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2010 Archives

Weyl and Schrödinger -- Posted by wostraub on Friday, December 24 2010
The Austrian physicist Erwin Schödinger came to the Swiss Federal Technical University in Zürich in late 1921, having been appointed chair of the theoretical physics department there. He soon met up with Hermann Weyl, chair of the mathematics department, and the two men became close friends and colleagues. And, according to Schrödinger's biographer, Walter Moore, even closer than that:
Those familiar with the serious and portly figure of Weyl at Princeton would have hardly recognized the slim, handsome young man of the 1920s, with his romantic black mustache. His wife, Helene Joseph, from a Jewish background, was a philosopher and literateuse. Her friends called her Hella, and a certain daring and insouciance made her the unquestioned leader of the social set comprising the scientists and their wives. Anny [Schrödinger's wife] was almost an exact opposite of the stylish and intellectual Hella, but perhaps for that reason [Weyl] found her interesting and before long she was madly in love with him. ... The special circle in which they lived in Zürich had enjoyed the sexual revolution a generation before [the United States]. Extramarital affairs were not only condoned, they were expected, and they seemed to occasion little anxiety. Anny would find in Hermann Weyl a lover to whom she was devoted body and soul, while Weyl's wife Hella was infatuated with [Swiss physicist] Paul Scherrer.
http://www.weylmann.com/erwin.pngHmm … well, Weyl was human after all. But what I want to describe here is how else Schrödinger and Weyl interacted. First of all, Schrödinger's discovery of the wave equation in 1925 could not have happened without Weyl's help in solving the differential equations, and Schrödinger (who shared the 1933 Nobel prize in physics with Dirac for this discovery) generously credited Weyl with his assistance. They also loved philosophy and mathematics, enjoyed hiking in the mountains, and had very similar political views. But, perhaps inevitably, the two men also shared a fascination with unified field theory.

Schrödinger was probably always interested in the problem of unifying gravitation with electromagnetism, but it was not until the early 1940s that he began to give the problem serious thought. Around 1946 he turned his thoughts to something that Einstein had become obsessed with, which was the idea of a non-symmetrical affine connection. In 1947 Schrödinger published several papers (The Final Affine Field Laws (Proc. Roy. Irish Acad., 51, 168-171), in which he tried to develop the most general affine connection, subject to as few constraints as possible.

In 1950, Schrödinger summarized some of this work in a little gem of a book called Space-Time Structure. In it he derives the following expressions, which he claimed met the most general necessary and sufficient conditions for a symmetric connection:

Here, Schrödinger's tensor Tilk is completely arbitrary, with the exception that it meet the "peculiar symmetry conditions" given by Eqs. 9.12 and 9.12a. Schrödinger then notes that for the unit tangent vector Ai = dxi/ds, Eq. 9.12a requires that

where η is a trivial constant. Thus, in Schrödinger's formalism the magnitude of any tangent vector is a true invariant. In deriving these expressions, Schrödinger assumed from the outset that the covariant derivative of the metric tensor gik (symmetric or otherwise) vanishes.

In his 1918 theory, Hermann Weyl took a fundamentally difference approach. He broke from Riemannian geometry by assuming that vector magnitude was not absolute but could vary under parallel transport. But this required that the covariant derivative of the metric tensor be non-zero, and in fact is was equal to gij;k = 2gijφk, where the semicolon denotes covariant differentiation and φk is a vector field that Weyl identified with the electromagnetic four-potential. Weyl's theory predicted that the magnitude L of an arbitrary vector Ai would change according to dL = φk L dxk, which he used with the parallel-transported vector magnitude law 2LdL = gij;k AiAjdxk

Weyl's theory thus predicted that the magnitudes of all vectors (tangent or otherwise) will change under physical transport in an arbitrary way. However, there are absolute vectors that cannot change; two examples are the relativistic four-momentum, which is given by gijpipj = m2c2, and the Compton wavelength, h/mc. Einstein was the first to point out this discrepancy in Weyl's theory, and for this reason he declared it unphysical. Weyl squirmed, but in the end he had no recourse but to abandon his idea.

What I now want to demonstrate is that Schrödinger's T-tensor provides a way out of Weyl's dilemma. Equations 9.12 and 9.12a show that the magnitude of any vector proportional to the unit tangent vector dxi/ds must be truly invariant. Consequently, we have two possibilities for the metric covariant derivative:

gij;k = 0                             which provides one definition of Riemannian geometry, and
gij;k + gki;j + gjk;i = 0       where gij;k ≠ 0

We see that the latter cyclic expression is completely equivalent to Eq. 9.12a if we assume that the metric tensor gij is symmetric in its indices (as in Eq. 9.12). Thus, we can immediately identify Schrödinger's T-tensor with the metric covariant derivative.

In my write-up on Weyl's 1918 theory, I show that Weyl's definition of the metric covariant derivative and his connection term must be replaced by

gij;k = 2gij φk - gki φj - gjk φi
Γijk = - {kij} - gkm gij;m

in order that Schrödinger's connection term be consistent; the latter expression is identical (disregarding the minus signs) to Schrödinger's 9.11. (The term with the brackets is the Christoffel symbol of the second kind.)

Most importantly, Schrödinger's T-tensor allows for the invariance of certain vector quantities. In fact, I am at a loss to identify a single physically significant vector quantity that is not proportional to the unit vector. The velocity, momentum, charge and current density four vectors are all such vectors, and these are the ones that Einstein was actually addressing when he criticized Weyl's theory.

A last point: it was exactly this time 85 years ago that Schrödinger sequestered himself in the Alps for several weeks (with a lady friend) and came up with his famous wave equation. Weyl referred to this event as a great moment of discovery "during a late erotic outburst" in Schrödinger's life. Philandering seemed to bring out the best in Schrödinger, as least as far as his creativity was concerned. To each his own.

Trig Smash -- Posted by wostraub on Sunday, December 19 2010
The Republicans are trying to defeat the strategic nuclear arms reduction treaty (START) with the Russians. Why? Because they want the American people to remain afraid and subject to their manipulation. And they also want the American people to remain stupid, which is why they CONTINUALLY say "nucular" instead of "nuclear." It's exactly the same deal when they continually speak of the "Democrat Party" instead of "Democratic Party." By intentionally slurring these words, the Republicans are in essence saying F**K YOU.

Yesterday on CSPAN-2 (yes, I have no life), Senator Jon Kyl (R-AZ) was arguing that the treaty isn't necessary because we already have a good relationship with the Russians. If that's true, then what did President Reagan mean when he said "Trust, but verify"? Then grumpy old John McCain (R-AZ) gets up and says just the opposite, arguing that we can't trust the Russians anyway so a treaty would only play to their advantage. His other remarks seemed to make it clear that he will only be happy when World War III gets underway.

And, right on schedule, the Republicans are bringing up the War on Christmas ruse again. I'm a Christian, but I think in this country religion makes people STOOPID.

Stupid, fearful people are DANGEROUS, but that's just what the Republicans want (provided the people remain submissive and pliable). I fear the day that we'll have President Sarah Palin rattling her sabers in the White House, while little Trig Palin plays on the Oval Office rug with the nuclear football. You betcha!

Momma, when I am prezdint I want a nucular war THIS big!

Update: "Harry Reid Has Eaten Our Lunch" — Lindsey Graham (R-SC)

I think it's a pity that the recent passage of DADT, the new START treaty and the 9/11 First Responders health provisions are being viewed as failures on the part of Republicans. They are victories for the American people, but I doubt they'll make much difference when the Republican House takes over in January.

Remember, Governor Haley Barbour's (R-MS) approval rating is still at 73% in his state, in spite of his recent disastrous racist remarks concerning civil rights (and Obama's ratings there are still an abysmal 37%). What does this mean? It means that the Republican Party will come back with a vengeance (literally) in 2011. They want to take us BACKWARD, against compassion and science and reason. They want to straight-jacket God and everything Jesus Christ taught and rewrite the Bible according to their racist, idiotic irrationality. As a conservative Republican relative of mine said recently, "I hope someone blows that nigger's [Obama] brains out."

Today's Question: Is everyone in Mississippi fat, racist and stupid?

The Weyl-Dirac Phase Factor -- Posted by wostraub on Sunday, December 19 2010
I said earlier that I would give Dirac a rest, but I need to bring him up one more time in order to describe how Hermann Weyl's 1918 gauge theory provided the basis for Dirac's Large Numbers Hypothesis (LNH).

Before I begin, you might want to take a look at this recent (13 May 2007) paper by Saibal Ray and his colleagues, which provides a very readable summary of Dirac's work (and which was motivated by Weyl himself).

Briefly, Dirac interpreted the seemingly magical, recurring appearance of the number 1040 in atomic and cosmological physics by assuming that Newton's gravitational constant G is actually an inverse function of cosmological time; that is, G ∼ t-1. Of course, Dirac realized that any variation in G would not be compatible with Einstein's gravity theory, which he wanted to preserve. In 1973, Dirac found that he could do this in principle by appealing to Weyl's 1918 theory (you can download Dirac's paper here).

But Dirac was all too aware of the fatal flaw in Weyl's theory, a flaw that was originally pointed out by Einstein. Basically, this involves the fact that Weyl's theory is based on the assumed invariance of physics under the local conformal (gauge) metric transformation gμν → λ(x) gμν. This requires that physical laws must also be invariant with respect to a regauged line element (or world interval)

ds2 = gμν dxμdxν ,
ds → λ1/2 ds

    Herr Dr. Cyclops: Weyl in middle age.

But as Einstein was the first to note, the line element ds can be made proportional to the ticking of a clock which, because λ varies in space and time, would mean that the spacing of atomic spectral lines would vary as well. Furthermore, the magnitudes of all physical vectors in Weyl's theory change as well. This means that quantities such as the Compton wavelength (h/mc) and relativistic mass (m2c2 = pμpμ) would vary from place to place and time to time as well. They do not, so Weyl's theory must be wrong.

Naturally, Weyl desperately tried to save his theory. He did this by proposing that there are in fact two types of intervals, one of "persistence" and one of "adjustment," each of which are dependent on how physical measurements are made. Weyl believed that laboratory measurements of atomic phenomena such as spectra would not change because at the atomic level the line element is truly invariant. Weyl's idea appeared to most physicists as straw-grasping, and the theory subsequently was abandoned.

In developing the LNH, Dirac had to salvage Weyl's theory, and he accomplished this using an argument very similar to Weyl's. Dirac postulated the existence of two types of metric intervals: dsA was called the "atomic metric," which involved quantities in the theory that preserved truly invariant physical numbers, while the "Einstein metric" dsE was presumed to apply to cosmological measurements and to solutions of Einstein's gravitational field equations. Dirac assumed that dsE could not be observed physically, but was responsible for equations of motion and classical mechanics described by Einstein's equations. Dirac noted the similarity of these intervals with Weyl's argument, although he proposed his dsE and dsA quantities for a very different purpose.

Dirac then needed a conformally-invariant action lagrangian to get the equations of motion:

I = ∫ (-g)1/2 W d4x

where W is a linear function of R and other terms. Dirac set the Ricci term equal to β2 R, where β was a time-dependent scalar quantity whose gauge suitably offsets that of (-g)1/2R. You can read the Saibal Ray article for further details as to how Dirac used this lagrangian in his LNH theory, but the upshot is this: Dirac's atomic interval turns out to be dsA = β (t) dsE, and with this he could show that this implies that

G = G0

where G0 is the usual Newtonian gravitational constant. With β a function of time, Dirac's G is thus also time-dependent, and this time dependency can be shown to be the source of the recurring large number 1040 in physical theories. In Dirac's theory, the gravitational "constant" gets weaker with time, so the expansion of the universe tends to overcome the pull of gravity.

There is a fascinating corollary to all this. In Weyl's 1918 theory, the electromagnetic four-potential φμ undergoes an infinitesimal gauge transformation equal to

φμ → φμ + ½ ε ∂μπ

where π is the gauge parameter and ε is a small constant. If we now regauge the metric tensor according to

gμν → exp[-½ ∫φαdxα] gμν

we easily see that the new metric tensor is fully gauge invariant. (The exponential term is non-integrable because it is path dependent. Non-integrability was the basis for Einstein's rejection of Weyl's original theory, because truly invariant physical quantities like mass can change unpredictably as they are transported from from point to point. Meanwhile, the concept of "path length" in the Weyl theory becomes meaningless.) Likewise, all quantities made up of the metric tensor and its derivatives, such as ds2, (-g)1/2 and R, are also gauge invariant. We can then write a simplified action lagrangian that involves the gauge-invariant quantity (-g)1/2R without resorting to scalar multipliers, as Dirac was forced to do. Furthermore, we don't even need to assume that the Weyl vector quantity φμ has anything to do with electromagnetism; it is now just some space-time dependent quantity that must be used if we want a conformally-invariant theory. The exponential used to ensure this invariance is the Weyl phase factor, although it is also known as the Dirac phase factor. In fairness, I propose that it be referred to as the Weyl-Dirac phase factor.

Although Weyl developed his 1918 theory as a means of unifying gravity and electromagnetism, his theory appears to have cosmological applications that now seem to outweigh the theory's original intent. Indeed, recent papers have shown that cosmological phenomena such as inflation, the observed acceleration of the expansion of the universe and the cosmological constant might be explained by Weyl's basic idea.

For additional information, see Nathan Rosen's 1982 paper Weyl's Geometry and Physics, which I've described previously on this site.

Mathematics Under the Nazis -- Posted by wostraub on Tuesday, December 14 2010
The old saying about neglecting history and thus being doomed to repeat it would make an appropriate epitaph for the human race, assuming someone will be around to write it after we've destroyed ourselves. But it isn't just history that we neglect—we often willingly discard our innate rationality as well.

University of Rochester professor Sanford L. Segal's 2003 book Mathematicians Under the Nazis provides a fascinating look at the state of German mathematics during the period 1933 to 1945. You may have heard about the Nazis' efforts to eradicate "Jewish physics" in favor of pure Deutsche Physik (German physics). Beginning in the mid-1930s, numerous German physicists (notably Johannes Stark and Philipp Lenard) undertook efforts to discredit Einstein's relativity theories solely on the basis that they had sprung from the mind of a Jew and must therefore be wrong (indeed, relativity books were being burned in Germany not long after Einstein left the country in the summer of 1933). Like physics, German mathematics survived the Nazi ordeal, but both suffered greatly, though for different reasons. Segal's take on mathematics under the Nazis is particularly interesting.

Up until 1933, the German University of Göttingen was arguably the unrivaled hub of the world's greatest mathematicians, being the home of greats like Gauss, Riemann, Hilbert, Courant and Weyl. But in April of that year all Jewish civil servants, teachers and professors were summarily fired, and Deutsche Mathematik entered a death spiral from which it never fully recovered. While Germans prided themselves on their achievements in pure mathematics, Nazi nationalistic irrationality quickly eroded the public's attitude toward the value of the field.

Segal describes a 1933 poll that was taken regarding the value of mathematical education in Germany. Surprisingly, even at this early date the majority of the respondents "found only elementary-school mathematics useful" within or without their professions. The poll, conducted by one Dr. Heinz George (a non-mathematician), led to his later conclusion in 1937 that
The judgment that mathematics has a high educational value is indeed widespread, nevertheless, it is a purely traditional and uncritically accepted prejudice, that one may not simply trust—rather, in this form of generality, it must be unconditionally rejected.
Although the "layman-perceived" virtues of higher mathematics were deemed "character-building for a good Nazi," the worth of pure mathematics to the average German of the time was severely questioned.

Segal notes that these attitudes had a perceptible effect on the German war effort, as there was a deep suspicion of academics by the political forces governing Germany at the time, a suspicion that persisted at least until 1943. Indeed, as Segal also notes, both the United States and Great Britain had formed "academic-military-industrial-political" complexes for the purpose of coordinating and executing their war efforts, while Germany undertook no such effort. It is probably safe to say that Hitler's emphasis on battlefield strategies and premature invasions over military scientific and engineering superiority prevented Germany's successful development of the atom bomb. In this sense, misguided attitudes toward academic disciplines aided in Nazi Germany's demise.

Segal's book is a historical look at the state of mathematics at a specific period in Germany's history, a history which includes fantastic advances in the field. It is informational, not prophetic. But I see many parallels today in America today with regard to our attitude toward science and math. The emphasis on simplistic thinking, the right-wing anti-science movement, our inclination toward hubristic American exceptionalism, and the precipitous decline in the American educational system are significantly contributing to the dumbing-down of this country at a time when political, military and financial lies and falsehoods are running rampant—a dangerous situation that now exists only because America's people have chosen to stop thinking rationally.

Weyl and Emigration -- Posted by wostraub on Saturday, December 11 2010
Hermann Weyl left Germany in November 1933, arriving at the Institute for Advanced Study (IAS) in Princeton, New Jersey as one of the new school's most sought-after scientists.

He left with some misgivings. As a loyal and esteemed German citizen and Christian, he had nothing to fear from the Nazis, but his wife was Jewish (as were his sons, by association), which put their lives in some danger. On the other hand, his mathematics, science and even soul were tied to Germany. Weyl had been made the offer by the IAS three years earlier (at then then-enormous annual salary of $15,000), but his decision-making faculties were in a state of disarray—at one point wanting to go, and at another wanting to stay.

Upon his departure from Germany, he joined many other emigres, some who left around the same time as Weyl (like Einstein and Noether), and others who delayed leaving for various reasons. By 1936, those who had stayed (especially Jewish scientists and mathematicians) had realized their mistake. For quite a few, it was indeed too late to get out.

I'm reading Mathematicians Fleeing from Nazi Germany by Reinhard Siegmund-Schultze, which traces the paths and fates of some 140 European (mostly German) mathematicians from roughly 1933 to the late 1940s.

I'd known about Weyl's efforts to assist friends and colleagues in emigrating from Germany, but I didn't know the extent to which he helped. Perhaps his biggest difficulty was recommending people he knew well or didn't know at all for academic positions without sounding like a broken record. I also knew nothing about the xenophobia and anti-Semitic attitudes of American academic institutions, attitudes that were joined by anti-German sentiments once America had declared war on Germany. Siegmund-Schultze's book goes into great detail describing Weyl's various correspondence with colleagues trying to get out of Germany and his pleas to American colleges and universities to accept them.

Several years ago, a number of signed Weyl letters became available on eBay that had been written around 1940 to the German mathematician Artur Rosenthal, himself a desperate emigre. I actually bid on some of these, but it was obvious that I wasn't going to outbid universities competing for these valuable documents (they sold for many hundreds of dollars).

We now know the efforts the US government made to secrete many German scientists who were outright Nazis, Nazi sympathizers and even Nazi war criminals out of the country for military purposes (Werner von Braun is probably the most infamous example). Siegmund-Schultze's book provides the stories of those scientists who were less well known but still important to American academic progress.

Obama—An Antic Disposition? -- Posted by wostraub on Thursday, December 9 2010
You may recall from your high school English classes that the real mystery of Shakespeare's Hamlet is the reason why the Danish prince delays acting against his uncle, King Claudius, who has murdered Hamlet's father in order to marry his mother and take the crown. For whatever purpose, Hamlet takes on "an antic disposition," acting insane while he bides his time in his long, wavering plot to avenge his father's death. This turns what might have been a short play into Shakespeare's longest work.

It must be obvious now to any idiot familiar with Hamlet that Barack Obama has taken on essentially the same role in his presidency. For almost two years, Obama has repeatedly blown off one progressive campaign promise after another and caved in to the minority Republican party. Like the other characters in the play, the Democrats have pretty much stood by dumbfounded, wondering why Obama has acted the way he has (some have even pondered the idea that Obama, like Hamlet, is actually crazy-clever and has had something up his sleeve all along). But now, with Obama's latest unexplainable acquiescence to the GOP on the tax cuts provision, his base has finally risen up like the ghost of Hamlet's father, chiding him This visitation is to but to whet thy almost blunted purpose. But in reality it may be too late for Obama.

Hamlet's father in this little scenario is, of course, played by the erstwhile Constitutional Republic of the United States, which has been taken over in recent years by corporate Republican greed, corruption and lies:
Ay, that incestuous, that adulterous beast,
With witchcraft of his wit, with traitorous gifts—
O wicked wit, and gifts that have the power
So to seduce!
Like the play's minor characters, the American people wander around as in a daze, without a clue as to what has really happened. One might think the actual act of murder took place on 9/11, but its seeds were planted thirty years earlier under another false king, Ronald Reagan, whose disastrous economic policies were the juice of hebona poured into the ears of a sleeping American people, who are now morally dead but don't know it. I suppose it's also possible that Americans play the role of the "seeming-virtuous" Queen Gertrude, who may once have been decent at one time but is now a moral wreck, totally under the sway of the "bloat king," Rampant Capitalism.

And where is the fair Ophelia? I see her as the Democratic Party itself, naive but hardly innocent, in love with the dispassionate Prince Barack who once adored her but now has more pressing concerns on his mind. Like Ophelia, the Democrats eventually rebel (Rich gifts wax poor when givers prove unkind) and commit suicide (Ophelia's comes as a result of madness born of unrequited love and repressed sexual tension, the latter of which hardly describes the Democrats, but what the hell). As for her father Polonius and brother Laertes (both of whom are corrupt agents of the King but nevertheless well-intentioned), I'd have them represent moderate Republicans: they too will die in the end, when the King's murderous actions are revealed.

Ophelia, like the Democratic Party, headin' on down the river.

Lastly, Hamlet's school chums Guildenstern and Rosenkrantz are played by Obama's financial advisers Timothy Geithner and Larry Summers (adders fanged), who are secret Republicans trying to play Obama like the proverbial flute in the play (however, in real life they succeed).

My favorite scene is the end, when the truth finally comes out. All the important players in the play die ghastly deaths, of course, with the exception of Horatio, Hamlet's true and trusted friend (I don't think Obama has any real friends, so perhaps I've taken the analogy too far). Anyway, at the very end all lay dead on the floor, while Fortinbras and his troops remain to pay homage to Hamlet with a volley of gunfire. A famous critic of Hamlet, whose name I've forgotten, once asserted that Fortinbras' guns are actually trained on the survivors, so everyone dies. And that would be America herself.

Hamlet's underlying theme involves rot, decay and moral corruption; a common element in the play is the idea of a barely scabbed-over festering sore that gets freshly exposed from time to time by the Truth. America herself is today corrupt and has blood on her hands, but she continually tries to scab it over with nonsensical distractions, like Lady Gaga, American Idol, mythic religiosity, and the Great American Dream of Getting Rich. But still, from time to time, the scab is ripped off, the blood comes to light again, and cannot be cleaned away.

"Out, out," we cry, but that's another play, and another story.

Good night, sweet prince. Like Hamlet, Obama's done for.

Magnetic Monopoles -- Posted by wostraub on Thursday, December 2 2010
I'm going to give Dirac a rest now, but before I leave him I want to tell you about another of his discoveries (actually, a prediction), which is the magnetic monopole, or bare magnetic charge. I bring it up because it also happens to involve Hermann Weyl's phase factor.

We all know that an electric charge can exist as an individual entity (of plus or minus charge), while all magnets exist as dipoles; that is, magnets necessarily consist of a north pole and a south pole which, to the best of our knowledge, cannot be separated. Indeed, to date nobody has ever found a magnetic monopole, although such a discovery would make Maxwell's equations perfectly symmetric in terms of the electric and magnetic fields. It would also likely win its discoverer a Nobel prize.

In 1931, Dirac set out to see if he could find a theoretical way of explaining why electric charge is quantized; that is, why all electric charges must consist of integer multiples of the basic electric charge (1.6×10-19 coulomb), which is the charge on a single electron or proton (Dirac's 1931 paper is here). Dirac started by expressing Maxwell's equations with a magnetic monopole charge density and current; this is quite straightforward—you just write them into the equations. Dirac then set about trying to derive a magnetic version of the familiar Maxwell electric density expression

∇∙E = 4π ρ

In doing so, Dirac immediately recognized that the related Maxwell expressions ∇∙B = 0 and B = ∇×A were no longer valid. Dirac's way around this was truly ingenious, but I will present two alternative (and mathematically non-rigorous) approaches to keep things a tad more understandable.

To begin, consider an elementary particle of charge e passing a magnetic monopole fixed at the origin. The particle moves horizontally with a velocity V at a fixed distance y = b from the monopole (b is called the impact parameter). In polar coordinates, the particle is then located at the point x = R cosθ, y = R sinθ, where R2 = x2 + y2. Assuming that the particle is located a sufficient distance from the monopole, we can ignore any deflection of the particle in the x,y plane.

In exact analogy with the electrostatic case, the monopole will set up a radial magnetic field B about the origin having a magnitude B = g/r2, where g is the magnetic coupling constant. The charged particle will then experience a force given by the Lorentz vector force law

F = eV×B/c

where c is the speed of light and × represents the curl operation. Calculating, we get a force acting on the particle perpendicular to the x,y plane given by

F = eVgb/cR3 = e Vgb/c(b2 + V2t2)3/2

where t is the time the particle moves from x = 0 to x. The force F results in a change in the particle's momentum given by Newton's second law, F = dP/dt, so that

dP = [eVgb/c(b2 + V2t2)3/2] dt

Integration gives the simple result

P = 2eg/c

In quantum mechanics, the momentum of an elementary particle is quantized in terms of an arbitrary integer N times the reduced Planck constant ℏ, so we have

2eg/c = Nℏ or

e = ½ Nℏc/g

where N = 0, ±1, ±2, etc. This result shows that, regardless of the numerical value of g, the charge of an elementary particle must be quantized.

The quantum-mechanical approach is much simpler. The wave function of a free particle is given by

Ψ = exp[i(p∙r - Et)/ℏ]

where E is the particle's energy. In the presence of an external magnetic vector potential field A, we have a condition known as "minimal coupling," which is pp - e/cA. Therefore,

Ψ → Ψ exp[-ieA∙r/ℏc]

We now have the particle exercise a single circular orbit about the source of the field A, which is the magnetic monopole. The phase term then becomes


(which is also known as the Weyl phase factor). In order for the wave function to be single-valued, this phase term must be equal to unity. This results in the condition

e∮A∙dl/ℏc = 2π N

where N is again any integer, including zero. Stoke's theorem guarantees that this can be expressed using a closed surface integral (using ∇×A = B),

e∮B∙dS/ℏc = 4eπ g/ℏc

so that

e = ½ Nℏc/g

which is the same result we got before. [I knocked this out as quickly as possible, and I hope all the terms are correct.]

We now note a few interesting aspects of this expression. One, the result is independent of the impact parameter b. Although we had assumed that b was sufficiently large to allow us to ignore any x,y deflection of the particle, we now see that this assumption was not necessary—the particle could be light-years from the monopole! Two, the result is independent of the particle's velocity V. This allows us to ignore any relativistic effects, which we certainly did in the first calculation.

While Dirac calculated this same result in 1931 using a somewhat different reasoning, the conclusion is inescapable:
If even a single magnetic monopole exists anywhere in the universe, then electric charge must be quantized.
Dirac's discovery is highly important, because it explains the quantization of all electric charges. The only snag is: do magnetic monopoles exist?

If they do, they should be very easy to find, as they would appear in even relatively insensitive electromagnetic detectors. But in spite of more than forty years' worth of dedicated effort, nothing's been found. Perhaps the most notable search was conducted in 1970 at the University of California Lawrence Radiation Laboratory by Luis Alvarez (the same guy who, with his son Walter, successfully predicted that dinosaurs were wiped out by an iridium-enriched meteorite). He used a superconducting niobium coil to scan a number of promising samples, including moon rocks brought back by the Apollo 11 astronauts. No monopoles were found.

Well, it was, to use a 1970s vernacular, a real bummer. As of today, the existence of magnetic monopoles is one of the great open questions of physics.

Food for thought: If you lower a bar magnet into a black hole until one end is barely sticking out of the event horizon, does the end become a monopole? Or does the black hole become magnetized? Or can you even know what's happening at the event horizon?

America and Egypt -- Posted by wostraub on Thursday, December 2 2010
I visited Egypt recently. The food is great, and the people are very friendly and hospitable, but they are desperately poor and the country itself is an economic disaster. The "elections" are over, and once again the Mubarak ruling party has "triumphed". And again, the Egyptian people are the real losers.

The drive from Egypt International Airport goes right through the area where Egyptian President Hosni Mubarak resides; it's also the area where the country's military headquarters are based. The area is simply beautiful—manicured lawns and gardens, trees, impressive buildings and clean streets everywhere, and very little traffic (but be warned not to take any photos or, as I CAN ASSURE YOU, you'll risk being taken in for questioning.)

A few miles further and you enter the city of Cairo. The pyramids and the Antiquities Museum are wonderful, but the city itself is a shambles; it literally looks like it's crumbling. The air is unbreathable, underpaid policemen are everywhere, the traffic is unbelievably crowded, and sections of the Nile are grossly polluted with floating trash (although the water itself looks surprisingly clean).

The view from the back of my hotel typifies the inescapable
poverty of the city of Cairo

America gives Egypt over three billion dollars annually in military and economic aid to perpetuate the incredible discrepancy between the lives of Mubarak and his military sycophants and everyone else. I thought the people's acceptance of poverty was based on the Islamic "will of Allah" that keeps everyone in line, but I was wrong. The real reason has to do with fear and intimidation—government repression is what keeps the people under control.

My hand on some Karnak heiroglyphics. I've always been
fascinated by ancient Egyptian history, which is
considerably more real than that of Israel.

Tourism has replaced agriculture as the nation's number one source of income. Tourists flock to the tombs and the pyramids and the antiquities, and rightfully so, but the country is preposterously poor and the difference between rich and poor is simply outrageous. Why we continue to subsidize the corrupt Mubarak regime is beyond me (he and his family are secretly caching billions of dollars outside of his country à la his predecessor, King Farouk), but it reminds me of what is happening today in my own country.

A typical daily wage in Egypt is about 20 Egyptian pounds, which is less than $4. It guarantees poverty, and no help is in sight from Mubarak; only the rich have health and dental insurance and pensions. Meanwhile, in America the Republicans want to give the rich yet another 6% tax break, while 15 million people have no work and 50 million have no health insurance. And the Republicans will most likely win out over a cowering President Obama, who is being walked like a dog by the GOP. I can't help but think that America is headed in the same direction as Egypt.

Is there a God in Heaven?

More Stars -- Posted by wostraub on Thursday, December 2 2010
Astronomers are in the process of revising the estimated number of stars in the observable universe.

Recent analyses of light spectra from stars in elliptical galaxies shows a larger amount of sodium and iron than previously thought. These elements are formed in older stars, leading scientists to believe that the number of red dwarfs is also larger, perhaps by a factor of nine. Since elliptical galaxies represent about one-third of the observable universe's total galaxy count, astronomers think that there could be as many as 300 sextillion (3×1023) stars in the universe.

What the reports aren't saying is that this number is remarkably close to Avogadro's number which, if you recall any of your high school chemistry, is about 6×1023, the number of atoms or molecules in a mol of any substance (6×1023 squirrels is also a mol of squirrels—a hell of a lot of squirrels, and roughly the same number as those in my back yard).

The British mathematical physicist Paul Dirac (see my previous post) was intrigued by certain large numbers in nature, and he developed his Large Numbers Hypothesis on the basis of his investigations. I think he would have been fascinated by this new development.

A large elliptical galaxy (upper left) and a smaller spiral galaxy (lower right).
Our galaxy is a spiral.

"God is a Mathematician of a Very High Order" -- Posted by wostraub on Tuesday, November 30 2010
I was writing the other day about Dirac's relativistic electron equation, whose negative energy solutions turned out to represent antimatter, a form of matter totally unknown in 1928 when Dirac's theory was first published. Dirac himself invented the term "antimatter" in 1931, about one year before the antielectron was discovered in the lab by Caltech's Carl Anderson, who himself received the Nobel Prize for his effort.

Dirac also proposed at the time that other forms of antimatter might exist, including antiprotons. This was a bold leap at a time when only four basic particles—the electron, proton, neutron and photon—were known to exist. We now know that there are antiparticles for every particle; some particles (like the photon) are their own antiparticle!

I am currently reading Helge Kragh's 1988 book Dirac: A Scientific Biography which, until several weeks ago, I was completely unaware of. I read Graham Farmelo's excellent Dirac biography last year and figured that was pretty much the last word on Dirac, but Kragh's book is also interesting, primarily because it addresses Dirac's professional relationship with Hermann Weyl. It also includes much more mathematical details than Farmelo's book.

Kragh reports that Dirac was much impressed not only with Weyl's mathematics (which he also found difficult) but with Weyl's approach to mathematical and physical truth as well. Dirac was especially impressed with Weyl's failed unified field theory of 1918, and even turned to it in a remarkable paper he wrote in 1973. In spite of the theory's unphysical aspects, Dirac viewed the basic idea as beautiful and profound at a fundamental level.
Dirac was fascinated by the unified field theory that Hermann Weyl had originally published in 1918 … Dirac's interest in Weyl's unified theory stemmed from his student days in Cambridge, when he had studied it thoroughly. In accordance with the consensus of the time, he had concluded that the theory, although mathematically appealing, was physically unsound, but Dirac continued to be fascinated by the mathematical structure of the theory and by Weyl's general approach to physics, which he saw as representative of his own ideal of a "powerful method of advance." —Kragh, p. 239
Dirac maintained a belief in Weyl's theory based on methodological reasons, including the sheer beauty of the theory's basic idea. In 1938, Dirac "felt it imperative to find some way to clear away the objections to Weyl's beautiful theory" and in 1938 attempted to connect it to his (Dirac's) Large Numbers Hypothesis (which I have discussed elsewhere on this site).

Dirac returned to Weyl's theory in 1973, when he further explored his earlier idea that there are two ways of looking at the world, one in which the line element ds is measured in "Einstein units" (dsE) and another in "atomic units" (dsA). This idea was akin to a similar concept Weyl himself had proposed in 1920, when he tried to defend his 1918 theory against Einstein's objections (which I detail in Weyl's 1918 Theory). "We should reintroduce Weyl's theory" Dirac wrote, adding "It is such a beautiful theory and it provides such a neat way of unifying the long range forces. And there really is no clash with atomic ideas when we have the two ds's".

The last chapter of Kragh's book is titled "The Principle of Mathematical Beauty." This principle was shared by both Weyl and Dirac, and has become known as the Dirac-Weyl doctrine.
At the University of Moscow there is a tradition that distinguished visiting professors are requested to write on a blackboard a self-chosen inscription, which is then preserved for posterity. When Dirac visited Moscow in 1956, he wrote "A physical law must possess mathematical beauty." This inscription summarizes the philosophy of science that dominated Dirac's thinking from the mid-1930s on. — Kragh, p. 275
Exactly the same could be written about Weyl. Indeed, writes Kragh, Hermann Weyl could have served as a model for Dirac's philosophy. In 1932, Weyl wrote in his three-part monograph The Open World "The mathematical lawfulness of Nature is the revelation of Divine reason … The world is not a chaos, but a cosmos harmonically ordered by inviolable mathematical laws," while Dirac wrote in 1975 that "One could perhaps describe the situation by saying that God is a mathematician of a very high order, and He used very advanced mathematics in constructing the universe."
I died for beauty, but was scarce
Adjusted in the tomb,
When one who died for truth was lain
In an adjoining room.

He questioned softly why I failed?
"For beauty," I replied.
"And I for truth — the two are one;
We brethren are," he said.

And so, as kinsmen met a night,
We talked between the rooms.
Until the moss had reached our lips,
And covered up our names.                     — Emily Dickinson, 1830-1886
Granted, beauty is a subjective term, and beauty alone does not necessarily mean truth, to contradict Dickinson's famous claim. Still, there is something in the human mind that knows whether something is beautiful or not, whether it is a sunset or a mathematical expression. I personally believe that, unlike the bending and warping of space and time, and the possible subjectivity of beauty, truth is, by itself, an absolute concept; it simply exists, is eternal, and cannot be altered. To know the Truth is to know the mind of God.

Dirac and Weyl—a couple of pretty smart guys, indeed.

The Shadow Side of Mythology is Fundamentalism -- Posted by wostraub on Saturday, November 20 2010
On 23 October 2005, two employees from Amnesty International boarded a train bound from Boston to Washington DC. During the trip, they discussed Abu Ghraib and other then-recent revelations involving America's decision to use torture to coerce information from suspected terrorists. One employee cynically expressed her hopelessness that Americans would ever care what was happening in their name, while the other optimistically encouraged continued humanitarian work. At the stop in Baltimore, the noted New York Times columnist and writer Sy Hersh (who exposed the My Lai massacre, among many other things) boarded the train. Aware of Hersh's liberal position on the issue, one of the Amnesty International workers asked Hersh where he believed America was headed with regard to fundamental human rights.

"We're fucked," was Hersh's terse reply.

I was thinking about this story while reading Stephen Larsen's 2007 book The Fundamentalist Mind: How Polarized Thinking Imperils Us All, which tries to present an understanding of the rigidity of those who take myths literally and attempt to impose their beliefs on others, by force if necessary. While the author tries to establish a basic difference between mythology and religion, I think he fails; to me, religion is simply canonized myth. However, Larsen points out that myth can also be secular (fascism, free-market capitalism) and scientific ("scientism," a new word for me), which he describes as a "hybrid reality cobbled together from Newtonian physics and tenth-grade science." Ouch, I sure hope that doesn't include me.

But of course in a way it does—we all live out our lives under one or several myths. Larsen relates how the great Swiss psychoanalyst Carl Jung realized late in life that his "task of tasks" was to discover what myth he himself was living, with all else being secondary. In this sense, our myths fashion the masks we wear throughout our lives.

I'd never heard of Larsen, who has written some 40 books (he was a close friend and colleague of the mythologist Joseph Campbell, who I have heard of and seen), and this book was a revelation to me. Important, too, as I am now trying desperately to "set my world aright" given the state of the United States today. Things are so confused that we are literally creating our own myths now. Behold Sarah Palin and her family, who are all really only uneducated, politically naive and inexperienced celebrities, yet are being mythologized through trivial and made-up media such as Dancing with the Stars, Sarah Palin's Alaska, and their seemingly endless Facebook and Twitter comments that are viewed as great wisdom by the unquestioning right wing.

What happens when myth becomes fact. Taken from C.P. Pierce's Idiot America.

In the book, Larsen rightly warns that myths taken literally become dangerous in the hands of frightened, confused or angry people. A scary potential example: President Sarah Palin pushing the Red Button that annihilates Teheran's seven million inhabitants, winking conspiratorially and uttering You betcha!

Dirac in the News (Inadvertently) -- Posted by wostraub on Saturday, November 20 2010
I am deeply in awe of Hermann Weyl's gauge theory, but in reality I am more impressed with the work of the British mathematical physicist Paul Adrien Maurice Dirac, whom I have referred to on numerous occasions on this site. If I had the chance to talk to just one famous physicist, it would be Dirac.

Dirac was the discoverer of many things in physics, but perhaps his most famous discovery is antimatter, a subject that is much in the news today.

Scientists have finally created anti-hydrogen, which is an atom composed of an antiproton surrounding by an orbiting antielectron, or positron. Only 38 atoms have been created so far, but the important thing is that the atoms have been stabilized with a life exceeding a tenth of a a second. Such relatively long lifetimes allows for the possibility that the atoms can be studied; quantum mechanics predicts that anti-hydrogen should behave exactly as its ordinary matter counterpart hydrogen. That is, the emission spectra of anti-hydrogen should be identical to that of hydrogen. If not, then the Standard Model of particle physics will have to be substantially revised.

All of this is important because in the very early universe, the amount of antimatter must have been very close to that of ordinary matter. It's still a mystery why only matter is around today, but it has been surmised that there was a slight surplus of matter when the universe was created. When matter and antimatter meet, they annihilate one another in a burst of light. The surplus of ordinary matter might explain why we don't detect antimatter, at least here on Earth (antiparticles can be manufactured with linear accelerators, but they don't last very long).

Although antimatter was first detected experimentally in 1932, it was theoretically predicted back in late November of 1927, when Paul Dirac experienced a brilliant flash of insight. For some time, the 25-year-old physics whiz had been seeking a variant of Schrödinger's wave equation that was fully relativistic:
According to a physics legend, apparently even true, Dirac was staring at a fire one evening when he realized that what he wanted, for reasons that are now no longer relevant, was a relativistic wave equation linear in space-time derivatives ∂μ = ∂/∂xμ.     — Anthony Zee, Quantum Field Theory in a Nutshell
In view of the enormous importance of Dirac's discovery, I believe that whatever approach Dirac used to find the equation is very much relevant. After much searching, I have not been able to track down Dirac's precise reasoning; perhaps, as Dirac's biographer Graham Farmelo suggests, the discovery was simply the result of a "happy guess."

According to Dirac himself, the physicist was fond of "playing around" with mathematical expressions, and in doing so he occasionally found equations that were physically meaningful. Here, I will try to reconstruct what Dirac's reasoning most likely was.

First of all, Dirac knew that any valid relativistic treatment of the electron would have to treat time and space on an equal footing; that is, to the same order with respect to differentiation (preferably first order). Second, since both he and Wolfgang Pauli had independently derived earlier non-relativistic versions of Schrödinger's equation for electron spin that involved 2×2 matrices, Dirac suspected that matrices might play a role in the relativistic version. Remember, at that time nobody had any idea why electrons should exhibit two states of spin, but 2×2 matrices seemed to play an important role in their description.

Dirac also recognized that Schrödinger's equation is linear in the time derivative

iℏ ∂Ψ/∂t = HΨ

(where Ψ is a one-component scalar wave function and H is the Hamiltonian), implying that the relativistic Hamiltonian should also be to first order in the space variables. Furthermore, Dirac knew that the Schrödinger equation gives a straightforward interpretation of the quantity Ψ*Ψ as the probability density for an electron, which must always be positive definite (that is, the probability must be real with a value never less than zero).

Dirac was also aware that the Klein-Gordan equation, which arises from the relativistically correct quadratic energy-momentum relation

(E/c)2 - p2 = (mc)2         (1)

yields a density that is indefinite (it can be positive, negative or zero) and so cannot be interpreted as a probability density. The reason for this, Dirac recognized, was due to the fact that the Klein-Gordan equation is quadratic in the time and space differentials.

Solving (1) for the energy, we have

E = ± c(p2 + m2c2)½

The negative energy term initially puzzled Dirac. What the hell is negative energy? As he was to find out, the negative-energy solutions represented antimatter.

In the following, note that in quantum mechanics we have the transcription properties E/c = iℏ ∂/∂x0 and px = -iℏ ∂/∂x, etc. This then gives

(E/c)2 - p2 → -ℏ2 ημνμν ≡ -ℏ22

where ημν is the flat-space metric tensor and ◻2 is the D'Alembertian operator.

In consideration of all these facts, Dirac surmised that a relativistic electron equation should indeed be linear if a consistent form for the probability density was to be derived. Thus Dirac famously wrote down the "square root" of (1):

iℏγ0∂/∂x0 + iℏγ1∂/∂x + iℏγ2∂/∂y + iℏγ3∂/∂z = mc

or, in elegant summation notation,

μ∂ψ/∂xμ = mc/͎ Ψ         (2)

(which can also be written neatly as i◻Ψ = mc/ℏ ψ, where ◻ = γμμ).

Here, the four γ quantities are 4×4 matrices with special commuting properties and Ψ is now a 4-component wave function. This, in all its beauty, glory and fantastic power, is Dirac's relativistic electron equation.

Specifically, the commutation relations are

0)2 = 1
1)2 = -1
2)2 = -1
3)2 = -1


γ0γk + γkγ0 = 0 (k = 1,2,3)
γiγk + γkγi = 0 (i,k = 1,2,3)

[Note that the gamma matrices are made up of combinations of the Pauli matrices, so it's no coincidence that the electron's description does indeed involve matrices.]

Using Eq. (2), Dirac derived a variant for the hydrogen atom that included the electromagnetic interaction term and set to the task of solving the four simultaneous partial differential equations. After a little work (actually, this is a very difficult problem in itself), Dirac's computed energy levels for the electron matched experiment to within a tiny fraction of a percent.

To his surprise, Dirac's equation also automatically provided for the spin and magnetic moment of the electron; these quantities were a consequence of the mathematics, and did not have to be forced into the formalism, as in Pauli's ad hoc theory. And, of course, the probability density of Dirac's equation was positive definite.

When Dirac published his two papers in January-February 1928, they astounded the physics community (you can download these papers from my post dated 23 March 2009). Most physicists were ecstatic, although the noted German physicist Paul Jordan, who had been working on the relativistic electron problem himself, went into a deep depression. Even the great German physicist Werner Heisenberg, who was also keenly involved in the electron problem, went into a kind of peevish denial that someone else had beaten him to the goal.

As I state in my write-up Weyl Spinors and Dirac's Equation, if Dirac had patented his discovery the electronics industry alone would have made the man a billionaire. As it was, he settled for a few thousand dollars and the Nobel Prize in Physics in 1933.

Dirac's electron equation still blows me away.

Sandage Dead at 84 -- Posted by wostraub on Wednesday, November 17 2010
Renowned Pasadena cosmologist Allan Sandage has died of pancreatic cancer. He was 84.

In 1953, Sandage took over the work that Edwin Hubble had started in 1929, when he discovered that the universe is expanding. In 1931, Einstein took a trip up the mountain here above Pasadena to visit Hubble, whose discovery destroyed Einstein's belief that the universe was static. Oddly enough, the discovery also made Einstein's cosmological constant irrelevant, although it may make a reappearance when we finally understand dark energy.

Sandage is best known for his work on the Hubble constant, a number that linearly connects the relative velocity of a galaxy with its distance from Earth.

The KKK in Tennessee -- Posted by wostraub on Sunday, November 14 2010
Having outgrown their current house of worship in Murfreesboro, Tennessee, in May 2010 the town's Muslim population obtained a building permit to construct a 52,000 ft2 mosque in the rural outskirts of that town. Christian residents were outraged and filed suit to block the project on the grounds that Islam is not really a religion, but a terrorist government. [Read the link's comments; they're priceless.]

There are roughly 1.6 billion Muslims in the world which, according to the Christian residents of Murfreesboro, are all terrorists. If that were true, I believe we'd have all been overrun a long time ago.

Murfreesboro is located about 100 miles from Dayton, another Tennessee town that has an interestingly Red-State neanderthal flavor, being the location of the famous Scopes Monkey Trial of 1925. Sounds to me like the people of Tennessee haven't made much progress in the past 85 years, although inexplicably the state is the political home of Vice President Al Gore, a noted Democrat and perhaps the world's most vocal proponent of anthropogenic global warming. I wonder if he still hangs out there.

I forget who made the observation that when fascism comes to America, it will come carrying a Bible and waving the American flag. But Canadian author Margaret Atwood made the same prediction when she wrote her 1984 book The Handmaid's Tale, a dystopian futuristic account of how fundamentalist Christianity teams up with the US military to take over America (I won't discuss the book here, but you can trust me that it isn't pleasant). It's in this book that the KKK makes its triumphal reappearance, although here it stands for Kinder, Kirche und Küche (German for children, church and kitchen), representing the final conquest and enslavement of American women by the Christian church (presumably Southern Baptists).
We're often taught in schools that the Puritans came to America for religious freedom. Nonsense. They came to establish their own regime, where they could persecute people to their heart's content just the way they themselves had been persecuted. — Margaret Atwood
As Atwood also warns, it can happen here. Tennessee is leading the way.

Gravity-Electromagnetism Link -- Posted by wostraub on Thursday, November 4 2010
British mathematician David Toms of Newcastle University has just published an article demonstrating a definite link between gravity and electromagnetism. It's perhaps not the link that Einstein and Weyl had looked for, but it's important nonetheless. It involves what is known as asymptotic freedom, the belief that the electromagnetic force weakens as gravity strengthens. Toms' paper is a follow-up to earlier work done by Frank Wilczek, David Gross and David Politzer, for which they were given the 2004 Nobel prize in physics.

In the very early universe, the total electromagnetic charge of elementary particles in the expanding matter sphere must have been essentially zero, so the electromagnetic force was likely very small or even zero. The same was probably true for the weak and strong interactions. Consequently, Wilczek and others posited the likelihood that the three interactions are exceedingly weak at very high energies. Wilczek also investigated the possibility that the gravitational interaction acts to weaken electromagnetism at high energies, but he lacked the mathematical wherewithal to conclusively prove this. Toms seems to have succeeded in providing the necessary mathematical boost to Wilczek's hypothesis.

Toms' paper starts out with the Einstein-Maxwell action coupled with the Dirac action, a fairly conventional approach. The math then gets pretty hairy, and I won't pretend that I can follow all of it. But the paper appears to show that as gravity gets stronger, electromagnetic charge gets weaker. As Toms points out, this could point the way to a final unified understanding of all four interactions.

The paper is 19 pages long and easy to read, if not easy to understand. Brush up on your Green's functions before downloading it. [Alternatively, you can read this paper by Robinson and Wilczek, which is a tad simpler.]

Oktoberfest -- Posted by wostraub on Thursday, November 4 2010
I was in Munich last month with family during Oktoberfest, which is probably not a great time to visit. The city is jammed with Germans from all over the country, especially as this was the 200th anniversary of the festival. Everyone was dressed in Lederhosen and other traditional outfits, singing drinking songs, yelling and generally carrying on, and that's before they'd even started any serious drinking. I don't really care for German beer, but it was a lot of fun.

A typical drinking bash at the Englischer Garten in downtown München. The Chinese-style tower held a traditional German Oompah band that played everything from age-old drinking songs to Hello Dolly. Everyone there (including the band members) seemed to be plastered but having a ball.

Right next to the Englischer Garten is Munich's Ludwig Maximillian University, which Hans and Sophie Scholl attended while they carried out their secret activities against the Third Reich in 1942-43. They are memorialized with ceramic tiles set into the adjacent Geschwister-Scholl-Platz sidewalk, with reproductions of the students' photos and their seditious anti-Nazi leaflets:

Here's a photo of my younger son Kurt in the university's huge atrium, where the Scholls were arrested immediately after Sophie dumped her last batch of leaflets on the atrium floor where Kurt is standing. There is a museum devoted to the couple on the ground floor of the atrium, which houses original letters, leaflets and photos. Very moving, at least for me.

Outside, a building undergoing renovation was sheathed with a protective tarp that also commemorates the Scholls. Europeans have the neat habit of covering up renovations of old churches and buildings with full-size reproductions of the original structures, so that the work doesn't look so intrusive.

Speaking of the Scholls, I just finished Chris Hedges' book War is a Force That Gives Us Meaning. It should be mandatory reading for everyone, especially Americans, and especially at a time when the newly-invigorated Republican Party will be seeking new countries to attack for the sake of hubris and profit.

Westminster Abbey -- Posted by wostraub on Saturday, October 23 2010
Just a note to say that I visited some of my heroes (at least what's left of them) at Westminster Abbey in London last month. Bunched around the central cloister are the floor crypts of Isaac Newton, mathematician George Green (whose Green's functions used to give me fits), William Thomson (Lord Kelvin), Charles Darwin and James Clerk Maxwell, along with a plaque commemorating my personal favorite, Paul Adrien Maurice Dirac (although his remains are actually in Tallahassee, Florida, of all places). Nearby lie Michael Faraday, Ernest Rutherford, either John or William Herschel (I can't recall which), and J.J. Thomson, the discoverer of the electron.

I had hoped to take some pictures, but photography was not allowed on the day I visited. As Westminster Abbey is still a functioning church, and preparations for a service were underway during my visit, that is understandable. Indeed, one of the deacons had to remind me to remove my UCLA hat. I felt like an impious idiot, and deservedly so—the 1,000-year-old Abbey also serves as a mass grave.

Sorry for the oversight, guys. And thanks for opening mankind's eyes and allowing us to glimpse some of the wonder of God's creation. I love you all.

Bill is Grumpy Today -- Posted by wostraub on Saturday, October 23 2010
While the gravitational potential thus consists of an invariant quadratic differential form, electromagnetic phenomena are governed by a four-potential of which the components φμ together compose an invariant linear differential form φμ dxμ. But so far the two classes of phenomena, gravitation and electricity, stand side by side, the one separate from the other
This quote from Hermann Weyl's 1918 paper Gravitation and Electicity was written nearly a hundred years ago, but its implicit forlorn plaint seems equally valid today.

I am reading Stephen Hawking's new book The Grand Design, which yet again takes a stab at that great mystery of physics, the seemingly irreparable gulf that exists between quantum mechanics and gravitation.

It is not a great book. It is not even a mediocre book. In my opinion, it's just another effort by Hawking (and co-author Leonard Mlodinow of Cal Tech) to cash in on that endless hot topic in physics, grand unification. And they're not alone.

Have you caught Michio Kaku on the Science Channel lately? Or Brian Cox? Or even (gasp) Morgan Freeman? They're all doing it, selling physics theories by the gross, using a lot of hot air, whoosh-bang CGI imagery, and not a lot else. Dang it, our recent transformation from a print-based culture to a media-based culture is dumbing us down to the point where even physics is sold as mindless entertainment.

We keep hearing how string theory is 10500 theories, or five, or maybe even just one (M-theory). But it's all untestable, and perhaps always will be. Damn it all, the electroweak theory of the 1970s conclusively unified the weak and electromagnetic interactions, yet we still blithely talk about the four fundamental forces of nature, not three.

An untestable theory is no different from a religious faith. To me, the relevance of string theory to science is becoming nil. For the past forty years tens of thousands of physicists have spent untold hours cranking out mathematical papers on string theory, and as of today they're still totally in the dark as to whether it means anything. The mathematics is so convoluted and difficult that it takes years to become even acquainted with it, and looming behind it all is the very real possibility that it does not describe reality. I strongly suspect that if some genius were to discover conclusively that string theory is all nonsense, the same scientists would form a priesthood and go right on working on it, their inner voices telling them that it is true nevertheless.

Skipping ahead in Hawking's book, I note that the great physicist is hinting that the long-sought Unified Theory of Everything itself may not exist, outside of several individual theories that can never be made mathematically compatible. Are we at long last reaching the point where physicists are throwing in the towel?

The October issue of Scientific American has a one-page essay on Hawking's book and what he has to say in it. My advice—read the essay and don't waste your time on the book. And while you're at it, you can skip Kaku, Cox and the others as well.

Ordinary Time Dilation -- Posted by wostraub on Sunday, October 17 2010
Einstein's special and general theories of relativity predict that clocks run differently at high velocities and in the presence of strong gravitational fields. Non-scientists tend not to be much impressed with this, however, as the Einstein time dilation effect isn't appreciable unless you're talking about velocities near light speed or gravitational fields typical of those near neutron stars and black holes, conditions common only in science fiction.

However, last month's issue of Science featured an article by C.W. Chou and colleagues titled Optical Clocks and Relativity which demonstrates that the relativistic time dilation effect can be observed under conditions of ordinary speeds and gravitational fields (10 meters/sec, meter-size gravitational gradients). They used aluminum atoms for clocks!

Disclaimer: I downloaded the entire journal from Scribd (a neat legal document site that I've mentioned before), then pared it down to the article pages. Hopefully all this was legit.

Fine-Structure Funnies -- Posted by wostraub on Saturday, September 11 2010
Physicists at the University of New South Wales are reporting evidence that the famous fine-structure constant of physics may vary across the observable universe. In fact, they are positing that the value of the dimensionless constant, typically expressed as α = e2/ℏc, appears to have a preferred alignment from smaller to larger from one end of the universe to another.

The fine-structure constant is an anomaly in quantum physics. Its inverse is very close to 137, but almost every other quantum constant is preposterously small (Planck's constant, for example is on the order of 10-34 joule-second). The fine-structure constant is also famous in the anthropic principle argument, which states that the universe appears to have been finely tuned for the existence of life, specifically humans. If α were to differ by only a few percent, for example, then carbon atoms would be unstable, making carbon-based life impossible.

This observation has been taken by some to be proof of the existence of God. Perhaps so, but the multiverse theory gets around that by positing the existence of an infinite number of possible universes. Imagine a beach on a hot sunny day, on which zillions of sunbathers have crowded together to get away from the heat. A lone seagull flies high over the multitude and relieves itself. While the chances of any one sunbather getting hit are miniscule, there is a 100% probability that one person will be the unlucky recipient. He/she will undoubtedly yell Why me?! and be tempted to think that God, fate or predestination had something to do with it. A scientist might say that wind direction and velocity, air temperature and density, and the gastrointestinal state of the bird were all insanely fine-tuned for the unfortunate sunbather, but the sunbather will probably not be buying it.

Since quantum field-theoretic calculations show that the currently-known value of α is accurate to better than 9 decimals places (α-1 = 137.035999084), any variation in the constant would have to be tiny indeed. But a confirmed variation would be big news (consider the butterfly effect of chaos theory).

On the other hand, the most famous constant of nature, the gravitational constant G, is known to only 5 decimal places. Possibilities for its variation are much greater, yet nobody seems to be overly concerned about that. Me, I think that the reported variation in the fine-structure constant is almost certainly wrong. I believe that when you're dealing with variations at the sub-part-per-billion level (given the stakes involved), it's best to wait until more definitive data are available.

By the way, the above news is being reported in the journal New Scientist, which I've sometimes believed is a tad right-wing. I may be wrong, but this isn't the first time the journal has highlighted a paper that says Everything You Thought You Knew About Conventional Science Is Wrong So Maybe God Is Doing It After All.

I personally don't believe that physical constants and isotopic decay rates change in time or space. If God can change physical constants at will, that's fine, but then, to me, there would be little reason to study science in the first place. God gave us brains for reasons other than to stand around googly-eyed and blindly accept everything as one long, continuous miracle. Instead, He gave us brains to think and reason with. And that's where science comes in.

Freedom and Liberty -- Posted by wostraub on Saturday, September 11 2010
The right-wing screams Freedom and Liberty! Freedom and Liberty! over and over and over. The words have become meaningless because we have given up our freedom and liberty in exchange for "security." Two quotes come to mind:
People who trade their freedom for security deserve neither and will lose both. — Benjamin Franklin

Freedom and Honor! For ten long years Hitler and his henchman have manipulated, squeezed, twisted and debased these two splendid German words to the point of nausea, as only dilettantes can, casting the highest values of a nation before swine. — Sophie Scholl (in her final leaflet, 1943)
War is truly a racket, sold to us American idiots on the pretense of defense, Christianity (and I am a Christian) and holy nationalism: US to Sign Biggest Arms Deal in History

Happy 9/11.

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 1913 at the Swiss Federal Technical Institute in Zürich, did not engage in any meaningful collaboration after their arrival in America. Though both had expressed a preoccupation with unification over the period 1916 to 1929, 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 + r22 + 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!

The Pseudotensor of General Relativity -- Posted by wostraub on Tuesday, June 22 2010
Anyone who has studied Einstein's gravitation theory has been struck by the fact that the theory cannot satisfactorily account for the conservation of mass-energy. This observation normally begins with the realization that the Einstein tensor with mass-energy term,

Rμν - ½ gμν R = Gμν = Tμν

is conserved; that is, it has a vanishing covariant divergence,

Gμν = Tμν = 0

where the semicolon stands for covariant derivative.

Unfortunately, the covariant divergence is not the same as the ordinary divergence, which is what actually guarantees conservation. This fact greatly bothered Einstein, Weyl, Schrödinger and others, and they all sought ways to fix up the mathematics so that Tμν = 0, where the comma now stands for ordinary partial differentiation.

It turns out that there is a geometric quantity tμν that can be added to the mass-energy tensor Tμν so that

(Tμν + tμν) = 0

The trouble, damn it, is that tμν is not a covariant quantity. It is a pseudotensor that does not transform like a second-rank tensor under a general change in the coordinates. Furthermore, the pseudotensor's mixed forms tμν and tμν are also problematic; they cannot be used with the corresponding mixed forms of the Einstein tensor to give a divergenceless result. Schrödinger went so far as to call the pseudotensor a total Schein (sham).

Weyl himself checked to see if his 1918 variant of general relativity could produce the desired zero-divergence result; it did not. To this day, general relativity, our most accurate description of gravitation, cannot adequately account for the conservation of mass-energy. I find this to be ironic in the extreme!

Perhaps this in itself is a clue that the theory is still incomplete. Many believe that only a final, consistent theory of quantum gravity will resolve the problem.

Ryder Again -- Posted by wostraub on Friday, June 18 2010
Last week I wrote about Lewis Ryder's 2009 book Introduction to General Relativity, noting that it addresses many topics not included in my earlier favorite, the similarly-titled 1975 text by Adler-Bazin-Schiffer. Lewis, a senior lecturer at the University of Kent, is a quantum field theorist who is also much influenced with (if not infatuated by) the geometrical aspects of quantum fields. Much of this is reflected in the book.

If you do nothing but read Chapter 11 (Gravitation and Field Theory) of Ryder's book you will understand why earlier researchers like Einstein, Weyl and Kaluza were so determined to unite the forces of electromagnetism and gravity and also why their efforts were doomed to failure from the start. For one thing, gravity acts on all bodies equally, regardless of their masses, while the electromagnetic force depends upon the charge to mass ratio q/m. This means that, unlike gravity, electromagnetism cannot be made to disappear in a locally "flat" reference frame. Consequently, to the dismay of Einstein and his pals, electromagnetism cannot be "geometrized."

Still, Ryder notes that the similarities between EM and gravitation are tantalizingly close, and that these similarities cannot be due to mere coincidence:

Ryder goes on to note that if electromagnetism and the weak and strong forces cannot be geometrized like gravity, then gravity will have to be "quantized" if all four interactions are to ever be described by a single unified theory. But, in spite of the above correspondences between gravity and abelian/non-abelian gauge theories, nobody has any clear idea as to how gravity can be brought into the quantum fold. Nevertheless, Ryder notes that we should we thankful for how far we have come:
At the end of an introduction to Einstein's theory, however, it is best not immediately to start thinking about the next challenge. Like a climber who has arrived at the top of his mountain, we should simply sit down and admire the view. Is it not absolutely remarkable that Einstein was able to create a new theory of gravity in which the geometry of space itself became a part of physics? Whatever would Euclid have thought?
I think Ryder's book is destined to become a classic at its level (undergraduate and beginning graduate). It sells on Amazon for about $40, and I encourage you to get it.

PS: In spite of the fact that electromagnetism and the weak interaction were successfully united by Weinberg, Salam and Glashow in the 1970s, nearly everyone continues to talk about the usual four fundamental forces of nature. To the best of my knowledge, Ryder is the first author to talk consistently about the three fundamental forces. Good for him!

PPS: Speaking of coincidences, I think it was our good friend Auric Goldfinger who said that "Once is happenstance, twice is coincidence, and three times is enemy action." If Nature's three fundamental forces do indeed display these coincidental similarities, Whose Hand lies behind it all? William Blake, anyone?

Memories -- Posted by wostraub on Monday, June 14 2010
The story of how Erwin Schrödinger discovered his famous wave equation in 1926 is well known: he ran off to a mountain lodge for two weeks with a woman who was not his wife and, in what Hermann Weyl called "an outburst of genius in a late erotic phase of his life," came up with the answer. What is not so well known is that he had Weyl to thank for much of the discovery (see my post of 23 May 2007 and earlier remarks). It would not be remiss to call the discovery the Schrödinger-Weyl equation.

I was made aware of the wave equation as a clueless undergrad in 1970. In those days, chemistry majors weren't expected to know how to actually derive and solve the damned thing, but Prof. Baine believed it would be good for our souls (in retrospect, he was right). My third semester of physical chemistry started out with some 15 students, and within a few weeks Baine had it down to 6. He took this remnant kicking and screaming through the mathematical details, replete with the relativistic and non-relativistic versions, and within a few weeks we were thoroughly confused but thankful that we weren't physics majors. I still remember, with photographic clarity, the professor's remarks when he handed back our midterm exams: "I am very disappointed with these scores. Evidently, none of you have much potential as scientists." Ouch.

Years later I learned that Schrödinger had similar problems with his discovery. The solution to his time-independent wave equation, a partial differential equation in three dimensions, presented some difficulties, and he appealed to his good friend and colleague for help. Weyl, a brilliant mathematician, immediately solved the equation. Schrödinger later wrote "For guidance in treating the equation I owe the deepest thanks to Hermann Weyl." As is well known today, this was quite an understatement. Schrödinger shared the Nobel Prize in physics (with Dirac) in 1933 for his discovery. Weyl got a handshake and a pat on the back. That's the way it goes sometimes. (Note: there is no Nobel Prize for mathematics.)

Interestingly, Schrödinger's relativistic wave equation didn't work very well when applied to the hydrogen atom, while the non-relativistic version did. In 1928, Dirac took the "square root" of Schrödinger's relativistic equation and it worked beautifully. Thus, the 1933 Nobel prize went, perhaps inadvertently, to these two variations of the wave equation.

You can download an English translation of this famous paper (and many others) here.

UCLA Commencement 2010 -- Posted by wostraub on Monday, June 14 2010
After six long years, my younger son Kurt received his PhD in Molecular Genetics/Immunology from UCLA. Congratulations to him, his three fellow PhD recipients and the Department's 110 undergrads. (I graduated from that other school down the road, so had to keep a low profile.) Go Trojans! Bruins!

Kurt and his advisor (Prof. Bradley) work out the intricacies of the damned doctoral hood

New Relativity Book by Ryder -- Posted by wostraub on Tuesday, June 8 2010
The sheer number of books now available on general relativity must surely bewilder the mathematically inclined non-expert. One of the latest, Sean Carroll's Spacetime and Geometry, has been getting great reviews but for whatever reason just left me cold. However, we are fortunate to have another book relatively (!) hot off the press, and I heartily recommend it.

In addition to his truly great book Quantum Field Theory, Lewis H. Ryder's Introduction to General Relativity is yet another success from the University of Kent physicist. This book, which wonderfully complements Adler-Bazin-Schiffer's 1975 text (now out of print), is mathematically on par with the latter but includes many more topics of current interest, such as gravitation in curved spacetime, the Weyl tetrad formalism, non-abelian gauge theory, the Higgs mechanism, and the five-dimensional Kaluza-Klein theory, all of which were not addressed by Adler et al. In view of this, I would now recommend Ryder's book to the interested non-expert rather than Adler's.

(Everybody makes mistakes, even Ryder!)

PS: Ryder has a third book out called Elementary Particles and Symmetries. I'm dying to see it, but I can't find a reasonably priced copy. Please let me know if you have the book and what you think of it.

Weyl and Time -- Posted by wostraub on Friday, June 4 2010
The objective world simply is, it does not happen. Only to the gaze of my consciousness, crawling upward along the life line of my body, does a section of this world come to life as a fleeting image in space which continuously changes in time. — Hermann Weyl
I believe this quote, from Philosophy of Mathematics and Natural Science (1949), is more a very loose idea rather than a firm belief on the part of Hermann Weyl. He was trying to picture all of reality as a fixed tableau of events that is only glimpsed piecemeal by the conscious mind, which then tries to make sense of what it perceives. This process, Weyl posited, gives the illusion that events are occurring in time.

Every now and then Scientific American puts out a truly great issue, with something for everyone, and the magazine's June issue is an example. There are articles on Neanderthals, neutrinos, Penrose twistors, synthetic life, and a promising new broad-spectrum antiviral drug (developed by a UCLA professor that my son worked with). It also has a great article on the nature of time by UCSD professor Craig Callender titled "Is Time an Illusion?"

The concept of static time is essentially what Weyl was talking about. Callender points out that the human mind interprets what it senses as time's arrow by way of personally experiencing external dynamism in the world. But, as Callender rightly acknowledges, physics is time-invariant; that is, physical processes could just as easily go backward without breaking any rules ("unscrambling" an egg omelet does not violate the Second Law of Thermodynamics if time is pointing in the opposite direction). If physics is time-invariant, then perhaps time has no independent existence or, more succinctly, it may simply not exist.

I urge you to read this article. When you do, try to think of your brain as a three-pound "universe" which takes sensory information from the external world and tries to make sense of it any way it can. Ask yourself if the external world actually might not exist, or at least in the way your mind interprets it. Then try to fit the concept of time into all of this. Callender thinks that time may only be a convenient artifice that our brains use to make the world easier to understand.

Also remember that the question of time and what it is has puzzled people for thousands of years. St. Augustine of Hippo famously remarked that he knew what it was when he wasn't thinking about it, but had no clue when he did. That pretty much sums it up for me.

Massive Neutrinos Detected? -- Posted by wostraub on Tuesday, June 1 2010
Scientists with the European Organization for Nuclear Research (CERN) working with muon neutrino beams have reported the detection of the tau neutrino in their experiments, a finding that confirms the decades-long speculation that neutrinos have mass and oscillate from one type to another. The finding is also being reported by the Los Angeles Times.

Massive neutrinos throw a monkey wrench into the Standard Model of physics. A fundamental aspect of the model is that neutrinos are massless and described by Weyl spinors, which effectively split the Dirac spinor into two uncoupled parts and thus describe left-handed neutrinos and right-handed antineutrinos. If neutrinos have mass, then they must be described by Majorana spinors, a close cousin of the Weyl variant.

If the CERN discovery is confirmed, all bets are off and it's back to the drawing board. On the plus side, massive neutrinos could help explain the mystery of dark matter.

"Curiouser and curiouser," cried Alice.

Weyl on Inertia -- Posted by wostraub on Saturday, May 29 2010

By the way, one of the papers cited in the Gorelik paper is Hermann Weyl's Massenträgheit und Kosmos: ein Dialog ("Inertia and the Cosmos: a Dialogue"), a 1924 article originally published in the German journal Natural Sciences. It's a short paper in which Weyl describes a hypothetical debate on the nature of inertia from the Machian and general relativistic points of view.

The debaters: Saint Peter and Saint Paul!!

It provides some fascinating insight on inertia, a hot topic at the time with Weyl, Einstein and others. I'll translate the article and post it here when I get around to it.

Sketch of Weyl late in life.
Courtesy Emilio Segrè Visual Archives

Hermann Weyl and the Coincidence of Large Numbers -- Posted by wostraub on Saturday, May 29 2010
In earlier posts I've touched on the so-called "large numbers hypothesis" of P.A.M. Dirac, who sought to explain why the dimensionless number 1040 seems to keep popping up in a variety of atomic and cosmological circumstances. However, in 1919 Hermann Weyl made the same observation but without the benefit of more modern quantities, such as the fine structure constant.

In 2002 Gennady Gorelik, a researcher at Boston College and author of numerous biographical articles on the Russian physicists Andrei Sakharov and Lev Landau, wrote an interesting paper on Weyl and large numbers. It's very readable, with just a few simple equations, along with some great paper citations. Here it is in pdf format.

Photo of a very young Weyl, undated but probably around 1910.

God is Left Handed -- Posted by wostraub on Wednesday, May 19 2010
The journal New Scientist has an article that posits the origin of the handedness or chirality of molecules that are important to life, such as amino acids and sugars. In stereochemistry, enantiomers are molecules that can occur in two forms, each a mirror image of the other, but otherwise identical. One example would be a person's hands: they're identical, but no rotation, twisting or other simple change of orientation can change a right hand into a left one. Only a mirror can do that.

Lactic acid exhibits two mirror-image forms

The point here is that many of life's metabolic processes can and do distinguish one isomer from another. For whatever reason, life seems to prefer left-handed enantiomers over their right-handed counterparts; indeed, while not toxic, the right-handed forms are not even recognized, and pass through the body untouched. Thus, the consumption of foods made only from right-handed amino acids, simple proteins and sugars would result in starvation (well, not in every case, but you get the drift).

Trace amounts of amino acids found in meteorites demonstrate a definite preference for left-handed forms, while analysis of light from extra-galactic nebulae reveals the presence of left-handed amino acids in gas clouds.

The New Scientist article suggests that this preference for left-handedness might have resulted from the simple fact that every neutrino in the universe is left-handed, while its antimatter counterpart is always right-handed. Exploding stars, the source of much of the matter that makes up many systems of suns and planets (like ours), produce copious amounts of neutrinos, and these particles may have induced a preference for left-handedness in the amino acids and other simple organics molecules that formed from these explosions.

In 1929, Hermann Weyl developed a massless, two-component spinor formalism that violated parity—the idea that equal right and left symmetry was thought to be preserved in all nuclear reactions. Weyl's spinors were later shown to describe right-handed and left-handed neutrinos when parity was disproved for the weak interaction in 1956.

Observe a neutrino moving in the upward direction. Its spin is clockwise, as indicated by the thumb of your left hand pointed up with the fingers curled in the same clockwise manner. You will never find a neutrino with a different spin orientation. This is a great mystery! Of all the possible particles in the universal particle zoo, why were neutrinos created with this peculiarity? And is the handedness of neutrinos the reason why you're even here to read this article?

Weyl to Yang-Mills and Beyond -- Posted by wostraub on Thursday, May 13 2010
In 1929 Hermann Weyl dusted off his old gμν → exp(λ) gμν metric gauge theory and applied it to the wave function of quantum mechanics á la Ψ → exp(iλ) Ψ, where λ is an arbitrary function of the coordinates. By demanding that the laws of quantum physics remain invariant with respect to this change, Weyl discovered why electric charge is a true invariant, and in doing so laid the foundation for modern quantum field theory.

But even Weyl could not have envisioned where all this would lead. Intrigued by Weyl's basic idea, the physicists C. Yang and R. Mills in 1954 took it a step further. In a ground-breaking paper, they investigated the physics behind the replacement of the scalar parameter λ with an N×N matrix. While Nature seems to have avoided utilizing the N = 2 case (Yang and Mills had thought it might show that the proton and neutron are symmetric and interchangeable), the case for N = 3 led ultimately to the standard SU(3) theory of quantum chromodynamics (quarks, gluons and all that). The Standard Model of physics today is based on SU(3)×SU(2)×U(1) symmetry, with the tail end, U(1), due to Weyl.

This story explains much of my fascination with Weyl and his gauge theory, but I've barely touched on it here. It underlies much of mankind's search for a fundamental theory of everything, stretching from Weyl to Kaluza-Klein to Einstein to Yang-Mills to today's string theory.

For a more detailed and very readable account, see this 2005 article by J. Chýla. Another good source, written at the undergraduate level, is K. Moriyasu's 1983 book An Elementary Primer for Gauge Theory. This book opens with a basic account of Weyl's gauge theory, then goes on to the geometry of internal spaces (a fascinating topic in its own right) and finishes with color gauge theory (a professor friend of mine liked the book so much he never returned it). Still another is David Griffiths' more recent Introduction to Elementary Particles, now considered an undergraduate classic.

Weyl's Wormhole in the News -- Posted by wostraub on Friday, May 7 2010
Physicist David Goldberg of Drexel University has an interesting article in today's Los Angeles Times called Time Travel? Maybe. It was written in honor of the 75th anniversary of the famous paper by Albert Einstein and Nathan Rosen on the possibility of wormholes, those hypothetical spacetime "shortcuts" connecting one part of the universe to another (or one time to another):

(Although Hermann Weyl was the first to propose the concept of a wormhole back in 1921, there's no mention of Weyl in Goldberg's article. Oh well.)

Roger Penrose and others have shown mathematically that wormholes would almost immediately pinch off on a traveler, so that the voyager would have to move faster than light if she wanted to get anywhere (or any "when"). Theoretically, a negative-energy "stent" could be used to hold open the throat of a wormhole long enough and wide enough to allow the journey, but we're already getting pretty hypothetical here, especially in view of the fact that there is absolutely no observational evidence that wormholes exist.

My post of 28 May 2009 (see 2009 archive) provides a little bit of background on Nathan Rosen, who was an interesting character all by himself. In 1935, he collaborated with Einstein and Boris Podolsky on another famous paper, (the EPR paradox) which deals with nothing less than the nature of reality (see my write-up on Bell's inequality). In 1981, Rosen wrote a neat paper called "Weyl's Geometry and Physics," which you can download here. Rosen died in 1995 at the age of 86.

Incidentally, Weyl, Einstein and Rosen came up with the wormhole idea while searching for a unified theory of gravitation and electrodynamics. Considering how mundane terrestrial gravity and electricity are, it's amazing how far afield they got.

Seven Wonders -- Posted by wostraub on Thursday, May 6 2010
New Scientist has listed its candidates for the seven wonders of the quantum world. Not surprisingly, the Aharonov-Bohm effect and spooky action at a distance (Einstein's actual words, not mine), are included. Go check out the other five for yourself.

Humans don't understand this wonderful world, and it's clear that we don't deserve it, either.

Weyl's Neutrino Today -- Posted by wostraub on Wednesday, May 5 2010
I was writing earlier about spinors, whose mathematics describes fermions like electrons and quarks. The first truly important discussion of the mathematics of spinors was Hermann Weyl's 1929 paper, Elektron und Gravitation, in which he developed the concept of two-component spinors.

Weyl was roundly criticized for his mathematical treatment by none other than Wolfgang Pauli, the brilliant curmudgeon of physics, primarily because Weyl's spinor violated a type of symmetry known as parity (by comparison, Dirac's 1928 spinor had four components, and did not violate parity). However, Weyl was later proven to have been correct all along, when his spinor was shown to describe the parity-violating particle known as the neutrino, whose existence Pauli himself first proposed.

There is a neat article by physicists Graciela Gelmini, Alexander Kusenko and Thomas Weiler in the May issue of Scientific American on neutrinos, dealing mainly with their role in the rapidly-developing field of neutrino astronomy. The ability of physicists to detect the three types of the ghost-like, nearly invisible neutrino has improved dramatically over the past decade. In particular, scientists now know that they have small but detectable masses and that they can metamorphose from one flavor to another as they fly through space (a fourth type, called the sterile neutrino, has been hypothesized). The article is informative and relatively non-technical, and it's recommended reading.

Elsewhere on my website I wrote about how Ray Davis, co-winner of the 2002 Nobel Prize in physics, helped work out the solar neutrino problem. Electron neutrinos are formed in the Sun's core as a consequence of the fusion reaction, and the Standard Model of physics predicts their rate of flux. However, when this flux is measured on Earth the number of neutrinos is only 33% of the predicted number. Davis' work aided the discovery of the mechanism behind neutrino metamorphosis and the trick Nature was playing on Earth's observers: in the Sun, all the neutrinos start out as electron neutrinos (νe) but, by the time they get to Earth, only one-third of them are still νe, accompanied by equal numbers of muon neutrinos νμ and tau neutrinos ντ. But these ratios can differ depending on the provenance of the original neutrino flux; instead of 1:1:1 (complete pion decay), we also see 4:1:1 (light neutrino decay) and others.

I laughed out loud when the Gelmini-Kusenko-Weiler article mentioned another revealing population statistic. A recent, highly-publicized random sampling of American scientists has shown that only 6% are registered Republicans.

This citation alone is worth the price of the magazine.

When a Body Meets a Body ... -- Posted by wostraub on Wednesday, May 5 2010
Stephen Hawking recently noted that, should an alien race stumble upon Earth, there's every likelihood that they'd treat humans as scum fit only for annihilation, and that we'd do the same if the tables were turned. In response, Robert Wright in today's New York Times writes
It turns out there’s reason to hope that, actually, we’d be kinder to a new world than Europeans were to the New World.
This statement upset me, because I think it's so much bird pucky, but I had to read the entire article to calm down. Wright acknowledges that we humans have a long way to go before we'd treat ET with more respect than indigenous peoples received when whites stumbled upon them.

While visiting my brother-in-law in San Francisco last week I read Wright's book, The Evolution of God, which Wright touches on in his article. In the book he notes that humans may be hard-wired through evolution to treat each other with respect, and that this hard-wiring is actually a survival tactic for which we humans developed religion to explain it. I'm not sure if this means that the "tiny voice" inside us urging us to do the right thing is our conscience, God or the Holy Spirit (or whatever), or just genetic wiring, but maybe the distinction is unimportant.

I watched a new episode of The American Experience on PBS last night which dealt with the My Lai massacre of March 1968, in which US soldiers in Vietnam murdered over 500 defenseless men, women, children and infants. When their leader, William Calley, was tried and convicted for the crime, people in every Red State in the Union threw a fit, screaming that Calley was being treated too harshly. I personally believe that Calley should have been strung up by his private parts, but that's beside the point. The point is that if we continue to treat our fellow human beings in such a callous manner, the only hope for ET is to come packing should he ever have the misfortune of stumbling upon this planet.

That said, I think Wright is still being too disingenuous with regard to human attitudes toward one another (and other living things, for that matter). The Old Testament is awash in bloodshed and genocide but, because Christians fear death so much, they suspend whatever benevolence they feel and swallow that crap whole, saying that either the Hittites or the Midianites or the Moabites or the Vietnamites had it coming or that God is a cruel God, but God nonetheless, so He can wipe out whoever He wants.

I believe that the New Testament is also full of holes, but its message of love transcends whatever mistakes it's burdened with, and it totally supersedes the stupid myths and bloody fairy tales of the Old Testament. Unless Christ's teachings are earnestly taken to heart, the inhumanity of Amalek, Midian and My Lai will be repeated forever.

By the way, here's a storyline for Spielberg's next movie:
Aliens arrive in armadas of spaceships and claim Earth as their own, explaining that their Holy Books have promised them all the resources of the Milky Way Galaxy. Humans resist and are wiped out.
I'm sure this idea has been thought of before (it's actually being enacted in the West Bank) but, if not, then it's mine, dammit, and I'll slaughter anyone who tries to steal it from me.

Darwin Note -- Posted by wostraub on Monday, May 3 2010
I mentioned physicist Charles G. Darwin in my last post. His short book, The Next Million Years, reflects his interest late in life on population dynamics, particularly Malthusian catastrophe theory. I'm reading it now, and it looks like Darwin's view of the world at this point was pretty bleak.

By the way, the website Scribd is a treasure trove of free online books and articles, mostly technical, and it is where I found Trautman's article on square root ideas. They can be read online, but if you want to print or download them in pdf format you'll need a free subscription to the site.

Damned Spinors -- Posted by wostraub on Monday, May 3 2010
It is truly baffling that the vast majority of ordinary matter in the universe—the stuff that we're most familiar with—is composed of half-integer-spin fermionic particles like electrons, protons and neutrons whose mathematical description was unknown until Dirac formulated his relativistic electron equation in 1928. It is perhaps even more baffling that this description, which is fundamentally simpler than the Schrödinger wave equation and tensor formalism of pre-1928 physics, was so difficult to grasp when it was first proposed. And it is probably accurate to say that the fermion description, which is based on the mathematical critter known as a spinor, continues to elude nearly everyone, including a sizeable percentage of physicists.

Shortly after Dirac's discovery, British physicist Charles Galton Darwin (the grandson of that Darwin) wrote
Relativity theory is based on nothing but the idea of invariance, and develops from it the conception of tensors as a matter of necessity; and it is rather disconcerting to find that apparently something has slipped through the net, so that physical quantities exist, which it would be, to say the least, very artificial and inconvenient to express as tensors.
The "something" that Darwin referred to was the spinor.

Dirac's spinor was four-dimensional, but in 1935 Hermann Weyl discovered that spinor representations could be written down for any number of dimensions using Clifford algebras and that tensor products of these representations were irreducible.

The upshot of the spinor concept is based on a very simple idea, and to demonstrate it I will use the familiar metric tensor gμν as an example. There is a very important theorem in differential geometry that says that any tensor of rank two or greater can be decomposed into into a sum of vector products with the same number of factors as the rank of the original tensor. This means that we can write the 4-dimensional metric tensor as

gμν = a0b0 + a1b1 +a2b2 + a3b3

where the a's and b's represent four sets of vectors each. Because gμν is symmetric, we can simplify this to

gμν = ½(γμγν + γνγμ)

where we've replaced the a's and b's with the four-dimensional gamma matrices. It is precisely these 4×4 matrices that Dirac used in his electron equation. Matrices are operators in quantum mechanics, and the gamma matrices need something to operate on. That something is the 4-component spinor ψμ, which now acts like a column vector.

Unlike Schrödinger's wave equation, which is a single expression in the scalar function Ψ(x), Dirac's equation is really four simultaneous equations in the spinor ψ(x). What is Dirac's spinor? It's basically the same as Schrödinger's Ψ, only there's four of 'em. And they're all interdependent, making Dirac's equation a real bitch to solve. So what's the payoff, if Schrödinger's equation works? It's the fact that spinors are what God uses to describe fermions accurately, whereas Schrödinger's equation, though relatively precise, is really just an approximation. In a very real way, it's like comparing Einstein's gravity theory to that of Newton's.

But I digress. The main point is that a gamma matrix can be viewed as the "square root" of the metric tensor, and this "square root" idea is what Nature or God or whatever uses to describe one hell of a lot of things in the universe. Taking the square root can lead to conceptual difficulties, including negative and imaginary quantities, but it is this feature that really makes it work in quantum mechanics. The gamma matrices are complex quantities (well, γ2 is), but in higher dimensions they're generally all complex. And quantum mechanics is a theory that literally lives on complex quantities.

Since the gamma matrices are complex, so too are the spinors. But that's not the real problem, which brings me around finally to why I've called this post "Damned Spinors."

Here it is: the gamma matrices serve as operators, and the spinors act as the vectors getting operated on. But, in view of the stated decomposition of gμν, the gamma matrices appear themselves as single-indexed vectors γμ. In addition, with its four components, the Dirac spinor ψμ should be a vector as well (or maybe it's a set of four scalars). But these quantities can't both be vectors, and the spinor cannot be a scalar!

And they are not. Under a coordinate transformation, a vector obeys a well-known transformation rule, while a spinor transforms in a completely different way. The gamma matrices, on the other hand, can be treated as fixed matrix representations, so they don't transform at all. And it is the transformation rule for spinors, primarily for Lorentz transformations, that makes these "square root" quantities so hard to comprehend.

In short, the spinor represents a mathematical animal that completely escaped detection until quantum mechanics was developed in the mid-1920s. And, with the advent of ever-more complicated quantum field theories (including string theory), life without spinors would be impossible.

But I still hate spinors!

For a far more interesting account of the "square root" idea in physics, see this article by Andrzej Trautman (it's 4.7 MB, so don't click if your connection is slow).

And for more mathematical details on spinors, you can look at my article on Weyl's spinor, but for a better (and shorter) overview, see Viktor Toth's online write-up here.

Also, as you might expect, Charles Galton Darwin had an interesting life, perhaps more interesting than you might imagine. You can check him out on Wikipedia here.

The Gulf Oil Mess -- Posted by wostraub on Saturday, May 1 2010
Who's to blame? The latest illegal "immigrant" from the Gulf of Mexico (or maybe we should blame the British) is predicted to be far worse than the Valdez disaster but, with the leak spewing uncontrolled from 5,000 feet underwater, no one has a clue how to stop it except former First Lady Laura Bush:

Cogent, articulate, and sharp as a tack.

Many Christian groups are blaming President Obama for the spill, noting that it occurred shortly after he talked Israeli Prime Minister Benjamin Netanyahu into stopping illegal settlement construction in Palestine, while Teabaggers in Louisiana and Mississippi blame the nation's godless liberals for the disaster, adding that God would make the mess disappear magically if only Nancy Pelosi were assassinated.

Meanwhile, Louisiana Governor Bobby Jindal doesn't have a clue how to react—he previously told Obama to go screw himself over federal Katrina disaster funding, but now he needs that money to keep his state from becoming an oil-soaked toilet (instead of just an ordinary toilet). On the other hand, ex-half-governor Sarah Palin is not concerned, saying that she's seen all this before from her house in Alaska and that Nature, along with $25 billion in cleanup efforts, lawsuits, lost jobs and millions of dead sea birds and marine animals, will fix the problem Herself in only 20 years or so.

At the same time, the nation's liberals are screaming "I told you this would happen" as they watch the catastrophe unfold from their new AT&T-powered iPads. And everyone is dreading the thought of a suggested new fuel tax to help pay for the cleanup (Republicans say and repeat: Tax and spend! Tax and spend!)

But this, too, shall pass, even if we have to construct ocean barriers to seal up the entire Gulf (which we may have to do, if the oil keeps gushing out). We'll do anything, except develop the technology to prevent future spills or find some alternative to our goddamned fossil fuel addiction.

Remember, if we end up with an open-air National Emergency Petroleum Reserve in the Gulf, I thought of it first.

Old San Francisco -- Posted by wostraub on Thursday, April 29 2010

Ho hum, just another gorgeous spring day in San Francisco.

Although the entire area is pockmarked with the remains of military barracks and old artillery and battery placements from long ago, this photo I took from Hendrik Point, just north of the bridge, says it all: God bless this beautiful world.

One-Way Conversations -- Posted by wostraub on Monday, April 19 2010
New Scientist has an interesting article on Caine Mutiny author Herman Wouk in which he recounts a discussion he had with the famous Caltech physicist Richard Feynman many years ago.

Feynman, neither entranced nor impressed by the noted writer, asked him if he knew calculus. "No" was the answer. "Then you had better learn it," replied Feynman. "It's the language of God."

Wouk, a devout orthodox Jew, shrugged off the remark.

It has always struck me as grossly unfair that scientists are expected to learn about, respect and even revere religion, but others, notably theologians, can't be bothered to learn science, or its universal language, mathematics.

Weyl and the Aharonov-Bohm Effect -- Posted by wostraub on Sunday, April 4 2010
You may be familiar with what is surely one of the most profound discoveries of physics in the past 50 years, the Aharonov-Bohm effect. But you may not be aware that there is a neat connection between this effect and Weyl's 1918 gauge theory.

Here is an overview of the Aharonov-Bohm theory and its relationship (probably purely incidental) with Weyl's gauge idea. Most references tend to gloss over or muddle the mathematical details of the effect (the phase shift calculation in particular), so I've tried to make them as clear as possible.

I'd like to think that Weyl would have been really impressed with A-B, if he'd lived to see it in 1959, when Aharonov and Bohm published their work. It's a truly profound discovery, which was experimentally verified in 1986. I think it even has religious implications, but I'll spare you my thoughts on that here.

The noted physicist David Bohm is gone, but I encourage you to read up on him. A victim of McCarthyism, he had a truly interesting and fulfilling life nevertheless.

Fortunately, Yakir Aharonov is still with us. Now pushing 79, he's a distinguished professor of physics at Chapman University here in Southern California. Coincidentally, my daughter Sheryl received her law degree from Chapman University Law School, so now I can say I have something in common with Aharonov besides a love of beautiful physics!

The Mathematica website has an animated version of the Aharonov-Bohm effect that you can watch online. You can also download it and play around with it, if you have Mathematica 7 on your computer. Or, you can look at this little clip I made:

ORCH OR -- Posted by wostraub on Wednesday, March 31 2010
There seems to be a war going on today between proponents of artificial intelligence (AI) and its variants and a relatively newer theory called orchestrated objective reduction (ORCH OR). AI, which has been around for some time, basically posits that sufficiently advanced digital machines and their attendant algorithms can attain human-like intelligence, which can be roughly described as the ability to perceive and understand one's environment, make logical decisions, and learn from one's successes and mistakes. The computer HAL in the movie 2001 is an example of AI.

ORCH OR, on the other hand, says that the important thing is really consciousness, not high-level digital computing capability, and that consciousness is an inherently human characteristic. It further stipulates that consciousness is most likely a quantum phenomenon in which neurons and related microstructures in the brain participate in wave function self-collapse.

In the Copenhagen interpretation of quantum mechanics, the wave function collapses onto a unique eigenstate when an observation or measurement is undertaken. The observer, sentient or otherwise, is generally taken to be outside the system he/she/it is observing. ORCH OR posits that the human mind, being self-aware, collapses wave functions within itself, something no computer could ever do. To me, this is really profound stuff!

The founders of ORCH OR are the renowned Oxford mathematical physicist Roger Penrose and the noted anesthesiologist Stuart Hameroff. Penrose is the author of The Emperor's New Mind (1991) and The Road to Reality (2005), two books that I have discussed off and on on this site, as well as the Weyl curvature hypothesis, which I've also addressed at some length. Hameroff has a neat website called Quantum Consciousness, which I have only begun to dig through.

Although I have a fondness for computers and electrical engineering, I have to admit that ORCH OR seems to be much closer to the truth (despite the fact that I know next to nothing about the neurological sciences). I suppose this preference is due in part to the fact that Penrose is one of my heroes, but as a Christian I tend to believe that phenomena like consciousness and free will are solely human attributes, and that they lie at the foundation of our relationship to a creator God. Questions pertaining to right and wrong, I believe, can never be fully comprehended by a digital machine, regardless of how advanced it might be.

Bolstering the ORCH OR approach is Penrose's view of Gödel's theorem, which basically states that any sufficiently reasonable set of arithmetic rules cannot be both consistent and complete. For example, the self-contradictory sentence This statement is false can never be arithmetically resolved. This is oversimplifying things, but Gödel's theorem, which first appeared in 1931, shook the mathematics community to the core. Penrose believes that the human brain has the ability to circumvent the purely arithmetic axioms, rules and algorithms (which are inherently self-limiting according to Gödel) that necessarily constrain a digital machine.

It should be noted that some pretty brainy scientists think that ORCH OR is all wrong (notably the brilliant MIT cosmologist Mark Tegmark), while folks like Deepak Chopra also have their own valid opinions.

I needn't remind you at this late date that I'm a complete idiot in these matters, but I find it fascinating and compelling and hope you will, too.

LHC at 7 TeV and Other Stuff -- Posted by wostraub on Tuesday, March 30 2010
The European Large Hadron Collider achieved 3.5 trillion electron volt (TeV) beam energy (a new record) on March 19, but the report wasn't picked up by the media until today, when two 3.5-TeV proton beams racing in opposite directions were actually brought together. The resulting 7-TeV experiment is yet another record, but it's still only half the LHC's full design energy of 14 TeV. That capability is still many months away.

It's interesting that much of the news (at least its fringe element) remains dominated by groups of people who fear the creation of a stable micro-black hole or strangelet particle that will destroy the Earth. These are typically the same people who believe that the LHC represents some kind of monstrous affront to God, and so should be destroyed (this reminds me of the 1997 movie Contact, in which a multi-billion-dollar wormhole machine is blown up by a religious fanatic). Apparently, the ability of the world's nuclear superpowers to destroy the Earth many times over is somehow okay with these folks. Perhaps if they could view the LHC as a weapon their fears would be alleviated. (But then the weapon would still be in Europe's hands, not ours, and so should be destroyed.)

By the way, 3.5 TeV roughly represents the kinetic energy of a hovering fruit fly; in quantum terms, this is a truly stupendous energy! At that energy, an LHC proton has a velocity of about 99.9999964% of the speed of light. The associated Lorentz factor is about 3,726, so that the time dilation effect is huge; 1 hour to us is only about 1 second to the proton.

Quite an achievement, but much more is yet to come. I thank God that I live in this day of science wonders, but I'm continually puzzled by the general public's lack of interest. Science breakthroughs like the LHC must compete against mundane phenomena like Sandra Bullock's philandering husband, which invariably capture the public's rapt attention in a nanosecond. To me, such ignorance is akin to blasphemy.

A little story that relates to all this:

My father saw Halley's Comet in 1910 and used to tell me about it when I was a kid. In many ways this was responsible for my lifelong interest in science. His father took him to Riverview Park in Quincy, Illinois, about a block from their modest home on Cherry Street, and it was there, at the age of five, that my father, along with a large crowd of people, stood stunned by the view of a bright, seemingly-motionless astronomical body soaring high above. Twenty miles down the Mississippi River lay Hannibal, Missouri, the hometown of many of my kinfolk, and of Mark Twain, who died as the comet flew overhead. Twain's birth in November 1835 and death in April 1910 coincided almost to the day with the comet's 75-year orbital cycle, a fact I consider almost a miracle in itself.

My father only saw the comet that one time, as he died five years before its return in February 1986. But this time it was my turn, and I dragged my home-built 12½-inch reflecting telescope out so my family and I could watch this once-in-a-lifetime spectacle. But in the intervening years the comet had off-gassed so much of its material that it appeared only as a dim, amorphous blob in the eyepiece. I was disappointed, but still thankful to have witnessed something that my father had seen so long ago.

Today, in an age of iPhone apps, Facebook and instant communication, such wonders seem to have also faded in the public's eye. Have we simply become too jaded to appreciate major science discoveries and the universe that wheels right above our heads? And just exactly what is it about celebrity gossip and the sexual habits of sports figures that people find so damned fascinating? I'm sure I don't know.

Enough seriousness. For a change of pace, here's a humorous take on the LHC.

Weyl and the Accelerating Universe -- Posted by wostraub on Sunday, March 28 2010
Over the past decade numerous research papers have appeared that have examined the possible role of Weyl's geometry in the observed acceleration of the expansion of the universe. Some of these have been quite interesting, but none have reported any definite relationship between Weyl geometry and acceleration.

There have been three primary explanations for acceleration. The first is that the observations are erroneous, due to misinterpretations of deep-space supernova data. This "explanation" has been pretty much ruled out. The second involves the existence of dark energy, a mysterious, unexplained field that is created as the universe expands. It is gravitationally repulsive, so in time it tends to overwhelm the gravitational weakening of ordinary baryonic matter that accompanies expansion; research has been quite active in this area. The third explanation involves the cosmological constant Λ, an oft-neglected but perfectly valid extra term in the Einstein gravitational field equations that can provide the observed acceleration effect. This explanation is also of interest, because if true it eliminates the need for dark energy, which many cosmologists believe may be a kind of imaginary "ether" similar to that which plagued the physics community prior to acceptance of Einstein's 1905 theory of special relativity.

Einstein's general relativity theory is expressed by the set of gravitational field equations given by

Rμν - ½gμνR + Λgμν = Tμν

(I assume you are already familiar with the individual terms.) By comparison, Weyl's theory is expressed by

R(Rμν - ¼gμνR) = Tμν

Note that the left side of Weyl's equations is traceless (in four dimensions), which conveniently mirrors the traceless Maxwell stress-energy tensor Tμν of electrodynamics. Also, the cosmological constant term Λgμν is omitted, because its divergence cannot be set to zero.

In a space-time in which Λ and the matter tensor Tμν are negligible, Weyl's field equations explain all the usual tests of general relativity (light bending, gravitational red shift, perihelion advance of the planet Mercury's orbit, etc.) just as well as those of Einstein's. But does Weyl's theory also provide an explanation for the observed acceleration of the universe?

A recent paper by John Miritzis of the University of the Aegean in Greece says that it does not. The paper is not only quite readable, but includes a derivation of the Weyl field equations using the same approach (Palatini's method) that is traditionally used to derive Einstein's equations.

Here is Miritzis' paper, which I think you'll find interesting as well as educational in its approach.

PS— It is amusing to note that Miritzis, who has published extensively in gravitational field theory, is affiliated with his university's Marine Sciences Department!

Microsoft Project Tuva -- Posted by wostraub on Wednesday, March 17 2010
Microsoft has what might be called a Feynman Video Player over at Project Tuva, which features a number of black-and-white videos of Caltech physicist Richard Feynman lecturing to some Cornell students in the 1960s.

[The site's photo of Nobel laureate Feynman juxtaposed with college dropout Bill Gates is unfortunate.]

Accessing the website requires a download, but it's free, quick and well worthwhile. The videos are also closed-captioned.

If you've never heard the story of Feynman and Tuva, Google around for it. It's an interesting tale of someone who wanted to go someplace a bit off the beaten track.

Wilder and Wilder -- Posted by wostraub on Wednesday, March 10 2010
The journal New Scientist reports that renowned Oxford mathematician Roger Penrose has a new book coming out—with another one coming out after that. Amazing, considering Penrose is pushing 80 now.

Sir Roger Penrose (born 1931)

As I think I've mentioned earlier on this site, Penrose is the author of the Weyl curvature hypothesis, which attempts to explain the relationship between entropy, the arrow of time and cosmological gravitation. It is really nothing more than the observation that entropy, which was next to zero at the time of the Big Bang, coincided with a vanishingly small Weyl curvature tensor, which describes gravitational tidal forces. Naturally, the tidal force at the universe's birth would have been miniscule; over the intervening 13.7 billion years, entropy has grown, while Weyl tidal forces have increased because of cosmological gravitation. Penrose believes that Weyl curvature will may even become infinite in the far distant future of our universe.

What does this mean? Penrose thinks that for the second law of thermodynamics to exist (and to have a universe resembling the one we now see), God (or Whoever or Whatever) would have had to choose a single cell of phase space out of the approximately 1010123 cells that make up the total volume of the observable universe (the total phase space is essentially the number of coordinates needed to describe the positions and momenta of all the particles in the universe). Anyone who has ever studied statistical mechanics will be blown away by such preposterous odds. If Penrose is right, it says to me that the likelihood that our universe came into being by statistical chance alone is practically nil. [On the other hand, an infinite multiverse could easily give rise to an initially low-entropy universe (actually, an infinite number of them!) so it cannot be taken as a proof of the existence of God.]

Penrose presented his hypothesis in two of his books, The Emperor's New Mind and (one of my favorites) The Road to Reality. The first can be read by anyone, while the second requires earnest curiosity and patience from non-experts.

As New Scientist observes, The Road to Reality was wildly popular in spite of the fact that it consists of more than 1,000 pages of fairly dense mathematics and convoluted diagrams. Perhaps it was popular because it is one of those books that are good for one's soul.

Penrose's upcoming book is Cycles of Time: An Extraordinary New View of the Universe. According to New Scientist, Penrose believes he has found a way around the cosmological heat death issue, which is thought to result from dark energy expanding the universe to the point of boring near-nothingness. Penrose thinks that the only particles in existence at that far-off time will be massless, traveling at the speed of light. If he is right, then the flow of time will cease to exist (think about it). According to Penrose, this will then allow for more creation events like the Big Bang—unending universes, again from nothingness.

And Weyl curvature—will it then somehow revert from being near-infinite back to zero? Or do we just start over with new universes? Surely Penrose will address these questions.

Amazon doesn't have the book yet, and I've seen no publication date. But I'll be one of the first to get it when it does come out.

Update: Found it.

Meitner and Noether -- Posted by wostraub on Tuesday, March 9 2010
Speaking of Lise Meitner, here is the July 12, 1933 testimonial that Hermann Weyl wrote on behalf of another prominent Jewish researcher, the noted mathematician Emmy Noether:
Emmy Noether has attained a prominent position in current mathematical research – by virtue of her unusual deep-rooted prolific power, and of the central importance of the problems she is working on together with their interrelationships. Her research and the promising nature of the material she teaches enabled her in Göttingen to attract the largest group of students. When I compare her with the two woman mathematicians whose names have gone down in history, Sophie Germain and Sonja Kowalewska, she towers over them due to the originality and intensity of her scientific achievements. The name Emmy Noether is as important and respected in the field of mathematics as Lise Meitner is in physics.

She represents above all "Abstract Algebra." The word "abstract" in this context in no way implies that this branch of mathematics is of no practical use. The prevailing tendency is to solve problems using suitable visualizations, i.e. appropriate formation of concepts, rather than blind calculations. Fräulein Noether is in this respect the legitimate successor of the great German theorist R. Dedekind. In addition, Quantum Theory has made Abstract Algebra the area of mathematics most closely related to physics.

In this field, in which mathematics is currently experiencing its most active progress, Emmy Noether is the recognized leader, both nationally and internationally.
It is interesting to note that Weyl's testimonial, which was subsequently submitted to the Ministerium in Berlin, failed in its attempt to save Noether from dismissal from her university teaching job by the Nazis (in April 1933, they initiated the summary firing of all Jewish teachers and public service workers in Germany).

Meitner and Noether were both secular, non-practicing Jews, but of course it was their race alone that motivated Nazi hatred against them and their kind. But what made Noether's tenure even more unstable involved her reputation as an anti-war pacifist.

I'm guessing, but what probably "saved" Meitner (if that's the right word) was the fact that she was a noted nuclear physicist and close friend and colleague of the brilliant radiochemist Otto Hahn, who also spoke to the Nazis on her behalf. Meitner remained in Berlin until 1938, when even the great Hahn could no longer protect her.

Emmy Noether (1882-1935) shortly before her death

In view of Weyl's mention of Meitner and his references to the implication that abstract algebra is not "practical," it seems also probable that the Nazis at the time simply valued nuclear physics more than mathematics, although in 1933 no one could have known about the fissile properties of uranium 235. That was not elucidated until 1939, and even then it is doubtful that Germany recognized its awesome potential as bomb material. It is ironic that, had the Nazis retained the services of the many brilliant Jewish scientists and mathematicians that they kicked out, Germany's atomic experiments might have had a much more frightening outcome.

I find it also ironic that Meitner routinely worked with radioactive materials but lived to be almost 90, unlike Marie Curie, whose exposure to radium was almost certainly responsible for her death by anemia at the age of 66. By comparison, Noether died at the relatively young age of 53, felled by infection following routine surgical removal of an ovarian cyst. In view of this, and speaking as a non-mathematician myself, I'm tempted to believe that mathematics is indeed more hazardous than physics!

Lise Meitner Book -- Posted by wostraub on Friday, February 26 2010

Lise Meitner as a PhD student, about 1905

Austrian-born Jewish nuclear physicist Lise Meitner was perhaps second only to Marie Curie in terms of sheer experimental brilliance. Author Ruth Sime has written an excellent biography of the scientist in her wonderful book Lise Meitner: A Life in Physics, and it's recommended reading.

If you don't have the time to watch the entire video, here's a little bit about Meitner's life—

After receiving her PhD in physics in 1906 at the University of Vienna (only the second woman to do so), Meitner went to Berlin, where she eventually went to work with the gifted German radiological chemist, Otto Hahn. When Hitler assumed power in Germany in January 1933, Meitner was doing cutting-edge nuclear research under Hahn who, though a Christian German patriot, protected Meitner as best he could against the summary firing of Jewish scientists by the Nazis. She was advised to leave Germany by many emigrating physicists, including Hermann Weyl, who notified her of a junior professorship in America at Swarthmore College. But the great German physicist Max Planck assured Meitner that the Nazi storm would soon blow over, so she remained in Berlin.

But Planck was wrong. By 1938, emigration of all remaining scientists and intellectuals was banned, and Meitner, lacking the exit visa she could have easily obtained earlier, and facing increasingly rabid anti-Jewish persecution by the Nazis, found herself trapped in Germany. At great personal risk, Meitner managed to flee to still-unoccupied Holland by bribing some border guards.

Just before Christmas 1938, Meitner went hiking in the snow with Otto Frisch, a fellow physicist who was also her beloved nephew. They had kept up with the work of Meitner's prior supervisor Otto Hahn, who was having difficulty understanding some of his lab results. In a brilliant flash of mutual insight, Meitner and Frisch did some calculations on a few scraps of paper and discovered what what actually happening in Hahn's lab. These calculations, which utilized Einstein's famous E = mc2 for the first time as an experimental tool, showed that Hahn's research had resulted in nuclear fission—the splitting of the uranium atom.

Meitner was subsequently invited to work on the Manhattan Project in Los Alamos, New Mexico. But she was vehemently opposed to the military application of nuclear fission, which she feared had the capacity to ultimately destroy the world. "I will have nothing to do with a bomb!" she declared.

Inexplicably, Hahn alone received the Nobel Prize in Chemistry for his work in 1944. Meitner's contributions, which were fully equal to Hahn's, were not even mentioned. Despite this enormous professional slight, Meitner remained a close friend with Hahn. Traveling to the United States after the war, Meitner was greeted as a hero, receiving numerous awards, citations and honorary degrees. This reception, to a great extent, helped her overcome the disappointment she felt for having been robbed of a Nobel prize.

Meitner traveled to Cambridge, England, where she died in 1968, a week before her 90th birthday. On her gravestone is engraved



What Time Is It? -- Posted by wostraub on Friday, February 26 2010
The objective world simply is; it does not happen. Only to the gaze of my consciousness, crawling upward along the life line of my body, does a section of this world come to life as a fleeting image in space which continuously changes in time. — Hermann Weyl
What Weyl meant in this famous quote is that time, which is necessary for all change to occur, may only be a phenomenon of the mind. Weyl thought that perhaps the external universe is a fixed tableau of space, energy and matter that only appears to evolve as our mental processes progress through it. In this sense, time is a purely human construct with no physical existence of its own.

Caltech physicist Sean Carroll is the latest scientist to tackle the question of time. In his new book From Eternity to Here: The Quest for the Ultimate Theory of Time (which I have not yet read), Carroll addresses the question of time and its relationship with entropy, which involves the tendency for disorder to increase with time.

String theory aside, physicists believe we live in a 4-dimensional world composed of ordinary 3-space and time. But unlike the familiar up, down, sideways and back and forth of space, time as a dimension escapes understanding, except as a mathematical construct. St. Augustine of Hippo was perhaps the first modern person to address the puzzle of time, when he noted (and I paraphrase) "I know what time is when I don't think about it, but when I think about it I don't know what it is."

Well, whatever time is, it may in fact prove to be something along the lines indicated in Weyl's quote. It would appear that sentience is related to time in a very intimate way, so it is possible if not probable that a mouse or a leopard does not truly perceive time in any meaningful sense—all time is pretty much the present as far as unconscious entities are concerned. Recall the final words of Robert Burn's poem To a Mouse:
Still you are blessed, compared with me!
The present only touches thee:
But oh! I backward cast my eye,
On prospects drear!
And forward, though I cannot see,
I guess and fear!
So it may be that time is a purely human phenomenon which, like much of physics, cannot truly be understood.

But I feel one thing is certain: when we die, we no longer exist, and whatever consciousness we may retain (if any) is instantly whisked to what the physicist Frank Tipler calls the omega point, the final destination of everything and everyone. I liken this to the Lorentz transformation taken to its ultimate extent, where v = c and the concepts of space and proper/coordinate time truly end or becomes meaningless. Perhaps eternity itself is nothing more than a space-time inhabited only by entities traveling on null geodesics.

If this point of view is anywhere near the truth, then it's no wonder we remain flummoxed by the nature of time. And, at the same time (!), it might also explain the essential ineffability of God.

Weyl's Cones -- Posted by wostraub on Monday, February 1 2010
Spinors are mathematical quantities that represent the middle ground between the vectors and tensors of ordinary geometry. Unlike the rotation of vectors, spinors are characterized by the fact that they require a rotation of 720o (4π radians) before they return to their original position; a rotation of 360o only reverses their direction, so that Ψ(θ) → − Ψ(θ+2π). In mathematical physics, spinors are used to describe particles with intrinsic spin; for example, electrons, protons, and neutrons are all spin-½ objects.

This odd behavior has surprising analogs in the ordinary world. For example, lay a cup in the palm of your right hand, with your right arm stretched in front of you. Now rotate your hand and arm toward your body and then away, twisting your hand back to the front again, keeping the palm up (your elbow is now also pointing up). The cup has undergone a rotation of 360o (2π radians), but your right elbow is now pointing painfully up as well. Now, sweep your arm counterclockwise over your head, and continue the rotation until your hand and arm return to their starting point. Getting things back to normal took a rotation of exactly 720o (4π radians).

There are other examples, but around 1929 Hermann Weyl came up with a far more interesting (and less painful) analog.

Take two identical cones with a vertex angle of φ and fix their tips together with a flexible cord so that the upper cone is free to roll without slipping on the other. After a full rotation, how many degrees has the base of the upper cone undergone?

Naively, one would think 360o. But this overlooks the fact that every point on the upper cone's surface undergoes both a rotation about its own axis and a translation about the fixed cone. To answer the question, we have to consider the movement of the base of the moving cone, which is located a distance r + R = R[1 + cos(φ)] = 2R cos2(½φ) from the center of the fixed cone's base. The ratio of the circumference of this radius to the cone's radius is therefore 2 cos2(½φ), which is also proportional to the actual circumference that the cone traverses. Note the presence of a half angle, which is a signature of spinors.

When φ = 90o, the upper cone's base is perpendicular to that of the fixed cone, and the cone rotates exactly 360o. But when φ approaches zero, the cones, which now resemble thin needles, describe a rotation approaching 2(360) = 720o. This is the spinor case.

Weyl also noted that when φ approaches 180o, cos (½ φ) → 0, and the cones are so flattened that the upper cone just wobbles atop the other, and a full rotation approaches zero degrees!

The noted British mathematical physicist Roger Penrose tells an abbreviated version of this story on Page 41 of Hermann Weyl, 1885-1985: Centenary Lectures. Another good reference is Christoph Schiller's mammoth (164 MB, 2,000 pages and counting) and free online physics text, Motion Mountain. If you're ever stranded on an island, this is definitely a book you'll want to have.

My article Weyl's Spinor and Dirac's Electron Equation describes the mathematics of spinors in much more detail.

Another Hermann -- Posted by wostraub on Saturday, January 23 2010
Recall your earliest days when you studied calculus in college (it could also have been high school). When you got to the subject of double integrals, you probably did not note that the quantity dx dy in the integrand was not the same as dy dx. As an engineering student, it sure as hell didn't get my attention, and I'll bet that few of my professors were aware of it, either. About as far as the professors would go was to say that we could write it either way, but we should always avoid writing dx dx and dy dy, as they were either meaningless or vanishingly small.

However, if you went to a really good school, your professor would have told you something along these lines: The quantity dx dy represents an infinitesimal area, and area is directional—that is, there is a unit vector n1 that points away from one side of the area, and there is another vector n2 that points away from the other side. Clearly, n1 = - n2, so it makes sense to think that dx dy = - dy dx. But can this be proved mathematically?

Born in 1809, Hermann Günter Grassmann was the son of a German minister who taught mathematics and physics (isn't it neat that enlightened people like that existed once?) In due course, Grassmann became an accomplished philosopher, linguist, physicist, humanist, and all-around scholar. He did not get interested in mathematics until his early 20s and, though he was brilliant in the field, his convoluted writing style prevented his ideas from being given much credence during his lifetime. Anyway, in 1844 Grassmann wrote his magnum opus, The Theory of Linear Extension, a New Branch of Mathematics. It introduced to the world a lot of new stuff, including 3+n-dimensional manifolds, tensors, and the concept of a vector space, which Hermann Weyl would later recall as an "epoch-making" endeavor. But it also introduced a new kind of number, which today we call a Grassmann number. Grassmann numbers are unusual in the sense that they are very simple to learn and use but are unlike anything you would have ever dreamed up yourself.

Hermann Grassmann, 1809-1877. Not just another pretty face!

Simply stated, Grassmann numbers are anticommutative under multiplication; that is, if A and B are Grassmann numbers, then AB = - BA (and consequently AA = 0). Perhaps you're saying that this is no big deal, since matrices can behave the same way. But it is a big deal, because while matrices may or may not anticommute, Grassmann numbers always anticommute. And strangest of all, they are just numbers, not matrices.

You may have been taught that dx is just a differential, but in a space of more than one dimension it magically becomes a Grassmann number. I will give you the simplest demonstration of how this comes about. Consider a coordinate transformation from polar to cartesian, where we have the familiar relationships x = r cos θ, y = r sin θ. Differentiating, we get

dx = cos θ dr - r sin θ dθ
dy = sin θ dr + r cos θ dθ

Now multiply dx times dy, keeping all the differential terms in their exact order. You get

dx dy = sin θ cos θ dr2 + r cos2θ dr dθ - r sin2θ dθ dr - r2 sin θ cos θ dθ2

Now, if all the polar differentials are taken as Grassmann numbers, we will have dr2 = dθ2 = 0 and dr dθ = - dθ dr, which leaves us with

dx dy = r dr dθ

which is the correct expression for differential area in cartesian coordinates. And this is the real reason why we don't put things like dx dx under the integral sign!

Because of their anticommuting nature, functions of Grassmann numbers are simple. For example, the exponential function exp(aB), where a is an ordinary number and B is Grassmannian, is just exp(aB) = 1 + aB. And any function of two Grassmann numbers can be written as f(A,B) = a0 + a1A + a2B + a3AB. They don't get any simpler than that!

If you're wondering where these wonderful numbers come from, don't ask me—I'm just as bewildered as you are. Maybe God got bored one day, and said "What the hell, I'll make those, too." But they're the basis of what is called external algebra, and they play an important role in the algebra of differential forms. Grassmann numbers also lie at the basis of fermionic quantum field theory, an important application that Grassmann himself could never have imagined.

I won't go into the details here (you can get them in my write-up on Gaussian integrals), but where else can you find an algebra where the integral is an operator, equal to its own derivative:

∫ dθ = ∂/∂θ

I was impressed when I first studied complex analysis, which shows that the imaginary number i = (-1)1/2 has as much to do with reality (and perhaps more) as any real number. But I think Grassmann numbers have it beat hands down.

Reinventing Gravity -- Posted by wostraub on Wednesday, January 13 2010
In 1918, Hermann Weyl modified Einstein's then-new general relativity theory in a failed effort to unite it with electromagnetism. Then Kaluza and Klein tried it, then Schrödinger, then Pauli, and then, ironically, Einstein himself, whose failed effort spanned the final 30 years of his life. Along the way, an untold number of lesser physicists also tried their hands at the problem.

In the 1980s, quantum theorists noticed that their evolving string theories actually demanded a spin-2 particle, which they obligingly identified as the graviton, the hypothetical quantum of the gravitational field.

More recently, University of Toronto gravity theorist John Moffat has also tackled Einstein's theory, albeit with a more modest (if that's the right word) goal, which is not to embed electromagnetism into the theory, but to explain the so-called "Pioneer anomaly" and maybe even the apparent acceleration of the expansion of the universe.

In his 2008 book Reinventing Gravity: A Physicist Goes Beyond Einstein, Moffat describes his effort to derive a variant of Einstein's theory by way of an action Lagrangian that includes a variety of scalar, vector and tensor terms alongside the usual Einstein-Hilbert term. Judging by the theory's seeming agreement with observation, it looks like a pretty good theory (see the August 2009 paper written by Moffat and co-author Viktor Toth). Perhaps best of all, the theory dispenses with the need for "dark matter."

While the Moffat-Toth paper is clear and readable (any undergraduate can follow it), in my opinion it succeeds by throwing everything into the Lagrangian. Dirac tried a similar approach, which included a scalar field term in the action, but in Moffat's theory we have it all: scalars, vectors and tensors, along with mass terms for the scalar fields. Even the gravitational "constant" G is a field, as is the mass μ of the vector field! By comparison, the current action for the standard model of quantum field theory has something like 40 terms in the Lagrangian, so perhaps Moffat's theory isn't so complicated after all. But still ...

And this is only part of it!

I can't help but think about the old practice of curve fitting, which involves fitting experimental data to empirical mathematical expressions that often have no theoretical basis (for example, one can describe the relationship between water vapor pressure and temperature fairly accurately with a simple parabolic curve). But curve fitting requires parameters that have to be adjusted by hand to fit the data. (In his book, Moffat states that his theory requires no parameters at all, so maybe he's on to something.)

Many physicists have expressed their hope of one day having a unified theory of everything, a theory that's conceptually so simple that it will fit into a single line of mathematics that can be worn on a T-shirt. Moffat's theory, if correct, would require a mighty big T-shirt (I take 42 large), whereas string theory will take one the size of Nevada.

By the way, I've communicated with Viktor Toth a number of times, and he's a really neat guy. He's a Hungarian-Canadian computer/software expert and author whose website reveals the same love of physics I have (be sure to check out his short physics and math articles).

Cracking the Code -- Posted by wostraub on Wednesday, January 6 2010
The subject of Fulvio Melia's 2009 book Cracking the Einstein Code: Relativity and the Birth of Black Hole Physics is really New Zealand physicist Roy Kerr, who in 1963 found an exact solution to Einstein's gravitational field equations for a massive spinning object. Mostly non-technical, the book's 150 pages can be read in a few hours, and it's worth the time and effort. (For related information, see my 10 October 2009 post.)

Melia's run-up to the subject of Kerr and black holes includes some truly fascinating history on the first tests that were performed on Einstein's theory, which was published in November 1915. The very first test was not a test at all, but an explanation for an astronomical puzzle that had vexed astronomers since Newton's day. We tend to visualize these early observers with their clunky, primitive reflecting telescopes, whose primary mirrors were made of polished speculum metal, but that view is wrong. The accuracy of the equipment and the veracity of the astronomer's calculations (laboriously done by hand) almost defy description: astronomers had to take great pains in making their observations and correcting for the gravitational effects of all the planets on one another to determine the true orbital motions. But by the mid-1700s, the Keplerian ellipticity of the orbits of the planets was amply confirmed, and the calculated positions of the planets from day to day and from year to year exactly matched observation. The Newtonian "clockwork universe" seemed to be assured.

But then, when improved observational equipment became available in the early 1800s, it was noticed that the observed position of the planet Mercury (which is closest to the Sun) did not quite match the calculations. Again, the astronomers checked their orbital calculations and compared the result to what they saw in the telescopes. By extrapolating the minute discrepancy between Mercury's observed and calculated position for a period of one hundred years, Le Verrier found that the planet's orbit was 35 arc-seconds off. Decades later, Newcomb's refined calculations showed a discrepancy of 43 arc-seconds. By comparison, the Moon subtends an angle of about 0.5 degree in the sky, or 30 arc-minutes, about 40 times Mercury's orbital discrepancy for an entire century!

Continued observation conclusively confirmed the 43 arc-second figure, and astronomers were at a total loss to explain it. Some postulated the existence of an unseen planet ("Planet X" or "Vulcan") that orbited in sync with Earth but always behind the Sun, perturbing Mercury's orbit but otherwise eluding detection. Others suggested that asteroids or dilute solar-system dust might provide the explanation for the perturbation of Mercury's orbit. But nobody dared think that there might be something amiss with Newton's physics.

Within a few months of Einstein's 1915 announcement of general relativity, the German physicist Karl Schwarzschild solved the Einstein equations exactly for the simple two-body orbital case, and discovered that planets did not rotate about the Sun in perfect ellipses, but in slightly precessed elliptical orbits. That is, after completing one orbit the planets' positions would be slightly advanced. Schwarzschild easily calculated the advancement for Mercury, and found it to be 43 arc-seconds. He had found the answer to Mercury's anomalous orbital behavior! When he communicated this finding to Einstein, Einstein was so giddy with joy and excitement that he could not sleep for several days. [Some say that Einstein himself, armed with Schwarzschild's metric, calculated Mercury's orbit. Either way, it's a relatively simple calculation, so I'm giving Schwarzschild the credit.]

Karl Schwarzschild (1873-1916). German physicist and a
tragic casualty of World War I

Schwarzschild's analysis did not include the effect of the Sun's rotation on surrounding space-time. Several researchers, notably H. Thirring and J. Lense, tried to incorporate the angular momentum of a spinning, gravitational mass into the field equations but, with the exception of a first-order approximation, none were successful.

Then in 1963, Roy Kerr, at the time a physics professor at the University of Texas in Austin, came up with an exact solution. By comparison with the Schwarzschild solution, which any undergraduate can now duplicate, the Kerr metric is devilishly more complicated. It describes space-time in the vicinity of a rotating mass, and as a result it postulates wholly-new and unprecedented insights into the nature and topology of space-time, angular momentum, and kinetic and potential energy. In particular, it can be shown that space-time itself is actually "dragged" around a rotating black hole, so that the concept of inertial rest becomes meaningless. Furthermore, it has been shown theoretically that the rotational energy of a rotating black hole can be extracted to do useful work, with the attendant depletion of the hole's mass, in exact accordance with Einstein's E = mc2 law.

Kerr's work also finally allowed the complete description of a black hole, whose only three parameters are mass, angular momentum, and electric charge (hence the adage "black holes have no hair," meaning that they're actually very simple objects*). In 1965, E. Newman and his collaborators used the Kerr metric to derive the metric of a charged rotating black hole, which predicts even more phenomena.

Kerr himself wrote the book's Afterword, where he shares some interesting anecdotes about his life in general and his discovery in particular. Highly recommended.

* It is said that when John A. Wheeler coined the term "black hole" in December 1967, French physicists were upset, believing that the term carried a sexual connotation. They were even more upset when the "no hair" phrase hit, which connoted even more sexual naughtiness. Funny, I always thought the French had the jump on everyone when it came to such matters!

The Weyl Tensor and Gravity Radio? -- Posted by wostraub on Saturday, January 2 2010
In 1831, Michael Faraday discovered his famous law of induction, which stated that wiggling a magnet in the presence of a wire will induce an electric current in the wire. As noted in my 5 December 2009 post, Faraday reasoned 20 years later that wiggling a massive object might induce a similar effect involving the object's gravitational field.

In 2003, the journal New Science reported a related effort by University of California at Berkeley physicist Raymond Chiao to detect gravitational waves using high-temperature superconductors. Chiao even constructed a home-made "gravity radio" to test his idea. While it hasn't yet been successful, some scientists think that he might be on to something.

There is a formal, if somewhat hypothetical, analogy (called gravitoelectromagnetism, or GEM) between Einstein's gravitational field equations and the Maxwell equations of electrodynamics. This analogy was unknown to Faraday but his basic idea pointed in the same direction. Chaio's initial efforts were based on calculations involving recent work on GEM but, as described in the New Science article, there was a conceptual flaw: the effect Chiao sought was of short range and hence useless as a marker for the detection of gravitational radiation. This effect is called the Lense-Thirring field phenomenon, and is related to the "dragging" of an inertial reference frame (actually, spacetime itself) by a rotating mass. But this effect is of very short range, and far too short to be of any practical use in a detector.

But is there any other gravitational "warping" effect that persists over large distances? Indeed there is, and it was discovered by Hermann Weyl around 1920. Weyl determined that the Riemann-Christoffel curvature tensor Rμνλβ could be broken up into two pieces called the Ricci term and what is today called the Weyl term. The Ricci term involves the compressive, volume-deforming effect normally associated with gravity; it is especially large very near the gravitating source. By comparison, the Weyl term involves what are known as "tidal effects," in which an object even very far from the source can be distorted in shape while the object's volume remains constant. It is the Weyl tensor that is responsible for the "spaghettification" of the unfortunate astronaut who wanders too close to a black hole.

A Berkeley colleague suggested to Chiao that the Weyl tensor might be what he was for looking for. Indeed it was, and Chiao's revised theory was accepted for publication in the prestigious journal Physical Review.

In my little article Weyl's Conformal Tensor you can read how this tensor is derived, along with some brief notes and references on its importance in general relativity.