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Who Was Hermann Weyl?

Wheeler's Tribute to Weyl (PDF)

Old Stuff
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Math Tools
Weyl's Spinor and Dirac's Equation
Weyl's Conformal Tensor
Weyl Conformal Gravity
Weyl's 1918 Theory
Weyl's 1918 Theory Revisited
Weyl v. Schrodinger
Why Did Weyl's Theory Fail?
Did Weyl Screw Up?
Weyl and the Aharonov-Bohm Effect
The Bianchi Identities in Weyl Space
A Child's Guide to Spinors
Levi-Civita Rhymes with Lolita
Weyl's Scale Factor
Weyl's Spin Connection
Weyl and Higgs Theory
Weyl & Schrodinger - Two Geometries
Lorentz Transformation of Weyl Spinors
Riemannian Vectors in Weyl Space
Introduction to Quantum Field Theory
Electron Spin
Clebsch-Gordan Calculator
Bell's Inequality
The Four-Frequency of Light
There Must Be a Magnetic Field!
Non-Metricity and the RC Tensor
Curvature Tensor Components
Kaluza-Klein Theory
The Divergence Myth in Gauss-Bonnet Gravity
Schrodinger Geometry
A Brief Look at Gaussian Integrals
Particle Chart

Einstein's 1931 Pasadena Home Today

Why I'm No Longer a Christian

Uncommon Valor

She did not forget Jesus!
"Long live freedom!"

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2007 Archives

Hermann Weyl and CPT Symmetry -- Posted by wostraub on Saturday, December 15 2007
As I noted earlier, I managed to locate a copy of Hermann Weyl's Zaum — Zeit — Materie: A General Introduction to His Scientific Work, and I'm still making my way through it. A considerable portion of the book is in German which — contrary to popular belief — I am only moderately fluent in, so it's taking some time.

Weyl seems to have been in love with group theory, especially the continuous Lie groups (SU(2), SU(3), and all that) and he appears to have concentrated the latter part of his life on this topic, along with other subjects in pure mathematics. His earlier interests in general relativity, cosmology and philosophy seem to have waned during this time of his life, and it leads me to wonder how his immigration to America in November 1933 might have affected his professional inclinations.

At any rate, I find Weyl's mathematics (unlike his physics) to be very difficult and hard to follow. But there is one thing that jumps out of the book at me, and that involves the following questions:
Why does mathematics describe the physical world so well? Why should Nature obey the laws of mathematical symmetries? Why does group theory govern so much of what goes on in the world?
These are hardly new questions, but the answers seem to be just as far from us today as they were in Weyl's time.

Even more intriguing is the fact that Weyl, in 1929, was able to deduce that Nature should also obey the discrete symmetries described by charge, parity and time (CPT) invariance. Even today, we haven't the slightest idea why God decided that these symmetries should carry so much power and influence over Nature. In a later edition of his 1928 book Group Theory and Quantum Mechanics, Weyl wrote
The problem of the proton and the electron is discussed in connection with the symmetry properties of the quantum laws with respect to the interchange of right and left [parity invariance], past and future [time invariance], and positive and negative electricity [charge invariance]. At present, no acceptable solution is in sight; I fear, that in the context of this problem, the clouds are rolling together to form a new, serious crisis in quantum mechanics.
Weyl's analysis of the Dirac and Maxwell equations in the context of combined CPT symmetry led him to the correct conclusion that the mass of the then newly-discovered positron (the anti-electron) should be identical to that of the electron. The "crisis" that Weyl referred to involved the then-prevailing opinion that the positron should be nothing more than the familiar proton, whose mass exceeds that of the electron by a factor of almost 2,000. Of course, we now know that there was no crisis at all. But the reason why CPT symmetry mandates these kinds of physical consequences remains a total mystery.
God and the Many-Worlds Interpretation of Quantum Mechanics -- Posted by wostraub on Monday, December 3 2007
I apologize for this overly-long post.

This month’s Scientific American (it’s becoming an oxymoron, isn’t it?) has a fascinating article about Hugh Everett III, the late physicist whose 1956 PhD dissertation formalized the idea of parallel universes. And once again we see the hand of the noted physicist John Archibald Wheeler (see my earlier post), who was Everett’s academic advisor at Princeton University. Wheeler, apparently enthralled by the multi-universe concept, went to Copenhagen to discuss the idea with the great Niels Bohr, no less, who, unfortunately, didn’t like the theory. Wheeler returned to Princeton and coerced Everett into paring down his dissertation to make it more conventionally acceptable. The PhD, which was finally awarded to Everett in 1957, was a shadow of its former self, but it nevertheless succeeded in bringing the idea of parallel universes into the scientific world.

Hugh Everett, 1930-1982

Everett’s idea (which is now called the many-worlds interpretation of quantum mechanics) is pretty bizarre. But for reasons I will explain shortly, it deserves serious consideration.

The basic idea is very simple. The Schrödinger equation says that the wave function Ψ of an object represents a superposition of possible physical states (rather like a complicated musical sound wave that consists of a linear combination of individual waves). The wave function is a complex-valued quantity, which means that it is essentially unobservable until a measurement is made. Then it collapses probabilistically but uniquely into one of its allowable states; the resulting state is a real-valued quantity, and is the one we humans can actually observe with our eyes and ears. Wave function collapse is one of the central pillars of what is known as the Copenhagen interpretation of quantum mechanics.

But since its development in the late 1920s, the Copenhagen interpretation has dogged physicists with the whole collapse idea. How does observation (which can be a very “gentle” process) bring about collapse? Does it take an intelligent entity to collapse a wave function? Can a mouse collapse a wave function? Why can’t we see a quantum-mechanical superposition of states? Why just one? These aren’t just idle musings; they involve the very foundation of that slippery thing we call reality.

Everett was also bothered by these questions, and he considered what might happen if a measurement doesn’t collapse a wave function. The conclusion was inescapable — incredulous as it sounds, if you flip a coin and get heads, then the universe must split off another universe in which a parallel YOU gets tails. Similarly, if you measure the energy state of a free particle (which has an infinite number of superposed energy states), an infinite number of universes is created, one for each possible observation. In both cases, the wave function remains intact and uncollapsed, and the only price we have to pay for this is our sanity!

(Everett went even farther than what I've sketched above. His original PhD thesis dealt with a continuously-evolving wave function for the entire universe, and included the observer as an intrinsic quantum participant in the overall observation process.)

The main reason this SciAm article aroused my interest is because I just finished reading Frank Tipler’s latest book, The Physics of Christianity. Tipler, a highly respected (and apparently even sane) mathematical physicist at Tulane University, wrote a similar (and more mathematical) book called The Physics of Immortality, and his latest effort is a slightly more refined follow-up. Basically, Tipler’s thesis is this: the Trinity of God, Jesus Christ and the Holy Spirit is all that Christianity says it is, and Everett’s many-worlds interpretation (MWI) proves it.

Assuming you are a relatively sane person, even a casual glance at Tipler’s writings will lift your eyebrows. I don’t for a minute accept a lot of what he says, despite really, really trying to follow his mathematical logic. But here’s something to consider: the majority (60%) of the world’s notable physicists believe (often grudgingly) that the MWI is either certainly true or probably true (see this reference). A few of their comments:
I think we are forced to accept the MWI if quantum mechanics is true. — Richard Feynman, Physics Nobel Laureate

I don’t see any way to avoid the MWI, but I wish someone would discover a way out. — Leon Lederman, Physics Nobel Laureate

I’m afraid I do [believe in the MWI]. I agree with John Archibald Wheeler, who once said that it is too much philosophical baggage to carry around, but I can’t see how to avoid carrying that baggage. — Philip Anderson, Physics Nobel Laureate

The MWI is okay. — Murray Gell-Mann, Physics Nobel Laureate

The MWI is trivially true. — Stephen Hawking

For what it is worth [I prefer the MWI over the Copenhagen interpretation]. — Steven Weinberg, Physics Nobel Laureate
To this list, Tipler adds another authority:
Jesus answered, “My kingdom is not of this world.” — John 18:36
Tipler could have also added:
“In my Father’s house are many mansions …” — John 14:2
I need to shut this down now, but lastly consider this: Tipler believes that God is not a magician, only (only!) an eternal and very clever physicist and mathematician who has figured everything out. If we can believe that God fashioned Eve out of a rib bone he yanked from Adam, we can surely believe in Everett’s many-worlds theory. I urge you to read The Physics of Christianity and decide for yourself.
Weyl, Wheeler and Wormholes -- Posted by wostraub on Sunday, December 2 2007
I've been reading Hermann Weyl's Raum-Zeit-Materie and a General Introduction to his Scientific Work, a neat collection of articles by noted Weylophiles Erhard Scholz, Skuli Sigurdsson, Hubert Goenner, Norbert Straumann, Robert Coleman and Herbert Korte. Having recently re-read Kip Thorne's book on wormholes, I was struck by a comment made by Coleman and Korte regarding Weyl's supposed "discovery" of the wormhole idea.

On Pages 198 and 199 of the book the writers provide a short list of Weyl's accomplishments, including his "invention of the wormhole concept in connection with his analysis of mass in terms of electromagnetic field energy." Since Thorne does not even mention Weyl in his book, I pulled Weyl's Space-Time-Matter off my shelf and went through it with a fine-toothed comb. Yes, Weyl talks at length there about electrodynamics and the problem of matter (and there's some discussion of "world canals" in Section 36), but I'll be damned if I can find anything remotely related to the wormhole concept.

Thorne has demonstrated that wormholes almost certainly cannot exist but, if they do, they would require a kind of negative-pressure exotic matter to keep them from collapsing. Nowhere in Thorne's book do I see any primary role for electromagnetism in relationship with this exotic matter.

I can imagine that, when Karl Schwarzschild wrote down the first exact solution to Einstein's gravitational field equations in 1916, the concept of a black hole (a term coined by John Wheeler in 1967) may have crossed his mind. However, black holes were quickly dismissed in those early days, and it is not hard to suppose that the idea of a wormhole (a term also coined by Wheeler in 1957) had not even been dreamed about.

I give Weyl credit for many wonderful ideas, but I don't think wormholes can be included on that list.
Not Even Wrong -- Posted by wostraub on Monday, November 5 2007
Wolfgang Pauli (1900-1958), the Austrian physics Wunderkind of the early-mid 20th century, often intimidated younger, inexperienced physicists by declaring their ideas ganz falsch, or "utterly wrong." Those who he really zeroed in on suffered the rather more blistering comment nicht falsch, or "not even wrong."

Not Even Wrong is the title of Peter Woit's poison pen-letter to string theory (and also the title of his fascinating website). Woit, a noted Columbia University physics lecturer who likens the untestable string theory to a kind of religion, feels that the theory's promise to unite the four fundamental forces of nature is nothing more than hope disguised as hyped progress.

Although the Standard Model of physics successfully unifies all of quantum theory with electrodynamics, it does so at the expense of assuming all kinds of physical constants that it cannot account for. But its most glaring oversight lies in the fact that it cannot incorporate Einstein's gravitation theory into the mix. To date, the Standard Model is 100% accurate in terms of its predictions of experimental quantum results, but it can tell us nothing about gravity. Over the past 90 years, gravity has steadfastly refused to associate itself with quantum theory despite the efforts of literally thousands of physicists, including Einstein himself (who spent the last 30 years of his life in the effort). The curmudgeonly Pauli himself also tried in vain, and finally declared that "what God hath put asunder, let no man join."

Woit's book is a great introduction to the Standard Model, including quantum field theory, but his description of the details of string theory is necessarily lacking, if only because the theory's mathematics is maddeningly difficult.

But as simplistic as it is, Woit's book has made me wonder if the ideas of Truth and Beauty, which I have always assumed to be identical, truly hold up. Although my own understanding of the mathematical details of string theory is limited, the parts I do understand are truly beautiful, and like many others I have tacitly assumed that string theory is too beautiful a concept for God to have overlooked.

But Woit warns us not to be overly impressed with Beauty alone, because it does not necessarily represent Truth. I had often suspected this, noting the concept of broken symmetry in quantum mechanics — if God's physical laws were perfect, then quantum symmetry could not be broken. It seems that although God started out with a great idea, he found it impractical — some imperfection is needed in the universe, if only to make things interesting. It goes without saying that God made mankind imperfect, but I believe he did this intentionally in order to give us free thought. Exactly why God gave us this gift or, for that matter, why he even gives a damn about us, is a profound mystery.

Woit considers string theory to be an "ossified ideology," and recommends that scientists now move on toward a fuller understanding of quantum field theory and its relationship to mathematics. Will string theory prove to be a waste of time and effort? Even if it is, it at least has given us a glimpse into the mind of God, which probably cannot be understood anyway.
Insects and Worldlines -- Posted by wostraub on Sunday, October 28 2007
2005, the "Year of Physics," brought about the appearance of I don't know how many more books on Einstein, no doubt inspired by the 100th anniversary of Einstein's annus mirabilis, 1905, the wonder year in which the 26-year-old Swiss patent clerk cum world-renowned scientist produced four papers that would forever change physics.

Now another book has appeared. Albert Einstein: The Persistent Illusion of Transience (edited by Ze'ev Rosenkranz and Barbara Wolff), is too slim (264 pages) to qualify as a coffee-table book, but its high-quality photographs of its subject more than make up for the book's brevity. I'm not sure that it really adds anything that we didn't already know about the man, but it's nice to see that people are still interested in him and his science.

Einstein used to quip that his fame grew out of his awareness of something that had escaped most people (and insects):
When the blind beetle crawls over the surface of a world globe, he doesn't realize that the track he makes [a "worldline" or geodesic] is curved. I was lucky enough to have spotted it.
In their very comprehensive (and, at nearly 1,300 pages, very long) 1973 foundation text Gravitation, Misner, Thorne and Wheeler also spotted it, this time using the analogy of an ant crawling over the surface of a piece of fruit ("The Parable of the Apple"):

It was the very first graphic in this book (above) that caught my eye one day in 1975, when I spotted the text on the shelves of the miniscule public library in Lone Pine, California. Widely viewed as the standard graduate-level text on general relativity, I wondered how in hell it had landed in a tiny town whose only claim to fame was that, as the portal to Mt. Whitney, it had hosted Humphrey Bogart and company during the filming of the 1940 classic High Sierra. The book was my companion on a day-long hike up the 14,000-foot mountain during that glorious summer that I discovered the miracle of general relativity. It also brought me closer to God, whose miracles and wonders I continue to marvel at.
The Fox and the Forest -- Posted by wostraub on Tuesday, October 2 2007
Roger and Ann Kristen are government scientists, developing leprosy bombs and other high-tech disease-culture weaponry for a war that never seems to end. Unlike most American patriots in the fascist country of the United States in the year 2155, they hate what they do. They hate the killing, the fear-mongering, the torture, the constant propaganda, the all-pervading culture of death.

But they play along, and eventually they are rewarded with a vacation courtesy of the government-sponsored Travel in Time, Inc. Of course, they have to put up a bond and leave all their assets in government hands as assurance they will return after their time vacation. But where to go? They decide on New York City, 1938.

But Roger and Ann have no intention of staying in New York, nor do they plan to return to their own time. They run off to Mexico City, where they adopt the names Bill and Susan Travis. They've managed to take with them a small fortune in travelers checks, and plan to live out their lives in total anonymity, away from the horrors of 2155. They carefully erase all evidence of their escape, hoping they'll find peace in another place, another time.

Unfortunately for these little rabbits on the run, there's also a fox in the forest of time.

This is the premise of Ray Bradbury's brilliantly disturbing 1950 short story, The Fox and the Forest, which is reprinted in his collection of short stories, The Illustrated Man. It may even be on the Internet somewhere. At any rate, you should read it.

Bradbury is now 87 years old. I saw him frequently at the old Vagabond Theatre in Los Angeles in the late 1960s, where they used to run science fiction movies and silent films. He was a friendly, approachable guy who clearly loved his medium, which was mostly science fiction, fantasy and eccentric horror. Bradbury's The Fox and the Forest, The Lake, and The Small Assassin are irreducible masterpieces. I wish there were more writers of his caliber today. Or, at the very least, more writers with a moral sense.

Where would I go? Probably Europe in the mid-1920s. Or Victorian England.

Dear God in Heaven, anywhere but here and now.
Wolfgang Panofsky Dead at 88 -- Posted by wostraub on Thursday, September 27 2007
Stanford University's Wolfgang Panofsky is dead. The father of Stanford's linear electron accelerator and one of the discoverers of the neutral π meson, Panofsky was also noted for his abhorrence of nuclear weapons and their proliferation.

I remember a soft-spoken, kindly, balding Panofsky at a 2004 lecture he gave in Los Angeles. His entire talk was about nuclear proliferation and ways of reducing or eliminating the spread of nuclear weapons, and I recall being touched by the compassion the man felt toward humanity and the underlying sadness he felt about the seeming inevitability of mankind's willingness to wage war and the role that nuclear weapons would ever play in that insanity.

Panofsky was born in Berlin in April 1919. A family of intellectual Jews, the Panofskys left Nazi Germany in 1935 fearing for their lives. Wolfgang's father, a noted art historian, took up teaching at Columbia University and the Institute for Advanced Study. Wolfgang and his older brother were of high school age in Germany but were accepted at Princeton, where Wolfgang majored in physics. Graduating at the age of 19, he then went to Caltech in 1938 after receiving a personal invitation from the school's president, Nobel Laureate Robert Millikan.

Panofsky received his physics PhD in 1942 but, being a native German, was declared an enemy alien under California's Alien Exclusion Law. Millikan came to his defense, however, and Panofsky was granted naturalized citizenship.

He then went on to consult for the Manhattan Project in New Mexico, where he personally witnessed the Trinity bomb test from a B-29 bomber on July 16, 1945.

During a stint at UC Berkeley during the McCarthy era, Panofsky abruptly resigned after being coerced into signing a loyalty oath. He subsequently made his way to Stanford, where he distinguished himself not only in groundbreaking accelerator physics but in world peace. He was instrumental in developing the Atmospheric Test Ban Treaty of 1963 and, in 1972, the Antiballistic Missile Treaty.

He leaves his wife of 65 years, Adele, five children, eleven grandchildren, and two great-grandchildren.

There are so few of the great physicists like Panofsky still alive today. God bless him, and may we all meet with peacemakers like him in heaven.
The Connection (Γ αμν) Again -- Posted by wostraub on Tuesday, September 18 2007
In 1918 Hermann Weyl tried to unify gravity and electromagnetism by a generalization of Riemannian geometry. He did this by eliminating the notion that the magnitude of a vector is invariant with respect to parallel transport. In doing so, he was forced to identify the electromagnetic 4-potential with a non-zero covariant derivative of the metric tensor.

Subsequent to this effort, numerous other prominent physicists tried their hand at the unification game, which at the time was simplified by the fact that only two forces were then known — gravitation and electrodynamics. Einstein, Kaluza, Eddington, Pauli and Schrödinger each took their turns and, ultimately, their lumps.

Weyl’s effort remains notable for the fact that the geometry that describes his unification is invariant which regard to a local gauge variation of the metric tensor; this idea failed, but in 1929 Weyl applied the gauge concept to quantum mechanics, where it found a home. But why does the gauge idea work for the wave function and not the metric tensor?

The most obvious answer has to do with the fact that the wave function Ψ(x,t) is a complex-valued quantity whose meaning is clear only when its conjugate square Ψ*Ψ is taken. Even then, this square (though real) can only be understood as a probability. By comparison, the wave function by itself is at best a probability amplitude. The metric tensor gμν, on the other hand, is a purely real quantity that needs no “squaring.” Similarly, the invariant line element ds2 = gμν dxμdxν, which measures the interval between events in spacetime, is also a real quantity.

On the basis of Einstein’s criticism that the line element itself should be invariant with respect to gauge variations (but isn’t in Weyl’s geometry), Weyl decided to adjust the metric tensor via an exponential scale factor

gμνexp [ k ∫φμ dxμ ] gμν

where k is a constant and φμ is the Weyl vector (which he associated with the 4-potential). Weyl knew that in quantum mechanics this vector was a complex quantity; consequently, the adjusted metric tensor and the line element could be made gauge invariant by a suitable choice of the constant k. Thus, it is (gμν*gμν)1/2, and not gμν, that must be taken as real. The Weyl scale factor makes for some interesting physics, but its presence in Lagrangian actions introduces an integral term that is hard to interpret (it actually prevents the derivation of classical, tried-and-true equations of motion).

Eddington was aware of this defect, and in response he decided that the metric tensor should not be taken as the fundamental quantity. Instead, he chose to develop a theory based on the affine connection, which defines the parallel transport of vectors (the concept of a connection was first proposed by Cartan, and later expanded by Weyl). In Weyl’s original theory the connection term has φμ embedded in it, which makes the connection complex-valued. (Indeed, the terms making up the connection are to a large extent arbitrary; the connection only collapses to the usual Christoffel definition when a Riemannian manifold is imposed.)

This renewed focus on the connection term motivated Einstein and others to consider a connection that is non-symmetrical in its two lower indices. Indeed, the so-called theory of the non-symmetrical field occupied Einstein for the last decade or so of his life. Most physicists today consider the theory to have been a tragic waste of the great scientist’s time and effort.

The connection term is still an open topic in mathematical physics and differential geometry. If we do not impose the demand of a Riemannian manifold, its precise makeup is largely arbitrary. Is this how quantum effects enter into gravitation, as Weyl and Einstein had hoped? Probably not, although it can be argued that a connection describing internal spaces, possibly in multiple spacetime dimensions obeying higher gauge symmetries, may yet find application in the description of a consistent quantum gravity theory.
Ramanujan -- Posted by wostraub on Saturday, September 8 2007
I just finished reading Robert Kanigel's award-winning 1991 book The Man Who Knew Infinity: A Life of the Genius Ramanujan. The book's great length stands in stark contrast to the very brief life of its subject, the largely self-taught Tamil mathematician Srinivasa Ramanujan, who died in 1920 at the age of 32.

Ramanujan's genius was saved from obscurity by the noted British mathematician Godfrey Hardy, who brought the 25-year-old to Trinity College in 1913 and served as the younger man's mentor until Ramanujan's death by tuberculosis seven years later. Although devoted to Ramanujan, the book is almost equally a tribute to Hardy who, unlike many other noted scholars in his circle, saw Ramanujan as an equal and not as a talented but inferior person of color.

The book does not overlook the profound tragedy of genius cut off at an early age, and the author ponders what heights Ramanujan might have attained if he had lived longer. Ramanujan was particularly adept at evaluating truly complicated improper integrals, and I could not help but wonder what luck the mathematician might have had with the infinite-dimensional path integral of quantum field theory, which can only be solved perturbatively.

A devout Hindu, Ramanujan saw a divine hand in all mathematical expressions. "An equation for me has no meaning," he wrote, "unless it expresses a thought of God."

UPDATE. There's a new book out based on the life of Ramanujan.
Why Gödel Thought US Dictatorship Possible -- Posted by wostraub on Tuesday, September 4 2007
In his 2005 book A World Without Time, Brandais University philosophy professor Palle Yourgrau writes
Years later, asked for a legal analogy for his incompleteness theorem, [Gödel] would comment that a country that depended entirely upon the formal letter of its laws might well find itself defenseless against a crisis that had not, and could not, have been foreseen in its legal code. The analogue of his incompleteness theorem, applied to the law, would guarantee that for any legal code, even if intended to be fully explicit and complete, there would always be judgments "undecided" by the letter of the law.
If this is indeed how Gödel felt, then he unequivocally predicted that an event like 9/11 could plunge the United States into a dictatorship, an outcome that the Founding Fathers simply could not have foreseen.

[Gödel's later years were plagued by paranoia and hypochondria. Fearing that he would be poisoned by hospital doctors, he stopped eating and died in 1978 of self-imposed starvation. At the end, he weighed 65 pounds.]

We Americans are always bragging about how brilliant the Founding Fathers were in drafting the US Constitution. But I believe that Gödel was absolutely right -- the Founders could not have foreseen that their country would utilize an event (largely brought upon by itself) as an excuse to give the president dictatorial powers. And this is exactly what has happened.
Kurt Gödel and the US Constitution -- Posted by wostraub on Friday, August 31 2007
I noted in my previous post that in 1949 the brilliant Austrian-American mathematical logician Kurt Gödel had discovered a solution to Einstein's field equations that allowed for time travel. His discovery was presented to Einstein on the occasion of the latter's 70th birthday party. (See my September 25, 2005 post for more info.)

Kurt Gödel and friend, early 1950s

I neglected to mention that a year earlier Gödel believed he had discovered a logical inconsistency in the US Constitution that allowed for the establishment of a dictatorship in America -- and told a federal judge about it!

The story, which is true, has Gödel traveling by car with his Princeton colleagues Albert Einstein and economist Oskar Morgenstern to Trenton, New Jersey, where Gödel was to be sworn in for his US citizenship. During the drive, Gödel expressed his concern that an inconsistency in the US Constitution allowed for a dictatorship to be imposed on the American people. Einstein and Morgenstern told him not to worry about it.

The attending federal judge had earlier sworn in Einstein, and he invited the distinguished trio into his chambers for a pre-swear-in chat. The judge happily informed Gödel that, unlike war-time Germany, a dictatorship could never happen in America. At this point an agitated Gödel blurted "Yes, it can! I've discovered a loophole in the Constitution that allows for a dictator to take over the country!" or words to that effect.

Einstein and Morgenstern were able to defuse the situation, however, and Gödel was duly sworn in.

I've heard this story many times, but I've never heard the basis for Gödel's argument. Some think it's Article 5, which allows for amendments. Others think it involves the establishment of executive powers. But I'm not a lawyer, and despite a careful reading of the Constitution I can't even imagine what might have concerned Gödel.

But I fear he was right all along. (I'll omit my usual anti-Bush rants, as you all probably know of which I speak.)

Anyone know more about this story? If you're an armchair Constitutional theorist, I'd be happy to hear from you.

UPDATE: Several readers directed me to this: New Yorker Article, but it still doesn't explain why Gödel thought the Constitution was flawed.
Good Bye to Clocks Ticking -- Posted by wostraub on Thursday, August 30 2007
I can't go on. It goes so fast. We don't have time to look at one another. I didn't realize. So all that was going on and we never noticed. Take me back — up the hill — to my grave. But first: Wait! One more look. Good-by, Good-by, world. Good-by Grover's Corners ... Mama and Papa. Good-by to clocks ticking ... and Mama's sunflowers. And food and coffee. And new ironed dresses and hot baths ... and sleeping and waking up. Oh, Earth, you're too wonderful for anybody to realize you! Do human beings ever realize life while they live it? — Every, every minute? ... I'm ready to go back ... I should have listened to you. That's all human beings are! Just blind people.

— Emily Webb to the Stage Manager in Thornton Wilder's Our Town
The newly-deceased Emily got her wish to travel back in time to witness her 12th birthday. Did Weyl ever wonder about time travel? Indeed, he did. Thirty years before Kurt Gödel's 1949 discovery that a rotating universe could enable travel backward in time, Weyl wrote
It is possible to experience events now that will in part be an effect of my future resolves and actions. Moreover, it is not impossible for a world-line (in particular, that of my body) — although it has a time-like direction — to return to the neighborhood of a world-line point which it already once passed through. The result would be a spectral image of the world more fearful than anything the weird fantasy of E. Hoffmann [an eccentric 19th-century German writer] has ever conjured up. In actual fact the very considerable fluctuations of the components of the metric tensor needed to produce this effect do not occur in the region of the world in which we live. Although paradoxes of this kind appear, nowhere do we find any real contradiction to the facts directly presented to us in experience.
No doubt, Weyl (like Einstein) did not believe in super-luminal velocities, so that mode of time travel to the past was verboten. Also, Weyl probably never heard of wormholes, so that idea was out, too. That left motion about the spacetime surrounding a rotating massive body. Although Weyl died eight years before the physicist Roy Kerr discovered the exact metric describing a spherical, chargeless rotating mass, he was aware of the theoretical work of Lense and Thirring, who in 1918 were able to deduce the approximate field of a rotating body. Today, this effect is called frame-dragging.

Weyl knew that the field of a sufficiently massive body undergoing a high rate of rotation would cause the light cones of a test particle moving in the direction of rotation to tip over in the same direction, thus creating what is known as a closed timelike curve. Timelike, because the body never travels faster than light, and closed because the rotating field brings the particle back into its own past light cone. The net result — backward time travel (maybe). Weyl thus realized, as far back as 1918, that matter not only warps spacetime, but that rotating matter "drags" spacetime along with it. Gödel's discovery only confirmed this effect.

But this is just science fiction, right? Many physicists today don't think so. The dynamics of an object in free-fall within the dragged spacetime of a massive spinning black hole are now well-known, and they are bizarre. What is not known is what ultimately happens to the object. Does it emerge from the black hole's ergosphere into another place and time? Or does it eventually fall into the singularity, to be crushed out of existence?

University of Connecticut physicist Ronald Mallett thinks that he might have a clue as to how a table-top time-travel device could be constructed using a circular rotating beam of laser light, which theoretically produces dragged spacetime within its interior (to see his short and very readable paper, go here. ).

Mallett with prototype device, circa 1960!

Mallett, whose father died at the age of 33 due to a heavy smoking habit, decided at an early age to become a physicist so he could go back in time and save his father. Mallett no longer believes this is possible, but his fascination with the concept of time travel has continued to this day unabated. So it is with many of us!

Since black holes result from the collapse of spinning stars and the accretion of rotating matter, it is hardly an overstatement to say that all black holes spin and so have angular momentum (neutron stars, the closest cousins of black holes, can have measured spin rates of hundreds and even thousands of revolutions per second). Therefore, frame dragging (and all its associated odd phenomena) is the rule rather than the exception in this wonderful, strange place that God created for us.
Weyl Letter with Autograph -- Posted by wostraub on Tuesday, August 28 2007
If you're interested in getting your own autograph of Hermann Weyl (I have several), have a look at this offering on eBay.

The letter was sent to Artur Rosenthal, a mathematician at Heidelberg University. Like other professors of Jewish descent, he was
summarily fired by the Nazis in 1933. By 1938 he was probably desperate to get out of Germany. Weyl tried to get him a job at
Princeton. I don't know what became of him.

It's going for about $40 now, but my guess is it will top $100 by auction's end. Good luck! [It sold for $158. Ouch.]

Expanding Spacetime -- Posted by wostraub on Thursday, August 23 2007
Some time ago I was contacted by Johan Masreliez, who has developed a theory of expanding spacetime somewhat along the lines of what Hermann Weyl had proposed. But while Weyl assumed that the metric tensor could be appended by a non-integrable 4-dimensional scale factor, Masreliez' theory assumes that the metric involves a factor that instead involves a global time factor alone.

General relativity is a classical theory, and one of its primary tenets says that there can be no global time marker. Nevertheless, cosmological models like the Robertson-Walker metric have provided theoretically important descriptions of the behavior, evolution and fate of the universe. So, I try to remain objective.

However, Masreliez' theory predicts that black holes do not exist. While it is important to keep in mind that black holes have never been directly observed, a universe devoid of these objects deviates so radically from current cosmological thought that it really makes me doubt that Masreliez is on the right track (also, I've been in love with black holes for 40 years). Still, Masreliez' theory leads to some pretty interesting things. So again, I try to remain open minded.

You can download Masreliez' book on his website. It's a fairly straightforward read, and I recommend it.
Ralph Alpher Dead at 86 -- Posted by wostraub on Friday, August 17 2007
Ralph Alpher, the George Washington University-trained physicist who was the first person to fully understand the beginnings of the universe, died August 12 in Austin, Texas.

Louise, his wife of 66 years, died in 2004.

When Alpher completed his PhD dissertation (actually his second, as the first had to be abandoned) his advisor, the noted cosmologist George Gamow, thought it would be fun to publish Alpher's results in the prestigious journal Physical Review with the equally-notable Hans Bethe as co-author (the names Alpher, Bethe and Gamow were a play on the first three letters of the Greek alphabet). But the little game backfired on Alpher, because the physics community mistakenly believed that he had made only a small contribution to what turned out to be an important paper.

Alpher's work proved that the early universe was composed of about one helium atom for every ten hydrogen atoms, a result that holds up today. Immediately after his dissertation paper was published, Alpher wrote another paper proving that the Big Bang's fireball would leave a background radiation having a temperature of about 5o Kelvin.

But in the mid-1940s to mid-1950s, scientists could simply not believe that the universe started out as a titanic explosion. Instead, they preferred to believe in what was called the "steady-state" theory, which held that the cosmos always existed (in spite of the observed expansion of the universe). Alpher could not get any traction on his Big Bang theories, so he left academia to work for General Electric. He stayed there until his retirement in 1987.

In 1964, Arno Penzias and Robert Wilson of the Bell Telephone Laboratory detected the background radiation that Alpher had predicted twenty years earlier. The radiation, which was measured at 2.73o K (still the modern value), established once and for all the validity of the Big Bang theory and put the final nail in the coffin of the steady-state theory.

But, incredibly, the Nobel Committee somehow overlooked Alpher's work and awarded the 1978 Nobel Physics Prize to Penzias and Wilson, with nary a mention of Alpher's ground-breaking theoretical research. Alpher was understandably distraught at the oversight, and even suffered a heart attack from the stress of fighting for recognition.

Alpher's is not the only hard-luck Nobel story, although more often than not they involve women scientists (hooray for Curie, but you've probably never heard of Lise Meitner or Rosalind Franklin, who both got royally screwed by the male-dominated Nobel Committee).

But I would like to think that, right at this moment, God is busy explaining everything about our wonderful universe to an awed and overjoyed Alpher. As the apostle Paul had it, the world to come is far better than the place we're in now.
Four Neutrino Flavors? -- Posted by wostraub on Tuesday, July 17 2007
Hermann Weyl was perhaps the first physicist to posit the existence of the neutrino. At first it was only a mathematical prediction. In 1930, Pauli proposed the neutrino in order to preserve mass-energy conservation. Twenty-five years later, it was found experimentally. Still later, two more types of neutrino were discovered following Weyl's death in 1955.

In the 1990s it was discovered that the three types of neutrino can oscillate into one another or "mix." That is, a muon neutrino could be "caught" as an electron neutrino, and so forth. Because neutrinos are now known to have small but different masses, they can exist as a superposition of three mass eigenstates.

That picture may now be changing. The July 2007 edition of Scientific American includes a summary of the efforts by Fermilab researchers and others to confirm very tentative evidence to date for a fourth neutrino.

The Standard Model currently allows for only three -- the electron, muon and tau -- all of which participate in the weak interaction. But there is some leeway for a fourth species (dubbed the sterile neutrino), with the provision that it not interact with the weak force.

If it exists, the sterile neutrino would interact only with gravity. This scenario is in line with current string theory predictions in which the sterile neutrino (like the graviton) can weave in and out of multi-dimensional branes. One result of this mobility allows the sterile neutrino to influence the flavor mixing of the other three, which are supposedly bound to the four-dimensional "braneworld" in which laboratory observations are made.

For a relatively simple explanation of neutrino mixing and how the sterile neutrino might fit into the scheme of things, see this article by Fermilab researcher B. Kayser.
Smolin on String Theory -- Posted by wostraub on Friday, July 13 2007
I just finished reading Lee Smolin’s The Trouble with Physics Amazon Books, in which the renowned quantum physicist bewails the impending failure of string theory. As a string-questioner myself (actually, I don't get most of the theory's math at all), I think it's a wonderful book!

Smolin is quick to point out that it’s not technically a theory, because it cannot be tested. It’s more like a hunch. Meanwhile, theoretical physics now finds itself in a desert, its greatest achievements well behind it, with little more than string theory to cling to.

And it all started with Hermann Weyl, to whom this often-annoying website is devoted.

Smolin credits Weyl as the originator of the “unified theory” craze that caught up Einstein, Pauli, Heisenberg, Schrödinger and many others from 1918 until about the 1960s. String theory then picked up where the old unified theories left off, and it has been just as unsuccessful.

The so-called Standard Model of physics, known more affectionately as SU(3)×SU(2)×U(1), reached its zenith in the 1980s and 1990s, when the predicted weak-interaction particles Z0, W+ and W- were discovered (1985) and the top quark was finally detected (1995). Since then: very little, with the possible exception of the notion (Smolin calls it a discovery) that neutrinos have mass. No wonder, he notes, that the world’s smartest physicists are hitching their stars to string theory.

Smolin, late of Yale and Pennsylvania State and now at the Perimeter Institute, is no less an accomplished string theory expert himself. But he sees little beyond the theory’s beautiful mathematics and the allure of extra dimensions (seven at last count, not including the 3+1 of good old spacetime). Without experimental verification, it’s really nothing more than a religion without even any Gospels to back it up. He quotes physics Nobelist Gerard t’Hooft:
Imagine that I give you a chair, while explaining that the legs are still missing, and that the seat, back and armrests will perhaps be delivered soon. Whatever I did give you, can I still call it a chair?
But Smolin isn’t just complaining. He points out that there are other ideas out there that might beat out strings as understandable and experimentally verifiable unified theories: loop quantum gravity, spin networks (see my post of earlier today) and various spacetime-background-independent approaches to quantum gravity. So there’s optimism to be had, but Smolin nevertheless regrets the thousands of physicists and untold academic resources that are currently being expended in the (possibly futile) search for strings.

Jesus Christ once said that there are many mansions in his father’s house (John 14:2). I still think he was referring to the many-worlds interpretation of quantum physics, in which there are an infinite number of universes awaiting us after death. I don’t personally see a need for many dimensions, and until string theory is completely played out (hopefully in my lifetime), I will side with Smolin.
Spin Networks -- Posted by wostraub on Friday, July 13 2007
For those of you who are interested in an easy introduction to spin networks, John Baez has posted a write-up by Roger Penrose on some of the simpler details.

You civil engineers out there who have done finite-element modeling (structural dynamics, groundwater transport, pipe networks, etc.) should find this easy going. Spin networks involve combinatorial methods that preserve certain quantities at each vertex, although the details are more complicated.

Here's a somewhat related problem for you engineers. If you can solve it, you will become famous and probably very well-off.

Large finite-element grids involve very sparse admittance or coefficient matrices whose components are based on the way the grid nodes are numbered (sparse matrices have many zeros in them). Sparse matrices are good, as they reduce computer storage requirements and computational effort. All solution algorithms involve some method of inverting these matrices in an efficient manner. If you take a square sparse matrix and invert it, chances are it will no longer be sparse. But by simply renumbering the grid, you can increase the sparseness of the inverted matrix. The sparseness of the inverted matrix will always be equal to or less than that of the coefficient matrix.

Example: the following graphs show the results of before-and-after vertex renumbering. The renumbering increases the sparseness of the inverted coefficient matrix by a factor of three (trust me):

Problem: develop an algorithm that produces the optimal renumbering of the grid nodes such that the sparseness of the inverted matrix is as large as possible.

Hint: Based on my playing around with the problem many years ago, the solution likely involves combinatoric extremalization of the pure number N = <x|A|x>, where A is a square coefficient matrix (aij = 1 if nodes i and j are connected, 0 otherwise) and x is the numbering vector, which starts out as [1, 2, 3 ...]. Warning: discrete extremalization is much more difficult than continuous extremalization. You can't just take a derivative and set it equal to zero!

Those of you who have investigated the "traveling salesperson" problem will see a parallel here. Bell Labs has worked on this problem for many years, as it's involved in how digital communications are routed efficiently. Thousands of brilliant scientists and mathematicians have not been able to come up with an optimal solution. Indeed, it is not known whether such a solution even exists. But maybe you can do it.

Oh, and yes, I have no life to speak of.
"We do not know what death is ..." -- Hermann Weyl -- Posted by wostraub on Monday, July 2 2007
Last week, Peter Roquette, Professor Emeritus of the University of Heidelberg, posted a comprehensive and very moving description of the personal and professional relationship between Hermann Weyl and Emmy Noether (whom you can read about in my Weyl-Higgs write-up). Roquette, a mathematician, has written extensively about the mathematical correspondence between Noether and the German mathematician Helmut Hasse. You can Google him if you want more information. Noether and Weyl Article

Included in Roquette's insightful article is the full text of Weyl's funeral dedication to Noether on April 18, 1935, which contains
We do not know what death is. But is it not comforting to think that our souls will meet again after this life on Earth, and how your father’s soul will greet you? Has any father found in his daughter a worthier successor, great in her own right?
[Note: Noether's father was himself an esteemed professor at the University of Erlangen, and justifiably proud of his daughter's substantially greater mathematical abilities.] It also includes Weyl's moving but fruitless petition to have Noether retained as a professor in Germany in the summer of 1933, when the Nazis summarily fired all scholars of Jewish descent or heritage.

In coming to Princeton as a German emigre himself in late 1933, Weyl selflessly endeavored to obtain a position for Noether at the Institute for Advanced Study as well. This was denied (possibly because of the school's antisemitic attitude), although she did find a position (at reduced salary) at Bryn Mawr.
Science and Religion, Again -- Posted by wostraub on Saturday, June 30 2007
I just finished watching BookTV on C-SPAN2, which featured science writer Natalie Angier talking about her new book The Canon: A Whirligig Tour of the Beautiful Basics of Science. Angier is also a recipient of the Pulitzer Prize, which she won in 1991 at the age of 33.

At one point she was asked about her interview with Dr. Francis S. Collins, the born-again director of the Human Genome Project. The question: how does Collins reconcile his Christian beliefs with his scientific beliefs? Angier provided the answer: Collins sees no ambiguity whatsoever. Angier then talks about how difficult it is for laypersons to understand how one could be both faithful and scientifically-minded.

While the program was on, I happened to be finishing Veltman's book on elementary particles (see my previous post). It suddenly dawned on me that Young's two-slit experiment provides an ideal way of demonstrating how faith and science not only can coexist, but also complement one another.

If you pass light waves through a very small hole or slit in an otherwise opaque barrier, the light spreads out on the other side, like the waves that result when a stone is dropped into a still pool of water. If you then pass light waves through two very closely-spaced slits, the waves from each slit again spread out, but they interfere with each other. The result is that the combined waves either reinforce themselves (constructive interference) or cancel each other out (destructive interference). All this is very straightforward and has been observed countless times.

If we now replace the beam of light with bullets fired at the slit, the interference pattern disappears. That's because light is a wave, while bullets are particles.

But now we fire a beam of electrons at the two slits. The interference pattern reappears. That's because electrons are so small they can exhibit wave-like properties. But now we fire the electrons one at a time, say, one every day. Over a period of weeks and months, we see the same interference pattern appearing. Somehow, a single electron is able to interfere with itself! (We can do exactly the same thing with light, where the intensity is reduced to one photon fired per week.) Again, all this has been demonstrated countless times.

You may now ask, what if we follow one of the electrons to see which slit it passes through? Maybe that will shed some light (no pun) on this mystery. But if this is done, the interference pattern disappears. It's as if Nature does not want us to really know what the hell is going on.

Veltman asks: how are we to understand this? How can a single object (like a photon or an electron) interfere with itself as it passes through the slits? And why does the interference pattern disappear when we try to determine which slit the particle passes through?

HIS ANSWER: The only thing that counts is what we observe. Until an observation is made, we can obtain NO INFORMATION WHATSOEVER about what is "really" going on.

And to me, this beautifully demonstrates the relationship between science and religion. A true scientist can wonder about what is really happening; she can formulate all kinds of theories involving infinite-dimensional propagators and probability amplitudes, and maybe some of what she proposes makes sense to other scientists, but she can never really know what is going on. A true Christian looking at this phenomenon can only say that this is the way God makes Nature behave. Neither is more correct than the other in the absolute sense.

Scientists make observations and try to come up with explanations for what they see. People of faith try to come up with explanations for what they do not see. Science does not disprove the existence of God -- it's just that God is not relevant much of the time. Accounting theory or mathematics does not need God, neither does the flow of electricity or the interaction of elementary particles. Similarly, religion does not disprove science. It may, however, try to get at WHY things are the way they are. Science does not do that -- it always asks how, not why.

Collins is right -- there's no ambiguity at all. But there is a hell of a lot of subjective, judgmental insanity going on in this world. Many Christians scream "Evolution is anti-God!", while many scientists yell "There is no need for a god!" In my opinion, they're all wrong.

Did God create the world in six days? All right, what is a day? Twenty-four hours? What was an hour when God forged the universe? Was it 24 hours, or 24.000000001 hours? What the hell significance does an hour mean to God anyway? It could have been 500 million years, the way we measure it today. No one knows, because nobody alive today was there to witness it.

Did you know that the Old Testament describes two different creation events (Genesis 1 and Genesis 2)? When you track the descendants of Adam through the Old Testament, how long did each ancestor live? Don't know? Then you cannot postulate when Creation occurred. Science says it's closer to 13.7 billion years ago, not 6,000. So where's the problem?

As for science, you only have to ask one question to stop any discussion about the existence of God. It is this -- WHY. That ends it, because science can never answer that question.

The division between science and religion is purely political, designed to drive the political parties in this country farther apart. The winner discredits the loser, but of course everyone loses in the end.

But there is one important difference between these warring camps today: even radically secular science will never demand that people be burned at the stake for not believing in quantum mechanics, whereas many people of faith today believe that President George W. Bush can turns lies into truth.
Veltman on Elementary Particles -- Posted by wostraub on Wednesday, June 27 2007
A non-scientifically-minded friend of mine recently pointed out to me that, in accordance with Einstein's E = mc2, the energy available to mankind must be nearly infinite. He reminded me of the scene in Back to the Future where Doc Brown replenishes the power source of his flying Delorean with a few banana peels and a shot of stale beer, throwing in the beer can for good measure.

I had to explain to him that Brown's act violates all kinds of conservation laws, not to mention the fact that nobody knows how to convert the energy of ordinary matter into pure energy. Instead, I asserted, Einstein's famous equation is useful mainly as a mass-energy accounting tool, not a prescription for free energy from trash.

By far the best book I've seen to date that explains all this in a straightforward and (mostly) non-mathematical manner is Martinus Veltman's 2003 book Facts and Mysteries in Elementary Particle Physics, admittedly not the kind of book my friend would be picking up at Barnes & Noble anytime soon. The 1999 Nobel Physics Laureate, Veltman (curiously, his Christian name is the same as that of my late aunt's!) is Professor Emeritus at the University of Michigan, although he originally hails from Utrecht University in the Netherlands, where he worked on weak-interaction physics. Veltman was the PhD advisor of Gerardus t'Hooft (co-recipient of the 1999 Nobel with Veltman) who, as a lowly Utrecht graduate student in 1971, proved that all gauge theories are automatically renormalizable. This would have made Hermann Weyl very proud, indeed.

I have one other book by Veltman, 1994's Diagrammatica: The Path to Feynman Diagrams (paperback). Mathematically, it's a readable, mid-level text that introduces canonical quantization from first principles using creation/annihilation matrices whose properties are so neat, they're actually fun. The Almighty Creator (who undoubtedly knows these matrices intimately), is not only the greatest physicist but is also entertainingly practical in the extreme.

Anyway, if you're interested in modern elementary physics and want the best resource available on the subject at the layperson's level, you can't go wrong with Veltman's book. It explains everything from quarks and gluons to hadrons and their antiparticles on up, their interactions and their conservation principles, along with brief but fascinating sketches of many famous physicists. Equally enjoyable is Veltman's rather strange and often hilarious use of the English language.
Krauss on Extra Dimensions -- Posted by wostraub on Tuesday, June 26 2007
Case Western Reserve University's Lawrence Krauss is a leading particle physicist and cosmologist, and he has written a number of excellent books (including The Physics of Star Trek, which I thought was rather silly, but that's another story). His most recent book, Hiding in the Mirror, discusses the subject of extra dimensions and why they hold so much allure nowadays.

Krauss with friend

Krauss ends his book with Hermann Weyl's "Truth/Beauty" quotation, and he graciously credits Weyl as the guy who essentially started the entire extra-dimensions craze. Krauss seems to be not so crazy himself about string theory, which proposes that we live in an eleven-dimensional "membrane" world. Krauss feels that, because string theory cannot (as yet) be demonstrated experimentally, it is really no different than a religious belief. I do not know what faith (if any) Krauss practices, but he is also a leading proponent of reason over nonsense (he is especially critical of early creationism and the right wing's continued attacks upon evolution), although that, too, is another story.

It is true that Weyl's 1918 geometric gauge theory, like Einstein's general relativity theory, involved only four dimensions, but his work provided the stimulus for Theodor Kaluza's five-dimensional theory, which was worked out in 1919. But all of these guys owed a tremendous debt to the German mathematician Bernhard Riemann, who in the 1860s developed the mathematical basis for all their work.

Krauss' book does not mention Riemann, a curious oversight in a book dealing with extra dimensions. As perhaps the greatest mathematician of the 19th century, Riemann was no stranger to multiple mathematical dimensions. Riemannian geometry, perhaps Riemann's greatest achievement, is the basis of modern geometrodynamics and, if Riemann had lived a few more years, he might have trumped Einstein and everybody else.

Sickly for most of his life, Riemann was born in 1826 and died of tuberculosis at the tragically-young age of 39, not long after developing his geometry. He was convinced that his was the "true geometry of the world," and believed it could be used to describe all physical phenomena. His initial efforts failed, but it was only because Riemann was stuck in three dimensions. If he had only been gifted with Einstein's foresight to view time as the fourth dimension, the general theory of relativity (gravitation) would have undoubtedly appeared around 1870, 45 years earlier than Einstein's opus of November 1915.
Adieu to Schrödinger -- Posted by wostraub on Wednesday, May 23 2007
I finished Moore's book on Schrödinger and found it to be a fascinating account of not just Schrödinger's life and work but a glimpse of how the physicists of his day struggled to make sense of the emerging quantum theory of the mid-1920s.

It's interesting to note that Schrödinger initially wanted to believe that the wave function Ψ was a purely real quantity, despite the fact that it was embedded in his complex wave equation (actually, it's a diffusion equation, but what the hell). It's also notable that Hermann Weyl, Schrödinger's best friend, helped enormously with the mathematics. In my opinion, it should have been called the Schrödinger-Weyl equation.

In early 1927, Erwin and his wife Anny were invited to Cal Tech in Pasadena. Anny found Pasadena "unbelievably beautiful, like a great garden." The sentiment was echoed by Schrödinger, who loved the Southern California climate. The great Dutch physicist Henrik Antoon Lorentz (Einstein's idol) was also visiting at the time. It's neat to think that these great scientists might very well have driven down my street (Orange Grove Boulevard) exactly 80 years ago.

Schrödinger remarked to his host, the noted Cal Tech Nobel laureate Robert Millikan, that he wished Pasadena were populated by Italians or even Spaniards, not Americans, although he felt they were considerate to a degree quite unknown in Germany at the time. This is understandable, as Schrödinger, who had recently visited New York City, hated the place and thought Americans to be uncultured.
Schrödinger: Life & Thought -- Posted by wostraub on Tuesday, May 22 2007
I managed to find a library copy of Walter Moore's Schrödinger: Life and Thought and am in the process of reading it. Erwin Schrödinger was Hermann Weyl's best friend (from their days together at the ETH in Zürich until Weyl's death in 1955), and I thought this book would provide additional information on Weyl. Yes, it did.

Moore relates the notoriously open relationship that the otherwise devoted Schrödinger and Anny (his wife of 41 years) practiced, which was due primarily to Erwin's predilection for extramarital affairs. Schrödinger's intellectual abilities seems to have been matched only by his libido, and he had many lovers, even into his old age. Anny herself had her share of paramours, including Weyl (whom she called Peter):
Anny would find in Hermann Weyl a lover to whom she was devoted body and soul, while Weyl's wife Hella was infatuated with Paul Scherrer [another ETH physics professor].
This relationship was confirmed in, of all things, a friendly letter from Anny to her husband Erwin in 1936:
Even if the love between Peter and me should sometime come to an end, I would always be blessed that it had formerly existed, as I know that fate has given me the greatest happiness that a person can ever be given.
But the best part of the book (so far) is the story behind Schrödinger's famous wave equation, and how he came across it late in 1925 (and even this story involves an amorous romp between Erwin and an unknown former love in Arosa, a secluded Alpine resort).

Amazingly, Erwin and Hermann remained best of friends until Weyl's death in 1955. And when Schrödinger's heart finally stopped at age 73 on 4 January 1961, Anny was there to give him a farewell kiss. Go figure.

If I find anything else interesting in the book, I'll report on it later.
Biggest Supernova Ever Seen -- Posted by wostraub on Tuesday, May 8 2007
A team of astronomers from the University of California at Berkeley has discovered an enormous supernova in the galaxy known as NGC 1260. It exploded in September last year, producing the most massive outpouring of energy ever witnessed. (Actually, because this galaxy is 240 million light years away, the star blew up 240 million years ago.)

The supernova, designated as SN 2006gy, had an estimated energy output of 1045 joules, enough to outshine the star's entire galaxy of perhaps 200 billion stars. It's bigger than anything ever seen, and its output has been remarkably persistent:

The pre-nova mass of SN 2006gy is estimated to have been about 150 solar masses. That's truly enormous, because stars that big are notoriously unstable and have extremely short lives. But most supernovas blow off only a fraction of their total mass into space, leaving a neutron star or black hole behind. Scientists believe SN 2006gy blew up completely, which would explain why the explosion's energy was so great.

Any chance of such a supernova occurring in our Milky Way? Well, there's an unstable, 100-solar-mass star known as Eta Carinae about 7,500 light years away from us that scientists say will probably do the same thing. Its light output would be so great that the supernova could be seen during the day, but it would pose no hazard to life on Earth. On the other hand, star explosions known as gamma-ray burstars are far more dangerous; if one went off within several hundred light years, most life on Earth would be extinguished (President George W. Bush was recently overheard saying "We just gotta get one a them things fer the Department of Defense").

Here's the Berkeley paper. It's about 10 pages long and somewhat technical, but very readable.

Eta Carinae underwent a colossal false nova event in 1843 which almost destroyed the star. It survived, but remains the best candidate to date for a SN 2006gy-like explosion. The dumbbell-shaped ejecta cloud streaming out of the region now dwarfs the central star itself. Frightening.
Feynman's Thesis -- Posted by wostraub on Wednesday, May 2 2007
I finally got around to reading Laurie Brown's book Feynman's Thesis, which, to the best of my knowledge, is the only publicly available version of Feynman's 1942 PhD dissertation. I was astonished to find that Feynman's thesis, which details his discovery of the path integral, is understandable, fun to read, and short -- incredibly, the document's only 68 pages long. (Mine was the exact opposite -- at 225 pages, it was incomprehensible, boring, and long.)

I have long been fascinated by Feynman's idea, which represented an entirely new approach to quantum mechanics. In fact, it represents another quantum theory altogether.

Basically, Feynman said that a particle goes from Point A to Point B along an infinite number of different paths, or "histories." It can travel forward and backward in time, at any velocity, do loops, interact with virtual particles, and cross the entire universe an infinite number of times. Every path, no matter how improbable or illogical, is just as important as any other path. Each path is assigned a probability amplitude* that by itself says nothing. But when you combine these amplitudes, you find that the infinite number of paths available to the particle shrinks enormously. In classical physics, the path that a particle settles into is the classical path -- a straight line in flat space, or a parabolic arc in the presence of gravity.

Several years ago I tried to explain this in my write-up Introduction to Quantum Field Theory. I wrote the path integral part off the top of my head, and today I was pleased to find that it was remarkably similar to Feynman's treatment in his dissertation.

Now if only I could understand all that Feynman did in the 46 years following his Princeton thesis!

* The probability amplitude of an event is a complex number whose norm (its "square") is a real number. The physicist Nick Herbert describes things this way: the wave function Ψ is a probability amplitude, also called a "possibility," while its square Ψ*Ψ is a real number, the "probability."
The Most Beautiful Thought -- Posted by wostraub on Saturday, April 28 2007
Right after his discovery of special relativity in 1905, Einstein was looking for a more general application of his theory -- one that included gravitation.

I think it was in 1908 or thereabouts that he had what he later described as the most beautiful thought of his life: if a person were to to fall off the roof of her house, she would not feel her own weight during the fall. Einstein instantly realized that the phenomenon known as gravity could be transformed away by a suitable change of coordinates, and he began to look for a description of relativity that was independent of any particular coordinate system. This led him ultimately to his 1915 theory of general relativity, which is cast in the coordinate-invariant language of tensor calculus.

I was instantly reminded of this little story of Einstein's when I read about Stephen Hawking's recent experience with weightlessness. The 65-year-old Lucasian Professor of Mathematics at Cambridge University boarded the Vomit Comet and undertook half a dozen parabolic "plunges," each of which rendered him weightless and floating for 20 to 25 seconds. He remarked later that he greatly enjoyed a brief chance to escape his wheel chair and experience Einstein's beautiful thought first hand.

Although I believe that purely recreational flight (like billionaires paying millions to go into orbit) is a ludicrous waste of resources, I'm happy for the guy (and I'm sure Einstein was with him in spirit).
Weyl's 1929 Paper -- Posted by wostraub on Tuesday, April 24 2007
I once remarked that I planned to review Weyl's 1929 paper "Electron and Gravitation," which formally presented Weyl's gauge idea in the context of the then still-emerging quantum theory. While I've always found Weyl's physics to be straightforward, his mathematics tends to be rather obtuse and difficult to follow (at least for me). Complicating the matter is the fact that Weyl's notation is unfamiliar to many physicists (for example, Weyl expresses the tetrad as ea(β) rather than the purely tensorial index form eaβ). Maybe this is no big deal, but I have a hard time following stuff when the notation is odd.

Lochlainn O'Raifeartaigh's excellent 1997 book The Dawning of Gauge Theory includes a translation and detailed clarification of Weyl's paper (with modernized notation), but it's still a tough read.

Prof. Wulf Rossmann of the University of Ottawa recently contacted me on a different matter but then put me onto his own take of Weyl's paper. He sticks with Weyl's tetrad notation but to his credit clarifies Weyl's paper to a greater extent than O'Raifeartaigh. The interested reader will definitely want to visit Rossmann's website (which includes his book on differential geometry in downloadable pdf format, along with other online papers).

Rossmann also points out a reference in Weyl's paper that has always puzzled me. In the the paper's first section, Weyl notes that his 2-component spinor ψ is unable to accommodate left-right parity; he correctly surmises that parity would therefore require another, independent 2-component spinor (Weyl even suggests that this spinor describes the proton). Weyl cryptically notes that he will address this issue in a "Part II," but he never mentions it again. Rossmann believes that Weyl never wrote Part II because Anderson's 1932 discovery of the positron (not to mention Chadwick's discovery of the neutron in the same year) compromised Weyl's ψ-connection in its relationship with the Weyl current j μ*σ μψ (σi are the Pauli matrices; σ0 is the 2×2 unit matrix). Thus, Weyl's hope for a theory of everything (which in those days was just gravity and electrodynamics) was quashed for good.

Finally, Rossmann offers the following quote from Weyl's book Space-Time-Matter, which is all too true (but gives non-mathematicians like me little solace):
Many will be horrified by the flood of formulas and indices which here drown the main idea of differential geometry (in spite of the author's honest effort for conceptual clarity). It is certainly regrettable that we have to enter into purely formal matters in such detail and give them so much space; but this cannot be avoided. Just as we have to spend laborious hours learning language and writing to freely express our thoughts, so the only way that we can lessen the burden of formulas here is to master the tool of tensor analysis to such a degree that we can turn to the real problems that concern us without being bothered by formal matters.
Weyl's Pedestal Shaken a Bit -- Posted by wostraub on Tuesday, April 17 2007
In 1970 I took three courses in physical chemistry at university. The book we used was Walter Moore's text, appropriately entitled Physical Chemistry. Now a classic, it was, and still is, a tough book, especially the chapter on quantum chemistry. As I recall, we undergrads didn't like Moore very much.

Over thirty years later, in 2001, Moore wrote a biography of the great Austrian physicist Erwin Schödinger called Schrödinger: Life and Thought, which was reviewed for the New York Times by Richard Teresi, the author of The Three-Pound Universe. Here is an excerpt from that review:
Schrödinger's wave equation, published only a few weeks later, was immediately accepted as "a mathematical tool of unprecedented power in dealing with problems of the structure of matter," according to Mr. Moore. By 1960, more than 100,000 scientific papers had appeared based on the application of the equation. Schrödinger lavishly thanked his physicist friend Hermann Weyl for his help with the mathematics. (He was perhaps indebted to Weyl for an even greater favor: Weyl regularly bedded down Schrödinger's wife, Anny, so that Schrödinger was free to seek elsewhere the erotic inspiration he needed for his work.)
Ouch. I'm not a prude, but this aspect of Weyl's personal life troubles me. I have read in numerous places that post-World War I Germany was sexually liberated, and I already knew that Einstein had more than a few sexual skeletons in his closet, but this hits close to home.

I have not yet read Moore's Schrödinger book, primarily because a) I don't want to pay Amazon $50 for the thing; b) it's not available from my library's interlibrary loan program; and c) I'd already had enough grief from Moore almost 40 years ago. But if I can lay my hands on a copy I'll you all know what I think of it -- and whether my opinion of Weyl changes any as a result.
Arrhenius and Climate Change -- Posted by wostraub on Monday, April 9 2007
Svante Arrhenius was a Swedish chemist who, around 1895, was the first to quantify the relationship between chemical reaction rates and temperature. He won the Nobel Prize in Chemistry in 1903 for his work on electrolyte theory.

Every undergraduate chemistry major learns how to derive Arrhenius' rate equation, but what most don't realize (as I didn't realize at the time) was that Arrhenius was the world's very first climate modeler. In 1894 he derived a remarkably accurate relationship between atmospheric carbon dioxide levels and global temperatures. Using only a slide rule, he calculated that a doubling of CO2 concentrations would raise the Earth's average temperature by 9.9oF. By comparison, today's modern supercomputers and vastly more accurate climate models predict an increase of 10.4oF.

Arrhenius' story is recounted by environmental writer and journalist Fred Pearce in his new book With Speed and Violence: Why Scientists Fear Tipping Points in Climate Change. An erstwhile skeptic of doomsday climate-change scenarios, Pearce looks at all the evidence from all the angles, and comes up with a prediction: we're all in very big trouble.

Even Arrhenius could not have foreseen the day when humans would be pumping out 8.2 billion tons of CO2 into the atmosphere annually, an amount far in excess of the planet's ability to absorb without global climatic consequence.

From CO2 buildup and the breakdown of the Atlantic Conveyor to deforestation and the unavoidable release of tens to hundreds of billions of tons of frozen Siberian methane, Pearce paints a bleak picture for 21st century Earth and its inhabitants.

While reading the book, I was at once struck by the words of Chris Hedges in a recent edition of New Statesman, where he notes that the real motivation of today's fundamentalist Christians is not religiosity but despair:
... The danger of this theology of despair is that it says that nothing in the world is worth saving. It rejoices in cataclysmic destruction. It welcomes the frightening advance of global warming, the spiraling wars and violence in the Middle East and the poverty and neglect that have blighted American urban and rural landscapes as encouraging signs that the end of the world is close. Those who cling to this magical belief, which is a bizarre form of spiritual Darwinism, will be raptured upwards while the rest of us will be tormented with horrors by a warrior Christ and finally extinguished. The obsession with apocalyptic violence is an obsession with revenge. It is what the world, and we who still believe it is worth saving, deserve.
If true, then may God forgive us all.
Hermann Weyl in Göttingen -- Posted by wostraub on Monday, April 2 2007
Here are some photos of Weyl talking at two separate colloquia taken in the early 1930s. The original caption on the first photo indicates that the lecture hall was filled to capacity, as the students at Göttingen did not know how much longer Weyl would be staying in Nazi Germany (Weyl's wife was Jewish, which jeopardized Weyl's entire family; they ultimately emigrated to America in November 1933). The second shows Weyl with the great German mathematician Richard Courant (far left).

Permission to use photos courtesy of the Los Alamos Laboratory, Emilio Segrè Visual Archives (American Institute of Physics Photo Archive).
Relativity for the Masses -- Posted by wostraub on Saturday, March 17 2007
The act of Taylor & Wheeler (MIT's Edwin F. Taylor and Princeton's John A. Wheeler, that is) has produced two books that you need to read if you want to learn special and general relativity quickly.

The first is 1992's Spacetime Physics, a 300-page fun read that covers pretty much everything on Einstein's special relativity. The book's side bars feature two iconic characters (Rodin's The Thinker and a wise-cracking black crow) who help guide the reader through special relativity's often confusing concepts (these guys are much like Simplicio and Sagredo from Galileo's Dialogues).

This book will also show you how to do some amazing calculations using only the flat-space metric ds2 = c2dt2 - dx2. For example, if you want to go to the Andromeda Galaxy in one year, all you have to do is build a spaceship capable of traveling at 0.999999999999875 times the speed of light*. Of course, your friends on Earth will be 2 million years older when you get there, but you can always make new friends in Andromeda. Also, President Bush will almost certainly be dead by the time you arrive. Break out the champagne!

The second book is 2000's Exploring Black Holes: An Introduction to General Relativity. It follows the same format but deals with general relativity, the theory of warped spacetime and gravitation. The book focuses primarily on simplified presentations of the Schwarzschild and Kerr metrics (which respectively describe static and rotating black holes), which are really all you need to know about gravity. Thinker and Crow are back again to make sense of the mathematics, which requires a knowledge of elementary calculus.

Wheeler, who will be 96 this year, is one of the few still living great physicists from the days of Einstein, Dirac, Wigner, von Neumann and Pauli. He knew them all personally, and he also knew Weyl well (you can read Wheeler's tribute to Weyl elsewhere on this site). Wheeler coined the term black hole in December 1967. Why he never won a Nobel Prize is a great mystery to many people.

Interesting tidbit: Wheeler graduated from Johns Hopkins University in 1933 with a PhD in physics. And that's all he got -- he skipped getting his BS and MS. Talk about being focused!

* The energy needed to bring every pound of your spaceship up to this speed is equivalent to approximately one billion Hiroshima-size atom bombs.
The Natural Gauge of the World -- Posted by wostraub on Friday, March 16 2007
Sometime ago I wrote about Eddington and his unified field theory of 1921, the one that Weyl called "not worthy of discussion." But it was Eddington who came up with the phrase "natural gauge of the world," and in was in the spirit of Weyl's 1918 theory that he proposed it.

In Eddington's theory the Ricci tensor Rμν, like Weyl's version, is constructed from a symmetric metric gμν and affine connection Γλμν. He then separates the Ricci tensor into its symmetric and antisymmetric components; the symmetric part is then made proportional to the metric tensor using a gauge scalar that, for all intents and purposes, is the cosmological constant. This gauge, asserted Eddington, represents the natural gauge of the world.

The great Austrian physicist and all-round curmudgeon Wolfgang Pauli was appalled. He denounced Eddington's theory as having "no significance to physics" and expressed his resentment of having a mathematician poke his nose into the realm of the physicists (actually, Eddington was both), a criticism that Pauli later directed at Weyl regarding the latter's application of the gauge concept to quantum mechanics. But Weyl won his argument, with Pauli apologizing and admitting that Weyl had been right all along.

Still, Eddington raised an important question: if the world permits (or demands) that metric spacetime be conformally rescalable from point to point, then what is the nature and consequence of that symmetry?

Indeed, the British mathematical physicist Roger Penrose is very much enamored of conformal invariance, and he asserts that the Weyl conformal tensor Cλαμν (which lies at the heart of the Weyl Curvature Hypothesis, or WCH) is essentially responsible for entropy and the assumed asymmetry of the arrow of time. Penrose even goes so far as to say that the WCH must be an essential element of a successful quantum gravity theory, which itself must be time asymmetric. This results from the assertion that the Big Bang was a topologically distinct event in the history of the universe -- the Weyl curvature tensor was identically zero at the time, whereas it is positive now and will remain positive (or even become infinite) regardless of whether the universe continues to expand forever or eventually falls back on itself in a Big Crunch.

You can learn a little more about Weyl's conformal tensor (and how to derive it) in my write-up on the menu to the left.
Hermann Weyl -- The Centenary Lectures -- Posted by wostraub on Saturday, March 10 2007
I just finished reading Hermann Weyl, 1885-1985: The Centenary Lectures, three talks that were given by C.N. Yang, R. Penrose and A. Borel on the 100th anniversary of Weyl's birth.

I've hunted all over for this book. Caltech didn't have it, or Berkeley, or UCLA, or Stanford; it finally turned up at Cal State Fullerton!

Anyway, it's interesting because it includes a full bibliography of Weyl's works, which total 167 scientific and mathematical papers, 17 books, and a dozen or so lecture notes. Also interesting is the fact that he published only a handful of physics papers after his move to the Institute for Advanced Study at Princeton in 1933, the year he left Germany (by his own admission, Weyl's interests changed as a result of his emigration).

The book includes transcripts of some of the speeches that were made at the Centenary Dinner held on October 24, 1985 at the Swiss Federal Technical Institute (the university where Weyl taught from 1913 to 1930). I was struck by the depth and breadth of Weyl's literary, poetic and philosophical interests, subjects that he knew intimately from the likes of Democritus, Leibniz, Kant, Mann, T.S. Eliot, Husserl, Russell, Kierkegaard, Nietzsche and Heidegger. How the guy ever had time to read all this stuff, while maintaining a reputation as one of the world's leading mathematical physicists, is simply beyond me.

Best of all, the book includes the speech given by Weyl's son Michael, also a PhD, whom I've been trying to locate for some time now. It is a touching and heartfelt reminiscence of a son who fully appreciated having a father who was not only a "mathematician-father with the soul of a poet" but a truly learned man who passed on his passions to Michael and his mathematician brother, Dr. Joachim Weyl.

In his speech, Michael included this poem by the minor poet Anna Wickham (1884-1947):
God, Thou great symmetry,
Who put a biting lust in me
From whence my sorrows spring,
For all the frittered days
That I have spent in shapeless ways
Give me one perfect thing.
Hermann Weyl included this poem in his last book, Symmetry, in which he remarks that the sphere in space represents true perfection -- so perfect, in fact, that it inspires not only awed admiration but sorrowful longing as well, because it is a reflection of the perfect symmetry and unattainable perfection that is God himself.
Weyl's Gauge Factor, Again -- Posted by wostraub on Monday, February 26 2007
Several people have written me asking how Weyl’s geometry might be modified using the nonintegrable gauge factor that I mentioned in my December 19 post. There I suggested that the line element ds2 could be made gauge invariant (or conformally invariant) via a “phased” metric tensor:

gμνexp[k∫φλdxλ] gμν

where k is a suitable constant. In addition to the line element, all tensor quantities constructed from the new metric (the metric determinant, the Riemann-Christoffel tensor and its contractions, the Christoffel symbols, etc.) would then be automatically gauge invariant, as would all quantities raised and lowered using the new metric.

Well, I had never considered this possibility, so I did a few calculations to see if it leads to anything interesting. For one thing, we can now define an action Lagrangian that is linear in the Ricci scalar density to get the free-space field equations:

S = ∫√(-g) R d4x

where the Ricci scalar R is constructed solely from the new metric. Here we are on familiar ground again, as the above quantity is the old Einstein-Hilbert gravitational action we all know and love. Thus, we neatly kill off the two objections Einstein held against Weyl’s theory: a non-gauge-invariant ds, and a fourth-order action Lagrangian.

However, if we try to append the Maxwell action terms Fμν Fμν and sμ φμ (where sμ is the electromagnetic source four-vector) to the above Lagrangian density, we immediately run into a problem: sμ φμ is not gauge invariant! Of course, it’s not gauge invariant in any other theory, either, but here it’s particularly problematic. Making matters worse is the fact that the Weyl vector φμ does not exhibit a true gauge weight (instead, it's a gradient, making an integration by parts necessary, which really messes things up).

This brings up an issue I’ve thought about for a long time: Just what the hell is the source vector, anyway? Classically, it’s just

sμ = ρ(x) dxμ/ds

where ρ is the electromagnetic 3-density. It seems like the only way to make this quantity gauge invariant is to consider its interaction with φμ and some quantum field, along the lines of Ψ*sμΨ, etc., and then impose gauge conditions on the wave function.

I'll think about it, but I don't believe it goes anywhere. Suggestions?
Squid Memories -- Posted by wostraub on Thursday, February 22 2007
Is it my imagination, or are giant squids (not Superconducting Quantum Interference Devices, but the scary ones with tentacles) showing up more frequently nowadays?

New Zealand fishermen looking for fish in Antarctic waters accidentally hooked a Colossal squid weighing almost 1,000 pounds and measuring over 30 feet in length. Article

The colossal squid, known even by elementary school children as Mesonychoteuthis hamiltoni (just kidding), is the larger (!) cousin of the more familiar giant squid (Architeuthis), which may grow to longer lengths but is rather more slender.

I've had a life-long fascination with these things. When I was five, my father took me to see Disney's 20,000 Leagues Under the Sea (that was in 1954), and several years later my older sister took me to see It Came From Beneath the Sea. By then I was hooked, so to speak. Disneyland used to have a full-sized model of a giant squid in its 20,000 Leagues Exhibit in Tomorrowland. It was so realistic that as a child I had a really hard time going in there:

In the 1980s, while scuba diving in extremely murky water off the Coronado Islands, some enormous, shadowy thing came up behind me. I turned, and as I did it quickly swam off, leaving me spinning head over heels in the backwash. The ascent back to the surface (I had been down about 100 feet) seemed to take forever. I never really got a look at it, but on the boat the divemaster told me it was only a harmless grey whale that had been seen swimming nearby. Nevertheless, that was the last dive I made that day.
Lorentz Symmetry -- Still Asking -- Posted by wostraub on Friday, February 16 2007
It was Hermann Weyl who showed that quantum mechanical gauge invariance is the continuous symmetry responsible for the conservation of electric charge. Lorentz invariance (invariance of the Lagrangian with respect to Lorentz transformations) is also a continuous symmetry, so what conservation principle does it represent?

I've addressed this problem before, but nobody was able to help me with it. Baez has been asked this same problem, but his answer is less than illuminating.

In his excellent book The Road to Reality: A Complete Guide to the Laws of the Universe, British mathematical physicist Roger Penrose provides the answer. In non-relativistic mechanics, the quantity

N = pt - mx

where t is time, p is the 3-momentum and x is the position vector, is (obviously) invariant with respect to time. Penrose notes that Lorentz invariance is responsible for making the center of mass of a particle (or that of a collection of particles) move in a straight line, with velocity p/m.

Pretty simple, isn't it? But I'll be damned if I know how to derive it! (short of using the Lorentz generators, as this guy does it). Penrose doesn't derive it either, but only says that Lorentz symmetry is a tad less common than rotational symmetry. Well, that's a big help.

Penrose obviously doesn't take into account idiotic readers like myself.
Juliet or Esmerelda? -- Posted by wostraub on Wednesday, February 7 2007
Unlike her happier fate in motion pictures, the gypsy Esmerelda in Victor Hugo's classic novel The Hunchback of Notre Dame dies by hanging. Her lovelorn admirer, the hideous hunchback Quasimodo, dispatches the corrupt Frollo but is unable to save the love of his life. Heartbroken, he disappears. Years later, grave diggers accidentally unearth the skeleton of Esmerelda, which is inexplicably embraced by that of a grossly deformed man. [And] when they tried to detach the skeleton which he held in his embrace, it fell to dust.

Cut to February 2007. Archaeologists working in Mantua, Italy (about 25 miles south of Verona) uncover the skeletons of two Neolithic people embraced in death. The remains, those of a young man and a young woman, are estimated to be from 5,000 to 6,000 years old. Neolithic burials nearly always involve only a single skeleton. Evidently, there was something very special about these two people from long ago. Sadly, their story is lost to us. Article

Coincidentally, Shakespeare placed his Romeo and Juliet in Verona, Italy. The ending to that story wasn't very pleasant, either. Something about a happy dagger that finds its sheath ...

Famous Papers -- Posted by wostraub on Wednesday, January 31 2007
The website Trivial Anomaly provides links to a dozen or so sites where you can read or download seminal papers of famous scientists. Want to see a translation of Einstein's original 1915 paper on the general theory of relativity? How about Schrödinger's 1926 paper announcing his discovery of wave mechanics? It's pretty neat stuff.

I'm also pleased that it includes my write-up of Weyl's 1918 gauge theory, perhaps because it's more detailed than the paper Weyl originally wrote (and because you don't have to know German to read it).
Hermann Weyl in Exile -- Posted by wostraub on Wednesday, January 24 2007
I just finished reading Forced Migration and Scientific Change: Emigre German-Speaking Scientists and Scholars After 1933 (1996), a collection of articles edited by Mitchell Ash and Alfons Söllner. Of particular interest is the chapter Physics, Life and Contingency: Born, Schrödinger and Weyl in Exile by Sküli Sigurdsson, whose 1991 PhD dissertation on Hermann Weyl I'm still trying to run down.

The book is not so much an overview of how emigrating German and Austrian scientists dealt with Hitler's rise to power in 1933 but a brief history of how their views on science, mathematics and philosophy were altered in the years immediately following the end of World War I, up to the time they left Germany.

Sigurdsson's article provided me with a little more information on Weyl's state of mind in the years 1918-1933 than I'd seen previously. For example, his decision to accept the mathematics chair in 1930 at Göttingen when the great David Hilbert retired was not an easy one. He seems to have accepted it more out of nationalistic pride than for any other reason, believing that he needed to help Germany maintain its "thought collective" and promote its tradition of highest-quality science. Nevertheless, he announced his resignation in October 1933 after becoming depressed and disillusioned with the Nazis, the overall political and economic climate of Germany, and the resulting restrictions on scientific inquiry.

During his years at Göttingen, Weyl's productivity had waned considerably and he was dissatisfied with the quality of science at the school. At his previous post at the Swiss Technical University in Zürich (where he was Mathematics Chair from 1913 to 1930), Weyl was the highest paid professor. His salary was increased at Göttingen, but was soon cut because of the school's rapidly deteriorating finances, the result of Nazi-imposed state cutbacks in higher (and mostly theoretical) education.

In January 1933 Weyl received an offer from Princeton's new Institute for Advanced Study. But he was so depressed that he could not muster the strength to make the decision to accept (he seems to have had difficulty throughout his life making life-altering decisions, an observation that was later confirmed by the great Göttingen mathematican Richard Courant). However, Weyl's wife was Jewish, which placed both her and their two sons in jeopardy with the Nazis. This forced Weyl to accept Princeton's invitation, and the family left for America in November 1933.

For Weyl, Born and Schrödinger, their forced emigrations (Born was Jewish and thus barred from teaching, while Schrödinger quit in protest over Born's firing; both men emigrated to England) brought about significant changes in their attitudes and philosophies regarding science and mathematics. Weyl retreated more and more into pure mathematics and pretty much abandoned his earlier interest in mathematical physics, particularly unified field theory. His book Classical Groups came out in 1939, while his earlier interest in physics and philosophy waned. And when he finally returned to Europe after his retirement in 1952, Weyl went to Zürich, not his beloved Germany.

Sigurdsson's article is rather dry and humorless, but there is one funny anecdote worth repeating. Weyl suffered from asthma and hay fever, and Sigurdsson notes that Weyl's decision to go to the Institute for Advanced Study was compromised by the fact that he could not get affordable health insurance in America!

Plus ça change, plus c'est la même chose ...
Eddington -- Posted by wostraub on Tuesday, January 23 2007
Earlier I remarked that God cannot create a new integer between the numbers 1 and 10. Here's a little story about how one man tried to do essentially the same thing.

Sir Arthur Stanley Eddington (1882-1944) was a British astrophysicist who, like Hermann Weyl, tried to develop a unified theory of gravitation, electromagnetism and quantum mechanics. His 1921 book The Mathematical Theory of Relativity (a copy of which I happen to own) praises the work of Weyl, whose ideas Eddington used to advance his own theory. But the physics community at the time roundly criticized Eddington's ideas; Weyl himself even went so far as to call them "not worthy of discussion" (undiskutierbar) in 1923. But it was also Eddington, who, on a solar eclipse expedition in 1919, took photographs of the sun and nearby stars and verified, rather sloppily, Einstein's prediction that gravity can bend starlight. Thus it was Eddington who made Einstein into an overnight scientific superstar.

Anyway, as brilliant as he occasionally was, Eddington made one famous goof. There happens to be a fundamental, dimensionless constant in quantum physics known as the fine structure constant, which is defined as

2πe2/hc = ≈ 1/137

where e is the electron charge, h is Planck's constant, and c is the speed of light (the fact that this constant is very nearly the reciprocal of the prime number 137 has profoundly disturbed physicists for over 80 years). But in Eddington's day, uncertainties in the values of Planck's constant and the electronic charge made this number closer to 136. With characteristic aplomb, Eddington set out to prove that it was exactly the integer 136.

By considering the magnitudes of certain quantities in an abstract phase space, Eddington came up with the number function

ƒ(n) = ½ n2(n2 + 1)

and, using some kind of reasoning, Eddington believed that ƒ(4) = 136 was the fine structure constant (Eddington used a similar argument to "prove" that the ratio of proton mass to electron mass was also an integer, 1836).

Only several years later, it was determined that the fine structure constant was actually closer to 137. Not to be outdone, Eddington, admitting to an earlier algebraic oversight, revised the above formula by adding +1 to the right hand side, thus recovering the correct value for the constant.

But the world's physicists were not to be taken as fools. They renounced Eddington's preposterous theory and, in mild rebuke, jokingly dubbed him "Sir Arthur Stanley Adding One."

Note: Interestingly, the late, great German Nobel laureate physicist Max Born (who happened to be Olivia Newton-John's grandfather!) noticed that Eddington's formula reproduced two numbers from the New Testament Book of Revelation, Chapter 13:
And I saw a beast coming out of the sea having ƒ(2) = 10 horns ... [and] his number is ƒ(6) = 666.
I feel fairly certain that God did not use Eddington's formula when he inspired John to write Revelation!

PS: Wolfgang Pauli (Nobel physics prize, 1945) was also fascinated by the fine structure constant, and he devoted much time and thought to its provenance. He passed away from cancer in 1958, and the number of the hospital room where he died was ... 137. Good one, God!
Twenty Eighty -- Posted by wostraub on Monday, January 22 2007
My older son's girlfriend loaned me a copy of James Surowieki's book The Wisdom of Crowds. It's really interesting -- it pretty much destroys the idea that irrational mob rule is the norm, and shows how crowds of people may have differing points of view but, when the average is taken, it tends to be pretty close to the truth. The book's one caveat is that a crowd must have a fairly firm grasp of reality, otherwise mob rule does indeed take over, with disastrous results.

You've probably heard about the 20/80 rule: 20% of the workforce does 80% of the work; 20% of people are difficult, while you can get along with the other 80%; 20% of the population is outright irrational, while 80% seem to have some grasp of what's going on, etc.

Well, that seems to apply only to the rest of the world. Here in America, we have the 33/67 rule: 33% of all Americans are out of their friggin' minds.

The most recent polls show that Bush's approval rating is now at 33%, the lowest for a sitting president since "I am not a crook" Richard Milhous Nixon occupied the White House. This is awful, but the flip side of the coin is that 33% of Americans still think Bush is doing a great job.

I know such people. While perhaps not certifiably insane, they all seem to have the mindset for nonsensical and/or dogmatic thinking. They are also very suspicious of things they do not want to believe, while leaving themselves open to outright falsehoods that they do want to believe. And they tend to believe what they are told to believe.

I once asked one such person (with two M.S. degrees in engineering, yet) if she believed it was possible for God to create a new integer between the numbers 1 and 10, or if scientists had somehow overlooked an undiscovered chemical element between sodium and magnesium in the periodic table. "Yes, of course," she replied, "because sin has blinded us from the truth." How does one begin to argue with such nonsense?

The trouble is, I like a lot of these folks. Most are decent people, and share the same Christian values that I hold to. But many have allowed their values to become warped by political opportunists and liars.

If Bush's popularity was 20%, he would undoubtedly be exposed as the monster he truly is. Impeachment and prosecution as a war criminal would probably follow. But at 33% he can hang on.

20/80 works, but 33/67 does not.

America cannot survive when 33% of its people are crazy. And I fear that neither can the rest of the world.
Weyl and von Neumann -- Posted by wostraub on Saturday, January 13 2007
There's a story that Hermann Weyl, when talking about his work at a conference or lecture hall, would become extremely agitated whenever colleague John von Neumann was in the audience. His nervousness was presumably due to the fact that von Neumann was widely viewed as a genius, and Weyl was afraid he'd make a fool of himself.

A better word would be "respectful," because in actuality the two men were friends as well as colleagues. And while it is quite true that von Neumann was a mathematical genius, his brilliance extended into physics, economics and linguistics as well. For example, at the age of six he was fluent in Greek (along with his native Hungarian), and could divide two 8-digit numbers in his head within seconds. Later, von Neumann did pioneering work in computer science, and today is known as the father of the digital computer (see my December 12 post).

There is a famous story involving von Neumann, apparently even true, that he was approached by the hostess of a party he was attending and given the following puzzle to solve:
Two bicyclists on a road are 100 miles apart. At a predetermined time they begin pedaling toward each other, each with a uniform speed of 10 mph. At the moment they start out, a fly sitting on the wheel of one of the bicycles starts flying toward the other bicycle at a speed of 20 mph. Upon reaching the other bicycle, it instantaneously turns around and starts flying back to the first bicycle. It does this repeatedly until the bicycles meet in the middle of the road, squishing the fly between the tires. How many miles does the fly travel?
[This story is so old that I am almost ashamed to repeat it.] There are two ways to solve the problem, but one way is immediate: the bicyclists meet at the midpoint after 5 hours of pedaling. The fly has been flying constantly during this time, so it flies a total of 5×20 = 100 miles. The second method involves calculating the infinite series L = 100/3 ∑ (2/3)n, where the sum is taken from n = 0 to n = ∞. Again, L = 100 miles.

At the party, when asked for the answer, von Neumann instantly said "100 miles." The hostess smiled and said, "Oh darn, you know the trick." To which von Neumann replied, "What trick? I got it by doing the infinite series."
Herman Weyl and Yang-Mills Theory -- Posted by wostraub on Thursday, January 11 2007
In his great 1929 book The Theory of Groups and Quantum Mechanics, Hermann Weyl jokingly remarked that It is rumored that the ‘group pest’ is gradually being cut out of quantum mechanics. Weyl was referring to objections that many physicists were voicing at the time about the use of group theory (which was viewed as pure mathematics) in the then-emerging field of quantum mechanics. Of course, as both a mathematician and physicist Weyl could already see the importance of group theory in quantum physics, and his seminal paper Electron and Gravitation (also in 1929) introduced the group SL(2,C) into the physics of 2-component spinors (which Dirac demonstrated are the basis of the relativistic theory of spin-1/2 particles). Weyl’s interest in group theory likely reached its zenith in 1939, when he published The Classical Groups: Their Invariants and Representations.

Cut to 1949. In that year, the Chinese-American physicist Chen Ning Yang (born 1922), having recently received his PhD, took a position as junior scientist at the Institute of Advanced Study, where Weyl had been a senior member since his emigration to American in 1933. He chatted occasionally with Weyl, and the two had lunch several times in the IAS commissary, but they never discussed physics or math.

By 1954, Yang was at the Brookhaven National Laboratory, sharing an office with the younger physicist Robert L. Mills (1927-1999), who had not yet received his PhD. In that same year, the two published a paper that would later represent the beginning of all modern gauge theories for particle physics. Entitled Conservation of Isotopic Spin and Isotopic Gauge Invariance, Yang and Mills worked out a non-Abelian gauge theory for the group SU(2), which today is recognized as the jumping-off point for SU(3) and quantum chromodynamics (quarks, gluons and all that). Yang-Mills theory is beautiful, but it is really nothing more than Weyl’s 1929 gauge idea taken one logical step further. However, Yang-Mills languished because it initially attempted to describe the proton and neutron as isotopic mirrors of one another (when in fact the two particles are composite, not elementary). Consequently, nobody really recognized the great leap the theory had made.

Yang went on to win the 1957 Nobel Prize in Physics (with T.D. Lee) for his work on parity violation in the weak interaction. But in 1980, he looked back on his days with Weyl at the IAS and wondered why the two of them had not bothered to collaborate on gauge theory:
I had met Weyl when I went to the IAS as a young member. I saw him from time to time in the next years, 1949-55. He was very approachable, but I do not remember having discussed physics or mathematics with him at any time. Neither Pauli nor Oppenheimer ever mentioned it. I suspect they also did not tell Weyl of the 1954 paper of Mills’ and mine. Had they done that, or had Weyl somehow come across our paper, I imagine that he would have been pleased and excited, for we had put together two things that were very close to his heart: gauge invariance and non-Abelian Lie groups.
This is very strange indeed, and a terrible loss to physics, in my opinion, because Weyl was only to live one more year after Yang’s departure from the IAS in 1955. I would like to think that the physics community’s recognition of the importance of Yang-Mills theory would have occurred if Weyl had only been aware of it and championed its cause.

A sad example of a missed opportunity. Weyl suffered a heart attack and died unexpectedly on December 8, 1955. His last spoken word was Ellen, his wife's name.
Information -- Posted by wostraub on Monday, January 8 2007
A few nights ago the Discovery Channel re-aired a one-hour documentary on the so-called information paradox, which involves the question of whether or not information (in bits or bytes or however you want to call it) can be destroyed (or is eternal in some sense).

Until 2003, British physicist Stephen Hawking believed that information could in fact be destroyed. Take a 1-kg stone and a 1-kg book (say, Nabokov's Lolita) and toss each into a black hole. Hawking's famous (if offhand sexual) remark that black holes have no hair simply states that a black hole has only three possible attributes: mass, angular momentum and electric charge. Thus, according to Hawking, any kind of matter -- be it rocks or books -- simply adds to a black hole's mass, and maybe also some angular momentum and charge, so a black hole is an exceeding simple thing. As far as a black hole is concerned, mass is mass, whether it's a rock or a book.

But other physicists -- notably Caltech's John Preskill -- believed otherwise. A book contains information (at least to us humans), and it seemed anathema that this information would be destroyed in its journey into a black hole. Their argument went like this. Take any physical system in a pure quantum state and drop it in a black hole. Over time, the black hole will evaporate via Hawking radiation. Eventually (and this may take eons), the black hole evaporates completely. Thus, the pure quantum state is converted to a random thermal state, in violation of quantum theory. Preskill and others believed that leaked Hawking radiation must somehow preserve the information that's tossed into a black hole.

In 2003, Hawking famously announced that he had been wrong all along, though he used an argument involving parallel universes to explain why. To date, most physicists have been dissatisfied with Hawking's reasoning, if not his conclusion. At any rate, information seems to be preserved.

This started me thinking about the nature of information. Does it always exist, and we humans merely discover it? Does the sentient mind create information? Of what use is information to the universe in the grand scheme of things?

Claude Shannon, the noted American engineer/physicist (1916-2001) proved that information (or knowledge) is related to probability according to the simple (and beautiful) equation

K = -∑ p log2 p = ∑ log2 (p-p)

where K is the information (measured in bits) gained by the observance of some event and p is the probability of the occurrence of that event (if the event gives rise to more than one possible observation, the probabilities have to be summed over as indicated). Thus, the less likely some thing is, the more information that can be gleaned from it. This (according to Imperial College's Igor Aleksander) is the "surprise" factor: a big surprise (the occurrence of a low probability event) conveys a lot of information.

Similarly, one of Shannon's colleagues, John Kelly, came up with the "gambler's advantage" equation

M = eS

(see John Baez's "translation" for Week 243), where S represents a wager's "inside information" and M is the average expected growth of wagered earnings. In simpler terms, the more you know, the more money you can make. Wall Street insiders have known this for years.

Kelly's equation assumes that the inside information S is not known by anyone else (otherwise it would not be inside information!) It seems very suggestive to me that the applicability of this simple equation somehow depends on the extent of the state of knowledge of one or more people. In this sense, it is reminiscent of Schrödinger's Cat.

From these somewhat different but related points of view, information is "good" in some fundamental sense, at least to sentient beings. (As a Texas oilman, President Bush failed in everything he tried, leading me to suspect he is not sentient at all, an assertion that his presidency seems to have confirmed.)

In consideration of all this, I've led myself to the conclusion that God is somehow the ultimate repository of all information (knowledge) -- past, present and future -- and that the concept of what we call "evil" is somehow related to the corruption or deliberate obfuscation of information. I don't know if time travel to the past is possible, but I'd like to think that all information -- whether it's the missing 18 minutes of the Nixon tapes, the facts behind JFK's assassination, the knowledge contained in the burned Alexandria library, or all the stories that Neanderthals used to tell around their campfires -- can never be truly lost to us. This seems to be confirmed by Shannon's equation: there's no way that the information K can be made negative. But it can be corrupted through obfuscation.

Perhaps this is why God has always expressed such an aversion to lies and falsehoods, which tend to circumvent the truth and lead us down the wrong paths. I just can't help but see a profound connection between information and truth, and how it continues to elude us as a consequence of our lying nature.