2009 Archives

Dirac, Once More -- Posted by wostraub on Monday, December 7 2009
My post on Michael Faraday reminded me of another story involving Paul Dirac, which is reported in Graham Farmelo's book (see my post dated 19 November).

In his mind's eye, Faraday saw the lines of force emanating from an electric charge or the pole of a magnet as actual field lines in space, the flux of which in any region being directly proportional to the field strength in that region. This view was subsequently replaced by the field concept, which then gave way to the particle exchange concept that is used today in quantum field theory.

Dirac detested the idea of a bare electron, separate from its field, and he never accepted the technique known as renormalization, which is used to remove the mathematical infinities associated with bare particles in quantum field theory. In 1956, Dirac considered Faraday's concept of lines of force and tried to reinterpret them in a quantum-mechanical sense; that is, Dirac believed that the field lines would become discrete in some sense when the field is quantized. Dirac expressed this idea thusly:
We may assume [that] when we pass over to the quantum theory the lines of force become all discrete and separate from one another. Each line of force is now associated with a certain amount of electric charge. This charge will appear at each end of the line of force (if it has ends) with a positive sign at one end and a negative sign at the other. A natural assumption to make is that the amount of charge is the same for every line of force and is just the [size of the charge of the electron]. We now have a model in which the basic physical entity is the line of force, a thing like a string, instead of a particle [my emphasis]. The strings will move about and interact with one another according to quantum laws.
Dirac thus anticipated string theory by several decades, and he accomplished this by appealing to Faraday's long-discredited view of electric lines of force.

Incredible. Edward Witten once remarked that string theory is a bit of 21st century physics that fell into the 20th century. By the same logic, Dirac must have come from the 25th century.

Faraday's Unified Field Theory -- Posted by wostraub on Saturday, December 5 2009
Hermann Weyl's 1918 theory of the combined electromagnetic-gravitational field was an almost logical extension of the Riemannian geometry that Einstein had relied upon when developing his own theory of gravity in 1915.

The generalization or abstraction of a successful approach to a physical problem often meets with success, but for the unification of gravity and other forces such as electromagnetism this has been far from the case. Historically, Riemannian geometry has been extended primarily via three different approaches: 1) increasing the number of spacetime dimensions beyond four; 2) eliminating the presumption of index symmetry in the metric tensor gμν and associated quantities; and 3) eliminating the restriction of constant vector length (or magnitude) under physical transplantation in spacetime. Weyl's 1918 effort was based on the third approach, and while it and other attempts led to interesting mathematical ideas (the notion of torsion in tensor calculus, for example), nothing concrete has ever panned out.

Michael Faraday, 1791-1867. Copyright 2009, The Royal Society

In view of this, it is interesting to note that the great English physicist Michael Faraday once attempted to link gravity and electromagnetism himself. In 1831, Faraday discovered the electromagnetic law of induction, which states that a time-varying magnetic field will induce a current flow in a nearby wire (this is the basis of mechanical electrical power generation and transformer operation). Then in 1851, noting the similarity between the inverse-square force laws for electric charge and gravity, he considered the possibility that a law of induction might hold for changing gravitational fields as well. Being perhaps the greatest experimentalist of his time, he set out to find out if there was such a law by conducting some bench-top experiments, which he reported in a paper published by the Royal Society.

Earlier, I posted a little story about Dirac's Large Numbers Hypothesis, in which the ratio of the electric and gravitational forces between an electron and proton (about 1039) plays a key role. Even in 1851, Faraday must have had at least some familiarity with this ratio, and this awareness alone would have told him that it would be hopeless to induce an electric current by moving fist-sized lumps of uncharged materials (his source of gravity!) to and fro in the vicinity of a metal wire connected to a galvanometer. But it was, conceptually at least, an interesting experiment into the possibility that gravity and electromagnetism might somehow be related.

And even more interesting is the fact that Faraday's investigation preceded James Clerk Maxwell's successful (and profoundly historic and important) unification of the electric and magnetic fields in 1864 by a dozen years!

Beautiful -- Posted by wostraub on Wednesday, December 2 2009
In the 1980s I became fascinated by Mandelbrot diagrams. There were a number of 2-D fractal computer programs at the time that allowed users to make their own (admittedly primitive) Mandelbrot graphics under the MSDOS operating system. The diagrams on based on a simple iterative equation (usually involving complex numbers) that defines each point in the graph. As iteration progresses, the points generate exceedingly complicated figures that can be magnified infinitely.

Anyway, at the time I wondered if such diagrams could ever be generated in 3-D, and what they would look like. Now I know, and they're beautiful almost beyond description:

The Cave of Lost Secrets, Copyright Daniel White 2009

The website Skytopia (courtesy of a heads-up by John Baez' website) deserves more than a casual look. The pictures are gorgeous, but the mathematics behind them is perhaps even more beautiful. Skytopia also hints at the promise of 3-D video flyovers (I'm drooling now).

The Weylon?! -- Posted by wostraub on Sunday, November 29 2009
Last year saw the publication of the arXiv article Standard Model and SU(5) GUT with Local Scale Invariance and the Weylon by Hitoshi Nishino and Subhash Rajpoot. I mention this article for a number of reasons:

1. Prior to submitting a paper, authors should always run a spell check. Otherwise, embarrassing words like "persuits" go undetected except by unproductive idiots like myself.

2. The article is one of numerous papers (not all of which are available at arXiv) indicating that Weyl's geometry (and scale symmetry) is enjoying something of a renaissance. The above paper posits the existence of a hypothetical vector field associated with a particle known as the "Weylon" in conjunction with the electroweak theory. This is good, although I don't believe such a particle exists.

3. ArXiv articles let authors get away with statements like this:

This is also good—physicists should be allowed to express their emotions from time to time. But Weyl's gauge idea was not in the least futile (its 1929 resurrection is perhaps the most profound idea in physics). And it's Hermann, not Herman, dammit.

And what kind of language is "his gauge idea may turn be out"?

Recommended reading, but don't take it too seriously. By the way, Profs. Nishino and Rajpoot seem to have advanced their theories to the point where others are treading on their intellectual toes. Hence this arXiv diatribe:
We point out that the works described by Foot et al. in arXiv:0706.1829 [hep-ph] and arXiv:0709.2750 [hep-ph] are derivatives of our work described in arXiv:hep-th/0403039, the extended version of which was published in "Standard Model and SU(5) GUT with Local Scale Invariance and the Weylon", AIP Conf. Proc. 881 (2007) pp. 82, Melville, New York, 2006. We are wondering how many motions (and publications!) they will go through before finally admitting that they have re-discovered our model, and of course, as is the usual practice these days, claiming afterwards to the world of their independent arrival at our model. Reference to our original work is long overdue.

1039 -- Posted by wostraub on Tuesday, November 24 2009
Paul Dirac explains his Large Numbers Hypothesis. Audio only, undated (early 1970s); about 9 minutes.

More on Dirac -- Posted by wostraub on Friday, November 20 2009
Farmelo's book on Dirac is a revelation, far and away better than any other Dirac biography available. Hermann Weyl is mentioned numerous times, although only in connection with the early (1929-1931) debate as to whether the negative-energy solutions of the Dirac relativistic electron equation involved protons or some kind of new, positively-charged particle. After discarding the proton possibility, it was Weyl who seems to have first noticed that the new particle would have to have the same mass as the electron. Caltech physicist Carl Anderson discovered it experimentally in 1932, when it was named the positron—the first particle of antimatter ever detected. It looked exactly like the electron, but spiraled the wrong way in the detector's magnetic field.

For students of the arcane (like me), here is Dirac's modest Tallahassee, Florida home as it appears today, courtesy of Google Maps:

The house at 223 Chapel Drive. The Google Maps camera van caught someone mowing the lawn!

There is also a picture of the house in Farmelo's book, evidently taken some years ago. I wonder if the current residents are aware that a giant of science (and perhaps the greatest physicist who ever lived) used to reside there.

And directly across the street (at 224 Chapel Drive) is the Chapel Hill Baptist Church. I think Dirac would have thought it poetic in some sense, given that Wolfgang Pauli once exposed Dirac's ambiguous religious nature with his famous quote "There is no God, and Dirac is His prophet."

"I Owe Him Absolutely Nothing" -- Posted by wostraub on Thursday, November 19 2009
Although I remain fascinated by the work of Hermann Weyl, my favorite physicist has always been Paul Adrien Maurice Dirac, the theorist who discovered antimatter and the relativistic electron equation (though Dirac once admitted that even he could not understand Weyl).

I'm reading one of the few biographies of Dirac (and one of the even fewer good biographies), this one called The Strangest Man: The Hidden Life of Paul Dirac, Mystic of the Atom by Graham Farmelo.

Dirac was a good-looking guy, but here he looks like Tesla

In the book's introduction, Farmelo describes how Dirac, after a lifetime of brilliant achievement and awards, happened to leave Cambridge and take up a research position at—of all places— Florida State University (the stunned but elated head of the physics department noted that getting Dirac was like the English department getting Shakespeare). He goes on to describe how the school's top cellular biologist, Kurt Hofer, became acquainted with Dirac and his wife, and the discussions he had with them at the Dirac's home on 223 Chapel Drive in Tallahassee.

The always taciturn and reticent Paul Dirac would rarely say anything during these visits, as was his particular custom. It was not out of disinterest, shyness or aloofness; that was just the way he was. Hofer vividly recalls one visit in which Dirac stopped staring at the fireplace and began to talk about his childhood in Bristol, England. His father, a Swiss linquist, was stern and cold, and treated his wife like a doormat and his three children the same. But for whatever reason he insisted that his son Paul speak to him only in French, the father's native language. And when Paul made any mistakes in his speech, he was severely punished.

Dirac as a young child, in 1907. Sans hair, and near life's end in 1984, he looked the same!

Paul, not particularly gifted in any language other than English, reacted by saying nothing. He then began an uncontrolled habit of vomiting during dinners with his father.

Hofer noted with shock how the normally quiet and dispassionate Dirac became agitated during this talk, but was even more shocked when Dirac added angrily "I never knew love or affection as a child," and then, speaking of his father, burst out contemptuously with "I owe him absolutely nothing."

Dirac's taciturn nature has always been one of the key aspects of the lore of Dirac, who won the Nobel prize in physics in 1933 at the age of 31. But Farmelo's book demonstrates that there was a sad pathology behind Dirac's "absent-minded professor" persona, one that I was not previously aware of, and one that I now find heart wrenching in the extreme.

I haven't finished the book yet, but so far it has been one of the best scientific biographies I've ever read. Published in September 2009, my local library has four copies of the book and they're all checked out. Obviously, I'm not the only person in Pasadena who's impressed with it!

How Weyl Came to America -- Posted by wostraub on Thursday, November 12 2009
In 1929, a wealthy American department store owner and his wife thought it would be nice to have a renowned institute of top-notch scientists and mathematicians to rival the great European centers of learning. So they hired a medical administrator, Abraham Flexner, to get one for them.

Steve Batterson's Pursuit of Genius: Flexner, Einstein, and the Early Faculty at the Institute for Advanced Study brilliantly documents Flexner's efforts to recruit the world's greatest scientists and mathematicians and place them in the finest institute money could buy. It's the story of how Flexner wooed the likes of Albert Einstein, Hermann Weyl, Oswald Veblen, Kurt Gödel and others to come to America at what turned out to be a critical time in European history. It's also the story of how Flexner, by a combination of salesmanship and cash, outwitted Caltech president Robert Millikan in enticing Einstein to forgo sunny Pasadena and come instead to Princeton.

That's Weyl in the upper left corner.

Weyl's story is particularly interesting. By 1929, Weyl was the preeminent mathematician at the ETH in Zürich, Switzerland, a renowned university in its own right. But the German-born Weyl had emotional ties to Göttingen, Germany, where he had obtained his PhD in mathematics in 1908 (at the age of 22) under the guidance of the 20th century's greatest mathematician, David Hilbert. Zürich's climate was better for Weyl's asthma than Germany, but with Hilbert's impending retirement from Göttingen in 1930 Weyl was lured into taking the mathematics chair at the famous school.

Shortly before his arrival, Weyl began to have second thoughts. Germany's hyperinflationary problems were still evident in 1930, and Adolf Hitler was becoming more and more popular with the disgruntled German populace. In addition, though Weyl was a Christian his wife was of Jewish descent, a bad heritage in the eyes of the upcoming Nazi Party. Weyl accepted the job, but when Hitler was appointed Chancellor in January 1933, Weyl knew his days in Germany were probably numbered. By early April of that year, almost all of Weyl's Jewish colleagues had been summarily fired by the Nazi regime (of course, much worse was to follow).

However, with not one but several invitations from Flexner to join the fledgling Institute for Advanced Study, Weyl could not quite make up his mind to leave Göttingen (Batterson ironically notes how Weyl could simultaneously work on multiple mathematical problems, but could not handle the binary problem of accepting a job!) Although his salary in Princeton would be more than generous, Weyl worried about cultural and language issues as well as his ongoing asthmatic condition. Just as importantly, Weyl was a native German, and he loathed the idea of leaving his native country.

Batterson's book includes similar stories of how other notable European scientists and mathematicians were brought together under often trying circumstances. Though the noted American mathematical genius George Birkhoff was one of his recruits, Flexner was concerned about the overwhelming European (predominantly German) complexion of the other recruits and how the new institute would suitably incorporate these men and women into the Depression-era city of Princeton, New Jersey.

A fascinating look at the founding of the IAS and its greatest faculty members.

Another Side of Weyl -- Posted by wostraub on Wednesday, November 11 2009
Earlier on this site I spoke of Walter J. Moore's 1992 book Schrödinger: Life and Thought, in which the rather sordid extramarital affairs (Seitensprünge, literally, leaps to the side!) of Erwin Schrödinger and Hermann Weyl are described. Sordid for me, but according to Moore such escapades were no big deal in the Weimar 1920s, at least for brilliant scientists and mathematicians.

Erwin Schrödinger, 1887-1961

Weyl once described the 1926 discovery of the wave equation by his close colleague Schrödinger as the result of an exceptional two-week burst of erotic activity in the Alps, when the seemingly indefatigable Schrödinger and an unknown lady friend desisted just long enough for Erwin to write his Nobel-winning paper on the wave equation (which is actually a diffusion equation, dammit).

Weyl is given credit with assisting Erwin with the mathematics, but Weyl was not exactly preoccupied with his work either, as he was busy with none other than Schrödinger's wife, Anny. Indeed, though Schrödinger was a lifelong devoted husband to Anny (he once described her as a great friend but a lousy lover), he didn't seem to mind Weyl's adulterous behavior, due I imagine to the liberal attitudes toward this sort of thing as mentioned in Moore's book. And it seems that Weyl's own wife, Hella, was herself enamored with another scientist, Peter Scherer (sorry, I'm getting dizzy already).

Here are just a few of the now-forgotten women involved in this sad tale. On the left is Ithi Junger, yet another of Erwin's conquests which, if Moore is to believed, is just one of dozens who fell under his spell. In the middle is Hildegrund (Hilde) March, the wife of Arthur March, another physicist colleague of Schrödinger's (Hilde was the mother of Erwin's first out-of-wedlock child); and finally we have Anny, Weyl's heartthrob, at least for a number of years in the 1920s.

Feet of clay, indeed.

Weyl and Philosophy -- Posted by wostraub on Saturday, November 7 2009
While rattling off the names of numerous German philosophers in my previous post, I did not mean to give the impression that I am at all knowledgeable in philosophy. Quite the contrary; I find the field fiendishly difficult, and to me the only thing more difficult than Weyl's mathematics is his philosophy, of which he wrote extensively as it evolved over his lifetime.

However, so as not to give the alternative impression that I am a complete philosophical dunce, I recently finished Simon Critchley's The Book of Dead Philosophers, which barely covers any philosophy but instead describes how some famous philosophers died. To me, the only real value of philosophy is that it teaches us how to die; or, to paraphrase Montaigne, "to philosophize is to learn to have death in your mouth." In this context, I think Weyl would have appreciated the following words from the introduction to Critchley's book:
... what defines human life in our corner of the planet at the present time is not just a fear of death, but an overwhelming terror of annihilation. This is a terror both of the inevitability of our demise with its future prospect of pain and possibly meaningless suffering, and the horror of what lies in the grave other than our body nailed into a box and lowered into the earth to become wormfood.

We are led, on the one hand, to deny the fact of death and to run headlong into the watery pleasures of forgetfulness, intoxication and the mindless accumulation of money and possessions. On the other hand, the terror of annihilation leads us blindly into a belief in the magical forms of salvation and promises of immortality offered by certain varieties of traditional religion and many New Age (and some rather older age) sophistries. What we seem to seek is either the transitory consolation of momentary oblivion or a miraculous redemption in the afterlife.
And therein lies the crux of all our problems: the fear of death is the motivation behind both our country's trillion-dollar military budget and the fundamentalist Christian, Jewish and Muslim insistence that the world become a universal theocracy—by choice, if possible, or by force, if necessary. Just as misery loves company, fundamentalist faith requires the uniform and universal application of religious thought and action; does not an inescapable ubiquity of faithful behavior make us feel that our beliefs—which cannot, after all, be proven mathematically or scientifically—are all real, all true? For example, the Mormon Church strives to maintain an atmosphere or "world" in which all entertainment, social interaction and religious activity take place within the oversight of the church to constantly encourage the idea that the church's teachings are true. In this sense, most of what we call "fellowship" in the world's churches can be viewed as nothing more than a kind of communal brainwashing in which conformity is the rule and independent thinking is outlawed.

It is therefore lamentable to me that Jesus Christ—man, prophet and God—today has to preach His beautiful, unfathomable teachings on wisdom, truth, and the love of one's fellow man under a preposterously absurd load of miracles and other magical nonsense, so that otherwise rational human beings are forced to struggle not only with His divine message, which is difficult enough to live by, but with a heap of unnecessary metaphysical baggage as well.

Weyl and the LHC? -- Posted by wostraub on Saturday, November 7 2009
At the risk of appearing obsessive over the seeming purely metaphysical problems of the Large Hadron Collider, and in view of the pressing need to keep the memory of Hermann Weyl at least somewhat active on this site (!), I'd like to quote Weyl from a manuscript he prepared in 1949 that seems curiously apropos:
Here some words of Aristotle come to mind which, to be true, refer to metaphysics rather than mathematics. Stressing its uselessness as much as Hardy does in his apology of mathematics, but at the same time its divinity, he says "For this reason its acquisition might justly be supposed to be beyond human power; since in many aspects human nature is servile; in which case, as Simonides says 'God alone can have this privilege', and man should only seek the knowledge which is of concern to him. Indeed, if the prets are right and the Deity is by nature jealous, it is probable in this case they would be particularly jealous and all those who step beyond are liable to misfortune."
(This quote also appears in Erhard Scholz' excellent article Philosophy as a cultural resource and medium of reflection for Hermann Weyl, which explores the influence of philosophers such as Hegel, Heidegger, Fichte and Husserl on Weyl's approach to mathematics and physics.) Of course, I don't imagine for a moment that Weyl ever believed that some aspects of universal knowledge are considered "off limits" by a jealous God who acts to sabotage our efforts in protecting certain of His secrets. Instead, Weyl goes on to what is perhaps a more realistic (if cynical) view of mankind's scientific striving:
For who can close his eyes against the menace of our self-destruction by science; the alarming fact is that the rapid progress of scientific knowledge is unparalleled by a congruous growth of man's moral strength and responsibility, which has hardly chance in historical time.
The dangers of mathematics and physics inherent in the ability of mankind to wage modern wars (as a German, Weyl was uncomfortably close to two of them) and to fashion weapons of mass destruction (Weyl observed with more than a note of sadness the development of the atomic bomb and, in 1949, that of thermonuclear weapons) is indeed not in keeping with our lagging moral progress, as Weyl asserted.

Lest one think I'm being overly dark here, I'm not. To prove it, here's a take on the Large Hadron Collider over at the pseudo-wiki Unencyclopedia. Enjoy, all ye who enter there.

And one more bird joke --

© Kristofer Straub, 2009

For the Birds -- Posted by wostraub on Saturday, November 7 2009

Just when we thought it couldn't get any weirder, the European Large Hadron Collider gets shuts down by a bird dropping a piece of bread into the works.
Are not two sparrows sold for a penny? Yet not one of them falls to the ground apart from the will of your Father. — Matthew 10:29

A bird! A very bird! — Humbert Humbert, Lolita
At least the LHC was not operating when the bird strike occurred, so the planned start-up will proceed as scheduled later this month. But, given the sheer complexity of the machine and the number of things that can go wrong, I'm beginning to wonder if there isn't something to those claims that, if not God, then the Universe itself is sabotaging the collider from the future. Where is all this going?

NOW IT'S WAR—Birds from the future attack the LHC in earnest.

It Was Like That When I Got Here -- Posted by wostraub on Tuesday, October 27 2009
In a lecture given at Yale University in 1931, Hermann Weyl remarked
Many people think that modern science is far removed from God. I find, on the contrary, that it is much more difficult today for the knowing person to approach God from history, from the spiritual side of the world, and from morals; for there we encounter the suffering and evil in the world, which it is difficult to bring into harmony with an all–merciful and almighty God. In this domain we have evidently not yet succeeded in raising the veil with which our human nature covers the essence of things. But in our knowledge of physical nature we have penetrated so far that we can obtain a vision of the flawless harmony which is in conformity with sublime reason.
Years earlier, in 1919, Weyl mused over the disparity of the classical electromagnetic and gravitational forces acting between a proton and an electron. Noting the enormity of the dimensionless ratio of the two forces (roughly 2.3 × 1039), Weyl suggested that it must point to some profound meaning. As more accurate values of fundamental physical constants became available, more and more physicists began to have the same notion.

Then in 1937, Paul Dirac proposed his "Large Numbers Hypothesis" (LNH), which states that all large dimensionless numbers in nature are somehow connected. Though Dirac was a quantum physicist, he noted that the LNH seemed to extend into the cosmological realm as well. For example, the ratio between the age of the universe (about 13.7 billion years) and the purely atomic unit e2mc2 (where e is electronic charge, m is electron mass, and c is the speed of light) is about 7 × 1039. Similarly, the ratio between the radius of the observed universe and the classical electron radius is roughly 1040. Other derived ratios involve squares and cubes of this number: the value of the cosmological constant in units of the Planck length squared is about 10-120; the estimated number of protons in the observable universe is 1080; and the total action of the universe in terms of the Planck constant is about 10120. (Not all these figures are ratios, but you get the idea.)

In looking over these numbers, Dirac considered two possibilities. One, they are coincidences; and two, they result because of some conspiracy of nature. In a 1937 paper he wrote
Now, you might say, this is a remarkable coincidence. But it is rather hard to believe that. One feels that there must be some connection between these very large numbers, a connection which we cannot explain at present but which we shall be able to explain in the future when we have a better knowledge both of atomic theory and of cosmology.
Dirac's summary comments included
It is proposed that all the very large dimensionless numbers which can be constructed from the important natural constants of cosmology and atomic theory are connected by simple mathematical relations involving coefficients of the order of magnitude unity. The main consequences of this assumption are investigated and it is found that a satisfactory theory of cosmology can be built up from it.
Dirac's thoughts on the LNH fueled later suppositions that at least some of the fundamental constants of nature were changing with time. In the 1937 paper he noted
Let us assume that [numbers involving the size of the universe] are connected. Now one of these numbers is not a constant. The age of the universe, of course, gets bigger and bigger as the universe gets older. So the other one must be increasing also in the same proportion.
These ideas culminated in a 1973 paper by Dirac (available here), in which he examined a weakening of the gravitational constant G through an appeal to Weyl's 1918 theory of the unified gravitational/electromagnetic field.

So if it's a conspiracy, like Dirac imagined, just what the hell is going on? What's it all about? Does God have anything to do with it?

In 1973, the British physicist Brandon Carter came up with the anthropic principle, which essentially states that conditions in the universe (including the values of the fundamental physical constants) must be such that intelligent life is possible. He further postulated that the existence of intelligent life presupposes a "privileged" set of space-time conditions that may exist only at certain ages of the universe. The seeming existence of such privileged conditions has been used by many theologians to justify the belief in God. Such an approach is largely illusory, however, because there are alternative explanations, the most notable being the "many worlds" interpretation of quantum mechanics and the associated multiverse theory. But I think it might be safe to say that there really are only three logical views: (1) The universe and its set of physical constants are what they are because God made them that way to make life possible; (2) The set of physical constants is one hell of a coincidence ("it was like that when I got here"); and (3) The number of possible multiverses (and the set of physical constants) is infinite, and we just happen to be in one in which life is possible ("it was like that when I got here" also applies in this case).

Whatever or whichever is the truth (and it may even be some combination), one has to admit that none of the above alternative views has anything to say about the teleology of the universe, intelligent or otherwise. Teleology has to do with the ultimate purpose of things; in this sense, it is even more fundamental than the concept of driving force, which can always be explained on the basis of non-equilibrium. I think it really makes no sense to talk about teleology without invoking God, because the subject is so far removed from human understanding that one is forced to "pass the buck" on to a supernatural entity who supposedly knows all the answers. The question "Why is there anything?" is very much on the minds of physicists these days, and it continues to resist all attempts at an answer. I admit that by bailing out on the teleological aspect of all existence I am simply copping out, but I'm pretty sure that the theologians don't know any better, either. I have always loved asking pastors and the like questions such as 'What will we do in Heaven? What will our purpose be there?" only to hear them hem and haw with non-answers like "We'll just bask in God's presence, sing songs of praise and kind of look around and admire the place." For eternity?! Please, even God would be bored out of His mind with that.

This brings us back to Weyl's opening remarks, in which he implies that memorizing scripture, studying dubious (if not outright contradictory) biblical histories and accounts, and otherwise ignoring the vast ocean of physical and mathematical truth all around us is a colossal waste of our God-given intellectual capabilities. I think Weyl would insist that developing a consistent, workable theory of quantum gravity would bring us closer to God and His plans (whatever they may be) is a far better human undertaking than occupying our minds with nonsense such as talking donkeys, sticks turning into snakes, parting seas, and millions of Israelites camping out at 42 sites for 40 years in the Sinai without leaving a single trace. And I think that when the time does come for God to reveal His plans (we'll all have died), our souls will be in a far better position to understand it.

Roy Kerr -- Posted by wostraub on Saturday, October 10 2009
In 1963, Cambridge-trained mathematician Roy Kerr solved a fiendishly difficult problem in gravitational physics: the field of a massive, rotating body. The Kerr metric describes the spacetime surrounding a spinning black hole, a body that is in fact much more common and natural than its non-rotating counterpart (as interstellar gas condenses to form a star, the gas cloud's angular momentum must be conserved, and this momentum persists throughout the star's entire life, including final gravitational collapse to the black hole state).

Roy Kerr, 1934-

Now a book has appeared that describes Kerr's mathematical odyssey in detail for the first time. Cracking the Code: Relativity and the Birth of Black Hole Physics by University of Arizona physicist Fulvio Melia takes us through Einstein's field equations, the Schwarzschild description for non-rotating bodies, and the early tentative work of Lense and Thirring to show how Kerr arrived at his solution. While the mathematics is minimal, the interested reader can turn to Adler, Bazin and Schiffer's book (described elsewhere on this website) for a complete mathematical treatment (but be warned—though elementary, it is not an easy read).

The history of black holes is as fascinating as the subject itself. For a variety of reasons, Einstein and many others did not believe black holes could exist. As a star undergoes gravitational collapse at the burn-out phase (exhaustion of hydrogen, helium and other light elements undergoing fusion), nature intercedes against complete collapse via degenerate electron pressure, in which the Pauli exclusion principle acts to prevent further compression of electrons. But gravity has a way of bypassing this pressure, and collapse of the star proceeds by fusing electrons with protons to form neutrons. The resulting neutron star represents the star's last bastion of hope against further collapse, and indeed many such stars have now been observed and studied in detail. But if the star's mass M exceeds a certain critical amount, then no known force of nature can forestall continued gravitational contraction; when this occurs, the star collapses to zero volume and essentially "winks out" of existence. All that's left is a singularity (a point of infinite density) hiding behind an event horizon of radius R = 2GM/c2. This is the so-called Schwarzschild solution.

For a Kerr black hole, the resulting spacetime geometry is far more interesting. In addition to an event horizon, other surfaces arise as a result of spacetime itself being dragged around the spinning central singularity. Whereas the journey of a cosmonaut into a Schwarzschild black hole is predicted to be brief and fatal, the consequences of a similar plunge into a Kerr black hole are uncertain; the voyager may even find herself being flung into another time or part of the universe.

Although the mathematics of black holes are not questioned, there are those who are still not comfortable with the idea of a gravitational singularity, especially in view of the fact that a quantum theory of gravitation still eludes us. The October 2009 issue of Scientific American includes an article describing black stars, theoretical bodies that share many of the exterior properties of black holes but whose interiors are not singularities, just fantastically-dense stellar cores. But such models are strictly hypothetical, because equations of state for gases and solids of even nuclear density are so well known today that the avoidance of complete collapse is highly improbable.

I don't know if the mathematics of the Kerr model have been explored by researchers to date in semiclassical gravity theories or in attempts to formulate realistic quantum gravity theories. Perhaps there is some kind of quantum-level phenomenon that prevents total collapse and the formation of a singularity. But even so, I doubt that it would rob us of the intrinsic wonder of the mysterious object we today call a black hole.

Unreal -- Posted by wostraub on Wednesday, October 7 2009
Here is the damnedest optical illusion I've ever seen, courtesy of John Baez' website:

Are the colors (or shades) of squares A and B the same or different?

Unbelievably, they're the same. I didn't believe it, either, so I downloaded the picture and used Adobe Photoshop to sample A's color and compare it with B's. They are exactly the same! I then went over to the Wikipedia site to find out why.

It mainly involves the eye's (or rather the brain's) ability to detect patterns in order to form meaningful images. It turns out that the eye is not a very good light meter. Whatever.

I'm still shaking my head.

On the Invisibility of the Lorentz Contraction -- Posted by wostraub on Thursday, September 24 2009
I got an email this morning asking me if the Lorentz contraction is really observable. Just exactly what this has to do with Hermann Weyl (the presumed honoré of this website) I really couldn't say, but what I do know is that I'm no expert on Lorentz, and I tend to be just as puzzled by it as the woman who emailed me. I wrote back telling her to read through the back issues of John Baez' posts, as he probably explains it far better than anyone else can. But I'll share what little I know here.

I recall a problem I had on a physics exam from long ago involving the Lorentz contraction of a moving sphere. It is easy to prove that, in order for light from the trailing edge of the sphere to reach the observer at the same time as the front edge, then the light from the back end has to have a head start. The net result is that the sphere appears to be rotated, not contracted; consequently, a moving sphere still looks like a sphere, and not the oblate spheroid that might have been imagined when Einstein first wrote his famous 1905 paper on special relativity. (For the same reason, a rod moving in the positive x-direction would appear to rotate counterclockwise about the z-axis, giving an observer the impression that it has actually contracted.)

See Marion and Thornton's Classical Dynamics (1988) for a nice description of this situation

After the exam, I asked my professor about all this. Imagining a charged sphere with a uniform charge density, I wondered how the electromagnetic four-vector could remain consistent with a Lorentz transformation if the sphere remains a sphere. That is, the charge density of a moving charged rod would contract, forcing the charge density to increase in accordance with special relativity. A moving sphere would not contract, so the charge density would be invariant, in violation of relativity. The professor told me that the Lorentz contraction is real in both cases, but you can't actually observe it for a moving sphere. Seeing my confusion, he referred me to a paper written by James Terrell in 1959. Well, the paper didn't help much, and I remain confused to this day on the optical visibility of the Lorentz contraction.

If all you need is a little elementary background on special relativity, my write-ups on the four-frequency of light and the magnetic field may be of help to you.

Anyway, here is Terrell's 1959 paper on the visibility/invisibility of the Lorentz contraction. I re-read it this morning, hoping that 40 years' worth of mental "maturing" would make it an easier read for me. But no go—I still don't get it.

Misner - Wheeler Again -- Posted by wostraub on Tuesday, September 15 2009
In response to several emails I received this morning, here's a quick follow-up on the Misner-Wheeler "already unified" field theory I mentioned in the last post. I will be intentionally terse because the details are too involved. Instead, you are referred to the original paper:

C. Misner and J. Wheeler, Classical physics as geometry: gravitation, electromagnetism, unquantized charge and mass as properties of curved empty space. Annals of Physics, 2:525-603 (1957).

Basically, Misner and Wheeler derived a set of conditions that must be imposed on the Ricci tensor Rμν and the contracted Ricci scalar R for these geometric quantities to be consistent with a spacetime in which a sourceless electromagnetic field exists; that is, it must satisfy not only Einstein's field equations but Maxwell's equations for the electromagnetic field. The remarkable thing about the MW theory is that these conditions do not involve anything that looks like electromagnetism whatsoever: the conditions are all purely GEOMETRIC! It is in this sense that the MW theory describes gravitation and electromagnetism as "already unified."

The MW theory assumes a special coordinate system in which tensor indices can be ignored; the tensor quantities in question can then all be treated as 4×4 matrices, which simplifies things enormously.

The Einstein field equations for a source-free electromagnetic field are given by (sorry for the crappy typesetting; I'll find something better someday):

Rμν - ½ Rgμν = FλμFνλ - ¼ gμν FαβFαβ

If we contract both sides with gμν, we immediately get the first MW condition, which is R = 0.

Two other conditions (which I will not bother deriving here) are

00 0

RμνRνβ = ¼ gμβRλαRλα

The above three conditions were also derived by Rainich in 1925 using an approach different from that of MW.

MW derived a fourth, purely-geometric condition that is too complicated for me to express here; you'll have to read the paper if you're curious.

The MW equations are fourth-order, highly non-linear and difficult to solve. Wheeler suggested that in order to solve the equations, one might introduce a fictitious antisymmetric tensor Fμν that satisfies Maxwell's equations, use it to build a fictitious stress-energy term, and stick this into the right-hand side of Einstein's field equations. In this sense, the Maxwell tensor Fμν becomes just a convenient artifice for solving the field equations. If this view is correct (and remember that nobody has yet solved the MW equations) then one may view the electromagnetic field tensor Fμν as a purely mathematical construct that for years has been given physical meaning!

It is interesting to note that Hermann Weyl's 1918 gauge theory produced a set of field equations in which the "½" coefficient in Einstein's theory is replaced by ¼. Weyl's field equations with a sourceless electromagnetic field energy term are therefore completely traceless, thus eliminating the R = 0 condition.

Einstein once remarked that the left-hand side of the gravitational field equations, which consist of purely geometric quantities, is a beautiful mathematical edifice that is "made out of marble," while the right-hand side, which holds the matter and energy terms, is an ugly construct that is "made out of wood." What he meant by this is that, while the pure gravity terms can be derived by straightforward mathematical arguments, there is a considerable amount of guesswork and arbitrariness associated with the matter/energy terms. For example, one can easily express the right-hand side for cold, catalyzed, non-interacting matter in terms of a mass density term (no collisions, no pressure, no gravitational/electromagnetic interaction, etc.), but ordinary matter does not behave this way. In particular, pure matter terms have to essentially be forced into the field equations because we don't really know where they come from, and the interaction terms have to be guessed at. This was why Einstein, Weyl and others had hoped that the then-new theory of general relativity would somehow provide an explanation for the structure of matter (and maybe even the "why" of matter).

However you may want to look at all this, I think that the philosophical and religious aspects of mankind's ongoing quest for the "why" and "how" of things is profound beyond words.

The 1993 book by Earman et al. has a nice summary of the MW unified theory, along with those of Weyl, Einstein, Eddington, Kaluza-Klein, Mie, Cartan and others.

Weyl on Matter and Singularities -- Posted by wostraub on Monday, September 14 2009
It has always amazed me how quickly advancements are made to fundamental physical discoveries. Example: In 1916, just months after Einstein's publication of his gravity theory, the German physicist Karl Schwarzschild used the theory to calculate the relativistically correct description of planetary orbits, which immediately provided the explanation for the planet Mercury's anomalous 43 arc-second discrepancy compared with the Newtonian result. (Schwarzschild sadly died months after his discovery from injuries received on the German front). And in that same year, the German aeronautical engineer Hans Jacob Reissner (1874-1967) discovered an equation similar to that derived by Schwarzschild but describing the field of a charged mass point. (Oddly, it took almost another 50 years for the metric of a massive, spinning, charged object to be discovered.) Reissner was something of an odd duck: he was not a scientist at all, but rather an engineer who dabbled in mathematical physics; he won the German Iron Cross in 1918 for his work in airfoil design and was highly regarded by the Nazis in the early 1930s, but left abruptly in 1938 for a teaching position at the Illinois Institute of Technology.

Anyway, in 1918 Hermann Weyl presented his own derivation for the metric of a charged sphere in his book Raum-Zeit-Materie (Space-Time-Matter), and he came to the conclusion that matter represents a true singularity of the combined gravitational/electromagnetic field. Singularities in Weyl's day were viewed with much disdain because they represented a breakdown in the continuum view of space-time (nowadays they are still problematic, but for a wide range of other reasons). Weyl also noted that the then-classical view of electron structure would have to be modified in view of the nature of this singularity, adding that it was probably meaningless to talk about "forces of cohesion" holding the electron together or any direct relationship between electron mass and its surrounding electromagnetic field.

It must be remembered that, in those early days of relativity, the nature of matter was a preoccupation (if not outright obsession) with Weyl, Einstein and others, and it was hoped that not only the structure of matter but the "why" of matter might actually be describable by Einstein's physics. We now realize how naive this hope was, if only for the fact that in 1918 just two forces — gravitation and electromagnetism — were known (remember that this was only a few years after the "plum pudding" model of the atom had been discarded). But the sheer beauty of Einstein's mathematics and its interpretation in terms of space-time "warping" was powerful and compelling, and many scientists (including Einstein and Weyl) truly believed that a unified theory of the world was close at hand. Well, 90 years have now gone by, and we're still waiting for that theory!

A novel solution to the problem of singularities in the geometry of space-time was proposed by the late, great physicist John Archibald Wheeler and his colleague Charles Misner in 1957. Drawing upon work done in 1925 by George Rainich, the so-called "already unified" field theory of Wheeler and Misner is still a fun read, and the mathematics used in its development is a joy to behold (but that is a story I will go into another time). Wheeler dispenses with singularities by endowing space-time geometry with a novel topology. Imagine an empty, spherical, four-dimensional world; the introduction of a particle of matter now does not produce a singularity but instead produces a deformation of space-time which, according to Wheeler, resembles a beer stein:

Lines of gravitational and/or electromagnetic force produced by the particle at point P1 may now flow along the handle of the beer stein and reenter the world at point P2 without the introduction of any singularity. The addition of zillions of particles would follow the same scheme, resulting in a four-dimensional world whose geometry resembles a Gordian knot. Non-breaking of the associated zillions of handles might be made physically possible by going over to spacetimes of higher dimension.

Pretty neat stuff.

Hermann Weyl on the Relevance of Mathematics -- Posted by wostraub on Friday, September 11 2009
In his lecture given at the Princeton Bicentennial Conference in 1946, Hermann Weyl talked about the apparent irrelevance of pure mathematics in the real world:
I have wasted much time and effort on physical and philosophical speculations, but I do not regret it. I guess I needed them as a kind of intellectual mediation between the luminous ether of mathematics and the dark depths of human existence. While, according to Kierkegaard, religion speaks of "what concerns me unconditionally," pure mathematics may be said to speak of what is of no concern whatever to man. It is a tragic and strange fact — a superb malice of the Creator — that man's mind is so immensely better suited for handling what is irrelevant rather than what is relevant to him.
But Weyl went on to add that mathematics needs to be recognized as a practical tool:
In the intervals between the brain tortures of mathematical problems we must seek somehow to regain contact with the world as a whole. The probing of the foundations of mathematics during the last decades seems to favor a realistic conception of mathematics: its ultimate justification lies in its being a part of the theoretical construction of the real world.
Given the fact that Weyl gave this lecture in 1946, the "dark depths of human existence" he refers to must have had at least some footing in the tragic application of atomic weapons on Japan just one year earlier. It is ironic that the seeming irrelevance of mathematics on the real world is in fact a preposterous misconception in this case: the atom bomb is actually a very simple device in which two or more subcritical masses of Uranium-235 (or Plutonium-239) are quickly brought together to create a supercritical mass. But in order for a detonation to occur, the precise process by which the masses are mechanically brought together (TNT implosion) involves extremely complicated mathematics, and it was this problem that occupied most of the mathematicians who worked on the Manhattan Project from 1942 to 1945.

Weyl in his office, Zürich, 1927

Appropriately, Weyl did not address this aspect of practical mathematical application in his lecture, but instead touched on the need to recognize the purer aspects of the field in the development of a better theoretical understanding of the world.

Still, I think Weyl missed his chance. He did not work on the Manhattan Project at all, preferring to remain at the Institute for Advanced Study at Princeton to pursue more mundane mathematical research. I have not been able to fully elucidate Weyl's true feelings about nuclear weapons except to note that he was far more pacifist than Edward Teller, the Manhattan Project physicist whose abiding post-war dream was to build the "Super," the thermonuclear fusion bomb that has now threatened the world for 60 years. But it seems very strange to me that Weyl did not at least touch on the moral aspect of "practical" mathematics at a time when other noted scientists, including Einstein and Oppenheimer, were left aghast and outraged by the previous year's incineration of over 200,000 Japanese civilians. These scientists were not afraid to openly voice their views, even when persecuted by the FBI and the House Un-American Activities Committee.

And it is doubly strange in view of Weyl's spiritual quests and admitted lifelong preoccupation with philosophical issues, his love for which going back to his post-graduate years at Göttingen.

For a dog or a mouse or a microorganism, all mathematics is irrelevant. But for humans, math has two primary purposes: one, to assist us in inventing the tools we need to help us survive or improve our world; and two, to help us understand the universe we live in. The first is indispensable, while the second satisfies a craving that only humans can experience — the need to discover and to know. It is an enduring sadness that for some of us — primarily the Republicans and others of their kind wholly incapable of higher thought — this second aspect of mathematics fulfills no purpose whatsoever.

I freely admit that if I had never studied science, I would be an avowed atheist today, struggling to comprehend a seemingly absurd universe that propagates itself in accordance with undeniably beautiful physical and mathematical laws while totally devoid of purpose. Instead, as a consequence of my very limited understanding of science and math and, in particular, what I know about the action principle, it is my unwavering belief that a benevolent God exists. I would like to think that Weyl shared these same views.

Hermann Weyl on Mind, Nature and God -- Posted by wostraub on Thursday, August 27 2009
Another rant, this time disguised as a comment on Weyl's view of God —

I occasionally go over to Google Books to look for some book of interest. Often, most (if not all) of the book can be read there free, although graphics and tables usually don't come out right.

Today I was looking for Princeton University Press' new (April 2009) book on Hermann Weyl called Mind and Nature — Selected Writings on Philosophy, Mathematics and Physics, and lo and behold I found it here. It's basically a collection of excerpts and such from some of his longer writings, along with an informative introduction by the book's editor, Peter Pesic. The book's only $28 at Amazon, but since I've pretty much read everything in it already, I'll pass. Anyway, much of what Weyl has to say in these writings deals with philosophy and, in a roundabout way, faith.

I've never seen this picture of Weyl before.

For whatever reason, of late I've become fascinated with Judeo-Christian religious historicity. Although I consider myself to be a Christian, I long ago gave up on most of the fabulous stories and miracles of the Old Testament. They're simply ridiculous, particularly when one compares the book's historicity with the archaeological evidence of the past 30 years. (To see what I'm getting at, read Finkelstein and Silberman's The Bible Unearthed: Archaeology's New Vision of Ancient Israel and the Origin of its Sacred Texts or, if you're feeling more scholarly, you can look at Donald Redford's authoritative Egypt, Canaan, and Israel in Ancient Times. If you then still believe there was an actual Exodus event, you have my sympathy.) My last book (it was dense and took me 8 hours yesterday to get through the damn thing), was Lindsey's A Gathering of Saints: A True Story of Money, Murder and Deceit, which relates the detailed story of Mark Hofmann's Mormon document forgeries (if you know the stories behind the Kinderhook plates, the "Book of Abraham" and Joseph Smith's seer stones, and you are still a Mormon, then you're a complete idiot).

But, as Kelly Bundy would say, I digest. Einstein came to terms with his Jewish religious heritage at the tender age of 12, when he realized that many of the Bible's stories simply could not be true. His God, as is well known, was the God of Spinoza, indistinguishable from Nature herself. But I've never been able to fully discern Weyl's view of God, which is not as straightforward as Einstein's, if only because he never seems to have come right out and said it. In some ways, Weyl's vision of God was similar to Einstein's in the sense that Weyl viewed God not as Nature but as "a mathematician and mathematics itself, because mathematics is the science of the infinite" (The Open World, 1932). He also spoke of God as the "completed infinite" (and I won't pretend that I really understand that one). But to Weyl God was simply incapable of being comprehended by anyone. This I understand, because I fully agree with it.

While Weyl was shocked by the infinite capacity of man's self-deception (and I believe he meant that in terms of most religious belief), he seemed resigned to it, although by the time he had turned 69 (a year before he died) he was still trying to find his own way on the issue. As I mentioned, Einstein achieved his own peace with religion at a very early age, and he noted later that it had much to do with a strong sense of suspicion against authority (he once famously remarked that as punishment for his rebellious attitude against authority figures, God made Einstein an authority himself).

I'm now 60 years of age, and I find that while I have come to terms with God the Father and Jesus Christ the Savior (I am a believer but, like Jefferson, I find it necessary as well as easy to toss out all those silly miracles and stuff), accepting the ocean of mindless human self-delusion that I see all around me remains frustrating in the extreme. For example, I have nothing personal against Mormons (I worked with a number of them, and I truly liked them), but their continued belief in what has been proved beyond all doubt to be nothing but preposterous nonsense demonstrates that man's dread of death and oblivion will always outstrip his reasoning ability. I think the fact that Utah, the Mormon's ancestral home, is the Number One Republican state in the union, with something like an 80% approval rating for George W. Bush, proves my point.

Not to overuse that old chestnut of Voltaire's (If you can be made to believe in absurdities, you can be made to commit atrocities), but it's quite true. The United States was attacked on George W. Bush's watch (he was actually more asleep than at watch), and we lost some 3,000 souls to Islamic terrorists. For the next 7 years, I watched helplessly as terrified Americans abandoned their professed trust in the teachings of Jesus Christ and instead embraced radical militarism, torture and imprisonment of innocents as well as suspects, domestic spying, and the destruction of habeas corpus and a host of other civil liberties. Christians I thought I knew well screamed for blood and vengence, and not a few of them advocated the nuclear annihilation of all the Arab countries. Others were simply terrified that we would get hit again, and as recent as this year I knew people who would not venture into downtown Los Angeles.

But when reminded of the apostle Paul's assertion that he would rather die and be with Christ than go on living, all of the Christians I talked with said something to the effect that "I am not ready to die, I still have too many things to see and do." The hypocrisy of this attitude absolutely astounds me — if they were truly Christians, would not God allow them to "see and do" in the vast infinite spaces and worlds of an eternal afterlife? Again, the subconscious dread of death overcomes reason and the fear of future attacks becomes the rationale for the murder, torture and abuse of fellow human beings, while religious faith is comfortably maintained by following the advice of right-wing monsters waving flags and Bibles.

I now feel 90% certain that humankind will eventually destroy itself, primarily because of differences involving nationalistic and religious antagonisms coupled with a fervent desire to see the return of an end-times Messianic figure. Time will tell.

Physics v. Math -- Posted by wostraub on Saturday, August 1 2009
Einstein announced his theory of general relativity in November 1915, following roughly four years of intense effort. A little over a year later, Hermann Weyl was teaching a course on the subject at the Swiss Federal Technical Institute in Zürich, and a year after that he published the first edition of his great book Space-Time-Matter (Zaum-Zeit-Materie), which delved far deeper into relativity than Einstein ever ventured. Einstein did publish his own book on the subject in 1921-22 (The Meaning of Relativity), but it was vastly inferior to Weyl's in terms of depth, scope and style.

Historians of physics know well the story of Einstein's efforts to complete his 1915 theory. What is perhaps not as well known is the fact that Einstein was ridden with anxiety over being the first one to publish it. The great Göttingen mathematician David Hilbert dogged Einstein right to the very end, and in some respects it could be said that the theory could have been called Hilbert's. It was Hilbert who, in 1915, noticed that the gravitational field equations (which Einstein labored to derive by way of a number of inefficient avenues), could be derived in a matter of a few minutes's worth of scribbling using only the Ricci scalar density and the variational principle.

David Hilbert, 1862-1943

And it was Hilbert who, though a friend and esteemed colleague of Einstein's, once remarked that "Every schoolboy in the streets of Göttingen knows more about higher-dimensional geometry than Einstein, but chalk is cheaper than gray matter."

Einstein's response to Hilbert

Indeed, Einstein's theory was based heavily on tensor calculus, a field of mathematics that had been developed fifty years earlier through the work of Christoffel, Ricci, Levi-Civita and others. In his frantic search for a coordinate-invariant theory, Einstein had turned to a close friend (it was either Michele Besso or Marcel Grossmann), who put Einstein onto the idea of a purely tensor formulation of relativity. It is entirely possible that Einstein was not even aware of the existence of tensors until sometime in 1912!

And, if the great German mathematician Bernhard Riemann had thought of the idea of basing his differential geometry in terms of four dimensions (where the fourth dimension is time) instead of only three, he might very well have discovered general relativity sometime prior to 1866.

Georg Friedrich Bernhard Riemann, 1826-1866. His tragic death at 39 (tuberculosis) ended the life of the second-greatest mathematician who ever lived (Gauss is still No. 1)

Einstein is nonetheless given his due nowadays, by mathematicians and physicists alike. But one should keep in mind the attitude that was prevalent with mathematicians in the years following Einstein's publication of the 1915 theory, and that was that physics was too important to be left to the physicists!

I do not think that Weyl shared this view, however. Weyl is a classic example of a "crossover" researcher who waded into physics and found an entire new world to explore. Weyl was perhaps the first to fully realize the great import of Einstein's discovery, and he made fundamental contributions to the theory's advancement and extension. Conversely, it is said that Einstein did not really like mathematics; he once likened it to the "common enemy" of all physicists.

I think Einstein would be totally confused by the mathematics of today's string theory, while Weyl, I would venture to say, would at least see the beauty if it.

Nostradamus and the Large Hadron Collider -- Posted by wostraub on Friday, July 31 2009
After interminable delays, the European Large Hadron Collider is expected to start up again in mid-November 2009. This event, according to a family member of mine who will not be named here, will bring about the planet's destruction via the creation of an uncontrollable black hole that will swallow the Earth. On the plus side, it will also trigger The Rapture© just prior to this catastrophe, allowing faithful Christians to join Christ in the air and escape destruction. Yay!

The "Chosen Ones" get raptured away in 2009's Knowing before this nasty event comes about. Nicholas Cage is not one of them.

How does my relative know this? Because, you see, it was predicted by the French seer Nostradamus in the 16th century. So it must be true, of course!

The percentage of people in this country who claim to be Christians but at the same time believe fervently in superstition, healing magnets, dowsing rods, zodiacal astrology and the occult continues to amaze and disappoint me. It's all a load of nonsense, but these people will only believe what they want to believe. Besides, it takes years to learn quantum mechanics but only two hours to read all the crap Nostradamus ever wrote.

It's no wonder our science and math scores are so pathetic.
If you can be made to believe absurdities, you can be made to commit atrocities. — Voltaire

The End of Einstein -- Posted by wostraub on Monday, June 15 2009
Albert Einstein died on April 18, 1955 at a hospital in Princeton, New Jersey. When admitted, he knew the end was near; he had been diagnosed months earlier with an aortic abdominal aneurysm, and warned that he would die when it ruptured. He was also told that his death would likely be a horrible one.

While lying in the hospital bed, Einstein asked for his pencil and most recent calculations. They were given to him, and he continued working on what turned out to be his last thoughts on Earth — a unified theory of the gravitational-electrodynamic field.

When the end came, Einstein was in and out of consciousness. The attending nurse reported that he muttered something in German, a language she did not understand, then passed away. On the floor next to his bed lay several sheets of paper, covered with calculations that looked like gibberish to the nurse. Here is one of those sheets:

Herman Weyl would have instantly understood these calculations, as they involve a mathematical object he helped create: the coefficient of affine connection. Einstein hoped that a non-symmetrical connection term (also known as a connection with torsion) would provide a unified description of both gravitation and electromagnetism.

As Einstein lay dying, did he dare hope that he was on the right track, or was he merely exercising what tortured faculties he had left? We'll never know.
Linger yet, thou art so fair! — Goethe

Weyl and Einstein -- Posted by wostraub on Monday, June 15 2009
Hermann Weyl's April 1918 metric gauge theory brought rave reviews from the world's physicists, Einstein included. "It is a work of highest genius," wrote Einstein to his colleague shortly after the theory's publication. But then Einstein caught something amiss — the line element ds of both Weylian and Riemannian geometry was not gauge invariant, so that the lengths of all vectors would depend on their histories. In particular, atomic spectral lines would vary from point to point in spacetime, quite in contradiction with experience.

But Weyl protested, indicating that God would not have missed such a golden opportunity to embed all of electrodynamics into the geometry of the world, as Einstein had done with gravitation. Einstein responded with
Could one really accuse the Lord God of being inconsistent if He passed up the opportunity you discovered to harmonize the physical world? I think not. If He had made the world according to your plan, I would have said to Him rather reproachfully: "Dear God, if it did not lie within Thy power to give an objective meaning to the vector equality of separated rigid bodies, why hast Thou, Oh Incomprehensible One, not refrained from preserving their shapes?
In his communication to Weyl, Einstein was reminding his friend that what we perceive as perfect beauty may not be what God actually had in mind. If disproven, string theory will certainly end up being the perfect example of such a disconnect between our minds and God's.

Particle physicists talk about something called "broken symmetry," which can allegorically be taken to mean that perfect beauty sometimes has to step aside so that the physical world can actually occur (for the mathematical details, see my write-up on Weyl and Higgs Theory). For example, think of the "punt" of a wine bottle, the symmetrical hill of raised glass on the bottom that results from the manufacturing process. Now imagine a marble balanced at the very top of this punt. It's a perfect symmetry, but a fragile one — equilibrium demands that the marble roll down the side and spoil everything.

I see this also as an example of the world God made for us. The birth of a newborn child is conceptually beautiful, but the actual process of birth (not to mention what brought it about nine months earlier) is a rather sticky, ugly mess.

It also serves to disprove the prettified worldview of conservatives, who see the world and its problems strictly in black-and-white, good-and-evil terms. Instead, God painted the universe in grey tones, a fact that seems to have eluded the country's Republicans.

Jean-Lou Chameau on American Science -- Posted by wostraub on Saturday, June 13 2009
Jean-Lou Chameau is Caltech's 8th president, a French civil engineer who's interested in earthquakes (S. California is a good place to study them). He was interviewed recently by the Los Angeles Times, of which the following is a portion:

America generates some of the most important science in the world, and yet many Americans don't believe in evolution, are skeptical of science. How do you explain this paradox?
I do not know — I don't have a background in sociology. But if you look at the major universities and research, there is no doubt that we have the greatest work. On the other hand, we have a K-through-12 system where, for some reason, either science is not valued enough or it is not well communicated to the students. So you have a significant majority of the public that has a relatively limited knowledge of science, and it may also be afraid of it because it was not really a major part of the education. And we are always afraid of the unknown. The issues that are facing us — energy, global change, health, water — science and technology are key to all those solutions. But when we need it most, we have a large segment of the population with limited knowledge of them.
Alas, all too true. He could also have been asked how American science/mathematics could ever compete with Twitter, Facebook, MySpace, American Idol and Carrie Prejean, but we all know the answer: it can't. Ignorance can be fixed, but stupidity is forever.

My favorite retort from the conservative anti-evolutionists is that evolution is just a theory. I tell them quantum mechanics, gravitation, thermodynamics and electrodynamics are also just theories, but it's these theories that drive their world. And if nuclear physics is also just a theory, why does America base its global security on 10,000 theoretical thermonuclear weapons?

I also tell them that there are no laws in science — it's theory all the way down. But if America's moronic 43rd president could convince them that 1 + 1 = kat, how on Earth are they ever going to learn the truth?

Rosen Again -- Posted by wostraub on Monday, June 8 2009
A few posts ago I related efforts I'd made to get a 1982 paper by Nathan Rosen ("Weyl's Geometry and Physics"), as abstracts of the paper promised a novel new approach to the Weyl-Dirac theory. Thanks to my local library's inter-library loan program, I received the reprint today. I've converted it into a pdf file here for those who may want to read it themselves.

It's an easy read, even if it is a trifle long at 36 pages. Rosen seems to have borrowed heavily from Adler-Bazin-Schiffer's book, at least in terms of notation and style, although he doesn't reference it. Otherwise, Rosen's approach is very original.

Like Dirac, Rosen approaches the problem of non-integrability of vector length by using a subterfuge that's actually better than the one Dirac devised. Basically, Rosen redefines the Weyl affine connection for both contravariant and covariant vectors and combines them into a new connection that produces conformally invariant scalar, vector and tensor quantities. And like Dirac again, he uses a variation of the Weyl-Dirac action Lagrangian to show that the Newtonian gravitational constant G might indeed vary inversely with universal time.

It was Dirac's original idea to produce a theory in which G gets weaker as the universe gets older. Long ago, Dirac had noticed that the magnitudes of certain dimensionless ratios (like that of the electric to the gravitational force between the electron and proton) appeared again and again in certain mathematical expressions he was playing around with. Sensing this could not be a coincidence, Dirac was subsequently led to believe that this could only be the case if the gravitational constant varied slowly with time.

Dirac could not adequately explore this idea using Einstein's theory of general relativity because Einstein's theory presupposes that G is a true constant. In 1973, Dirac turned to Weyl's original 1918 gauge theory and found it was much better suited to his purposes. Rosen references Dirac's paper, which I had posted here several years back. It's well worth wading through.

Idiot America -- Posted by wostraub on Saturday, June 6 2009
I just finished reading Bart Ehrman's Jesus, Interrupted and Charles Pierce's Idiot America. Pierce's book is a bit of a rant, but it speaks the truth and is enormously entertaining as well. Conversely, Ehrman is a highly respected Bible scholar who has published extensively on the provenance of the New Testament gospels and epistles. I've now read a total of three of his 20 or so books.

The commonality between these two books involves human reasoning or, more accurately, non-reasoning. Pierce proposes that in America today we have created and embraced a belief system that is based on hucksterism and entertainment, not facts or truth. By comparison, Jesus, Interrupted examines the fallacies behind the true authorship of the New Testament and why we believe what we believe irregardless of the historical facts.

There is an old Arab proverb that says when God made us humans, we immediately began to complain about our supposed physical imperfections (our ears and noses are too big, etc.). But the one thing we are completely satisfied with is our minds and the way we think — we're all perfectly happy with our thought processes and the validity of our belief systems. How many of us has said "I wish I had the same opinions as my neighbor"?

Sadly, Americans have reached the point where they can no longer reason anymore.

Pierce states that he was motivated to write his book after visiting the Creationism Museum in Petersburg, Kentucky, which is the work of a group called Answers in Genesis. When he saw the throngs of people attending the museum, which features a life-size fiberglass Triceratops outfitted with a saddle (proving that humans and dinosaurs had coexisted 6,000 years ago), he knew he had to address just how idiotic Americans had become.

Was Jesus 30 feet tall, or is this a mini-Diplodocus?

I myself thought I had seen it all, but now I learn that the AiG people have explained dinosaur fossils as either the work of Satan (to deceive us) or God (to test our faith). They now readily admit (albeit reluctantly) that dinosaur fossils are real, and that radiometric potassium-argon dating methods are accurate. But they insist that we mustn't believe our own eyes and the results we behold, lest the Tempter succeed in deceiving us, or God be disappointed that our faith is weak. I guess this explains why Satan was "walking up and down" in the Earth in the Old Testament book of Job — he was planting fossils!

[For a wonderful overview of radiometric dating from a Christian perspective, see this article by Dr. Roger Wiens, a Los Alamos physicist.]

Pierce claims that there are two types of eccentric folk in America: the harmless, lovable crank and the dangerous con artist. The con artists — the Hannitys, the Limbaughs, the O'Reillys, the AiG people, the Bushes, the Cheneys and their ilk — have won over America because Americans have dumbed themselves down to the point of utter stupidity.

Ehrman reaches the same conclusion but is loathe to admit it. Examples: Why do people believe that Matthew (a tax collector), and Peter (a fisherman), along with the other apostles, all illiterate, uneducated, Aramaic-speaking men, could write so brilliantly in the Greek language? And why is it that all the New Testament books and letters we have today are copies of copies of copies of copies written hundreds of years after Christ was crucified? Ehrman provides innumerable other examples of why the New Testament is largely a human book. It may indeed have been inspired by God, but the nearly uncountable omissions, additions and modifications contained in the New Testament indicate that God had no intention of preserving His words. If He did, He wouldn't have left it up to us errant humans.

And Ehrman does not neglect the Old Testament. For example, in Psalm 137 the writer (presumed to have been Asaph, or David, or even Solomon) claims that the smashing of Babylonian infants against rocks brings happiness. Now, did God inspire this pornographically-awful thought, or was it just the later musing of a vengeful, intertestimental Jewish scribe still pissed off over Israel's defeat by the Babylonians in 586 BC? Much of the Old Testament is nothing more than a protracted bloodbath. I find it hard to believe that God could have ever been so heartless toward His creation.

I am a Christian, primarily by choice. Much of what the Bible says makes sense to me, while a lot of it doesn't. I believe that by giving us sentience and free will, God had to provide a means of salvation for us, which is Jesus Christ. But I engage this belief out of choice, not because of what some authority tells me to believe. And my belief does not give me the right to physically threaten others for believing otherwise, or to encourage my political leaders to invade, occupy or nuke non-believing nations.

As Ehrman points out in his book, the Bible has to be read horizontally (by simultaneously comparing the books in detail) as well as vertically (one book at a time). Most Christians don't do this; in fact, they don't even read the Bible — they peruse it. And they generally aren't even aware that other writings even exist. What kind of belief system is this?

By dumbing ourselves down to the extent we have, we've chosen to ignore the real proof of God's existence and benevolence — His creation of the universe and all its laws — and instead have elected to rely on our very imperfect thought processes and nonsensical belief systems.

Like Pierce warns at the end of his book, things don't look good for us unless we make some changes. And I really don't see that happening.

Nathan Rosen -- Posted by wostraub on Thursday, May 28 2009
Today a reader asked me if I'd ever heard of the Weyl-Dirac theory. I replied that I have, and I wrote about it on this site some time ago; I think it's over in one of my post archives.

Anyway, the theory really is nothing more than Weyl's action Lagrangian with a modification made by Paul Dirac in 1973. To me, it's an ugly theory and one unbecoming the great Dirac, who once remarked that good equations have to be beautiful.

But the Weyl-Dirac "theory" still pops up occasionally, the latest being an otherwise excellent overview of the idea provided by the German Weyl-expert Eric Scholz. (Here's what appears to be an overhead presentation he gave in February of this year.)

The article includes a reference to a paper on the theory by Nathan Rosen (1909 - 1995), the noted Israeli physicist who was once the young colleague of Einstein. The paper is entitled "Weyl's Geometry and Physics," which appeared in the journal Foundations of Physics in 1982. (Yes, this is the paper I alluded to in my previous post.)

Since I wasn't able to find the paper anywhere, let me just use this occasion to mention how strangely circular theoretical physics often is.

Rosen was the co-author with Einstein (along with Boris Podolsky) of their famous 1935 paper "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?" (Yes, the title's grammar seems to be wrong, but it was Einstein, you see). In an even earlier post I mentioned how this paper, and the famous "EPR paradox" it presented to the world, was voted as one of the top five physics papers of all time.

Thanks again, Wikipedia!

It was also Einstein and Rosen who came up with a mathematically consistent gravitational metric that describes a Lorentzian wormhole (or "Einstein-Rosen bridge"). The wormhole, as every kindergarten kid knows, is a tubular, traversable deformation in spacetime that can be used to overcome the local speed limit of light; two points in space (or time!) can thus be connected superluminally, but only if you happen to own a wormhole.

Unfortunately, in 1962 John Archibald Wheeler (a good buddy of Weyl) proved that wormholes are unstable and would collapse at the speed of light. Thus, something is needed to hold the wormhole open long enough for me to get through (yes, I desperately want to get out of here). It has been theorized by Kip Thorne and others that such a "Samson-like" substance might actually exist in the form of some kind of exotic, negative-energy matter. Alas, Home Depot doesn't carry this stuff yet.

So again we see the familiar tag team of Weyl, Dirac, Einstein and Wheeler, and now we can add Rosen to the list.

If anyone out there has a copy of Rosen's 1982 Weyl paper, or if you happen to have a spare non-collapsing wormhole lying around, please let me know.

Some Holes in the Theory -- Posted by wostraub on Saturday, May 23 2009
Earlier I posted the two papers Dirac published in early 1928 on the relativistic electron. Dirac's electron equation soon became the cornerstone of modern quantum theory, but it took several years to iron out its full interpretation. One stumbling block had to do with the fact that the equation predicted negative energy states for the electron. Another involved the seeming over-abundance of solutions: whereas Schrödinger's equation resulted in a single expression for the electron's wave function, Dirac's gave four — two with positive energy, and two with negative energy.

Paul Adrien Maurice Dirac, 1902 - 1984

It was soon recognized that the two positive-energy solutions had to do with the electron's two spin states (up and down), but the negative-energy states continued to perplex even the brightest physicists, Dirac included. In his 1929 paper Elektron und Gravitation, Hermann Weyl proposed that these extra energy solutions might have something to do with the proton, the only other "elementary" particle then known (other than the photon). But Weyl subsequently discovered that the particle associated with these extra solutions had to have the same mass as the electron. Upon reading Weyl's paper and considering his ideas, Dirac was motivated to come up with another answer, that of holes in an electron sea.

Dirac's basic idea was this: an ordinary, free, positive-energy electron was fully allowed to collapse into a negative energy state, but if all those states around it were filled it would be prevented from doing so by the Pauli exclusion principle. Dirac envisioned a universe bathed in a "sea" of negative-energy "holes," which ordinarily prevented a garden-variety electron from collapsing into a negative energy state. But these "holes" also acted like particles having the same mass as an electron, and they were also allowed to propagate into the overlying positive-energy sea, though the exclusion principle still held.

Although Dirac's hole theory was praised at the time as a remarkable achievement in itself, Dirac was still bothered by its tenuous interpretation. In May 1931, he published a paper in which the hole theory was abandoned. In its place, Dirac proposed that the negative-energy states represented ordinary particles identical to electrons with the exception that they had positive charges. This bold prediction became the basis for the concept of antimatter. While Dirac had little to support his idea except for some fancy mathematics, he intuitively felt it to be correct. Then in 1932 the Caltech physicist Carl Anderson announced the discovery of the antielectron, which today is known as the positron. The positron, along with antiprotons, antineutrons and a veritable zoo of other forms of antimatter, have all been produced countless times in high-energy particle experiments and are an indisputable and inescapable consequence of modern quantum field theory. Dirac is therefore rightfully called the father of antimatter.

Years later, the famed Caltech physicist Richard Feynman provided a brilliant interpretation of antiparticles that is completely consistent with the mathematics he used to describe them. Noting that the term exp(-iEt/ℏ) invariably accompanies positive-energy particle expressions (where E, t are the energy and time and ℏ is Planck's constant divided by 2π), Feynman discovered that the term was invariant when describing negative-energy antiparticles propagating backward in time. (I recall a talk show many years ago — it may have been Dick Cavett — in which Feynman discussed this idea, completely flabbergasting his host.)

Dirac won the Nobel Prize in Physics in 1933 at the age of 31. He is inarguably the equal of Isaac Newton as the greatest physicist who ever lived (Dirac's powers far exceeded those of Einstein), and it is a great wonder to today's physicists that Dirac's name is largely unknown to the general public.

Dirac knew Hermann Weyl well and was often intimidated by Weyl's own prowess, although he considered Weyl to be primarily a mathematical genius with a gift for physics. Nevertheless, he admired Weyl's tendency to see physical truth in mathematics alone, and once admitted that he would have dismissed his hole theory immediately if he had maintained absolute faith in the mathematics of his electron equation. Dirac once wrote
Weyl was a mathematician. He was not a physicist at all. He was just concerned with the mathematical consequences of an idea, working out what can be deduced from the various symmetries. And this mathematical approach led directly to the conclusion that the holes would have to have the same mass as the electrons. Weyl did not make any comments on the physical assertions. Perhaps he did not really care what the physical implications were. He was just concerned with achieving consistent mathematics.
Like Weyl, Dirac was quick to give credit to others. You may be interested to know that the famous Heisenberg equation of motion

i ℏ dA/dt =
[A, H] + i ℏ ∂A/∂t

which describes the total time dependence of an operator A in terms of its commutation with the Hamiltonian energy operator H, was in fact first derived by Dirac. Dirac ascribed the equation to Heisenberg in his 1930 book Principles of Quantum Mechanics solely out of professional deference to an esteemed colleague that he did not always see eye to eye with!

More on Weyl -- Posted by wostraub on Monday, May 4 2009
I've been perusing Katherine Brading's 2001 PhD dissertation Symmetries, Conservation Laws and Noether's Theorem, which is available online. Brading is a professor of philosophy at Notre Dame University in Indiana and, while I don't really understand philosophy at all, her mathematical arguments are not only understandable but insightful as well.

Brading has given the best description of why local gauge symmetry is not only beautiful but necessary, although the basic idea has been addressed by many researchers over the years. Essentially, it's this:

In the action Lagrangians for quantum theories, one always finds terms such as Ψ*Ψ, either as written here or in combination with their first derivatives with respect to spacetime. If one replaces Ψ with exp(-iΛ) Ψ (where the coefficient Λ is a constant), the Lagrangian does not change because the mulitplier exp(-iΛ) and its complex conjugate cancel one another. Because the coefficient is a constant everywhere, one therefore says that quantum physics is invariant with respect to a global change of gauge. But if we demand that the coefficient be a constant over all spacetime, it has to be set up simultaneously as one moves from one point to another. This violates the spirit of special relativity, which says that a physical effect cannot occur faster than the speed of light.

Consequently, one has to abandon the idea of a constant Λ and replace it instead with a coefficient that is allowed to change continuously from point to point. The multiplier then becomes exp[-iΛ(x)]. Since this change has to be completely arbitrary (otherwise it would still violate special relativity), the coefficient Λ(x) must itself be totally arbitrary and structureless. This gives maximum freedom to the gauge, and we now have what is called a local gauge symmetry. (In modern quantum theory, the coefficient Λ is proportional to a square matrix, which sets up the action for the strong nuclear force.)

However, derivatives such as ∂μ exp(-iΛ)Ψ will now bring down derivatives of the gauge multiplier into the Lagrangian as well and, unless these cancel somehow, the action will no longer be gauge invariant. We are thus obliged to introduce additional terms into the Lagrangian that will provide this cancellation. These terms essentially serve as electrodynamic interactions, while the terms involving Ψ and its derivatives act as the kinetic terms. The completed action is now fully gauge invariant, and its variation with respect to the gauge parameter leads immediately to the familiar law of conservation of electric charge. The action is also automatically invariant with regard to position, momentum, angular momentum, energy and other conserved quantities. Beautiful beyond belief!

Of course, Hermann Weyl was not aware of the quantum wave function Ψ when he developed his first gauge theory in 1918, as quantum mechanics had not yet been discovered. But his basic idea that nature should be gauge invariant was essentially correct, and he eventually carried the idea over into quantum physics in his 1929 foundation paper Elektron und Gravitation.

Eighty Years -- Posted by wostraub on Sunday, May 3 2009
Eighty years ago this month Hermann Weyl published a seminal paper (Elektron und Gravitation, Zeit. f. Physik 330 56) that forever changed how we view nature. The paper introduced numerous new and/or novel applications of mathematical physics, several of which were vigorously attacked at the time of publication on the basis of their interpretation within the then still-evolving quantum theory. While full vindication of these efforts came about only shortly after his death in 1955, I believe Weyl witnessed enough progress in modern physics to have been more than satisfied with his contributions.

Although Weyl had more or less abandoned his earlier 1918 metric gauge theory (also known as conformal invariance), by 1929 he was still intrigued by the deep mathematical symmetries he sensed between gravity and electromagnetism. Weyl also thought that a true unification of these forces would shed light on the problem of matter, which was another subject of great interest to him from the previous decade.

The primary accomplishment of Weyl's 1929 paper was his derivation of the formal relationship between charge conservation and the gauge (phase) invariance of the quantum mechanical wave function. But what is so fascinating about the paper is Weyl's orthogonal approach to the problem. He first developed a 2-component spinor formalism which established the basic mathematical physics behind neutrinos, parity violation and time reversal — ideas that were to stun later physicists when they realized the full extent of their importance in the 1950s. Weyl then formalized the use of local tetrads (also known as vierbeins) as a means of transcribing quantum physics into curved manifolds, particularly spinors in non-flat space. In doing so, he discovered the spin connection, a kind of affine connection for spinor space akin to the ordinary connection term found in Riemannian geometry. With his tetrad formalism, Weyl then established a profound similarity between the Riemann curvature tensor R and the electromagnetic field tensor F. Today's physicists still shake their heads in awe at this similarity.

It was not until the very last section of his paper that Weyl established the connection between the gauge principle and electrodynamics. It is here that Weyl took the basic idea of global phase invariance and brilliantly extended it to the non-local case. Weyl thus established the abelian U(1) symmetry property of modern quantum mechanics. Weyl's 1929 paper also acted the the impetus for subsequent generalization of this symmetry to the non-abelian case. It is unfortunate that Weyl (who died in 1955) appeared to have been unaware of the 1954 Yang-Mills theory, a seminal paper in it own right which established the non-abelian approach to the description of the strong nuclear force.

[For a very readable contemporary discussion of these ideas, see Katherine Brading's excellent overview of Weyl's charge conservation principle and Noether's theorem.]

Although primarily a mathematician, Weyl was one of the earliest to also carry out fundamental investigations in mathematical physics. He was therefore a somewhat more enlightened individual than his friend and colleague Einstein, who was a brilliant physicist but rather plodding mathematician (the great German mathematician Hilbert once remarked that any Göttingen schoolboy knew more mathematics than Einstein). The source of Weyl's fascination in physics was of course his deep recognition of the profound mathematical symmetries that lie in nature.

Indeed, in 1960 the Nobel physicist Eugene Wigner wrote a paper expounding on the "unreasonable effectiveness" of mathematics in the natural sciences. In his paper Wigner addresses the completely inexplicable success of man's physical theories, not only as described by mathematics but as a consequence of mathematics. We now know that mathematical symmetries are largely (if not solely) responsible for this success, and it is through the action principle (or variational principle) that these symmetries translate into all the known physical and conservation laws.

Symmetry is a form of beauty, and beauty speaks for truth. Who but God could have set down these mathematical principles?

The Greatest -- Posted by wostraub on Monday, March 23 2009
Several years ago Discover magazine asked its readers to name the greatest physics papers of all time. The top five were Newton's Principia (which was actually a book); Einstein's 1915 general theory of relativity; the Einstein-Podolsky-Rosen paper of 1935 (see my write-up on Bell's Inequality for more information); Noether's 1918 paper on symmetry and conservation laws (see my write-up on Weyl and Higgs Theory); and Dirac's two-part paper from 1928 on the relativistic electron equation (see my write-up on Weyl Spinors and the Dirac Equation).

I don't remember if any single paper actually won top honors, but I voted for Dirac.

In my opinion, Dirac's electron equation represents the greatest intellectual achievement of humankind. The paper, published in the Proceedings of the Royal Society when Dirac was only 25 years and a few months of age, was universally hailed as a work of highest genius. The equation, and the wonderful matrices that Dirac discovered, quickly became the cornerstone of all modern quantum physics.

I've been asked on numerous occasions to post Dirac's two-part paper on this website, and here it is at last. My copy of Part I is of rather poor quality, but it's still quite readable.

Eighty-one years ago, God spoke to Dirac's mind. The Quantum Theory of the Electron is the result of that interaction. Enjoy.

Part 1
Part 2