Hermann
Weyl and CPT Symmetry -- Posted by
wostraub on Saturday,
December 15 2007

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

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

At any rate, I find Weyl's mathematics (unlike his physics) to be
very difficult and hard to follow. But there is one thing that jumps
out of the book at me, and that involves the following questions:

Why does mathematics describe the physical world so well? Why
should Nature obey the laws of mathematical symmetries? Why does
group theory govern so much of what goes on in the world?

These are hardly new questions, but the answers seem to be just as
far from us today as they were in Weyl's time.

Even more intriguing is the fact that Weyl, in 1929, was able to
deduce that Nature should also obey the discrete symmetries
described by charge, parity and time (CPT) invariance. Even today,
we haven't the slightest idea why God decided that these symmetries
should carry so much power and influence over Nature. In a later
edition of his 1928 book Group Theory and Quantum Mechanics,
Weyl wrote

The problem of the proton and the electron is discussed in
connection with the symmetry properties of the quantum laws with
respect to the interchange of right and left [parity
invariance], past and future [time invariance], and positive and
negative electricity [charge invariance]. At present, no
acceptable solution is in sight; I fear, that in the context of
this problem, the clouds are rolling together to form a new,
serious crisis in quantum mechanics.

Weyl's analysis of the Dirac and Maxwell equations in the context of
combined CPT symmetry led him to the correct conclusion that the
mass of the then newly-discovered positron (the anti-electron)
should be identical to that of the electron. The "crisis" that Weyl
referred to involved the then-prevailing opinion that the positron
should be nothing more than the familiar proton, whose mass exceeds
that of the electron by a factor of almost 2,000. Of course, we now
know that there was no crisis at all. But the reason why CPT
symmetry mandates these kinds of physical consequences remains a
total mystery.

God and
the Many-Worlds Interpretation of Quantum Mechanics -- Posted by
wostraub on Monday, December
3 2007

I apologize for this overly-long
post.

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

Hugh Everett, 1930-1982

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

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

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

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

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

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

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

I think we are forced to accept the MWI if quantum mechanics is
true. — Richard Feynman, Physics Nobel Laureate

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

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

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

The MWI is trivially true. — Stephen Hawking

For what it is worth [I prefer the MWI over the Copenhagen
interpretation]. — Steven Weinberg, Physics Nobel Laureate

To this list, Tipler adds another authority:

Jesus answered, “My kingdom is not of this world.” — John 18:36

Tipler could have also added:

“In my Father’s house are many mansions …” — John 14:2

I need to shut this down now, but lastly consider this: Tipler
believes that God is not a magician, only (only!) an eternal and
very clever physicist and mathematician who has figured everything
out. If we can believe that God fashioned Eve out of a rib bone he
yanked from Adam, we can surely believe in Everett’s many-worlds
theory. I urge you to read The Physics of Christianity and
decide for yourself.

Weyl,
Wheeler and Wormholes -- Posted by
wostraub on Sunday, December
2 2007

I've been reading
Hermann Weyl's Raum-Zeit-Materie and a General Introduction to
his Scientific Work, a neat collection of articles by noted
Weylophiles Erhard Scholz, Skuli Sigurdsson, Hubert Goenner, Norbert
Straumann, Robert Coleman and Herbert Korte. Having recently re-read
Kip Thorne's
book on wormholes, I was struck by a comment made by
Coleman and Korte regarding Weyl's supposed "discovery" of the
wormhole idea.

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

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

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

I give Weyl credit for many wonderful ideas, but I don't think
wormholes can be included on that list.

Not Even
Wrong -- Posted by wostraub
on Monday, November 5 2007

Wolfgang Pauli (1900-1958), the
Austrian physics Wunderkind of the early-mid 20th century,
often intimidated younger, inexperienced physicists by declaring
their ideas ganz falsch, or "utterly wrong." Those who he
really zeroed in on suffered the rather more blistering comment
nicht falsch, or "not even wrong."

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

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

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

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

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

Woit considers string theory to be an "ossified ideology," and
recommends that scientists now move on toward a fuller understanding
of quantum field theory and its relationship to mathematics. Will
string theory prove to be a waste of time and effort? Even if it is,
it at least has given us a glimpse into the mind of God, which
probably cannot be understood anyway.

Insects
and Worldlines -- Posted by
wostraub on Sunday, October 28 2007

2005, the "Year of Physics,"
brought about the appearance of I don't know how many more books on
Einstein, no doubt inspired by the 100th anniversary of Einstein's
annus mirabilis, 1905, the wonder year in which the
26-year-old Swiss patent clerk cum world-renowned scientist
produced four papers that would forever change physics.

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

Einstein used to quip that his fame grew out of his awareness of
something that had escaped most people (and insects):

When the blind beetle crawls over the surface of a world
globe, he doesn't realize that the track he makes [a "worldline"
or geodesic] is curved. I was lucky enough to have spotted
it.

In their very comprehensive (and, at nearly 1,300 pages, very long)
1973 foundation text Gravitation, Misner, Thorne and
Wheeler also spotted it, this time using the analogy of an ant
crawling over the surface of a piece of fruit ("The Parable of the
Apple"):

It was the very first graphic in this book (above) that caught my
eye one day in 1975, when I spotted the text on the shelves of the
miniscule public library in Lone Pine, California. Widely viewed as
the standard graduate-level text on general relativity, I wondered
how in hell it had landed in a tiny town whose only claim to fame
was that, as the portal to Mt. Whitney, it had hosted Humphrey
Bogart and company during the filming of the 1940 classic High
Sierra. The book was my companion on a day-long hike up the
14,000-foot mountain during that glorious summer that I discovered
the miracle of general relativity. It also brought me closer to God,
whose miracles and wonders I continue to marvel at.

The Fox
and the Forest -- Posted by
wostraub on Tuesday, October 2 2007

Roger and Ann Kristen are
government scientists, developing leprosy bombs and other high-tech
disease-culture weaponry for a war that never seems to end. Unlike
most American patriots in the fascist country of the United States
in the year 2155, they hate what they do. They hate the killing, the
fear-mongering, the torture, the constant propaganda, the
all-pervading culture of death.

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

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

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

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

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

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

Dear God in Heaven, anywhere but here and now.

Wolfgang
Panofsky Dead at 88 -- Posted by
wostraub on Thursday,
September 27 2007

Stanford University's Wolfgang
Panofsky is dead. The father of Stanford's linear electron
accelerator and one of the discoverers of the neutral π
meson, Panofsky was also noted for his abhorrence of nuclear weapons
and their proliferation.

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

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

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

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

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

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

There are so few of the great physicists like Panofsky still alive
today. God bless him, and may we all meet with peacemakers like him
in heaven.

The
Connection (Γ^{ α}_{μν}) Again --
Posted by wostraub on
Tuesday, September 18 2007

In 1918 Hermann Weyl tried to
unify gravity and electromagnetism by a generalization of Riemannian
geometry. He did this by eliminating the notion that the magnitude
of a vector is invariant with respect to parallel transport. In
doing so, he was forced to identify the electromagnetic 4-potential
with a non-zero covariant derivative of the metric tensor.

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

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

The most obvious answer has to do with the fact that the wave
function Ψ(x,t) is a complex-valued quantity whose meaning
is clear only when its conjugate square Ψ^{*}Ψ is
taken. Even then, this square (though real) can only be understood
as a probability. By comparison, the wave function by itself is at
best a probability amplitude. The metric tensor g_{μν},
on the other hand, is a purely real quantity that needs no
“squaring.” Similarly, the invariant line element ds^{2}
= g_{μν} dx^{μ}dx^{ν}, which measures
the interval between events in spacetime, is also a real quantity.

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

g_{μν} → exp [ k ∫φ_{μ} dx^{μ}
] g_{μν}

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

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

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

The connection term is still an open topic in mathematical physics
and differential geometry. If we do not impose the demand of a
Riemannian manifold, its precise makeup is largely arbitrary. Is
this how quantum effects enter into gravitation, as Weyl and
Einstein had hoped? Probably not, although it can be argued that a
connection describing internal spaces, possibly in multiple
spacetime dimensions obeying higher gauge symmetries, may yet find
application in the description of a consistent quantum gravity
theory.

Ramanujan -- Posted by
wostraub on Saturday, September 8 2007

I just finished reading Robert
Kanigel's award-winning 1991 book The Man Who Knew Infinity:
A Life of the Genius Ramanujan. The book's great length
stands in stark contrast to the very brief life of its subject, the
largely self-taught Tamil mathematician Srinivasa Ramanujan, who
died in 1920 at the age of 32.

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

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

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

UPDATE. There's a new
book out based on the life of Ramanujan.

Why
Gödel Thought US Dictatorship Possible -- Posted by
wostraub on Tuesday,
September 4 2007

In his 2005 book A World
Without Time, Brandais University philosophy professor Palle
Yourgrau writes

Years later, asked for a legal analogy for his
incompleteness theorem, [Gödel] would comment that a country
that depended entirely upon the formal letter of its laws might
well find itself defenseless against a crisis that had not, and
could not, have been foreseen in its legal code. The analogue of
his incompleteness theorem, applied to the law, would guarantee
that for any legal code, even if intended to be fully explicit
and complete, there would always be judgments "undecided" by the
letter of the law.

If this is indeed how Gödel felt, then he unequivocally predicted
that an event like 9/11 could plunge the United States into a
dictatorship, an outcome that the Founding Fathers simply could not
have foreseen.

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

We Americans are always bragging about how brilliant the Founding
Fathers were in drafting the US Constitution. But I believe that
Gödel was absolutely right -- the Founders could not have foreseen
that their country would utilize an event (largely brought upon by
itself) as an excuse to give the president dictatorial powers. And
this is exactly what has happened.

Kurt
Gödel and the US Constitution -- Posted by
wostraub on Friday, August 31
2007

I noted in my previous post that
in 1949 the brilliant Austrian-American mathematical logician Kurt
Gödel had discovered a solution to Einstein's field equations that
allowed for time travel. His discovery was presented to Einstein on
the occasion of the latter's 70th birthday party. (See my September
25, 2005 post for more info.)

Kurt Gödel and friend, early 1950s

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

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

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

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

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

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

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

UPDATE: Several readers directed me to this:
New Yorker Article, but it still doesn't explain why
Gödel thought the Constitution was flawed.

Good Bye
to Clocks Ticking -- Posted by
wostraub on Thursday, August
30 2007

I can't go on. It goes so fast. We don't have time to look
at one another. I didn't realize. So all that was going on and
we never noticed. Take me back — up the hill — to my grave. But
first: Wait! One more look. Good-by, Good-by, world. Good-by
Grover's Corners ... Mama and Papa. Good-by to clocks ticking
... and Mama's sunflowers. And food and coffee. And new ironed
dresses and hot baths ... and sleeping and waking up. Oh, Earth,
you're too wonderful for anybody to realize you! Do human beings
ever realize life while they live it? — Every, every minute? ...
I'm ready to go back ... I should have listened to you. That's
all human beings are! Just blind people.

— Emily Webb to the Stage Manager in Thornton Wilder's Our
Town

The newly-deceased Emily got her wish to travel back in time to
witness her 12th birthday. Did Weyl ever wonder about time travel?
Indeed, he did. Thirty years before Kurt Gödel's 1949 discovery that
a rotating universe could enable travel backward in time, Weyl wrote

It is possible to experience events now that will in part be
an effect of my future resolves and actions. Moreover, it is not
impossible for a world-line (in particular, that of my body) —
although it has a time-like direction — to return to the
neighborhood of a world-line point which it already once passed
through. The result would be a spectral image of the world more
fearful than anything the weird fantasy of E. Hoffmann [an
eccentric 19th-century German writer] has ever conjured up. In
actual fact the very considerable fluctuations of the components
of the metric tensor needed to produce this effect do not occur
in the region of the world in which we live. Although paradoxes
of this kind appear, nowhere do we find any real contradiction
to the facts directly presented to us in experience.

No doubt, Weyl (like Einstein) did not believe in super-luminal
velocities, so that mode of time travel to the past was verboten.
Also, Weyl probably never heard of wormholes, so that idea was out,
too. That left motion about the spacetime surrounding a rotating
massive body. Although Weyl died eight years before the physicist
Roy Kerr discovered the exact metric describing a spherical,
chargeless rotating mass, he was aware of the theoretical work of
Lense and Thirring, who in 1918 were able to deduce the approximate
field of a rotating body. Today, this effect is called
frame-dragging.

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

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

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

Mallett with prototype device, circa 1960!

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

Since black holes result from the collapse of spinning stars and the
accretion of rotating matter, it is hardly an overstatement to say
that all black holes spin and so have angular momentum (neutron
stars, the closest cousins of black holes, can have measured spin
rates of hundreds and even thousands of revolutions per second).
Therefore, frame dragging (and all its associated odd phenomena) is
the rule rather than the exception in this wonderful, strange place
that God created for us.

Weyl
Letter with Autograph -- Posted by
wostraub on Tuesday, August
28 2007

If you're interested in getting
your own autograph of Hermann Weyl (I have several), have a look at
this offering on
eBay.

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

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

Expanding Spacetime -- Posted by
wostraub on Thursday, August
23 2007

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

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

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

You can download Masreliez' book on his website. It's a fairly
straightforward read, and I recommend it.

Ralph
Alpher Dead at 86 -- Posted by
wostraub on Friday, August 17
2007

Ralph Alpher, the George
Washington University-trained physicist who was the first person to
fully understand the beginnings of the universe, died August 12 in
Austin, Texas.

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

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

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

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

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

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

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

But I would like to think that, right at this moment, God is busy
explaining everything about our wonderful universe to an awed and
overjoyed Alpher. As the apostle Paul had it, the world to come is
far better than the place we're in now.

Four
Neutrino Flavors? -- Posted by
wostraub on Tuesday, July 17
2007

Hermann Weyl was perhaps the
first physicist to posit the existence of the neutrino. At first it
was only a mathematical prediction. In 1930, Pauli proposed the
neutrino in order to preserve mass-energy conservation. Twenty-five
years later, it was found experimentally. Still later, two more
types of neutrino were discovered following Weyl's death in 1955.

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

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

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

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

For a relatively simple explanation of neutrino mixing and how the
sterile neutrino might fit into the scheme of things, see this
article by Fermilab researcher
B. Kayser.

Smolin
on String Theory -- Posted by
wostraub on Friday, July 13 2007

I just finished reading Lee
Smolin’s The Trouble with PhysicsAmazon Books, in which the renowned quantum physicist
bewails the impending failure of string theory. As a
string-questioner myself (actually, I don't get most of the theory's
math at all), I think it's a wonderful book!

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

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

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

The so-called Standard Model of physics, known more affectionately
as SU(3)×SU(2)×U(1), reached its zenith in the 1980s and 1990s, when
the predicted weak-interaction particles Z^{0}, W^{+}
and W^{-} were discovered (1985) and the top quark was
finally detected (1995). Since then: very little, with the possible
exception of the notion (Smolin calls it a discovery) that neutrinos
have mass. No wonder, he notes, that the world’s smartest physicists
are hitching their stars to string theory.

Smolin, late of Yale and Pennsylvania State and now at the Perimeter
Institute, is no less an accomplished string theory expert himself.
But he sees little beyond the theory’s beautiful mathematics and the
allure of extra dimensions (seven at last count, not including the
3+1 of good old spacetime). Without experimental verification, it’s
really nothing more than a religion without even any Gospels to back
it up. He quotes physics Nobelist Gerard t’Hooft:

Imagine that I give you a chair, while explaining that the legs
are still missing, and that the seat, back and armrests will
perhaps be delivered soon. Whatever I did give you, can I still
call it a chair?

But Smolin isn’t just complaining. He points out that there are
other ideas out there that might beat out strings as understandable
and experimentally verifiable unified theories: loop quantum
gravity, spin networks (see my post of earlier today) and various
spacetime-background-independent approaches to quantum gravity. So
there’s optimism to be had, but Smolin nevertheless regrets the
thousands of physicists and untold academic resources that are
currently being expended in the (possibly futile) search for
strings.

Jesus Christ once said that there are many mansions in his father’s
house (John 14:2). I still think he was referring to the many-worlds
interpretation of quantum physics, in which there are an infinite
number of universes awaiting us after death. I don’t personally see
a need for many dimensions, and until string theory is completely
played out (hopefully in my lifetime), I will side with Smolin.

Spin
Networks -- Posted by
wostraub on Friday, July 13 2007

For those of you who are
interested in an easy introduction to spin networks,
John Baez
has posted a write-up by Roger Penrose on some of the simpler
details.

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

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

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

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

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

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

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

Oh, and yes, I have no life to speak of.

"We
do not know what death is ..." -- Hermann Weyl -- Posted by
wostraub on Monday, July 2
2007

Last week,
Peter
Roquette, Professor Emeritus of the University of
Heidelberg, posted a comprehensive and very moving description of
the personal and professional relationship between Hermann Weyl and
Emmy Noether (whom you can read about in my Weyl-Higgs write-up).
Roquette, a mathematician, has written extensively about the
mathematical correspondence between Noether and the German
mathematician Helmut Hasse. You can Google him if you want more
information. Noether and Weyl Article

Included in Roquette's insightful article is the full text of Weyl's
funeral dedication to Noether on April 18, 1935, which contains

We do not know what death is. But is it not comforting to
think that our souls will meet again after this life on Earth,
and how your father’s soul will greet you? Has any father found
in his daughter a worthier successor, great in her own right?

[Note: Noether's father was himself an esteemed professor at the
University of Erlangen, and justifiably proud of his daughter's
substantially greater mathematical abilities.] It also includes
Weyl's moving but fruitless petition to have Noether retained as a
professor in Germany in the summer of 1933, when the Nazis summarily
fired all scholars of Jewish descent or heritage.

In coming to Princeton as a German emigre himself in late 1933, Weyl
selflessly endeavored to obtain a position for Noether at the
Institute for Advanced Study as well. This was denied (possibly
because of the school's antisemitic attitude), although she did find
a position (at reduced salary) at Bryn Mawr.

Science
and Religion, Again -- Posted by
wostraub on Saturday, June 30
2007

I just finished watching
BookTV on C-SPAN2, which featured science writer Natalie Angier
talking about her new book The Canon: A Whirligig Tour
of the Beautiful Basics of Science. Angier is also a recipient
of the Pulitzer Prize, which she won in 1991 at the age of 33.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

But there is one important difference between these warring camps
today: even radically secular science will never demand that people
be burned at the stake for not believing in quantum mechanics,
whereas many people of faith today believe that President George W.
Bush can turns lies into truth.

Veltman
on Elementary Particles -- Posted by
wostraub on Wednesday, June
27 2007

A non-scientifically-minded
friend of mine recently pointed out to me that, in accordance with
Einstein's E = mc^{2}, the energy available to
mankind must be nearly infinite. He reminded me of the scene in
Back to the Future where Doc Brown replenishes the power source
of his flying Delorean with a few banana peels and a shot of stale
beer, throwing in the beer can for good measure.

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

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

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

Anyway, if you're interested in modern elementary physics and want
the best resource available on the subject at the layperson's level,
you can't go wrong with Veltman's book. It explains everything from
quarks and gluons to hadrons and their antiparticles on up, their
interactions and their conservation principles, along with brief but
fascinating sketches of many famous physicists. Equally enjoyable is
Veltman's rather strange and often hilarious use of the English
language.

Krauss
on Extra Dimensions -- Posted by
wostraub on Tuesday, June 26
2007

Case Western Reserve University's
Lawrence Krauss is a leading particle physicist and cosmologist, and
he has written a number of excellent books (including The
Physics of Star Trek, which I thought was rather silly, but
that's another story). His most recent book, Hiding in the
Mirror, discusses the subject of extra dimensions and why they
hold so much allure nowadays.

Krauss with friend

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

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

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

Sickly for most of his life, Riemann was born in 1826 and died of
tuberculosis at the tragically-young age of 39, not long after
developing his geometry. He was convinced that his was the "true
geometry of the world," and believed it could be used to describe
all physical phenomena. His initial efforts failed, but it was only
because Riemann was stuck in three dimensions. If he had only been
gifted with Einstein's foresight to view time as the fourth
dimension, the general theory of relativity (gravitation) would have
undoubtedly appeared around 1870, 45 years earlier than Einstein's
opus of November 1915.

Adieu to
Schrödinger -- Posted by
wostraub on Wednesday, May 23 2007

I finished Moore's book on
Schrödinger and found it to be a fascinating account of not just
Schrödinger's life and work but a glimpse of how the physicists of
his day struggled to make sense of the emerging quantum theory of
the mid-1920s.

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

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

Schrödinger remarked to his host, the noted Cal Tech Nobel laureate
Robert Millikan, that he wished Pasadena were populated by Italians
or even Spaniards, not Americans, although he felt they were
considerate to a degree quite unknown in Germany at the time. This
is understandable, as Schrödinger, who had recently visited New York
City, hated the place and thought Americans to be uncultured.

Schrödinger: Life & Thought -- Posted by
wostraub on Tuesday, May 22
2007

I managed to find a library copy
of Walter Moore's Schrödinger: Life and Thought and am in
the process of reading it. Erwin Schrödinger was Hermann Weyl's best
friend (from their days together at the ETH in Zürich until Weyl's
death in 1955), and I thought this book would provide additional
information on Weyl. Yes, it did.

Moore relates the notoriously open relationship that the otherwise
devoted Schrödinger and Anny (his wife of 41 years) practiced, which
was due primarily to Erwin's predilection for extramarital affairs.
Schrödinger's intellectual abilities seems to have been matched only
by his libido, and he had many lovers, even into his old age. Anny
herself had her share of paramours, including Weyl (whom she called
Peter):

Anny would find in Hermann Weyl a lover to whom she was devoted
body and soul, while Weyl's wife Hella was infatuated with Paul
Scherrer [another ETH physics professor].

This relationship was confirmed in, of all things, a friendly letter
from Anny to her husband Erwin in 1936:

Even if the love between Peter and me should sometime come to an
end, I would always be blessed that it had formerly existed, as
I know that fate has given me the greatest happiness that a
person can ever be given.

But the best part of the book (so far) is the story behind
Schrödinger's famous wave equation, and how he came across it late
in 1925 (and even this story involves an amorous romp between Erwin
and an unknown former love in Arosa, a secluded Alpine resort).

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

If I find anything else interesting in the book, I'll report on it
later.

Biggest
Supernova Ever Seen -- Posted by
wostraub on Tuesday, May 8
2007

A team of astronomers from the
University of California at Berkeley has discovered an enormous
supernova in the galaxy known as NGC 1260. It exploded in September
last year, producing the most massive outpouring of energy ever
witnessed. (Actually, because this galaxy is 240 million light years
away, the star blew up 240 million years ago.)

The supernova, designated as SN 2006gy, had an estimated energy
output of 10^{45} joules, enough to outshine the star's
entire galaxy of perhaps 200 billion stars. It's bigger than
anything ever seen, and its output has been remarkably persistent:

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

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

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

Eta Carinae underwent a colossal false nova event in 1843
which almost destroyed the star. It survived, but remains the best
candidate to date for a SN 2006gy-like explosion. The
dumbbell-shaped ejecta cloud streaming out of the region now dwarfs
the central star itself. Frightening.

Feynman's Thesis -- Posted by
wostraub on Wednesday, May 2 2007

I finally got around to reading
Laurie Brown's book
Feynman's Thesis, which, to the best of my
knowledge, is the only publicly available version of Feynman's 1942
PhD dissertation. I was astonished to find that Feynman's thesis,
which details his discovery of the path integral, is understandable,
fun to read, and short -- incredibly, the document's only 68 pages
long. (Mine was the exact opposite -- at 225 pages, it was
incomprehensible, boring, and long.)

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

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

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

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

^{*} The probability amplitude of an event is a complex
number whose norm (its "square") is a real number. The physicist
Nick Herbert describes things this way: the wave function Ψ
is a probability amplitude, also called a "possibility," while its
square Ψ^{*}Ψ is a real number, the "probability."

The Most
Beautiful Thought -- Posted by
wostraub on Saturday, April
28 2007

Right after his discovery of
special relativity in 1905, Einstein was looking for a more general
application of his theory -- one that included gravitation.

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

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

Although I believe that purely recreational flight (like
billionaires paying millions to go into orbit) is a ludicrous waste
of resources, I'm happy for the guy (and I'm sure Einstein was with
him in spirit).

Weyl's
1929 Paper -- Posted by
wostraub on Tuesday, April 24 2007

I once remarked that I planned to
review Weyl's 1929 paper "Electron and Gravitation," which formally
presented Weyl's gauge idea in the context of the then
still-emerging quantum theory. While I've always found Weyl's
physics to be straightforward, his mathematics tends to be rather
obtuse and difficult to follow (at least for me). Complicating the
matter is the fact that Weyl's notation is unfamiliar to many
physicists (for example, Weyl expresses the tetrad as e_{a}(β)
rather than the purely tensorial index form e_{a}^{β}).
Maybe this is no big deal, but I have a hard time following stuff
when the notation is odd.

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

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

Rossmann also points out a reference in Weyl's paper that has always
puzzled me. In the the paper's first section, Weyl notes that his
2-component spinor ψ is unable to accommodate left-right
parity; he correctly surmises that parity would therefore require
another, independent 2-component spinor (Weyl even suggests that
this spinor describes the proton). Weyl cryptically notes that he
will address this issue in a "Part II," but he never mentions it
again. Rossmann believes that Weyl never wrote Part II because
Anderson's 1932 discovery of the positron (not to mention Chadwick's
discovery of the neutron in the same year) compromised Weyl's ψ-connection
in its relationship with the Weyl current j^{ μ}=ψ^{*}σ^{ μ}ψ
(σ^{i} are the Pauli matrices; σ^{0}
is the 2×2 unit matrix). Thus, Weyl's hope for a theory of
everything (which in those days was just gravity and
electrodynamics) was quashed for good.

Finally, Rossmann offers the following quote from Weyl's book
Space-Time-Matter, which is all too true (but gives
non-mathematicians like me little solace):

Many will be horrified by the flood of formulas and indices
which here drown the main idea of differential geometry (in
spite of the author's honest effort for conceptual clarity). It
is certainly regrettable that we have to enter into purely
formal matters in such detail and give them so much space; but
this cannot be avoided. Just as we have to spend laborious hours
learning language and writing to freely express our thoughts, so
the only way that we can lessen the burden of formulas here is
to master the tool of tensor analysis to such a degree that we
can turn to the real problems that concern us without being
bothered by formal matters.

Weyl's
Pedestal Shaken a Bit -- Posted by
wostraub on Tuesday, April 17
2007

In 1970 I took three courses in
physical chemistry at university. The book we used was Walter
Moore's text, appropriately entitled Physical Chemistry.
Now a classic, it was, and still is, a tough book, especially the
chapter on quantum chemistry. As I recall, we undergrads didn't like
Moore very much.

Over thirty years later, in 2001, Moore wrote a biography of the
great Austrian physicist Erwin Schödinger called Schrödinger:
Life and Thought, which was
reviewed for the New York Times by Richard Teresi, the
author of The Three-Pound Universe. Here is an excerpt from
that review:

Schrödinger's wave equation, published only a few weeks later,
was immediately accepted as "a mathematical tool of
unprecedented power in dealing with problems of the structure of
matter," according to Mr. Moore. By 1960, more than 100,000
scientific papers had appeared based on the application of the
equation. Schrödinger lavishly thanked his physicist friend
Hermann Weyl for his help with the mathematics. (He was
perhaps indebted to Weyl for an even greater favor: Weyl
regularly bedded down Schrödinger's wife, Anny, so that
Schrödinger was free to seek elsewhere the erotic inspiration he
needed for his work.)

Ouch. I'm not a prude, but this aspect of Weyl's personal life
troubles me. I have read in numerous places that post-World War I
Germany was sexually liberated, and I already knew that Einstein had
more than a few sexual skeletons in his closet, but this hits close
to home.

I have not yet read Moore's Schrödinger book, primarily because a) I
don't want to pay
Amazon $50 for the thing; b) it's not available from my
library's interlibrary loan program; and c) I'd already had enough
grief from Moore almost 40 years ago. But if I can lay my hands on a
copy I'll you all know what I think of it -- and whether my opinion
of Weyl changes any as a result.

Arrhenius and Climate Change -- Posted by
wostraub on Monday, April 9
2007

Svante Arrhenius was a Swedish
chemist who, around 1895, was the first to quantify the relationship
between chemical reaction rates and temperature. He won the Nobel
Prize in Chemistry in 1903 for his work on electrolyte theory.

Every undergraduate chemistry major learns how to derive Arrhenius'
rate equation, but what most don't realize (as I didn't realize at
the time) was that Arrhenius was the world's very first climate
modeler. In 1894 he derived a remarkably accurate relationship
between atmospheric carbon dioxide levels and global temperatures.
Using only a slide rule, he calculated that a doubling of CO_{2}
concentrations would raise the Earth's average temperature by 9.9^{o}F.
By comparison, today's modern supercomputers and vastly more
accurate climate models predict an increase of 10.4^{o}F.

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

Even Arrhenius could not have foreseen the day when humans would be
pumping out 8.2 billion tons of CO_{2} into the atmosphere
annually, an amount far in excess of the planet's ability to absorb
without global climatic consequence.

From CO_{2} buildup and the breakdown of the Atlantic
Conveyor to deforestation and the unavoidable release of tens to
hundreds of billions of tons of frozen Siberian methane, Pearce
paints a bleak picture for 21st century Earth and its inhabitants.

While reading the book, I was at once struck by the words of Chris
Hedges in a recent edition of
New
Statesman, where he notes that the real motivation of
today's fundamentalist Christians is not religiosity but despair:

... The danger of this theology of despair is that it says that
nothing in the world is worth saving. It rejoices in cataclysmic
destruction. It welcomes the frightening advance of global
warming, the spiraling wars and violence in the Middle East
and the poverty and neglect that have blighted American urban
and rural landscapes as encouraging signs that the end of the
world is close. Those who cling to this magical belief, which is
a bizarre form of spiritual Darwinism, will be raptured upwards
while the rest of us will be tormented with horrors by a warrior
Christ and finally extinguished. The obsession with apocalyptic
violence is an obsession with revenge. It is what the world, and
we who still believe it is worth saving, deserve.

If true, then may God forgive us all.

Hermann
Weyl in Göttingen -- Posted by
wostraub on Monday, April 2
2007

Here are some photos of Weyl
talking at two separate colloquia taken in the early 1930s. The
original caption on the first photo indicates that the lecture hall
was filled to capacity, as the students at Göttingen did not know
how much longer Weyl would be staying in Nazi Germany (Weyl's wife
was Jewish, which jeopardized Weyl's entire family; they ultimately
emigrated to America in November 1933). The second shows Weyl with
the great German mathematician Richard Courant (far left).

Relativity for the Masses -- Posted by
wostraub on Saturday, March
17 2007

The act of Taylor & Wheeler
(MIT's Edwin F. Taylor and Princeton's John A. Wheeler, that is) has
produced two books that you need to read if you want to learn
special and general relativity quickly.

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

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

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

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

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

^{*} The energy needed to bring every pound of your
spaceship up to this speed is equivalent to approximately one
billion Hiroshima-size atom bombs.

The
Natural Gauge of the World -- Posted by
wostraub on Friday, March 16
2007

Sometime ago I wrote about
Eddington and his unified field theory of 1921, the one that Weyl
called "not worthy of discussion." But it was Eddington who came up
with the phrase "natural gauge of the world," and in was in the
spirit of Weyl's 1918 theory that he proposed it.

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

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

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

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

You can learn a little more about Weyl's conformal tensor (and how
to derive it) in my write-up on the menu to the left.

Hermann
Weyl -- The Centenary Lectures -- Posted by
wostraub on Saturday, March
10 2007

I just finished reading
Hermann Weyl, 1885-1985: The Centenary Lectures, three talks
that were given by C.N. Yang, R. Penrose and A. Borel on the 100th
anniversary of Weyl's birth.

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

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

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

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

In his speech, Michael included this poem by the minor poet Anna
Wickham (1884-1947):

God, Thou great symmetry,
Who put a biting lust in me
From whence my sorrows spring,
For all the frittered days
That I have spent in shapeless ways
Give me one perfect thing.

Hermann Weyl included this poem in his last book, Symmetry,
in which he remarks that the sphere in space represents true
perfection -- so perfect, in fact, that it inspires not only awed
admiration but sorrowful longing as well, because it is a reflection
of the perfect symmetry and unattainable perfection that is God
himself.

Weyl's
Gauge Factor, Again -- Posted by
wostraub on Monday, February
26 2007

Several people have written me
asking how Weyl’s geometry might be modified using the nonintegrable
gauge factor that I mentioned in my December 19 post. There I
suggested that the line element ds^{2} could be
made gauge invariant (or conformally invariant) via a “phased”
metric tensor:

g_{μν} → exp[k∫φ_{λ}dx^{λ}]
g_{μν}

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

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

S = ∫√(-g) R d^{4}x

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

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

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

s^{μ} = ρ(x) dx^{μ}/ds

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

I'll think about it, but I don't believe it goes anywhere.
Suggestions?

Squid
Memories -- Posted by
wostraub on Thursday, February 22 2007

Is it my imagination, or are
giant squids (not Superconducting Quantum Interference Devices,
but the scary ones with tentacles) showing up more frequently
nowadays?

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

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

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

In the 1980s, while scuba diving in extremely murky water off the
Coronado Islands, some enormous, shadowy thing came up behind me. I
turned, and as I did it quickly swam off, leaving me spinning head
over heels in the backwash. The ascent back to the surface (I had
been down about 100 feet) seemed to take forever. I never really got
a look at it, but on the boat the divemaster told me it was only a
harmless grey whale that had been seen swimming nearby.
Nevertheless, that was the last dive I made that day.

Lorentz
Symmetry -- Still Asking -- Posted by
wostraub on Friday, February
16 2007

It was Hermann Weyl who showed
that quantum mechanical gauge invariance is the continuous symmetry
responsible for the conservation of electric charge. Lorentz
invariance (invariance of the Lagrangian with respect to Lorentz
transformations) is also a continuous symmetry, so what conservation
principle does it represent?

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

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

N = pt - mx

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

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

Penrose obviously doesn't take into account idiotic readers like
myself.

Juliet
or Esmerelda? -- Posted by
wostraub on Wednesday, February 7 2007

Unlike her happier fate in motion
pictures, the gypsy Esmerelda in Victor Hugo's classic novel The
Hunchback of Notre Dame dies by hanging. Her lovelorn admirer,
the hideous hunchback Quasimodo, dispatches the corrupt Frollo but
is unable to save the love of his life. Heartbroken, he disappears.
Years later, grave diggers accidentally unearth the skeleton of
Esmerelda, which is inexplicably embraced by that of a grossly
deformed man. [And] when they tried to detach the skeleton which
he held in his embrace, it fell to dust.

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

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

Touching.

Famous
Papers -- Posted by wostraub
on Wednesday, January 31 2007

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

I'm also pleased that it includes my write-up of Weyl's 1918 gauge
theory, perhaps because it's more detailed than the paper Weyl
originally wrote (and because you don't have to know German to read
it).

Hermann
Weyl in Exile -- Posted by
wostraub on Wednesday, January 24 2007

I just finished reading
Forced Migration and Scientific Change: Emigre German-Speaking
Scientists and Scholars After 1933 (1996), a collection of
articles edited by Mitchell Ash and Alfons Söllner. Of particular
interest is the chapter Physics, Life and Contingency: Born,
Schrödinger and Weyl in Exile by Sküli Sigurdsson, whose 1991
PhD dissertation on Hermann Weyl I'm still trying to run down.

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

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

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

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

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

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

Plus ça change, plus c'est la même chose ...

Eddington -- Posted by
wostraub on Tuesday, January 23 2007

Earlier I remarked that God
cannot create a new integer between the numbers 1 and 10. Here's a
little story about how one man tried to do essentially the same
thing.

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

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

2πe^{2}/hc = ≈ 1/137

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

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

ƒ(n) = ½ n^{2}(n^{2} + 1)

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

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

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

Note: Interestingly, the late, great German Nobel laureate physicist
Max Born (who happened to be Olivia Newton-John's grandfather!)
noticed that Eddington's formula reproduced two numbers from the New
Testament Book of Revelation, Chapter 13:

And I saw a beast coming out of the sea having ƒ(2) = 10
horns ... [and] his number is ƒ(6) = 666.

I feel fairly certain that God did not use Eddington's formula when
he inspired John to write Revelation!

PS: Wolfgang Pauli (Nobel physics prize, 1945) was also fascinated
by the fine structure constant, and he devoted much time and thought
to its provenance. He passed away from cancer in 1958, and the
number of the hospital room where he died was ... 137. Good one,
God!

Twenty
Eighty -- Posted by wostraub
on Monday, January 22 2007

My older son's girlfriend loaned
me a copy of James Surowieki's book
The Wisdom of Crowds. It's really interesting -- it
pretty much destroys the idea that irrational mob rule is the norm,
and shows how crowds of people may have differing points of view
but, when the average is taken, it tends to be pretty close to the
truth. The book's one caveat is that a crowd must have a fairly firm
grasp of reality, otherwise mob rule does indeed take over, with
disastrous results.

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

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

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

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

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

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

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

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

America cannot survive when 33% of its people are crazy. And I fear
that neither can the rest of the world.

Weyl and
von Neumann -- Posted by
wostraub on Saturday, January 13 2007

There's a story that Hermann Weyl,
when talking about his work at a conference or lecture hall, would
become extremely agitated whenever colleague John von Neumann was in
the audience. His nervousness was presumably due to the fact that
von Neumann was widely viewed as a genius, and Weyl was afraid he'd
make a fool of himself.

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

There is a famous story involving von Neumann, apparently even true,
that he was approached by the hostess of a party he was attending
and given the following puzzle to solve:

Two bicyclists on a road are 100 miles apart. At a predetermined
time they begin pedaling toward each other, each with a uniform
speed of 10 mph. At the moment they start out, a fly sitting on
the wheel of one of the bicycles starts flying toward the other
bicycle at a speed of 20 mph. Upon reaching the other bicycle,
it instantaneously turns around and starts flying back to the
first bicycle. It does this repeatedly until the bicycles meet
in the middle of the road, squishing the fly between the tires.
How many miles does the fly travel?

[This story is so old that I am almost ashamed to repeat it.] There
are two ways to solve the problem, but one way is immediate: the
bicyclists meet at the midpoint after 5 hours of pedaling. The fly
has been flying constantly during this time, so it flies a total of
5×20 = 100 miles. The second method involves calculating the
infinite series L = 100/3 ∑ (2/3)^{n}, where the
sum is taken from n = 0 to n = ∞. Again, L =
100 miles.

At the party, when asked for the answer, von Neumann instantly said
"100 miles." The hostess smiled and said, "Oh darn, you know the
trick." To which von Neumann replied, "What trick? I got it by doing
the infinite series."

Herman
Weyl and Yang-Mills Theory -- Posted by
wostraub on Thursday, January
11 2007

In his great 1929 book The
Theory of Groups and Quantum Mechanics, Hermann Weyl jokingly
remarked that It is rumored that the ‘group pest’ is gradually
being cut out of quantum mechanics. Weyl was referring to
objections that many physicists were voicing at the time about the
use of group theory (which was viewed as pure mathematics) in the
then-emerging field of quantum mechanics. Of course, as both a
mathematician and physicist Weyl could already see the importance of
group theory in quantum physics, and his seminal paper Electron
and Gravitation (also in 1929) introduced the group SL(2,C)
into the physics of 2-component spinors (which Dirac demonstrated
are the basis of the relativistic theory of spin-1/2 particles).
Weyl’s interest in group theory likely reached its zenith in 1939,
when he published The Classical Groups: Their Invariants and
Representations.

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

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

Yang went on to win the 1957 Nobel Prize in Physics (with T.D. Lee)
for his work on parity violation in the weak interaction. But in
1980, he looked back on his days with Weyl at the IAS and wondered
why the two of them had not bothered to collaborate on gauge theory:

I had met Weyl when I went to the IAS as a young member. I saw
him from time to time in the next years, 1949-55. He was very
approachable, but I do not remember having discussed physics or
mathematics with him at any time. Neither Pauli nor Oppenheimer
ever mentioned it. I suspect they also did not tell Weyl of the
1954 paper of Mills’ and mine. Had they done that, or had Weyl
somehow come across our paper, I imagine that he would have been
pleased and excited, for we had put together two things that
were very close to his heart: gauge invariance and non-Abelian
Lie groups.

This is very strange indeed, and a terrible loss to physics, in my
opinion, because Weyl was only to live one more year after Yang’s
departure from the IAS in 1955. I would like to think that the
physics community’s recognition of the importance of Yang-Mills
theory would have occurred if Weyl had only been aware of it and
championed its cause.

A sad example of a missed opportunity. Weyl suffered a heart attack
and died unexpectedly on December 8, 1955. His last spoken word was
Ellen, his wife's name.

Information -- Posted by
wostraub on Monday, January 8 2007

A few nights ago the Discovery
Channel re-aired a one-hour documentary on the so-called information
paradox, which involves the question of whether or not
information (in bits or bytes or however you want to call it)
can be destroyed (or is eternal in some sense).

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

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

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

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

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

K = -∑ p log_{2} p = ∑ log_{2} (p^{-p})

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

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

M = e^{S}

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

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

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

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

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