©William O. Straub, fl. 2004-2016



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Who Was Hermann Weyl?

Wheeler's Tribute to Weyl (PDF)

Old Stuff
2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016

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!"

Visitors since November 4, 2004:

2008 Archives  

Keep Off The Grass -- Posted by wostraub on Monday, July 28 2008
I spent the better part of yesterday reading Leonard Susskind's latest book, The Black Hole War: My Battle with Stephen Hawking to Make the World Safe for Quantum Mechanics. Susskind, Felix Bloch Professor of Physics at Stanford University, is one hell of a smart guy, although this book has been dumbed down so much that I'm now wondering why I plunked down $20 for it.

Susskind's basic thesis is that information falling into a black hole is conserved after all, thus preserving a basic tenet of quantum theory. For years, the famed Stephen Hawking believed that information was irretrievably lost, and the book describes the ongoing "war" between the two physicists, who have nevertheless always maintained a close friendship and mutual respect for each others' ideas. However, inexplicably, Hawking's now-famous change of mind in 2004 is presented in Susskind's book as almost an afterthought, and it's very anticlimactic, in my opinion.

There are very few equations in the book, although Hawking and Bekenstein's famous black hole temperature formula is given, but its derivation is too simplified to be of much interest. But Susskind's overview of the interplay between entropy and information is very interesting, and I came away with a renewed appreciation for information theory. Also, Susskind's Holographic Principle, which states that the information swallowed by a black hole exists as a two-dimensional "film" residing on the hole's event horizon, has the ring of plausibility to it.

In Chapter 17, Ahab in Cambridge, Susskind kind of goes to the Bahamas and takes a breather from physics and all that to give us some insight into the soul of this notable Stanford physicist, who's also an avowed atheist. He shares with us some thoughts he has while sitting in the King's College Chapel, where he muses over the likes of Joseph Smith (the founder of the Mormon Church), Darwin, Dawkins and even Melville's Captain Ahab. Susskind goes so far as to admit a certain hollowness he feels over a lifelong belief in nothing but electrons, protons and neutrons.

But then Susskind puts into words a feeling I've had for many years. He describes "Cathedralitis" as the awe inspired by a cleverly assembled pile of stones and colored glass. Though I am a devout Christian, I also have been uncomfortable with such monuments to religiosity — the Sistine Chapel, Notre Dame, even the Crystal Cathedral in Orange County — because, while they were built intentionally to awe, intimidate and bolster the idea that God really exists (and make their owners rich), they are really nothing more than man-made idols of stone, mortar and glass intended to encourage an unquestioning, unreasoning and uncritical belief in their own version of the Almighty. The early Christian church, by comparison, was nothing more than a relative handful of people who met in each others' homes to worship and share ideas. That was the church of Jesus Christ's time, and it's much more appealing to me today. To spend untold millions on fancy megachurches while there is widespread poverty and injustice in the world is, to me, the height of hypocrisy. Anyway, it's a great chapter, and in it you'll also learn why Susskind can walk across the grass at Cambridge today without being molested.

Two criticisms of the book. Susskind describes only two options for the fate of information falling into a black hole: it's either preserved or lost forever. But this assumes that you believe in black holes in the first place, and it isn't until Page 248 that Susskind gets around to that. My other criticism is the book's lack of explanation as to where information comes from in the first place. Nature? The sentient human mind? Susskind is silent about this, and the book would have been better if he had addressed it.

BTW: Susskind has recorded ten lectures on quantum mechanics at Stanford that you can watch for free on YouTube, with each being about an hour and fifty minutes. The lectures are pretty elementary, but the man is fascinating to watch.
Weyl and Dirac, Again -- Posted by wostraub on Wednesday, June 18 2008
There is one strong reason in support of the [Weyl] theory. It appears as one of the fundamental principles of Nature that the equations expressing basic laws should be invariant under the widest possible group of transformations. — P.A.M. Dirac, 1973
It is well known that the great British mathematical physicist Paul Adrien Maurice Dirac was fascinated by the universal physical constants, particularly the gravitational constant G, and he spent a great deal of time in his later years trying to understand them.

Paul Dirac, 1902-1984. He gets my vote as the greatest
physicist who ever lived.

Dirac was also fascinated by the apparently accidental similarity of certain large dimensional constants in nature. For example, the ratio of the electromagnetic force to the gravitational force between a proton and an electron, which is about 2 × 1039, is roughly the same as the ratio of the age of the universe (about 13.7 billion years) to the basic atomic unit e2/mc3, where e and m are respectively the charge and mass of the electron. Dirac was convinced that these similarities were not coincidences at all, but consequences of a deep connection between cosmology and atomic theory. He called this connection the Large Numbers Hypothesis. Dirac’s consideration of these ideas led him to also believe that the numerical values of these constants might actually be changing with time.

Thirty-five years ago this month, Dirac published a paper* in which he attempted to explain this connection using Weyl’s original gauge theory of 1918. That the renowned Dirac would return to this theory nearly fifty years after Weyl himself had abandoned it is testimony to the beauty of the idea (and, as anyone who is familiar with the guy knows, mathematical beauty was everything to Dirac). Dirac was particularly intrigued by the possibility of parallel transport of an arbitrary vector in a Weyl space in which transport takes place in time only. Thus, Weyl’s basic differential equation for the variation in vector magnitude for a charged particle, as given by

dL = φμ L dxμ


dL = φ0 L dx0

where dx0 = c dt and φ0 is the Coulomb potential. Dirac noted that if vector magnitude is taken to increase with increasing time, then it would naturally decrease when dL is shifted into the past. Thus the symmetry of spacetime is broken by Weyl’s geometry, and it can be fixed only if one simultaneously changes the sign of the charge of the particle while interchanging future and past. It was precisely this kind of symmetry breaking that led Dirac to believe that the numerical value of the gravitational constant G might actually be decreasing as the universe gets older.

Unfortunately, G is one of least precisely known fundamental constants. While most atomic constants have been determined to at least six or seven decimals, G's current value of 6.67428 ± 0.00067 × 10-11 m3 kg−1 s−2 hinders observational efforts regarding variation with time. Dirac himself was aware of this problem but he expressed hope that improvements in technology (e.g., interplanetary radar measurements) would answer the question within a few years of his paper.

Alas, Dirac's hope appears to have been in vain. Will we ever know G to greater precision? It is almost as if God decided to make the most obvious force in the universe the least well known. Don't ever think God doesn't have a sense of humor!

* P.A.M. Dirac, Long range forces and broken symmetries, Proc. R. Soc. Lond., A, 333, 403-418, 26 June 1973.
Weyl's Grave, Zürich -- Posted by wostraub on Monday, June 9 2008
Weyl Again -- Posted by wostraub on Sunday, June 1 2008
Here's a photo of a 39-year-old Hermann Weyl, taken in 1925. I have no idea where it was shot. It came on a photo CD I picked up back in 1999 at a conference at the University of Missouri at Columbia, of all places.

God's Country -- Posted by wostraub on Sunday, June 1 2008
Beautiful Washington State, May 2008. This is the Columbia River from Cathlamet, where I got attacked by some pelagic cormorants patrolling the ferry to Westport. I think they're trained to spot and go after Southern Californians.

Another Giant is Gone -- Posted by wostraub on Wednesday, May 21 2008
Willis E. Lamb, the physicist who verified the hyperfine energy shift in the hydrogen atom, is dead. He was 94 years old.

In 1928, English physicist Paul Dirac developed his relativistic electron theory, which predicted with extraordinary accuracy the allowed energy levels of the hydrogen atom. But the theory did not allow for tiny radiative corrections that were later deduced from quantum field theory. In that sense, Dirac's electron theory can be viewed as a single-particle classical theory, as it does not take into account the creation and destruction of virtual particles and antiparticles.

Curiously, Dirac was the first physicist to predict the existence of antimatter. He was also one of the founders of quantum field theory, and he realized early on that his electron equation was not the end of the story.

Immediately after the end of World War II, Lamb applied his radar research expertise to the hydrogen atom and, using an amazingly small apparatus by today's standards, discovered the energy difference. That difference, which has become known as the Lamb Shift, garnered Lamb the Nobel Prize in Physics in 1955.

The significance of Lamb's discovery lies in the fact that it immediately motivated advancements in theoretical quantum field theory, particularly the efforts of theorists Richard Feynman, Julian Schwinger and Sin-Itiro Tomonaga. All three went on to share the Nobel Prize in 1965 for their related work on quantum electrodynamics.
Weyl's Descendants, Again -- Posted by wostraub on Sunday, May 11 2008
I was recently contacted by Ulf Persson, Professor of Mathematics at the Chalmers University of Technology in Goetborg, Sweden. Dr. Persson says that he has contacted the son of Michael Weyl, the younger son of Hermann Weyl. I was glad to hear that Michael, who was born in 1918 (the year his father discovered gauge theory), is still very much alive.

I've asked Dr. Persson to forward whatever information he gets to me. I'll post whatever news I receive here.
Weyl and Noether -- Posted by wostraub on Saturday, May 10 2008
I found an Internet document today dated January 2008 by Peter Roquette, Professor Emeritus at the University of Heidelberg, Germany (by a strange coincidence, my daughter just happens to be visiting Heidelberg right now).

Anyway, Roquette writes about the personal and professional relationship between Hermann Weyl and Emmy Noether. It includes a translation of Weyl's appeal to the Nazi authorities to let Noether remain in Germany to teach, along with Weyl's funeral address on the occasion of Noether's death at Bryn Mawr College, Pennsylvania, where Noether was teaching at the time of her death in 1935.

I'm pretty sure I've posted Weyl's funerary address here before, but I couldn't find it in my archives. So, at the risk of being redundant, here it is again. It's quite moving, and from it you can get an idea of Weyl's selfless, humanitarian soul.
Weyl's Descendants — Where Are They? -- Posted by wostraub on Friday, May 9 2008
In the span of three months, I have received five requests for information on the descendants of Hermann Weyl. My answer: I have precious little information to give out.

Weyl had two sons, Joachim and Michael, who both went on to become PhDs. Joachim distinguished himself in mathematics. I know they had children themselves, along with grandchildren. Three years ago I was contacted by one of them, Elizabeth T. Weyl. After exchanging several emails, she stopped writing. Perhaps she passed away, or perhaps she is a Republican who takes umbrage with my political views (although I always thought Massachusetts was fairly liberal).

Since the centenary of Einstein's annus mirabilis, there has been much research into the fate of little Liserl, Einstein's love child (with Mileva Marić), who was born in 1902 and immediately given up for adoption. The rest of the Einstein family we know about quite well. But I have drawn a blank on the whereabouts of Weyl's descendants. This is a pity, because from what little I know Weyl enjoyed a close relationship with his sons. Einstein, on the other hand, was a miserable husband and father.

Perhaps one or more of Weyl's descendants knows about this website. If so, I wish they would contact me.
John Wheeler is Gone -- Posted by wostraub on Monday, April 14 2008
John Archibald Wheeler died yesterday in New Jersey at the age of 96. The inventor of the words "black hole" and "wormhole," Wheeler was teacher, colleague, mentor and friend to generations of physicists. He knew Hermann Weyl personally, along with Einstein and many other world-class physicists.

Wheeler (center) receiving the Enrico Fermi Award from President Johnson

After graduating from high school at 15, Wheeler received his PhD in physics in 1933 at Johns Hopkins University at the age of 21. That was it — no bachelor's or master's degrees, he focused straightaway on the PhD. As a professor at Princeton, he advised dozens of notable doctoral students, including Richard Feynman, Kip Thorne, Charles Misner, and Hugh Everett.

Despite having an encyclopedic knowledge of physics, hundreds of research papers and many fundamental discoveries to his credit, Wheeler never won the Nobel Prize, but many of his students and admirers did.

Wheeler will always be remembered for his contributions to general relativity, which he usually referred to as "geometrodynamics." In 1973, with co-authors Kip Thorne and Charles Misner, Wheeler wrote Gravitation, still the de facto standard text for physics graduate students.

Wheeler was one of the very few still-living scientists who saw it all — the rise of quantum mechanics and quantum field theory in the 1930s, the creation of the atom bomb (which he actively participated in), and the development of modern nuclear physics and cosmology. He will be missed.
Weyl Movie Clip -- Posted by wostraub on Friday, April 4 2008
I spotted this on You Tube this evening. It's a short home movie shot in 1947 at Fuld Hall at the Institute for Advanced Study in Princeton, New Jersey. You'll see Einstein, Gödel, Dirac, and (at about the 1:40 mark) a 61-year-old Herman Weyl talking with mathematician Paul Erdos. It's the only movie clip of Weyl I've ever seen.

Unfortunately, the picture quality is poor and there's no sound.

Einstein and Besso -- Posted by wostraub on Thursday, April 3 2008
Earlier I mentioned Michele Besso, Einstein's best friend since their very earliest days as students, who died only one month before Einstein's passing in April 1955.

I have since come across a fascinating account of how Einstein and Besso together tackled the problem of gravitation in mid-1913. The two collaborated on Einstein's general theory of relativity (gravitation) at that time, producing a set of notes that has since become known as the Einstein-Besso manuscript. It is one of only two extant manuscripts that document Einstein's frustrating struggle with the theory in the few years prior to his final triumph in 1915.

The 52-page manuscript is described in detail by Michel Janssen of the University of Minnesota in his wonderful article "The Einstein-Besso Manuscript: A Glimpse Behind the Curtain of the Wizard," which you can download from my site Here.

Einstein had already formulated the groundwork of his general relativity theory, and he knew he was on the right track, but there were problems that continued to plague him. Besso stepped in and the two men tried to resolve the difficulties, with Besso acting mostly as a sounding board to Einstein's efforts.

Janssen's article includes reproductions of actual pages of calculations in the hands of both Einstein and Besso. It is remarkable that the brilliant Einstein was capable of making truly simple mathematical mistakes, a trait that should make the rest of us mortals feel a tad better.
Einstein and Weyl -- Posted by wostraub on Wednesday, April 2 2008
British physicist Stephen Hawking has written a number of popularized science books (like The Universe in a Nutshell), but he has also edited a number of books dealing with the history of physics. One of these is 2007’s A Stubbornly Persistent Illusion: The Essential Scientific Works of Albert Einstein, which by itself perpetuates the seeming endless number of Einstein books that have appeared since the 100th anniversary of Einstein’s special relativity theory in 1905. The book’s title comes from a letter of condolence Einstein wrote to the family of his dear and lifelong friend, Michele Besso, who died in March 1955, just before Einstein’s own death a month later:
Now he has departed from this strange world a little ahead of me. That means nothing. People like us, who believe in physics, know that the distinction between past, present, and future is only a stubbornly persistent illusion.
The book is really nothing more than a collection of articles that Einstein wrote from 1905 to 1955, most of which are translated from the original German. I have read all of these articles over the years, not only for the physics but also to get a grasp of the mind of Einstein. In comparison with contemporary physics articles, Einstein's stuff is fairly easy to follow, and many of his philosophical ideas are a revelation, so I encourage you to pick the book up (your local library probably has it) and read it for yourself.

The special theory of relativity was but a prelude to Einstein’s greatest work, which was his general theory of relativity. Published in November 1915, it probably would have stunned the world even more if it hadn’t appeared at the height of World War I. Physicists prior to the advent of the theory had been concentrating on advancing physics via the special theory. They all made some progress, and a number of them were actually attempting to explain the nature of matter through the theory. These attempts all failed, but when the general theory arrived they took up the cause with renewed vigor.

The first of these efforts was probably that proposed by Gustav Mie in 1913, whose attempts to explain matter as a consequence of electrodynamics resulted in one of the first unified field theories. Mie and others, including Hermann Weyl, renewed this failed effort using the physics of Einstein’s curved spacetime, perhaps the most notable aspect of his general theory. But these efforts also failed.

In 1919, Einstein himself attempted to reconcile the nature of matter with the gravitational field equations of the general theory. But he immediately noted that the presence of matter in his equations was "gravely detrimental to the formal beauty of the theory." Einstein's equations in the presence of matter can be written as

Rμν - ½ R gμν = k Tμν

where Tμν is the matter tensor. Einstein described the left hand side of the equation as being made of "beautiful marble," as it perfectly described the motions of the planets and a host of other physical phenomena. But Einstein described the matter term as being made of "wood," because it consisted of almost empirical quantities that did not depend on the variability of the metric tensor gμν. Even though Einstein was able to deduce the form of the matter tensor for electrodynamics and a field of moving, non-interacting, incoherent matter, it brought him no closer to any real understanding of the nature of matter.

Einstein’s beloved colleague Hermann Weyl was also obsessed with the nature of matter, and his signature book Space-Time-Matter was written primarily to address this very question. In doing so, Weyl came across his own unified field theory, which at first seemed to successfully incorporate electrodynamics into the geometrical construct of the general theory of relativity. However, in time this effort also failed, and over the ensuing years Weyl moved away from his obsession with the nature of matter and became more interested in group theory and quantum mechanics. Erhard Scholz, an expert on the history and mathematics of Weyl’s work, eloquently describes Weyl’s interest in matter in his 2007 article "The Changing Concept of Matter in H. Weyl’s Thought, 1918-1930," which you can download from my site here.

Einstein famously stuck with his theory for the last thirty years of his life, trying to develop a consistent field theory that would unify gravitation and electrodynamics and, later, quantum mechanics. Just as famously, he failed utterly in all of these efforts. But late in life he never expressed any bitterness or frustration over his failures, and he remained convinced that the Truth was in the equations, and that it was his own lack of intellectual and mathematical ability that had prevented the sought-after breakthrough. Today, we realize that Einstein can be forgiven on two accounts: one, that today's mathematical physicists, who are far better trained than Einstein (and probably a lot smarter as well), aren’t getting anywhere, either; and two, that Einstein’s ideas may yet lead to a successful theory (perhaps via the mathematics of strings), albeit in a form that the great scientist could have never dreamed of.

Two more final tidbits for the more mathematically inclined among you: In his 1919 article "Do Gravitational Fields Play an Essential Part in the Structure of the Elementary Particles of Matter?" Einstein noted that a slight change in his field equations as given by

Rμν - ¼ R gμν = k Tμν

would resolve numerous problems with the theory of matter in the presence of an electromagnetic field. The ¼ coefficient now makes the left hand side of the equation traceless, as is the matter term for the electromagnetic energy tensor Tμν. Although Einstein was forced to make the change without any real justification, it is interesting to note that the field equations resulting from Hermann Weyl’s 1918 gauge theory provide this very coefficient automatically.

Lastly, Einstein's equations are of second order in the metric tensor and its partial derivatives, while Weyl's are of fourth order. This fact has been used to criticize the theory as having more solutions than are necessary. However, Weyl's equations for free space force the Ricci scalar R to be a non-zero constant; this not only removes the fourth-order difficulty but leads to the same predictions (bending of starlight, gravitational red shift, perihelion shift of Mercury, etc.) that Einstein's equations have long been noted for.
Weyl Was Also Human -- Posted by wostraub on Tuesday, March 25 2008
Jean Eisenstaedt's The Curious History of Relativity relates how the simple calculation of the gravitational red shift tripped up a lot of very smart people, including Einstein (its discoverer!), Pauli, Eddington, von Laue, and even Hermann Weyl.

As brilliant as he was, Pauli was subject to occasional computational lapses. Henrik Casimir relates one such instance involving Pauli's derivation of the red shift formula in class:
... I remember that once when he was speaking about the so-called red shift, he obtained an expression with the wrong sign, which meant a shift towards the violet instead of the red. He then began to walk up and down in front of the blackboard, mumbling to himself, wiping out a plus sign and replacing it by a minus sign, changing it back into a plus sign, and so on. This went on for quite some time until he finally turned to the audience again and said: 'I hope that all of you have now clearly seen that it is indeed a red shift.'

What student has not witnessed a similar scene, what teacher has not suffered a comparable agony?
As noted, Pauli was not the only one to get tripped up in the derivation; Einstein and Weyl stumbled too. How often have I also stood at the blackboard, momentarily blinded by my own stupidity!

You can look the derivation up on Wikipedia if you're so inclined — it's simple. Yet it provides some relief to know that at times we all make mistakes, we're all human.
The Real Jesus -- Posted by wostraub on Sunday, March 16 2008

Blood for Oil

From today's Daily Kos:

Damn you rich! You already have your compensation.
Damn you who are well-fed! You will know hunger.
Damn you who laugh now! You will weep and grieve.
Damn you when everybody speaks well of you!

A rant from a radical preacher? Without a doubt. Someone on the Obama campaign? Well, Sen. Obama says so. That's the Scholars Translation of Luke 6:24-26, and the speaker is Jesus of Nazareth.

In the King James Version, the first part of Luke 6:24 reads But woe unto you that are rich! That comes off as quaint and a lot less shocking to modern ears — not the kind of preaching that nets you space on Fox News. But Jesus meant his words to be shocking. He meant them to strike against the status quo and shake up the comfortable.

God damn America for treating our citizens as less than human!
God damn America for as long as she acts like she is God and she is supreme!

That's Jeremiah Wright.

Is the vision of a pastor standing in his pulpit shouting "God damn America" shocking? Yes. But don't mistake Wright's (or Jesus') statement for what some drunk in a bar would mean using the same phrasing. Wright isn't saying "FU America!" He's saying "These actions of America are worthy of God's condemnation." He's just saying it in a way that cuts through the Sunday morning sleepiness and makes people sit up in their pew.

From Gandhi to King, it's in the nature of spiritual leaders to grab their audiences by the throat and their nations by the short hairs. This was true at the time of the Civil War and during the Civil Rights movement. Martyrs did not become martyrs by appealing to the status quo.

Don't take this to mean that I agree with every word that Wright spoke (e.g. the United States did not create AIDS), but neither do I feel like his words require that "his church should lose its tax exempt status" that he's a traitor, or that he's an embarrassment to his church or to Senator Obama — all comments that have appeared on this site.

Do I think that 9/11 was the "chickens coming home to roost?" Yeah, I pretty much do. Of course the terrorists bear the personal responsibility for their actions and the deaths that resulted. But to pretend that decades of actions overseas had nothing to do with that terrible morning is far more delusional than anything said by Rev. Wright. If you jab a stick into a hornet's nest and shake it for fifty years, the hornets might do the stinging, but you can't blame only the hornets. Actions have consequences, and though we may pretend to both purity of motive and prescience about outcomes, the truth is that violence tends to generate violence in return. Or, as that radical I quoted above said "those who take up the sword, will die by the sword."

The purpose of a good sermon isn't to placate, ease, and make people comfortable. A dangerous religion isn't one that challenges people and makes them squirm. Makes them angry. A dangerous religion is one that is too amicable to what you already think, one that pats you on the head and sends you forth in assurance of your own righteousness. If you want to search for "traitors" in the pulpit, turn your eye toward those who never find anything wrong in the actions of this nation.

I understand why Senator Obama finds it necessary to distance himself from Rev. Wright. There were plenty of things in those sermons that I don't agree with, and I'm suspect many of the ideas that grate on my nerves also strike the Senator as either wrong or unsustainable politically. These days, three isolated words on the news seem far more important than context or intent. But I wish he didn't have to do so.

Because getting your personal beliefs regularly challenged, rather than reinforced, is important.

AMEN! — wostraub
Weyl Theory Redux -- Posted by wostraub on Saturday, March 15 2008
Unified field theories continue to permeate the literature. Why the interest? Because scientists want to know how the universe works, and the more philosophical ones among us hope that a unified theory will also tell us how God's mind works. It's an addictive, if probably wasteful, pursuit, but it's a hell of a lot of fun.

Here's a recent paper by Indian physicist S.C. Tiwari entitled Electron in the Einstein-Weyl Space. It's very typical of what you see nowadays — you propose a Lagrangian in 4-dimensional spacetime that is gauge invariant and contains the Ricci scalar (in this case Tiwari uses the scalar from Weyl's theory) along with other terms, you carry out an infinitesimal variation in the metric tensor, and voila! — you get the equations of motion for a brand-new unified field theory.

I suggest that you read the linked paper for two reasons. One, it provides a very simple and readable description of the unification game (accessible even to undergraduates); and two, it serves as an example of why many researchers are still interested in Weyl's 1918 theory.

As I've mentioned many times before, I find this interest curious because Weyl's theory is just plain wrong. On the other hand, Weyl's work captivated even the great Dirac, who in 1973 wrote a very interesting paper that tried to resurrect Weyl's theory in a modern context (Dirac's paper is truly beautiful, and I've sent out pdf versions of it to many people. I'll email it to you if you're interested).

Next month marks the 90th anniversary of Weyl's theory. Though it's wrong, it's still neat to see it popping up in the literature from time to time!
Heller Wins Templeton prize -- Posted by wostraub on Thursday, March 13 2008
Poland's Michael Heller, an ordained Catholic priest and cosmologist, has been named the recipient of the 2008 Templeton Prize. The 72-year-old Heller, the author of dozens of books and hundreds of scientific articles, struggled for years under Poland's Soviet era of anti-religious and anti-intellectualist thinking.

Science gives us knowledge, but religion gives us meaning. — Michael Heller

Heller won the $1.6 million prize on the basis of his distinctively different views on the origin of the universe and the causal relationship between physical laws and their underlying mathematics. A leading proponent of noncommutative geometry as a basis for the description of the early universe, Heller's question "Did the universe really need to have a cause?" points to mathematics as the basis of all reality, with God the author of all mathematics.

Heller was interviewed on National Public Radio this morning, and I was struck by his compassion, intelligence and patience with the state of the world today. When asked why conservatives in the United States tend not to believe in evolution, Heller described the reason as a "misunderstanding."

Well, in my opinion the reason is that they're idiots. Heller is obviously a much more reasonable and tactful man than I am, and no doubt a far better Christian. God bless him.

I try to learn but, boy, it's hard sometimes.
Day of Reckoning -- Posted by wostraub on Tuesday, March 11 2008
Der Dummkopf Führer, George W. Bush, spoke to a group of religious-themed broadcasters in Nashville, Tennessee today and his theme, as usual, was that the rest of the world is evil and has to be destroyed. And everything Bush said in his speech confirms what I just read in traditional conservative Patrick Buchanan’s latest book, Day of Reckoning: How Hubris, Ideology and Greed are Tearing America Apart.

Buchanan has occasionally driven me to tears with his conservative opinions, but the views he expresses in this book I can almost completely agree with. For one thing, Buchanan appears to be a true Christian; he’s a Roman Catholic, but, like me, he does not approve of abortion, homosexuality, and the rampant materialism, anti-intellectualism and anti-environmentalism that now defines this corrupt country.

At the same time, Buchanan manages to upset me with his supportive views regarding Arctic drilling and minority population control. The chapter Deconstructing America is actually very offensive, although Buchanan probably does not realize it. He admits that the colonization of America was founded not on God, freedom and equality but rather on greed and racism, though he is not apologetic in the least.

But Buchanan just nails it on everything else, which pretty much has to do with the disastrous ideology of George W. Bush. Writing about the run-up to the Iraq War, he says
"The only lesson we learn from history is that we do not learn from history," I concluded. This was written six months before Bush invaded Iraq.

But the president’s mind had been made up. Having named Iraq an axis-of-evil nation possessing weapons of mass destruction, having laid out his doctrine of preventive war, Bush, in March 2003, ordered the invasion. In three weeks, it was over. Yet the United States has never been able to find any evidence Iraq was plotting to attack us or its neighbors, has never found any solid tie between Saddam and Al Qaeda or the perpetrators of 9/11, has never found an Iraqi nuclear program, has never found any weapons of mass destruction. We attacked, invaded, and occupied a nation that had never attacked us or threatened us — to strip it of weapons it did not have.
President George W. Bush is a criminal. Most of those in his administration are criminals. It will aggravate me to my dying day that these monsters will receive no earthly punishment for their crimes and the misery and grief they have caused. And it aggravates me beyond words that the citizens of this country, and the Republican Party in particular, have chosen not only to ignore these crimes, but to support them.

Finally, for your viewing pleasure, here's Bush at last night's Gridiron Club dinner, actually boasting in song that convicted felon Scooter Libby has finally escaped the prosecutor! Watch it if your stomach can stand it:

Hard to believe that this moron is the President of the United States.
Planck and Hitler -- Posted by wostraub on Sunday, March 2 2008
In 1937, the great German physicist and Nobel Laureate Max Planck personally begged Adolf Hitler to end the dismissal of German scientists on the basis of race and religion. Hitler's response was resolute:
Our national policies will not be revoked or modified, even for scientists. If the dismissal of Jewish scientists means the annihilation of contemporary German science, then we shall do without science for a few years.
Fortunately for the world, Hitler's stubbornness set back the German heavy-water experiments to such an extent that the Nazi atom bomb never really had a chance.

Unfortunately, I see the exact same intransigence in President George W. Bush. He, like the backward party of Republican morons he represents, does not understand science and clings to the superstition that evolution is "just a theory." Well, jackass, so is electrodynamics, quantum physics, chemistry, gravitation and everything else that the modern world has adopted as fact.

A few weeks ago I noted here that high-energy physicists in this country were looking forward to the restoration of funding that seemed all but lost last year. But now Congress has reneged again, due in part because the country's spiraling down a fiscal rathole over the trillions we are (and will be) spending in Iraq. And now America really is becoming irrelevant with regard to basic scientific research.

Case in point: the European Large Hadron Collider will turn on sometime this summer, leaving Fermilab and SLAC in the dust. It will almost certainly confirm or reject the existence of the Higgs boson and extra dimensions, along with a host of other hypothesized findings. But we can expect Bush to remain intransigent to the end — to hell with all that "scientifical" nonsense, we got wars to fight and win, and dark-skinned enemies to be triumphant over.
Hermann Weyl or Albert Dekker? -- Posted by wostraub on Saturday, March 1 2008
Hermann Weyl wore rather heavy eyeglasses late in life, giving him a distinctive Dr. Cyclops-like appearance. Here he is in an undated photo from the mid-1930s.

Persistence -- Posted by wostraub on Monday, February 25 2008
Hermann Weyl, along with the likes of Wolfgang Pauli, Theodor Kaluza, Oskar Klein and even Paul Dirac, each spent several years on the development of a unified field theory of physics. These scientists initially thought they were on to something, but the theories were either rejected outright or made predictions that were proved to be wrong (or worse, they didn't predict anything, period).

Around 1925, the unification bug also bit Einstein. But unlike his contemporaries, Einstein spent decade after decade chasing one crazy idea after another until, as his friend Abraham Pais had it, "he laid down his pen and died." Einstein labored for thirty years on unification while essentially ignoring the real progress that was being made by others, most notably in quantum mechanics. Pauli and the others, all close friends of Einstein, could only sadly shake their heads, wondering why Einstein had wasted all that time and effort.

The situation can be compared today with the fact that roughly thirty years have now elapsed since string theory was first proposed. But although string theory has made no verifiable predictions and has offered no real insight into the nature of the physical world, it seems as if everybody is working on it. This turnaround from the futility of Einstein's efforts to the hope that string theory is the correct "theory of everything" is especially notable in view of the caliber of the scientists who are currently engaged in string research (most of these guys far surpass Einstein in terms of sheer brilliance and mathematical ability).

Columbia University's Peter Woit, along with anti-string comrade in arms Lee Smolin of Canada's Perimeter Institute, believes that it's time to give string theory a rest. They both feel that the world's most brilliant scientists should refocus their efforts instead on conventional quantum field theory or on promising new ideas like loop quantum gravity.

In his book Not Even Wrong, Woit laments the very real possibility that, like Einstein, we've wasted decades of research effort and money chasing a nonexistent mirage. Smolin's book, The Trouble with Physics, reiterates this same lament. Both scientists fear that, given the effort and expense already invested, researchers will feel they have no recourse but to follow string theory to the bitter end — that is, if there is one.

Several physicists have noted that the quest for a unified theory can be likened to a disease of the mind. Weyl himself was afflicted by it, but he wisely gave it up and pursued more productive avenues. Although Weyl's basic unification idea remained in the back of his head, he finally discovered its true (and very profound) application in quantum mechanics, not gravitation.

While watching glaciers melt on the Nature Channel last night, my younger son and I discussed the plight of the polar bears, whose disappearing ice floes are forcing them to do their hunting while swimming on the open sea. He wondered if the bears will survive the next few decades, and I replied that I had my doubts. At that moment, I wondered why it is that our brightest minds are playing with strings and not working on ways to ensure the survival of planet's inhabitants, including us.

In my moments of darkest despair, I wonder if these scientists already realize that mankind is determined to destroy this planet and itself, and that working on unified field theory simply provides a means of occupying their tormented faculties.
Crisis in California Education -- Posted by wostraub on Thursday, February 21 2008
While the high-energy physics community can look forward to significant funding increases in 2009 (as promised by President Bush, who knows damn well he won't be around then to fulfill them), California public schools are facing truly draconian budget cuts.

It seems that we go through this little exercise on a regular basis. While he was campaigning to boot out Gray Davis in 2003, Arnold Schwarzenegger promised that he would be an "education governor," dedicated to providing educational excellence in California's public schools. When reality began to set in in 2005, Schwarzenegger threatened to go back on earlier funding promises, but backed down under enormous pressure by the Democratically-held State legislature.

Now the state faces a gargantuan $16 billion budget shortfall. Unlike the federal government, which can borrow money or let deficits go on forever, the states are required to pass annual balanced budgets.

So how is Schwarzenegger proposing to deal with the crisis? That's right — by slashing the public education budget by almost $5 billion. And this time around, in view of the enormity of the state's budget crisis, Schwarzenegger means it. In response, school administrators around the state are already sending out notices of anticipated budget cuts, hiring freezes, and teacher layoffs.

This seems to be the way this country does things — when the money gets tight, our children's educational needs are abandoned.

Did I hear you say "raise taxes" to deal with the problem? Unlike his predecessor Davis, Schwarzenegger knows full well that would be political suicide. Although Schwarzenegger can sexually molest women, smoke pot and inject himself with illegal steroids without unduly arousing the hypocritical Republican ruling class in this state (or the fawning celebrity-watchers still entranced by having a Terminator for governor), raising taxes would be a terrible mistake. Why, he might even be recalled!

How about increasing class size? Well, I've done volunteer work for inner-city schools having 40-50 kids in each class, and I can tell you that very little education actually takes place under these conditions.

Then what about year-round school? Or expanded classroom hours? Or Saturday classes? Those options might work, but if you think the kids are educationally challenged now, I suspect they'd be little more than mindless idiots if these alternatives were implemented.

How about home schooling? That might also work, provided at least one parent is available to teach his/her kids during the day. I understand there's a lot of this in the South, where godless subjects like math, science and evolutionary biology have been replaced by good old-fashioned Biblical indoctrination. But in most places where both parents work, the folks are already pretty shagged out from working 70-hour work weeks, so that's probably out.

The only remaining solution seems to be to leave our children ignorant. Sure, we need more engineers and scientists, but we can always get them from India and China. Besides, those foreigners seem to be the only people interested in math and science anyway.
Google Sky -- Posted by wostraub on Thursday, February 14 2008
Google has just released its latest version of Google Earth, which now includes Google Sky. The basic software can be downloaded for free, but if you want to spend $400 you can get the same version that CNN uses.

Google Sky has thousands of high-definition photos of galaxies, planets, nebulae and other objects, along with their descriptions and histories and even their right ascension/declination coordinates. It's an incredible way to learn about God's universe.

My house from an altitude of 1,700 feet, along with God's house from about 2.2 million light-years.
His is much nicer.
Physics Pornography -- Posted by wostraub on Friday, February 8 2008
We can do the innuendo
We can dance and sing
And when it's said and done
We haven't told you a thing.
— Don Henley, Dirty Laundry

Columbia physicist Peter Woit, author of the best-selling anti-string theory book Not Even Wrong, is talking about the possibility that the European Large Hadron Collider, when switched on this year, will create a rift in spacetime and produce a mini-time machine. He and others in turn speculate that this is nothing more than an example of Big Physics Pornography. I agree.

Last night, the Science Channel aired a one-hour show on Einstein. Presumably focusing on Einstein's last few hours of life in the Princeton hospital where he died in 1955, it was instead just a montage of break-neck graphics designed to convince the viewer that s/he was watching something profound. I watched it for about thirty minutes, at which point the flashy, meaningless graphics and the truly annoying musical vocals had me running over to Animal Planet. Even CCNY quantum physicist Michio Kaku, the usually reliable populizer of modern physics, couldn't save this turkey. Indeed, I found even Kaku annoying in this one.

Einstein is an example of Physics Pornography, intended not to inform and educate but only to entertain. It also serves as an example of what has been spoon-fed to an increasingly disinterested and ignorant public for some time now: news and information as entertainment, or infotainment.

Woit cites the recent paper by I. Aref'eva and I. Volovich, which considers Weyl's conformal tensor in five dimensions, Tipler cylinders and Gödel's spacetime, but is otherwise unnotable (the authors could have at least gotten Kip Thorne's name spelled correctly).

... not for my love or lovely wool am I here, but to make some world a meal ...
Weyl and the Gravitational/Electromagnetic Field -- Posted by wostraub on Saturday, February 2 2008
In the first edition of his 1918 book Space-Time-Matter, Hermann Weyl derived the combined gravitational/electromagnetic field surrounding a spherical, non-rotating material point of mass M and charge Q. (I believe Weyl was the first to do this, but I'm not sure; the Finnish physicist Gunnar Nordström is usually credited with the derivation. At any rate it has nothing to do with Weyl's 1918 unified field theory.) Anyway, Weyl used the Schwarzschild line element

ds2 = eνc2dt2 - eλdr2 - r22 - r2sin2θdφ2,

where the quantities ν, λ are functions of the radial marker r (this is not the physical distance from the center, but it's close enough for discussion purposes). Using Einstein's equations for the space external to the mass point, he came up with

which is just the Schwarzschild solution with an extra term that accounts for the electric charge Q.

It is interesting to note that spacetime is flat (eν, eλ = 1) precisely at the critical distance marker r = Q2/(8πc2M). At this distance, the field effects of the mass and charge exactly cancel one another (ignoring any quantum contributions). Is this distance physically relevant?

For a proton, this distance works out to roughly 10-30 meters. This is only five magnitudes larger than the Planck scale, so it certainly cannot have any physical significance.

What about extended regions of charged matter in which Q becomes large? Well, that's a thought, but then the mass M must also become large, offsetting the charge effect. I can think of nothing that might yield an observable critical distance with the possible exception of a charged black hole.

When a star undergoes complete gravitational collapse (by whatever process you choose), any net electrical charge is almost certainly radiated away, leaving an uncharged black hole. But if this loss could be circumvented in some way, then it is just possible that a detectable critical distance could exist. Still, even for a black hole of one solar mass the hole's charge would have to be truly enormous (roughly 1024 coulombs) to set up a critical distance of only one meter. But the hole's crushing tidal forces would completely overwhelm any interest one might have at that point!

In 1965, the line element for a rotating, charged black hole was discovered by E. T. Newman and his collaborators. It can be shown that the external field of such a body is consistent with Dirac's relativistic electron theory, a purely quantum theory. In the 1970s, Cambridge physicist Stephen Hawking used quantum field theory to show that back holes can evaporate through the process of electron-positron pair formation at the event horizon. These findings give hope that the long-awaited unification of gravity and electromagnetism may one day be discovered, perhaps via major breakthroughs in black hole observation and research.

Lastly, it is interesting to note that Weyl, Einstein and other early relativists did not believe in black holes. [Note: the term black hole didn't come along until 1967; in Weyl's day, it was called a collapsed star.] They believed that as matter was compressed, it would eventually provide sufficient pressure to resist the star's final plunge past the event horizon. But it was eventually demonstrated that pressure acts as a kind of mass-energy that serves to increase gravity's compressional effect; thus, pressure actually hastens the star's collapse to a black hole.

Fascinating stuff, simply fascinating.
Whiz Kid -- Posted by wostraub on Saturday, February 2 2008
In Volume 6 of their book The Historical Development of Quantum Theory, authors Jagdish Mehra and Helmut Rechenberg describe the early concepts of atomic and quantum physics in the formative years of those theories (from the Bohr hydrogen atom to the Schrödinger equation). It includes a brief overview of Hermann Weyl's unified field theory along with the reaction of noted physicists in the first few years following its announcement.

Weyl's theory appeared about 2 years after Einstein's November 1915 general relativity theory. Since Einstein's theory of gravitation perfectly predicted the deflection of light and the perihelion advance of the planet Mercury, it was natural that Weyl's theory would also be investigated to see how well it could describe these phenomena.

It's hard to believe, but the book recounts how an 18-year-old Wolfgang Pauli was one of the first to investigate Weyl's theory. Using a variational principle for the Weyl action, Pauli showed the theory was in complete agreement with Einstein's, thus paving the way toward a fuller recognition of Weyl's work by the physics community.

However, Pauli also believed Weyl's theory to be deficient in several important areas. In 1919, Pauli wrote
The main difficulty ... is that the theory does not satisfactorily explain the asymmetry of both types of electricity [i.e., positive and negative charge]. That won't be easy to remove. I also do not want to forget about a physical-conceptual objection. In Weyl's theory one operates constantly with the field strength in the interior of the electron. For the physicist, however, this quantity is defined only as a force acting on a test body; and, since exist no smaller test bodies than the electron itself, the concept of an electric field strength in a mathematical point appears to be an empty, meaningless fiction. In physics, one would prefer to stick to introducing only those quantities which can, in principle, be observed. Might we not altogether be on the wrong track in considering continuum-field theories for the field in the interior of the electron?
It is absolutely remarkable that the teenage Pauli was not only able to analyze Weyl's theory in the way he did but to actually anticipate the need for extended (i.e., discontinuous or non-point) theories for the elementary particles. Almost 70 years later, Pauli's musings became the basis of modern string theory.
Yang, Mills and Weyl -- Posted by wostraub on Saturday, January 26 2008
The role of the scientist is not to dictate natural law but, rather, to uncover and elucidate it. However, if it were up to the physicists to choose the theoretical framework within which the physical world operates, gauge theory would be a promising candidate. It is through gauge theory that science makes its greatest inroad toward the reduction of the full spectrum of physical behavior into a single inevitable underlying principle of causation.

— Bruce Schumm, Deep Down Things, Chapter 8.

Someone wrote to me asking if gauge invariance hadn't actually been discovered by C.N. Yang and Robert Mills in the early 1950s, and not by Hermann Weyl. I suspect she had read Schumm's book, which focuses primarily on the contributions made by Yang and Mills. But in the book's end notes, Schumm rightly gives Weyl credit for the discovery in 1929.

Schumm goes on to say that Weyl saw U(1) gauge invariance primarily as a means of explaining the indispensable role of electrodynamics in quantum mechanics, while Yang and Mills took it to the next step, SU(2), in which the principle of gauge invariance becomes non-Abelian (non-commuting). SU(2) symmetry was subsequently expanded to SU(3), which is the symmetry behind quantum chromodynamics. For either symmetry, the gauge principle is an unavoidable requirement (in fact, these theories are absolutely reliant on the gauge concept).

There is no question that Weyl was aware of SU(2) symmetry late in his life, although he seems to have been unaware of the work of Yang and Mills. Yang himself expressed the opinion that Weyl would likely have been overjoyed by the findings in their work. Unfortunately, Weyl died a year after Yang and Mills published their first paper in 1954.

Are there symmetries higher than SU(3)? Indeed there are, and there has been much theoretical research into the so-called higher gauge theories for generalized SU(N) symmetry. But SU(3) is already pretty complicated, and theories like SO(10) are almost unimaginably so. So far, the Standard Model of particle physics requires only SU(3), but anticipated results from the Large Hadron Collider (see my January 24 post) may change all that.
"Sometimes I Dream of Higgs" — Tom Stoppard -- Posted by wostraub on Thursday, January 24 2008
The February issue of Scientific American has a couple of great articles focusing on the imminent start-up of the European Large Hadron Collider, now being completed after interminable delays. The collider will bring mankind into the 1-TeV (one tera electron-volt) energy range, which may be sufficient to detect the long-awaited Higgs particle (which is believed to be the agent that imparts mass to elementary particles) and any extra dimensions of spacetime.

BTW, here's a snippet of Tom Stoppard's 1962 play The Real Inspector Hound, which seems oddly relevant to the elusive Higgs particle:

Moon: It is as if we only existed one at a time,
combining to achieve continuity. I keep space warm
for Higgs. My presence defines his absence, his
absence confirms my presence, his presence precludes
mine ... when Higgs and I walk down the aisle together
to claim our common seat, the oceans will fall into
the sky and the trees will hang with fishes.

Where's Higgs?

That's pretty weird!

Dark Matter, Dark Energy -- Posted by wostraub on Wednesday, January 16 2008
Last night the History Channel aired a one-hour program called Dark Matter/Dark Energy, a pretty good layperson's overview of the "missing matter" problem now vexing cosmologists. The usual gang of experts, including Michio Kaku, Alex Filipenko and Caltech's Richard Ellis were on hand to try explain why it's a big deal. And it is a big deal — dark energy, a kind of anti-gravity field, is apparently causing the universe to expand at an ever-increasing rate. In only a few hundred billion years(!), the universal expansion rate will approach the speed of light, and all bits of matter remaining in the universe will exist only as lonely, cold, isolated islands of stuff. Ice death, as some cosmologists call it.

In the early 1920s, Einstein (who believed the universe must be static) famously introduced a "fudge factor" into his gravitational field equations to prevent them from producing either an expanding or contracting universe. When astronomer Edwin Hubble showed in 1929 that the universe was expanding, Einstein tossed out the factor, calling it the greatest blunder of his career.

But late in the History Channel program, Kaku describes how Einstein might have had it right all along. Since dark energy appears to be a kind of ether permeating all space, it might not be any kind of matter or field at all, but a quality of spacetime that is represented by by Einstein's fudge factor (the cosmological constant Λ).

It is interesting to note that Hermann Weyl's modified version of Einstein's theory produces an intrinsic acceleration effect without the artifice of a cosmological constant. If you vary the Einstein-Hilbert and Weyl actions respectively, you get

where the cosmological constant Λ has to be intentionally added in after the fact (this is allowed because the covariant derivative D of the metric tensor is zero). Weyl's action principle results in a traceless field term which, when applied to empty space (T = 0), gives the same results as Einstein's field equations (gravitational red shift, bending of starlight, perihelion advance of the planet Mercury, etc.) but with an acceleration effect that is proportional to a constant, non-zero Ricci scalar R. However, Weyl's field equations are not divergenceless. This means the theory cannot account for the conservation of mass-energy.

I've pondered previously if Weyl's theory (which has been proven wrong) was even more on the right track than Einstein's. But until scientists can get a better handle on the problem, we won't know for sure who, if anyone, was right.
Let's Go Back -- Posted by wostraub on Wednesday, January 16 2008
I remember when I started college in 1967 there was this guy in one of my classes who had a Curta calculator. He paid something like $150 for it, a fortune in those days, and I still remember how badly I wanted one:

The Curta was produced by some Austrian guy who worked up the design while in a Nazi concentration camp. It had the look, size and feel of a combination pepper grinder and hand grenade (which means that you'd win a free one-way trip to Guantánamo if you brought one into class nowadays). It had a bunch of sliders on the side of the cylinder with which you'd dial in the numbers you wanted to multiply or divide, and this awesome crank on top that you'd spin until the thing locked up, giving you the answer. I don't recall how many decimal places it went out to, but it was much more accurate than a slide rule.

The only thing equivalent to a Curta was the Frieden calculator, an electric typewriter-like thingy with numerical entry buttons. It weighed maybe 20 pounds (so it was hardly portable), but it could do calculations out to 14 decimals on the better models. The downside was that it would have to crank for quite a while if you wanted that kind of accuracy.

My physical chemistry professor was a stickler for accuracy, and he demanded that we do our thermodynamics problems using the Frieden. This meant queuing up outside the school's data-processing center (yes, that's what it was called then) to get some time on the thing. Either that, or go punch a lot of data cards in FORTRAN and run the problems on the school's IBM 360.

I recently looked for a Curta calculator on eBay and got the shock of my life. Yes, they're available, but they now run around $1,000 for a working unit. Dang it all, I still can't afford one!
Too Late Learning Why I'm Hung Here -- Posted by wostraub on Thursday, January 10 2008
Hermann Weyl, the accomplished German mathematical physicist and somewhat less accomplished philosopher, was also a lover of poetry, particularly Goethe. I've read some of Goethe's stuff (translated from the German), and I admit that I don't get it. In fact, I don't get a lot of poetry, although I know what I like. I never knew why until recently.

I have a crumbling edition of Oscar Williams' Immortal Poems of the English Language, a paperback that I acquired sometime during my early college years. It pretty much sums up what I know about poetry. My favorite is Alfred Hayes' The Slaughter-House, which is deep and moving but also easy to understand. But does poetry really have to be understood in the left-brain kind of way?

I attended my very first poetry workshop this evening. I didn't understand much of what I heard, but with each poem I was left with a feeling that I couldn't quite put into words. I'm beginning to think that this is what poetry is really all about. It's that feeling you take home with you that you can't quite express.

The great British physicist Paul Adrien Maurice Dirac once lamented that poets take something understandable and make it obtuse, while physicists try to do the exact opposite. Perhaps that is why equations always appealed to me — the meaning and application is usually obvious, and I don't have to think about it too much. But occasionally an equation comes along that is also beautiful. When I look at Dirac's electron equation, for example, it's as if God is speaking. I don't even see Dirac as the discoverer — he's just some guy who accidentally stumbled across something that God has known intimately for all time. Is it pretentious to liken great physical ideas with poetry?

Maybe poetry is how God allows humans to express their innermost feelings without being too obvious about it. Overtness appeals to me, but that's mainly because I want the punch line and don't want to work too hard to get it. But, as Einstein famously remarked, God is subtle, and perhaps subtlety lets us enjoy the end result that much more.

I remember the first time I read Eliot's The Love Song of J. Alfred Prufrock. I thought to myself, It's just some middle-aged guy who can't make up his mind about a girl or something. I read it again this evening, and still didn't understand it; however, I'm middle-aged myself now, and the poem induces a feeling I did not have when I was young. Yet I distinctly remember my son Kurt reading it when he was in high school, and he was moved and saw its greatness. So what's wrong with me?

High school, that great four-year desert through which I wandered without a clue in a kind of girl-crazy, hormone-induced lunacy, did produce some appreciation in me for poetry. I still recall, with photographic exactness, my freshman English teacher, Mr. James McNeely, reciting Henley's Invictus. McNeely, then thirtyish, bald, of slight build and perhaps all of 5 feet tall, had undoubtedly experienced more than his own share of life's travails, yet there he stood, unapologetically booming out the poem, especially the line My head is bloody, but unbowed.

Now some 45 years in the past, that experience was my first inkling that there may be some good in poetry after all.

I certainly don't have another 45 years to work on it, but maybe it's worth another look.