 |
My work always tried to unite the
Truth with the Beautiful, but when I
had to choose one or the other, I usually chose the Beautiful.
Hermann Weyl
|
I died for Beauty, but was scarce
Adjusted in the tomb,
When one who died for Truth was lain
In an adjoining room
Emily
Dickinson
|
Hermann
Klaus Hugo Weyl (1885-1955).
German mathematical physicist. In 1918, proposed an early form of gauge
symmetry in an attempt to unify electrodynamics and gravitation.
Subsequently applied a similar approach to quantum physics and
discovered what is today considered one of the most profound and
beautiful concepts in modern physics ― the principle of gauge
invariance.
Shortly
after Einstein announced his theory of general relativity (gravitation) in November 1915, Weyl
began an intensive study of the theory's mathematics and began
publishing related scientific papers dealing with its physical applications. In 1918 Weyl published his book Raum-Zeit-Materie
(Space-Time-Matter), which provided the first fully comprehensive
analysis of
the geometric aspects of the theory and its
relationship with spacetime physics. One of the topics covered in the
book was Weyl's idea that gravity and electromagnetism might both be
derivable from a generalization of Riemannian geometry, the
mathematical basis from which Einstein had developed his relativity
theory. Weyl's idea was based on a new mathematical symmetry that he
called gauge invariance.
I came across
Weyl's book in 1975, but it didn't impress me very much because I
didn't know general relativity. However, in the summer of that year I stumbled across Misner-Thorne-Wheeler's
massive Gravitation
during a one-week work assignment in the microscopic rural town of Lone
Pine, California (which then had a population
of perhaps 500 people). Miraculously, the town's tiny public library
somehow
had this book, which is now regarded as a classic graduate text on
general relativity. I checked out the
book and brought it back to the hotel room to read in the off-hours.
The
book took immediate and total possession of me, and motivated me to
learn everything I could about general relativity. (I got the chance to
talk to co-author Kip Thorne in 1994, and he was quite amused when he
learned where one of his books had ended up.)
But Gravitation is not an easy read, and I had to look for more introductory texts. I soon came across Adler/Bazin/Schiffer's Introduction
to General Relativity,
which besides being easier had a chapter on unified field theory, including
Weyl's 1918 theory of the combined gravitational-electromagnetic field. For whatever reason,
the theory's mathematical beauty absolutely
fascinated me. I had known about local and global phase invariance
from my studies of quantum mechanics,
but I was not aware that Weyl's theory was the origin of this powerful
symmetry principle in quantum physics.
I
have since read all of Weyl's books and
many of his papers. Although today I believe that my interest is now
based more on an appreciation of modern gauge theory (easily the most
profound and beautiful concept of physics), I credit Weyl for having
initiated the idea in 1918 and for his subsequent (1929) seminal
application of the idea to the then still-developing quantum theory.
In his 2002
biographical memoirs, the great contemporary mathematician Sir Michael
F. Atiyah praised Weyl as the discoverer of the gauge concept and as
the driving force behind the current emphasis of gauge
theories on modern theoretical physics:
The past 25 years
have seen the rise of gauge theories--Kaluza-Klein models of high
dimensions, string theories, and now M-theory, as physicists grapple
with the challenge of combining all the basic forces of nature into one
all embracing theory. This requires sophisticated mathematics involving
Lie groups, manifolds, differential operators, all of which are part of
Weyl's inheritance. There is no doubt that he would have been an
enthusiastic supporter and admirer of this fusion of mathematics and
physics. No other
mathematician could claim to have initiated more of the theories that
are now being explored. His vision has stood the test of time.
This
website is my feeble attempt to document (and in many cases expand on) Weyl's ideas and thoughts on
gauge symmetry in a manner that will be accessible to anyone with a
basic understanding of calculus. Not a lot has been written about the
original theory's underlying mathematics, and I wanted to provide a
fairly detailed and complete mathematical description for those who
want to learn about Weyl's ideas and to appreciate the beauty of his
gauge theory. As this site progresses, I will also include discussions
of other topics in mathematical physics (as well as some related
scientific philosophy) which exhibit a similar mathematical beauty and
elegance.
Please contribute to these worthwhile
humanitarian organizations.
| Completely Cuckoo -- Posted by wostraub on Tuesday, August 24 2010 |
 A new book on Einstein's effort to unify gravitation and electromagnetism is now out, written by Jeroen van Dongen, an assistant professor at the University of Utrecht in the Netherlands.
The book includes a considerable amount of material on Hermann Weyl's association with Einstein on the unification effort, but what particularly interested me was their relationship after both men had moved to Princeton's new Institute for Advanced Study (IAS) in 1933 (Einstein left Germany with his wife in the summer of that year, with Weyl and his wife coming over in November).
It has always struck me as odd that Einstein and Weyl, whose close friendship began back in 1912 at the Swiss Federal Technical Institute in Zürich, did not engage in any meaningful collaboration after their arrival in America. After all, both had expressed a preoccupation with unification over the period 1916 to 1929, although Weyl quickly ditched the subject when his gauge theory (originally part of his earlier unification efforts) found overwhelming success in quantum mechanics. Still, their friendship persisted, even though their interests took different paths. Why?
Unfortunately, I did not find a definitive answer in van Dongen's book, although it implies that Einstein's reputation as an "ostrich" with his head stuck in the sand of classical physics (he never did accept quantum theory, except perhaps as an approximation) was such that other scientists did not want to associate with him (at least professionally) for fear of being saddled with a similar label. The IAS director, J. Robert Oppenheimer, even went so far as to call Einstein "completely cuckoo."
Weyl's post-1933 interests were primarily involved with mathematical problems, and one of his more frequent collaborators was his son, Joachim. Weyl also maintained a lively interest in philosophy, something he was smitten with from a very early age (his wife Hella was a student of the noted German philosopher Husserl), but his interest in unification had pretty much been snuffed out.
Einstein freely acknowledged his own stubborn (and almost completely fruitless) persistence in attempting to unify gravitation with electromagnetism, but he stuck with it for three decades until, in the words of Abraham Pais, "he laid down his pen and died." Thirty years of largely wasted effort died with him.
Still, Weyl and his old colleague remained good friends, and Weyl gladly participated in the IAS's celebration of Einstein's 70th birthday in 1949. But, in this picture taken of the birthday celebrants, Einstein looks old beyond his years, dressed in his typical rumpled pants and pull-over, his famous hair going every which way, and very much unlike a still somewhat youthful-looking Weyl, dressed in his usual suit and tie (third from the left):
Einstein once famously remarked that his lifelong contempt for authority eventually attracted the consternation of God, who made Einstein an authority himself as punishment. But Einstein's "authority" late in life was really just an honorary thing, his relevance in physics having long been extinguished when his unrealistic dream of unification just wouldn't go away.
Still, van Dongen's book offers an insightful look into how and why the beauty of physical law so strongly resonates in today's scientists. |
| Eleven Pictures of Time -- Posted by wostraub on Thursday, August 12 2010 |
 What, then, is time? If no one asks me, I know. If I wish to explain it to someone who asks, I know it not. — St. Augustine of Hippo
The distinction between past, present and future is only a stubbornly persistent illusion. — Albert Einstein
The world simply is, it does not happen. Only to the gaze of my consciousness, crawling up the life-line of my body, does the world fleetingly come to life. — Hermann Weyl
Although I've thought about the nature of time since I was little, I haven't gotten very far. But now I'm reading The Eleven Pictures of Time by the Indian mathematical physicist and computer scientist C.K. Raju, and it's helping a lot. This guy has put one hell of a lot of thought into what time really is.
I think it's safe to say that most people believe in the concept of absolute time, the Newtonian idea that time moves at the same uniform rate everywhere in the universe (Newton himself questioned this uniformity, but eventually settled on it to keep his theories rational). It was Einstein in 1905 (or, if you believe Raju, Lorentz in 1904) who showed that time is relative, its rate depending on the relative velocity between two events or observers. Einstein's theory of special relativity therefore gave us the concepts of proper time (or "wristwatch time," the absolute time that an observer carries with himself) and coordinate time, the time that we measure on a clock that is moving apart from us with some relative velocity. Thus, the clock of a person fixed in some reference frame will differ from that of a moving person, in accordance with the rules of the Lorentz transformation. As every high school student knows, these rules have been verified countless times in the laboratory.
When the velocity of relative motion hits the speed of light, things get very weird. Massive bodies cannot travel at the speed of light in a vacuum, but light itself, in the form of photons, can (and indeed must). Because it travels on what is known as a null worldline, a photon exists everywhere in the universe instantaneously: space and time literally have no meaning to a photon. In your frame of reference, photons are "born" when you turn on a lamp, and "die" when they are absorbed in the retina of your eye. But the photons see things very differently: to them, they have always existed, and will always exist. The phrase "God is light" may have more meaning than we normally accord it.
Absolute time is easier to think about because we all perceive time as flowing at a constant rate, and it is logical to think that this rate is the same for everybody everywhere. So when a loved one dies, we say "They're with God now," or some such same-time rationalization. But this is almost certainly not true. It would be more correct to say that, from our reference frame, "That guy's dead," while in the dead person's reference frame things could be very different.
I bring up this issue of dead and alive because, like Raju (and probably many others), I feel that it's central to the very concept of time. The closest thing we'll ever get to death without experiencing the real thing is sleep (or being in a coma), during which we do not experience time. That is, a state of unconsciousness renders time immaterial, or at least irrelevant. It is tempting to think that extreme preoccupation with something can change our perception of time, as in "We were having so much fun. Where did the time go?", but this is a substantially different perception of time. When we die, time no longer exists, and if there is an afterlife, we move instantly to whatever end point awaits us. The physicist Frank Tipler called this the "Omega Point" which, in coordinate time at least, supposedly exists trillions of years in the future.
In his book, C.K. Raju talks about "cyclic" and "linear" time, and points out the effects these concepts have had on religious thought, policy and dogma. A Christian believes in linear time, in which time progresses through the Apocalypse and Judgment and then onward toward eternal Heaven and Hell. By comparison, a Hindu would believe in cyclic time, in which time progresses as a cycle of continued death and rebirth. These two extreme views of time, Raju shows, have important consequences regarding the concepts of free will and predestination. (Raju is careful not to get into purely metaphysical arguments in his book; he's simply pointing out how important time is in our perception of things, particularly when they involve religious issues.)
We tend to see life and death as polar opposites, with death representing a state of nothingness. But this point of view is anathema in quantum field theory, which says that there is no such thing as nothingness. The quantum vacuum is a region of space-time in which there is no matter or energy, no lines of force, no electric or magnetic fields, no scalar or magnetic potentials. But it is hardly empty: according to QFT, virtual particles and photons and their fields are constantly being created and annihilated in a vacuum in accordance with the "other" Heisenberg uncertainty principle, ΔEΔt ≥½ ℏ , where ΔE is the uncertainty (actually the standard deviation) in the mass-energy of the created particle, Δt is the uncertainty in the time span of its existence, and ℏ is Planck's constant divided by 2π. Tiny particles thus flash into and out of existence for unimaginably brief periods of time, but their existence is a fact nonetheless, and they can interact with one another and enjoy a "life" in spite of the fact that "nothing" is there.
This seemingly preposterous concept has to be accepted as true, because QFT is the most precise theory ever to come out of the mind of man. It predicts, to cite just one example, a gyromagnetic ratio for the electron that is accurate to 12 decimal places as compared with experiment. This is like measuring the distance between Los Angeles and New York to within a fraction of the width of a human hair. Even then, physicists refuse to call QFT a "fact;" it is and will remain just a theory for the simple reason that it is falsifiable. That is, QFT may someday be proven to be wrong in some fundamental way, or it may turn out to be just an approximation to an even more accurate theory (for instance, perhaps the calculated and experimental values of the gyromagnetic ratio will be found to differ at the 20th decimal place, necessitating a revision). This represents the essential difference between science and religion: religious belief is, by its very nature, not falsifiable because it cannot be tested.
So when fundamentalist Christians say that quantum mechanics is "just a theory," they're really being boneheaded. Indeed, they should embrace quantum physics wholeheartedly, because it represents the best hard evidence we have that God exists.
In his book The God Theory, astrophysicist Bernard Haisch writes that God's purpose is to take the potential of the universe He created and convert it into experience. Haisch uses the analogy of a person having a billion dollars to show that, although it is a vast sum of money, it is totally useless to its owner if it is not spent. In the same way, God converts the vast potential of matter and energy in the universe into experience, which for whatever reason He finds pleasing and interesting. (Does that sound implausible? Well, why should it? What else is God going to do with His time?)
Raju's book is very lengthy, and I'm still working my way through it. But he encourages the reader to "skip around" the book first and seek out bits of particular interest, then read it whole hog, and that's exactly what I'm doing. At about $75 on Amazon, it's pricy, but you can probably get it through your library. I encourage you to look at it. |
| "God's Number" Solved -- Posted by wostraub on Wednesday, August 11 2010 |
 Under the heading "God couldn't do it faster," this week's New Scientist reports that the maximum number of moves to solve any given Rubik's Cube combination has been solved. It's 20.
The computer programmer who worked 15 years to solve this conundrum used group theory to achieve the result. And I thought I didn't have a life.
I used to be pretty good with the cube, but I always had to use an established procedure to do it. I picked the thing up this morning out of a box in my closet, along with the instructions, and ... couldn't even follow the instructions anymore. The cube is now just sitting there, laughing at me.
Dang my 61-year-old brain. |
| Real and Unreal -- Posted by wostraub on Wednesday, August 4 2010 |
I've been watching the Science Channel's series The Wormhole hosted by Morgan Freeman, one of my favorite actors. It's OK, but for the past five or six years most of this stuff looks like it's just being cranked out of the same noise machine—the graphics and sound effects are neat, sometimes even inspiring, but the lack of explicit physics and math makes it all look a tad phony.
I don't doubt the sincerity of Mr. Freeman's enthusiasm, but he's just the latest guy to jump on board the cable channel high-tech express. Noted City College physics professor Michio Kaku has been doing it for years now (I think it's a full time thing, as I haven't seen any new textbooks from him lately), while British physicist Brian Cox (whose androgynous good looks make him appear like a cross between Rob Lowe and Keanu Reeves) is the latest guy to go the whoosh-bang route. Hey, I'm sure it pays better than a full professor's salary. But frankly, when Mr. Freeman gets done, I won't be watching this stuff anymore.
Nevertheless, one of the Wormhole episodes featured a pretty neat (if not entirely new) idea that seems to have been reworked from 1999's The Thirteenth Floor or maybe The Matrix. And, having just seen diCaprio's excellent film Inception, I believe that the basic idea still has something to say for itself.
It's this: As computers and digital simulation technologies get faster and more capable, they'll eventually be able to display 2D graphics that cannot be distinguished from the real thing (indeed, we're already pretty much there). The next step (and it's inevitable) will be 3D simulations that cannot be distinguished from reality. Outside of the undeniable entertainment value, there are a number of underlying issues involved with the unfolding of this technology that have rather severe religious, political and cultural implications.
The Wormhole episode posits the possibility that our very existence (and that of the universe we observe) is simply the output of a very powerful and technologically-advanced 4D computer simulation that will be developed and implemented by our descendants. An immediate corollary idea, actually proposed in the episode, is that the computer simulator is none other than God Himself. Along with this goes the idea that the program is so sophisticated that it can simulate beings that have (or "think" they have) free will. This idea is essentially the gist of The Thirteenth Floor (a favorite of mine), in which an aging scientist creates a simulated, pre-World War II world that he can actually drop in on and participate in (it involves young girls, natch). Things get complicated when the simulated beings discover they aren't truly real, and then get pissed off. Things get even more complicated when the simulators discover … well, you should rent the movie and see for yourself.
Inception is a bit different (it uses dreams in architecturally-fabricated dreamscapes) but just barely. Interestingly, it involves the "dream-within-a-dream" concept, which is a bit confusing but entertaining just the same. (Did you ever have a dream in which you consciously knew you were dreaming? If so, then next time ask yourself this question during the dream: Who's doing the dreaming?)
Well, none of this is entirely original, I know, and I know I'm hardly the first person to think about it. But there's another issue.
Imagine a day in the not-too-distant future when 2D and 3D simulation technology is perfected. Are you a big Humphrey Bogart fan like me, and wish he had made more films? With the new technology, and with the appurtenant ability of computers to "learn" a character and reproduce it, you can watch Bogie in any number of new, Oscar-level films (I'll go with the post-Casablanca sequel in which Victor gets bumped off and Ilsa is reunited with Rick). That's good, right? Well, it gets better. How about hitting a switch that changes the static and rather boring mountain view of your house's picture window into a sweeping, 3D panorama of the Grand Canyon, or a Cretaceous landscape with the occasional T. rex rambling by? What could be possibly wrong with any of this?
Well, when human beings are no longer able to distinguish computerized dreams or simulations from reality (and I believe Republicans have already achieved this), they will be subject to total control by others, and by that I mean the simulators. Imagine having a young, athletic, good-looking but completely non-existent, simulated President who is immune to the embarrassing, all-too-common gaffes and foibles of ordinary presidents. He (or she) is also smart and charismatic beyond words, and capable of uniting, mobilizing, coercing or convincing the nation's citizens toward some goal desired by his/her simulators. Imagine also that the media—television, radio, cable, newspapers—are also simulations.
Now things are not as pretty, but it gets worse, and this is it: Imagine now that, by accident, design, or the deliberate act of a newly-sentient computer, the simulators themselves are made to believe in their own simulated world; that is, the liars come to believe in their own lies, and become trapped in their simulated world without knowing it. This scenario seems preposterous, but it may in fact have already happened, as there would be no way to detect it.
Trapped in a world of one's own lies—it could be either Heaven or Hell, but the latter seems more likely.
The end of a bad day—the simulator in The Thirteenth Floor gets hoisted on his own petard. |
| Stranger and Stranger -- Posted by wostraub on Saturday, July 24 2010 |
In 1917, Hermann Weyl discovered the line element ds of a spherically symmetric gravitating mass in a vacuum in an isotropic coordinate system. This line element is often used in cosmological research because of its simplicity:
ds2 = A (dx0)2 - B(dr2 + r2dθ2 + r2sin2θdφ2)
where A and B are functions of the the radial coordinate r and the Schwarzschild mass (in a flat space, A and B both reduce to unity).
Earlier this year, University of Indiana physics professor Nikodem Poplawski wrote a paper demonstrating that a modified form of the isotropic metric can describe an Einstein-Rosen (wormhole) metric with interesting interior (within the event horizon) mathematical properties that are quite distinct from those of the simple Schwarzschild metric. In short, Poplawski posits the possibility that astrophysical black holes may actually be Einstein-Rosen bridges in disguise whose interiors hold entire universes of their own. Poplawski goes on to suggest that our own universe may be the interior of a black hole that resides inside yet another different universe.
According to Poplawski, a simple Schwarzschild black hole and the Einstein-Rosen variant would appear identical to exterior observers—only a plunge into the hole would reveal its true character. It is only in the hole's interior that the observer could determine if her universe were actually within a larger, separate universe. It is also possible that the black hole is just the usual Schwarzschild type, in which case the infalling observer would be torn apart by tidal forces and annihilated at the singularity.
Poplawski has now released a more recent paper (July 2010) that appears to confirm his ideas, although his approach utilizes an idea originally proposed by Einstein. It involves a connection term Γαμν that is not symmetric with respect to its lower indices. Poplawski shows that with such a connection the mass density of a rotating black hole is enormous but not infinite, a result due to the torsion (antisymmetry) of the connection term. Poplawski believes that this torsion is a consequence of the spin of elementary particles that make up the matter in the black hole.
Neither of Poplawski's papers is mathematically difficult, but his concepts and arguments are hard to follow (at least they are for me). The best I can advise is for you to look at these papers and decide for yourself if they make any sense. (The latter paper is mentioned in this week's issue of New Scientist, a fact that may or may not lend additional credence to Poplawski's proposals.)
The universe is stranger, and more wonderful, than anyone could have imagined. |
| The Truth Will Out ... Won't It? -- Posted by wostraub on Sunday, July 18 2010 |
I'm stepping out of my vacation (in the undisclosed location vacated by former VP Cheney) momentarily to post these two items.
The first is an article that appeared recently in the Boston Globe that reports on a study regarding how facts and truth affect people's beliefs. Conservatives tend to ignore the truth, while even well-educated and informed liberals (redundant, I know) can be expected to cling to their belief systems regardless of the facts about 10% of the time. Will we ever know the truth? It looks doubtful—cognitive dissonance is a b*tch.
Second, there's a paper written a few months ago by the noted University of Amsterdam physicist Erik Verlinde on the possibility that gravity is only a consequence of the entropy associated with the position and dislocation of physical bodies. It's 29 pages long, but written at the third-year undergraduate level, so it's easily accessible to idiots (like me). Verlinde derives Newton's second law (F = ma) from first principles (something I've never seen done before) along with Newton's law of gravitation (F = GMm/r2) and then goes on to derives Einstein's gravitational field equations. Verlinde's hypothesis is disturbingly simple but compelling, and many physicists today are thinking that the guy might be onto something.
And so, again, what is the truth? Does spacetime curve or does it not? What's going on? I'm beginning to despair that we'll ever know, but I still hold out the hope that God will explain it to me at some later date.
Extra: Globally, June 2010 was the hottest month ever recorded and 2010 is the hottest year on record so far. This information destroys Fox News' assertion that the previous winter's East Coast snow data "proved" global warming is a hoax. So what's the truth? And does it really matter anymore?It doesn't matter whether it's an otherworldly oilcano-style catastrophe, a gigantic explosion that kills dozens of [mine] workers that somehow prompts calls for even less regulation, the open acceptance of torture as an official practice, or wholesale electronic eavesdropping on domestic communications. As long as there are 31 flavors on the shelves and another season of American Idol coming, there's never going to be anything automatic about political passion. You're going to have to create it yourself, if that's what it takes to get you and your neighbors to hit the streets, vote, and generally do stuff. — David Waldman Have a great summer! |
Sophie did not forget Jesus
Long live freedom!
