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

 Hermann Weyl
 

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

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

I came across Hermann Weyl (pronounced vile) and his ideas on gauge invariance many years ago after finding Misner-Thorne-Wheeler's book Gravitation during a one-week work assignment in Lone Pine, California (population maybe 1,000). Miraculously, the town's miniscule public library somehow had this book, which is now regarded as a classic graduate text on geometrodynamics, also known as general relativity. I checked out the book and brought it back to my motel room to read in the off-hours. The book took immediate and total possession of me, and motivated me to learn everything I could about general relativity.

I then came upon another book, Adler/Bazin/Schiffer's Introduction to General Relativity, which included a chapter on unified field theory. Weyl's 1918 theory of the unified electrodynamic/gravitational field was one of the topics in that chapter. The mathematical beauty of the theory absolutely fascinated me. I had known about phase invariance in quantum mechanics, but did not know that Weyl's theory was the origin of this powerful symmetry in quantum physics. I have since read all of Weyl's books and many of his papers, and my fascination with the man and his theories continues to grow.

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

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

This website is my feeble attempt to document Weyl's ideas in a manner that will be accessible to anyone with a basic understanding of calculus. Not a lot has been written about the original theory's underlying mathematics, and I wanted to provide a fairly detailed and complete mathematical description for those who want to learn about Weyl's ideas and to appreciate the beauty of his gauge theory. As this site progresses, I will also include discussions of other topics in mathematical physics (as well as some related scientific philosophy) which exhibit a similar mathematical beauty and elegance.
 
 
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My Secret Book -- Posted by wostraub on Tuesday, June 30 2009
Although ignorance makes people believe what is false, they need to be both ignorant and arrogant to assert it so shamelessly. &mdash Petrarch, ca. 1347

I like your Christ; I do not like your Christians. Your Christians are so unlike your Christ. — Mahatma Gandhi

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! &mdash 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 587 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