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
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
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)
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
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
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
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
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!
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."
Weyl-Dirac Phase Factor -- Posted by wostraub on Sunday, December 19
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).
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/β
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.
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
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
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
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
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
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.
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
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
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
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
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]
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 p → p - e/cA.
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
(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
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
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
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
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?
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
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
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
Is there a God in Heaven?
Stars -- Posted by wostraub
on Thursday, December 2 2010
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
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
Our galaxy is a spiral.
is a Mathematician of a Very High Order" -- Posted by wostraub on Tuesday, November 30
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.
Shadow Side of Mythology is Fundamentalism -- Posted by wostraub on Saturday, November 20
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
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
What happens when myth becomes fact. Taken from C.P. Pierce's Idiot
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
in the News (Inadvertently) -- Posted by wostraub on Saturday, November 20
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.
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
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
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
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
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 ημν
∂μ∂ν ≡ -ℏ2 ◻2
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,
iγμ∂ψ/∂xμ = mc/͎ Ψ (2)
(which can also be written neatly as i◻Ψ = mc/ℏ ψ, where ◻ = γμ∂μ).
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
[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.
Dead at 84 -- Posted by wostraub
on Wednesday, November 17 2010
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.
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
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
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.]
-- 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.
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
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
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.
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
However, last month's issue of Science featured an article by
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.
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
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
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.
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)
Cuckoo -- Posted by wostraub
on Tuesday, August 24 2010
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
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.
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
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"
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.
Number" Solved -- Posted by wostraub
on Wednesday, August 11 2010
Under the heading "God couldn't do it faster," this week's
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.
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
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
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
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.
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
ds2 = A (dx0)2 - B(dr2 + r2dθ2
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
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
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
Have a great summer!
The Pseudotensor of General Relativity --
Posted by wostraub on Tuesday,
June 22 2010
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;ν = Tμν;ν = 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,ν
= 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),ν = 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
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
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
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
Still, Ryder notes that the similarities
between EM and gravitation are tantalizingly close, and that these
similarities cannot be due to mere coincidence:
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
At the end of an introduction to
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!
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
Memories -- Posted by wostraub on Monday, June 14 2010
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
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
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.)
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
UCLA Commencement 2010 -- Posted by wostraub on Monday, June 14 2010
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!
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
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
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
(Everybody makes mistakes, even Ryder!)
PS: Ryder has a third book out called Elementary Particles and
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
I believe this quote, from Philosophy of Mathematics and Natural
(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
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.
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.
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
with the European Organization for Nuclear Research (CERN) working with
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
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.
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
"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
The debaters: Saint Peter and Saint Paul!!
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.
Hermann Weyl and the Coincidence of Large Numbers
-- Posted by wostraub on
Saturday, May 29 2010
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
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
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.
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
Weyl's Wormhole in the News -- Posted by wostraub on Friday, May 7 2010
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
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.)
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
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.
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
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
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
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
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
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
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
This citation alone is worth the price of the magazine.
When a Body Meets a Body ... -- Posted by wostraub on Wednesday, May 5 2010
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.
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.
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:
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.
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
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
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
Shortly after Dirac's discovery, British physicist Charles Galton
Darwin (the grandson of that Darwin) wrote
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
The "something" that Darwin referred to was the spinor.
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
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.
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
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
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.
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
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
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:
Old San Francisco -- Posted by wostraub on Thursday, April 29 2010
Ho hum, just another gorgeous spring day in San Francisco.
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
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.
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.
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
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
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.
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
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!
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
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
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
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.
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
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.
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,
LHC at 7 TeV and Other Stuff -- Posted by wostraub on Tuesday, March 30 2010
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
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.)
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
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
A little story that relates to all this:
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.
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
Weyl and the Accelerating Universe -- Posted by
Sunday, March 28 2010
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
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
It is amusing to note that Miritzis, who has published extensively in
gravitational field theory, is affiliated with his university's Marine
Microsoft Project Tuva -- Posted by wostraub on Wednesday, March 17
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.
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
The journal New
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.
As I think I've mentioned earlier on this site, Penrose is the author
of the Weyl
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
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.
Meitner and Noether -- Posted by wostraub on Tuesday, March 9 2010
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.
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
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
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
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
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
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
Lise Meitner as a PhD student, about 1905
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—
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.
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.
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!"
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
A PHYSICIST WHO NEVER LOST HER HUMANITY
What Time Is It? -- Posted by wostraub on Friday, February 26
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
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.
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
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.
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.
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
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
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.
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
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.
Another Hermann -- Posted by wostraub on Saturday, January 23
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?
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
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
Hermann Grassmann, 1809-1877. Not just another
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
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
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!
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
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.
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
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."
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
Cracking the Code -- Posted by wostraub on Wednesday, January 6
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.
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
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.
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
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.
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
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.
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.
himself wrote the book's Afterword, where he shares some interesting
anecdotes about his life in general and his discovery in particular.
* 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
Saturday, January 2 2010
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
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.
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.
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
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