what I post in the future.
Ants — Posted Wednesday, 28 January 2015
When I was around ten years old I became fascinated with ants. Ants were everywhere in my world of 1959, my back yard in particular, where I would watch colonies of large black ants (friendly and harmless) and red ants (aggressive and stinging), always separated by a safe distance, as if they had polite, formal, established borders beyond which neither species ventured beyond. In view of their behavior, I often wondered about their intelligence — they were so wonderfully coordinated and ordered — and a few years later I actually wrote a short story about how ants had developed a rudimentary religion. I actually preferred the red ants, even though I got stung many times over the years of observing and playing with them.

I moved on, but in high school I had a friend (Dan E.) who was nuts about ants, and he resolved to major in entomology when he got to university. Entomology, the study of insects, is a lot broader than the study of ants (myrmecology), but ants are probably what factored in my friend's decision for a college major, which eventually took him to the University of California at Davis (he was a lot smarter than me — I could only get into Cal State Long Beach). But he ended up with a PhD in art history, of all things, while I left chemistry for physics and engineering, but that's another story.

But the real expert on ants today is undoubtedly E.O. Wilson, the noted Harvard entomologist, sociobiologist, evolutionary biologist and biodiversity advocate, whose latest book is The Meaning of Human Existence. That's a rather ambitious (or even pretentious) title for an ant expert, but quite appropriate for a brilliant 85-year-old scientist whose passions and expertise stretch far beyond anthropods. I only recently purchased the book and, having read only the chapters on religion and free will thus far (though it's a short book), I cannot predict how much success it will have with the general public. But what I have read makes a lot of sense.

Wilson ties much of the meaning of human life to religion (which he views as a colossal mistake), specifically humans' reverence regarding deific creation myths and tribalism. As Wilson notes, there are many creation stories, no two of which can be simultaneously correct, but it goes without saying that the universe did indeed somehow come into being. Of the twenty or so "eusocietal" organisms that have arisen on Earth — the ones that form caring, long term, multi-generational familial bonds — humanity is the only one that became self-aware. Yet even intelligent sentience was not enough to protect that single organism from self-deceit. Wilson believes that tribalism resulted purely as an evolutionary survival trait, especially when regional human populations grew to the extent that cut-throat competition became more advantageous to survival than cooperation, but humans even today do not recognize the fact that tribalism carries with it the same innate barbarity that its altruistic alter ego, religion, invariably has failed to correct.

Wilson's book tries to answer what meaning, if any, the emergence of sentient human life has on the totality of terrestrial existence, if not the universe itself. As noted earlier I have not read the entire book, but from the book's reviews I gauge that Wilson has not been very successful. It's a subjective issue, after all, one that to date no one has been able to overcome. To even pose the question of the meaning of human life is not only ambitious but probably self-defeating — every writer's effort is doomed from the start.

In my own life of rather shallow philosophical wandering I have come to liken humanity as a slave of ancient Roman times who finds himself permanently shackled to a Roman warship, ceaselessly pulling the oar while being both whipped and fed. It's a hopeless situation, true, but he still manages to find hope in the form of a miraculous release (recall the story of Ben-Hur) or, that failing, in the belief of an eternal afterlife of heavenly bliss. From that perspective Wilson believes mankind has utterly failed itself by retreating into the habit of wishful thinking, which to this day has overcome all attempts, including rational science, to defeat it. It's a resilient fall-back position because it requires no education, experience, logic or even intelligence to grasp it, and the fact that wishful thinking — like religion itself — cannot be disproved adds to its effectiveness.

I will finish Wilson's book, but I'm not optimistic that I will really learn anything from it.
Religion is regarded by the common people as true, by the wise as false, and by the rulers as useful. — Lucius Seneca, Roman statesman, 5 BCE - 65 CE
Math in the Media — Posted Friday, 23 January 2015
The movies The Theory of Everything and The Imitation Game are getting rave reviews from critics and viewers alike right now, and while it's neat to see the public getting interested in such things, the stories told in the films leave much to be desired.

Last night I saw The Imitation Game with Benedict Cumberbatch as the famous Bletchley Park mathematician Alan Turing, who was primarily responsible for cracking the seemingly impenetrable German Enigma cipher during World War II. The film of course focuses on Turing's homosexuality and his efforts to decipher Enigma, and what I thought was best about the film was how it related Turing's sexual "otherness" to how he approached mathematics — his thinking processes were quite different from those of the other (quite talented) Bletchley mathematicians, and this is certainly what gave Turing the edge. It rather reminded me of how people have described Caltech physicist Richard Feynman, whose thinking was "orthogonal" to everyone else's (there's being smart and there's being a genius, but some geniuses are just off the scale completely). In that regard, Turing and Feynman shared a true commonality.

However, as a film it doesn't match the acting, drama and storyline of 2001's fictionalized but highly underrated Enigma, with Dougray Scott and Kate Winslet portraying essentially the same characters (although Scott's character is not gay in the film, it's obviously based on Turing). Enigma's last scene is quite moving emotionally, and the music is heartrendingly beautiful (I often listen to the soundtrack while at the gym). Sadly, it was composer John Barry's (Midnight Cowboy, Dances with Wolves, Out of Africa) last film score. Enigma also has numerous scenes which explain how the Enigma code worked, along with scenes that attempt to explain the sheer beauty of mathematics — two important aspects of Bletchley Park that are completely lacking in The Imitation Game. Plus, in Enigma you can watch them manipulating an actual German Enigma machine, owned by the movie's co-producer, Mick Jagger.

The Theory of Everything is well acted, but the true sense of what Stephen Hawking accomplished mathematically in his problematic life is just not there. It rather reminded me of some episodes of Cumberbatch's Sherlock series, in which his character is shown rapidly going through logical combinatoric exercises that I found confusing and even pretentious. In the end, the purpose of The Theory of Everything seems to be to simply entrance the viewer with Hawking's brilliance without trying to explain what he actually did.

Still, these two films are far better than 2013's Gravity and last year's Interstellar, both of which were highly rated but relied more on fancy computer graphics than on story or science. Noted Caltech physicist Kip Thorne served as science advisor on Interstellar and he absolutely insisted on scientific accuracy (often to the annoyance of director Chris Nolan), but I found the film to be highly improbable on a number of questionable scientific points, not least of which being that I don't believe wormholes exist (Nolan also considered faster-than-light travel as an alternative, but Thorne quickly quashed that one). Meanwhile, I found Gravity to be just so much computerized eye candy (not to mention my opinion that actors Sandra Bullock and George Clooney are also used as such), while Bullock's miraculous survival at the end belies what would have certainly been a quick death for all the astronauts if it had been real life.

But again, it's great to see so much science and math on the big screen today.

The last scene of Enigma. It's Trafalgar Square in London, after the war, and Tom (Dougray Scott) is waiting for Hester (Kate Winslet). In the crowd he spots the erstwhile love of his life, Claire (Saffron Burrows), the British spy who broke his heart. But things have changed since Tom and Hester left Bletchley Park, and for the better.

(© Manhattan Pictures International, 2001)
Scientific Conservatism? — Posted Sunday, 4 January 2015
Harvard history professor Naomi Oreskes' recent New York Times article Playing Dumb on Climate Change talks about the dangers of conservative scientific thinking — conservative in the sense that even when evidence is overwhelming, scientists are often still loathe to accept a rational, evidence-based conclusion. This type of conservatism, she asserts, is making the threat of anthropomorphic climate change seem less dangerous than it really is.

Oreskes provides a very understandable definition of Type 1 and Type 2 errors: scientists make a Type 1 error when they conclude that an effect is real when it is not, and they commit a Type 2 error when they claim an effect is unreal when it actually exists. Whether scientists make either error has a lot to do with the incidental consequences of making a bad claim; thus, "extraordinary claims require extraordinary evidence" is the rule, and it is better to miss a real effect, since their careers will suffer less. And when scientists do make an extraordinary claim, they like to back it up with as much statistical support as they can muster; as Oreskes explains, this is typically done using "confidence limits." A statistical confidence limit of 95% means that the odds that a claimed effect is due to random chance alone is 5%, while that of a 99% CL is only 1%.

In the case of global climate change, Oreskes notes that atmospheric scientists already have all the physical and statistical evidence they need to confidently demand political and industrial change regarding human activities. So why don't they? Her answer:
Why don't scientists pick the standard that is appropriate to the case at hand, instead of adhering to an absolutist one? The answer can be found in a surprising place: the history of science in relation to religion. The 95 percent confidence limit reflects a long tradition in the history of science that valorizes skepticism as an antidote to religious faith.
Oreskes also notes that
Moreover, while vigorously denying its relation to religion, modern science retains symbolic vestiges of prophetic tradition, so many scientists bend over backward to avoid these associations. A vast majority of scientists do not speak in public at all, and those who do typically speak in highly guarded, qualified terms. They often refuse to use the language of danger even when danger is precisely what they are talking about.
With regard to what many scientists rightly consider the existential threat of global warming, we would seem to have a case of scientists choosing to "scrootch back" in the corner of the debate room either to avoid being noticed, to avoid a classic Type 1 error, or to avoid offending religious fundamentalists, who need offer no evidence whatsoever of any scientific claims they choose to make.

The Constitution talks about the tyranny of both the majority and the minority, but it implicitly assumes rationality of thought behind whatever issue is being discussed or whatever law is being implemented. As Oreskes implies in her article, scientists are committing a dangerous act by underestimating the threat of climate change out of conservatism, while the religious crowd is doing the same by denying it for totally irrational reasons.
Strange Bedfellows — Posted Saturday, 3 January 2015

Since the 1930s, George Washington University has convened a series of informal annual conferences on theoretical physics, each focusing on a specific topic. Hermann Weyl, then at the Institute for Advanced Studies in Princeton, was an invited attendee of several of these conferences. In November 1947, the topic at hand was gravitation and electromagnetism, so of course Weyl was there. Here's the group photo, with the 62-year-old Weyl seated at the left end. Also prominent in the photo are Edward Teller, John A. Wheeler, J.R. Oppenheimer and Richard Feynman, all fresh from Los Alamos, where they developed the atomic bomb. At this conference they discussed more mundane issues like quantum electrodynamics and unified field theory. The man with the enormous cranium standing fifth from the left is David R. Inglis of the University of Massachusetts. A nuclear physicist and ardent weapons-ban advocate, he seems to be glaring at Edward Teller (sitting next to Weyl), who was arguably more responsible for the widespread proliferation of nuclear weapons than any other scientist. It was also Teller who infamously betrayed Oppenheimer (seated, right center), a noted pacifist, thus destroying his career.

Much thanks to David Marans, the author of Logic Gallery, for alerting me to this photo. (David welcomes comments and suggestions for his online book. He can be reached at davidjan43@aol.com.)
Generalizing General Relativity — Posted Thursday, 1 January 2015
This year marks the 100th anniversary of Einstein's general theory of relativity (gravitation), and I still find it remarkable that efforts to generalize the theory remain in vogue to this day. The non-Riemannian theory of Hermann Weyl (1918) was the first of many such theories, notable today if only for the fact that its gauge aspects and application to quantum physics a decade later became the foundation for all modern quantum theories. Einstein himself resorted to non-Riemannian formalisms in his own 30-year-long effort to unify gravity with electromagnetism and quantum physics. And yet, sadly, the hoped-for unification of gravitation and quantum mechanics today remains as elusive as ever.

Ten years after Einstein announced his theory, mathematician Luther Eisenhart of Princeton University published an interesting book on non-Riemannian geometries, which to this day represent perhaps the oldest, purely-classical approach to generalizing the theory. Such geometries necessarily involve vector and scalar additions to the Riemannian connection term \(\Gamma_{\mu\nu}^\alpha\) and, while at times fascinating, the approach is best summed up by the noted physicist and Einstein biographer Abraham Pais, who wrote "the use of general connections means asking for trouble." Indeed, there seems to be no end of problems with generalized connection coefficients, which at the very least invariably wipe out many of the cherished symmetry properties of the Riemann-Christoffel tensor \(R_{\,\,\mu\nu\lambda}^{\,\alpha}\) while providing no discernible benefits.

Weyl's 1918 theory arguably came closest to describing a consistent, all-encompassing theory (although it involved only gravity and electromagnetism), but his death in 1955 preceded a perhaps more lucid approach to generalization, which was the 1961 theory developed by Princeton's Carl Brans and Robert Dicke, who used a combination scalar-tensor approach within a wholly Riemannian geometrical framework. While this avoided the non-Riemannian pitfalls previously alluded to, it was rather ugly and unable to successfully reproduce the predictions made by Einstein's original theory. Over the ensuing decades the Brans-Dicke theory itself has been generalized by numerous researchers to include scalar and spinor quantum fields, producing a variety of even uglier theories that have yet to produce any meaningful predictions.

As many people noted in 2014, scientific progress seems to be stuck in a rut. One hundred years after Einstein gave us his profoundly beautiful theory of gravity, there has been little real progress made on unified field theory or on any theory that describes gravity within the equally beautiful and profound mathematical formalism of quantum physics. Despite the ongoing brilliant work of scientists like Edward Witten, Stephen Hawking and others, we're no closer to a truly comprehensive description of Nature than we were in the days of Weyl and his fellow generalists. Think about that for a minute — scientists have routinely cracked the mysteries of Nature, often only after long and arduous searching, but in the case of gravity all of man's ingenuity has yet to provide any real progress. We still do not know if gravity involves extra dimensions, extra forces, strings, branes, supersymmetry, quantum loops or other fanciful notions, while the Large Hadron Collider, which supposedly disclosed the Higgs boson (or something very like it) in 2012, provided no useful clues.

Gravity, undoubtedly the first of Nature's forces that mankind became aware of (often painfully), has turned out to be far more mysterious and elusive than we might ever have imagined.
Exodus: Believe the Lie — Posted Thursday, 1 January 2015

In the new blockbuster movie Exodus: Gods and Kings, Moses is only forty (not eighty) and there are only 600,000 fleeing Israelites, not millions, but faithful viewers will still think they've witnessed actual history. Thanks to superior computer-generated imagery, the Exodus myth lives on.

Let us go way back in time, to about 1250 BC, when Egypt's 19th Dynasty, under the reign of Pharaoh Ramses II, ruled much of what we today call the Middle East, including all of the land known as the Levant (Israel and its immediate neighborhood). It included all of Egypt, portions of Nubia (Sudan) and Libya, the entire region of Canaan (what is today Israel, the West Bank and areas to the immediate west), and all the way up to the border of what is today called Lebanon, Syria and Jordan. The lands outside of Egypt were controlled by garrisons of armed Egyptian soldiers, numbering in the tens of thousands, overseeing vassal states runs by local puppet governors and kings, all beholden to Pharaoh. Ramses, the greatest pharaoh of all time, ruled Egypt, economically and militarily the most powerful country in the world, and was a ruler to be reckoned with.

Impose now if you will the story of several million Israelite slaves held in Egyptian bondage, somewhere in the eastern Nile Delta, where a great city known as Pi Ramses (House of Ramses) was reportedly constructed by forced labor. According the the Old Testament Book of Numbers, 603,550 Israelite men revolted under the leadership of an Israelite-Egyptian noble called Moses, who had received the call of God to lead his people out of Egypt into the Promised Land, which we know today as Israel and its immediate environs. The Israelite men took their wives, children, parents and others, numbering certainly no less than 2 million souls, along with livestock and plunder taken from their Egyptian captors, and headed out into the western Sinai, by way of the shallow Sea of Reeds (not the Red Sea). According to the Bible, they avoided a more direct route along the Mediterranean coast, no doubt because of the Egyptian garrisons posted there; undoubtedly, a smart move on Moses' part.

Some 40 years later, with Moses dead, the Israelite general Joshua took what was now a large army into the land of Canaan, west of the Jordan River. They conquered the land, and by the direct mandate of God they killed every living thing they encountered — men, women, children, infants, slaves, livestock, fruit trees and crops.

Question: What was Egypt doing while all this was going on? Pharaoh Ramses II was the most powerful ruler in the world at the time, and it seems doubtful that he would have restrained his armies while Joshua marauded the Levantine countryside, which was under Ramses' reign and protected by legions of Egyptian soldiers. The Bible gives no account of Egypt's reaction; indeed, it doesn't even mention Egypt at this point in time, nor does it identify the name of the reigning pharaoh. And indeed, in the last several books of the Hebrew Torah, also known as the Pentateuch, Egypt doesn't seem to even exist.

Fast forward to today. Israel occupies territories granted to it in accordance with the Partition Plan for Mandatory Palestine (a follow-up of the British Mandate for Palestine, first formulated in 1922 and implemented in May 1948). Arab countries adjacent to the region immediately went to war with Israel's army, which by some miracle happened to exist at the same time for just such an eventuality (the British, French and others supplied arms to Jewish guerrillas prior to 1948, but this hardly explains how they could have defeated half a dozen Arab states without help from those same countries). Israel won that war, and others that followed over the years, and today it continues to occupy disputed territories claimed by Palestinians displaced by the wars and by Israel's continued practice of seizing and building on what had historically and legally been Palestinian land (ultra-orthodox Jews, and many orthodox Jews, feel that Israel is still being cheated — according to the Old Testament, God granted Israel much more land than its occupies today, and they continue to agitate for greater Israeli territorial aggression).

More recently, the Palestinian Authority was denied nationhood by the United Nations, so today it applied for membership with the International Criminal Court (ICC), the Netherlands-based international judiciary that oversees crimes against humanity, war crimes and crimes of aggression, largely independent of the United Nations. To date, the United States, Israel, China, India and about 16 other nations have refused to join the ICC, primarily because of past or ongoing practices that would render those countries in violation of ICC humanitarian statutes. The Palestinian Authority is unquestionably guilty of war crimes and crimes against humanity, primarily against Israelis, so the Authority's joining with ICC would presumably help bring judgment against itself. But it could also petition the ICC to judge Israel for the same crimes, including crimes of aggression. It will be interesting to see how this all works out, since the United States and Israel have vowed to impose harsh sanctions against the Palestinians for seeking justice in the way they have.

But the upshot of all this is the fact that Israel's existence today is really based on the Exodus myth, which has never been substantiated archeologically, scientifically or even logically. I have written extensively on this topic elsewhere, but my conclusions are hardly my own, being backed up by many renowned archaeologists, scholars and investigators, notably Donald Redford, Israel Finkelstein and Neil Silberman. Even the noted Los Angeles Rabbi David Wolpe has admitted to his congregation that the story is allegorical myth, at best a metaphor symbolizing God's relationship with the Israelites, and not to be taken literally. To be brief: the Exodus story never happened.

Unfortunately, the Exodus story became the signature event of the Jewish faith, when the Pentateuch was penned around 600-800 BC, and the associated Passover observance has since become a sacred Jewish tradition, recognized even by Jesus of Nazareth. As is well known, his observance of the traditional Passover meal occurred during his tragic last visit to Jerusalem.

But it is far more tragic that the Exodus story is also the basis for Israel's presumed right to all the land it can get its hands on, which, with a population already at 8 million crammed into an area the size of New Jersey, will inevitably spur internal demands for more space (the Germans called it Lebensraum). The reason a two-state solution with Palestine will never occur is not because of political intransigence, religious differences or decades-long ethnic hatred, but because a fictional story written some 2,600 years ago has been taken literally by billions of people who should know better. It is ironic that the Palestinians, being predominantly Muslim, also believe in the Exodus tale, but I fear they can't see that the joke is on them.

America has a myth, too, along with its own Lebensraum. Called Manifest Destiny, it was employed in a similar manner to wrest land and resources from a supposedly savage race of inferior people who were also exterminated, and again God was used as the excuse. Acknowledgment of this atrocity was never formally made, much less any offer of reparations, but the few survivors were at least granted small allotments of unwanted land (provided nothing of value was ever later discovered there, in which case the vanquished were moved again).

Assuming the Palestinians are not exterminated outright by Israel, I wonder where they'll end up, as they sure as hell won't get their own nation as things stand today. God does work in mysterious ways, doesn't he?

The mummy of Ramses II (the Arabic pronunciation is rahm-SEIZE), born 1303 BC and died August 1213 BC, age 90. He ruled Egypt for 66 years, siring some 100 children along the way. I saw the guy a few years ago in his glass enclosure at the Egyptian Antiquities Museum in Cairo, and I wondered if this was the pharaoh who faced off with Moses. But no — Moses probably never existed, but this man certainly did.