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Who Was Hermann Weyl?
Wheeler's Tribute to Weyl (PDF)

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Weyl's Spinor and Dirac's Equation
Weyl's Conformal Tensor
A Note on Weyl Conformal Gravity
Weyl's 1918 Theory
Weyl's 1918 Theory Revisited
Weyl v. Schroedinger
Why Did Weyl's Theory Fail?
Weyl and the Aharonov-Bohm Effect
Weyl's Scale Factor
Weyl's Spin Connection
Weyl and Higgs Theory
Lorentz Transformation of Weyl Spinors
Riemannian Vectors in Weyl Space
Introduction to Quantum Field Theory
Electron Spin
Clebsch-Gordan Calculator
Bell's Inequality
The Four-Frequency of Light
There Must Be a Magnetic Field!
Kaluza-Klein Theory
A Brief Look at Gaussian Integrals

Kets, Bras and All That
General Relativity
God and Physics
God and Light

Particle Chart (Courtesy CPEP)

Einstein's 1931 Pasadena Home Today

Uncommon Valor

Sophie did not forget Jesus
Long live freedom!

226601 visits since 11/1/2004.

My work always tried to unite the
Truth with the Beautiful, but when I
had to choose one or the other, I usually chose the Beautiful.

Hermann Weyl

I died for Beauty, but was scarce
Adjusted in the tomb,
When one who died for Truth was lain
In an adjoining room

Emily Dickinson

Hermann Klaus Hugo Weyl (1885-1955). German mathematical physicist. In 1918, proposed an early form of gauge symmetry in an attempt to unify electrodynamics and gravitation. Subsequently applied a similar approach to quantum physics and discovered what is today considered one of the most profound and beautiful concepts in modern physics -- the principle of gauge invariance.

Shortly after Einstein announced his theory of general relativity (gravitation) in November 1915, Weyl began an intensive study of the theory's mathematics and was soon publishing related scientific papers dealing with its physical applications. In 1918 Weyl published his book Raum-Zeit-Materie (Space-Time-Matter), which provided the first fully comprehensive analysis of the geometric aspects of the theory and its relationship with spacetime physics. One of the topics covered in the book was Weyl's idea that gravity and electromagnetism might both be derivable from a generalization of Riemannian geometry, the mathematical basis from which Einstein had developed his relativity theory. Weyl's idea was based on a new mathematical symmetry that he called gauge invariance.

I came across Weyl's book in 1975, but it didn't impress me very much because I didn't know general relativity. However, in the summer of that year I stumbled across Misner-Thorne-Wheeler's massive Gravitation during a one-week work assignment in the microscopic rural town of Lone Pine, California (which then had a population of perhaps 500 people). Miraculously, the town's tiny public library somehow had this book, which is now regarded as a classic graduate text on general relativity. I checked out the book and brought it back to the hotel room to read in the off-hours. The book took immediate and total possession of me, and motivated me to learn everything I could about general relativity. (I spoke with co-author Kip Thorne about this in 1994, and he was quite amused to learn where one of his books had ended up.)

But Gravitation is not an easy read, and I had to look for more introductory texts. I soon came across Adler/Bazin/Schiffer's Introduction to General Relativity, which besides being easier had a chapter on unified field theory, including Weyl's 1918 theory of the combined gravitational-electromagnetic field. For whatever reason, the theory's mathematical beauty absolutely fascinated me. I had known about local and global phase invariance from my studies of quantum mechanics, but I was not aware that Weyl's theory was the origin of this powerful symmetry principle in quantum physics.

I have since read all of Weyl's books and many of his papers. Although today I believe that my interest is now based more on an appreciation of modern gauge theory (easily the most profound and beautiful concept of physics), I credit Weyl for having initiated the idea in 1918 and for his subsequent (1929) seminal application of the idea to the then still-developing quantum theory.

In his 2002 biographical memoirs, the great contemporary mathematician Sir Michael F. Atiyah praised Weyl as the discoverer of the gauge concept and as the driving force behind the current emphasis of gauge theories on modern theoretical physics:

The past 25 years have seen the rise of gauge theories--Kaluza-Klein models of high dimensions, string theories, and now M-theory, as physicists grapple with the challenge of combining all the basic forces of nature into one all embracing theory. This requires sophisticated mathematics involving Lie groups, manifolds, differential operators, all of which are part of Weyl's inheritance. There is no doubt that he would have been an enthusiastic supporter and admirer of this fusion of mathematics and physics. No other mathematician could claim to have initiated more of the theories that are now being explored. His vision has stood the test of time.

Weyl was an exceptionally gifted mathematician and physicist, but he was also a highly cultured man in the classical German tradition. He studied and wrote extensively on philosophy and was a serious student of German poetry and literature. His mathematical writing style could be exceedingly obtuse, but his other writings reveal a genuinely warm person who truly understood the human condition. Weyl was also very human himself; he could be overly thoughtful and cautious, often to the point of being unable to take action or make even basic decisions, and sometimes with the result that he became physically incapacitated. He was a devoted and loving husband and father, yet he could also be persuaded to stray, in accordance with the surprisingly liberal attitudes of post-World War I Weimar society.  

This website is my feeble attempt to document (and in many cases expand on) Weyl's ideas and thoughts on gauge symmetry in a manner that will be accessible to anyone with a basic understanding of calculus. Not a lot has been written about the original theory's underlying mathematics, and I wanted to provide a fairly detailed and complete mathematical description for those who want to learn about Weyl's ideas and to appreciate the beauty of his gauge theory (I'm even of the opinion that much of Weyl's work can be understood and appreciated at the high school/beginning university level). As this site progresses, I will also include discussions of other topics in mathematical physics (as well as some related scientific philosophy) which exhibit a similar mathematical beauty and elegance.

"Duck Soup, My Ass" — Posted Monda, April 21 2014

Speaking of lead poisoning, the latest episode of Cosmos tells the story of how Caltech scientist C.C. Patterson discovered the true age of the Earth (4.55 billion years) in 1956 by analyzing the relative concentrations of lead and uranium in zircon crystals. Patterson was also the guy who discovered the dangers of lead in the environment caused by the anti-knock gasoline additive tetraethyl lead. The discovery almost destroyed his career, since the chemical industry (primarily the Ethyl Corporation) was making billions with the additive. They tried hard to shut him up, but Patterson persisted, and by analyzing Arctic snow and ocean water samples he finally conviced the EPA (in 1973) to ban lead compounds in gasoline.

(Modern radiometric dating methods give the age of the Earth as 4.54 \(\pm\) .05 billion years, nearly identical to Patterson's finding.)

Besides showing some nice shots of Pasadena and Caltech, the Cosmos episode also relates how, for hundreds of years, the "official" age of the Earth was the one determined by Bishop James Ussher in the 1600s. By carefully studying all the "begats" in the genealogies of the Old and New Testaments, Ussher calculated that the Earth was created on Saturday night on 22 October, 4004 BC (and at 9 pm, by golly). Incredibly, there are still many Americans today, perhaps 20% of the population, who still believe the Earth is only about 6,000 years old. And that percentage is much higher on the Republican side of the House of Representatives.

An interesting story, and one showing that lead, which featured prominently in the fall of the ancient Roman Empire and the demise of the Franklin Expedition, can't be blamed for the insanity running amok in our country today.
America on a Sledge to Nowhere — Posted Monda, April 21 2014
After 34 years of working full time, my retirement in 2001 meant, to a great extent, having the time to read. There are many recent good and even great books out there today, and I'm having a hard time prioritizing, since I can't read 24 hours a day. But when an interesting (but very short) book comes out, it's easy to slip it in. The one I just finished is the revised and expanded version of the 228-page Frozen in Time: The Fate of the Franklin Expedition by Owen Beattie and John Geiger, originally published in 1989. Beattie, a noted professor of anthropology at the University of Alberta, supervised the 1982-84 investigations of the famed Franklin Expedition, a failed 1845 search for the Northwest Passage. Captaining the two British ships Terror and Erasmus, Royal Navy Officer Sir John Franklin initiated a planned three-year effort to discover a passable ocean route from the Atlantic to the Pacific oceans, thus opening up a more direct trade route to Asia. Admirably equipped and provisioned, the expedition was doomed by weather, sickness and disease, resulting in the deaths of all 129 of the expedition's officers and sailors.

When Franklin's ships failed to return as planned, numerous search and rescue parties were conducted to learn what had happened. The book describes many of those efforts in detail, some of which were as notable and tragic as the expedition itself. On the basis of what physical information could be gleaned from these searches (including a few sketchy written accounts by Franklin crew members), by 1930 it was believed that three successive cold summers had led to the expedition's becoming mired in ice shortly after arrival. The crew then underwent rapid debilitation via cold, sickness (especially tuberculosis and pneumonia) and starvation, and by 1849 all of the men were dead.

However, it was not until Beattie's 1984 exhumation of three marked graves found on Beechey Island that a serious contributing factor for the crew's demise was discovered. Assisted by a team of pathogists and other forensic experts, Beattie found high concentrations of the toxic metal lead in the nearly perfectly-preserved corpses of John Torrington (that's his blue-eyed countenance pictured above), William Braine and John Hartnell, who had all died and been buried in the frozen ground in early 1846. The team also found many hundreds of empty food tins that had once held canned meat, soup and vegetables. The tins had all been soldered using lead and lead alloys, and Beattie surmised that lead poisoning had also bedeviled the expedition, most likely from day one. Furthermore, Beattie noted that written accounts of the Franklin crew's ongoing battle with scurvy (even though ample supplies of lime juice were stored on the ship) also described the effects of lead poisoning, which often mimics the symptoms of other sicknesses.

But to me Beattie's account of the "boat place" was most revealing. Following the trail of what little physical evidence was still available in 1984 (pieces of wood, nails, tin cans, bone, etc.), Beattie's team came upon the scattered remains of from six to fourteen Franklin crew members and a lifeboat they had mounted on a sledge for land travel. The men had apparently struck out in a southerly direction, apart from the others, in a vain search for help. Inexplicably, the boat they dragged for some 120 miles contained a variety of heavy items they would have had no earthly use for — candlesticks, silverware, books, etc. — raising the very distinct possibility that the doomed men, incoherent and stupified by lead poisoning, were dragging the boat over land simply because they had gone insane. Surviving written eye witness reports, by local Inuits who lived in the area at the time, talk of a group of strangely-behaving white men dragging a boat near the shoreline. This account would seem to verify the story.

While precise details of the crew's demise will never be known, there is sufficient evidence (mostly bone cuts and other markings) to show that the crew had resorted to cannibalism to survive. Meanwhile, the book's photos of the exhumed corpses are both horrific and fascinating (I recall a National Geographic from the time detailing the investigation whose photos kept me from sleeping a night or two). But, being the obnoxious politically-motivated idiot that I am now, I could not help but see the picture of insane men uselessly dragging a dead weight as a metaphor for the state of my country today. For some time now I have believed that America's citizens have gone incrementally insane over the past few decades to the point where the country, like Franklin's mad crewmen, is indeed doomed.
Be The Friction That Stops the Machine — Posted Tuesday, April 15 2014

Some Things Never Change — Posted Tuesday, March 25 2014
Hermann Weyl and Paul Dirac both considered the possibility that the Newtonian gravitational constant G might be changing with time, as this would support theories they developed to explain why a certain large number ratio seems to appear again and again in Nature (for example, the ratio of the electromagnetic to gravitational force is about \(10^{40}\), as is the ratio of the radius of the observable universe to that of the electron). This idea became known as the Large Number Hypothesis, which assumed that G is getting slightly smaller over cosmic time periods.

The notion of a non-constant G has also been used by young-Earth creationists to argue that radioactive decay rates are getting smaller, which would (not really) explain why radiometric dating techniques appear to indicate a much older Earth than that indicated in the Bible.

But now new research has shown that, at least for the last 9 billion of the universe's 13.8-billion-year history, the gravitational constant has not varied more than (at most) one part in a billion. After an exhaustive study of 580 observed supernovae events, professor Jeremy Mould and his PhD student Syed Uddin at the Swinburne Centre for Astrophysics and Supercomputing and the ARC Centre of Excellence for All-Sky Astrophysics showed that the Newtonian constant G has not changed appreciably over cosmic time. Their research, which focused on Type 1a supernovae, demonstrated a constant G within an upper bound of \(\dot{G}/G\) of \(10^{-10}\).

The legitimacy of Type 1a supernovae studies was demonstrated in 1998 when three astrophysicists, Saul Perlmutter, Adam Riess and fellow Australian astronomer Brian Schmidt, used Type 1a supernovae to show that the universe is expanding at an accelerated rate, thus proving the existence of dark energy (see my post dated November 11, 2013 for more information). In 2011, the three scientists were awarded the Nobel Prize in phyics in recognition of their work.

A Type 1a supernova occurs when a white dwarf accretes sufficient matter from a companion star to attain critical mass. It then explodes, and the light given off by the explosion is used as a standard candle to measure the supernova's distance from the Earth. This critical mass is the same for all Type 1a events, and it depends on the value of G in the relativistic calculations used in its determination. Observations of billion-year-old Type 1a events demonstrated the constancy of G to high precision.

A less technical overview of the Swinburne study can be read here.

An absolutely fascinating discovery.
Black Friday — Posted Friday, March 21 2014
While I don't expect newscasters to be astrophysicists, I don't expect them to be complete idiots, either. CNN anchor Don Lemon, a nice enough guy, unfortunately represented the bulk of clueless humanity when he seriously asked a guest if the ill-fated Malaysian Flight 370 might have been the victim of a black hole. Also unfortunate was the fact that his guest 1) did not break out in hysterical laughter, and 2) reminded Mr. Lemon that a black hole would "suck up the entire universe." I guess she didn't know that the universe's countless black holes have somehow failed to do that just yet.

And while I understand that reality shows like Life After People and Strip the City might be entertaining in a fun if impossible sort of way, I'm waiting for someone at CNN to seriously suggest that if we have the technology to Drain the Ocean, then we should use that technology to find Flight 370 (and Captain Kidd's treasure). But then if Animal Planet and even National Geographic can seriously suggest that mermaids exist, perhaps humankind has already reached the end of its rope.
Primal Fear — Posted Friday, March 21 2014
Religion is the great salve that protects believers from death and the great unknown, right? If you believe that, then watch this short (2:42), award-winning video, Lights Out. And watch it ALONE, AND IN THE DARK.

(Duct tape helped get George W. Bush reelected. But it ain't gonna save you, lady.)

Sweet dreams.
C'est une croix qui de l'enfer nous garde. — Gounod, Faust
Lies My Teacher Told Me — Posted Wednesday, March 19 2014
This is Mr. Bernard York, my freshman year high school history teacher, who I believe defied the powers that be on numerous occasions when it came to telling the truth about American history. We used a textbook in which even I, in my extreme state of total naivete at age 14, could read between the lines. Something wasn't right — there was a lot of pro-America bullshit in those pages. And Mr. York echoed that sentiment, subtly if not overtly (this was the early 1960s, after all).

I vividly remember one glorious day in early June 1964 when Mr. York, no doubt as eager for the school year to end as his students, abandoned his lecture, wrote the scrambled letters of various Japanese cities on the blackboard, and asked for volunteers to unscramble them while he kicked back. One of them was "TOKYO." Mine was the only hand that went up, and I confidently strode to the blackboard and wrote "KYOTO". Round of applause from the students, "Well done!" from Mr. York, and I triumphantly took my seat. Truth be told, I was otherwise a total idiotic nobody in high school, with mediocre grades and an almost pathologically shy personality, and the next three years of school were traumatic and painful, socially awkward, and accompanied by a never-ending round of crushes on unsuspecting female students. In short, I hated high school. But I digress.

One of my all-time favorite books is James Loewen's Lies My Teacher Told Me: Everything Your American History Textbook Got Wrong, first published in 1995 with a second edition in 2007. I see it as almost a companion to Howard Zinn's far more acidic but all-too-accurate A People's History of the United States. Both books tell the real history of this country. And both books have garnered bimodal reviews on Amazon.com — you can easily tell the liberal lovers of the truth (however painful) from the neoconservative lovers of blatant falsehood ("Oh, why can't we just blindly love our country without thinking about all those evil things it did?")

I read the first edition when it came out and am now digesting the second, which includes the author's views on the Iraq War, which I personally view as one of the most heinous and criminal acts ever undertaken by this country. But the latest edition retains its devastating comments on the state of American history teaching, which even the most jaded reader will understand.
High school American history textbooks do not, of course, adopt or even hint at the American colossus [hegemonic] view. Unfortunately, they also omit the realpolitik approach. Instead, they take a strikingly different tack. They see our policies as part of a morality play in which the United States typically acts on behalf of human rights, democracy, and "the American way." When Americans have done wrong, according to this view, it has been because others misunderstood us, or perhaps because we misunderstood the situation. But always our motives were good. This approach might be called the "international good guy" view.
And that's the sanitized version. The rest of the book is a much more devastating overview of how America has deluded itself, not only in terms of history teaching, but in its views on its "noble" and "godly" role in world history itself.

Despite the intervening 50 since since I took his class, I can still recall Mr. York's take on "Manifest Destiny." It was not "God's will," he said, but a euphemism for something dark and destructive — the willful genocide of the indigenous Indian nations.
Improbable v. Impossible — Posted Tuesday, March 18 2014
Here's a highly improbable story, fictional but one that I absolutely guarantee will play out some day, and exactly as I've written it.

In the not too-distant future, computers will be able to calculate the transcendental number \(\pi\) out to previously unimaginable accuracy — to be precise, let's say \(10^{1,000,000}\) decimal places — in a matter of only a few weeks or so. Mathematicians, being a curious sort, will concurrently develop sophisticated programs to determine if there are any unusual or weird numerical patterns in \(\pi\), like the not too-unusual \(999999\) sequence around the 762nd decimal you see here:

Then, using the trivial code \(A=1, B=2, C=3\) etc., at some unbelievably-distant decimal place in \(\pi\) they discover the phrase
Upon publication of this discovery there is a momentous worldwide resurgence in religious belief, based on the apparent statistical impossibility of finding such a detailed, prescient message in the seemingly random infinite number \(\pi\). But upon further investigations at an even more distant decimal place, mathematicians find the message
Upon publication of this numerical discovery, the human race goes into a long period of self-reflection and doubt, not knowing what to think or believe anymore.

In 1985 I traveled to New Zealand on a two-week business trip. I caught a taxi at Auckland Airport and directed the driver to my hotel. It was a long drive, and along the way the cab driver and I struck up a conversation about various things, and at one point I asked him how long he'd been driving a taxi. He told me he only moonlighted as a cab driver to make ends meet, as the taxes in New Zealand, a socialist country, were pretty steep. He then told me that he worked as a water resources engineer at his day job. I informed him that I also worked in water resources, and was in New Zealand to inspect some water treatment equipment for possible purchase for my company. He replied that he had an appointment the next day to meet with a guy from Los Angeles regarding water quality issues and a potential sales contract. Upon exchanging names, we realized that he and I had been talking to each other by phone for the previous six months about water quality issues, the drought in nearby Melbourne, Australia, and life in general. We had a good laugh over this coincidence, which seemed pretty amazing at the time. But the next day in his office he introduced me to a consulting engineer for his firm, and I experienced another improbability: the consultant and I recognized each other immediately, as we had both majored in chemistry at California State University at Long Beach, graduating in 1971. He too had left chemistry and had gone into engineering.

I've experienced a number of coincidences like these, but they're insignificant compared to some of the stories that Imperial College mathematics professor David J. Hand relates in his new book The Improbability Principle: Why Coincidences, Miracles, and Rare Events Happen Every Day. He opens the book with an unbelievable incident that actor Anthony Hopkins experienced in London while preparing for a role. But Hand notes that, given the billions of human beings that have ever lived and the countless experiences they have shared, unbelievable incidents are simply par for the course, mathematically speaking.

In previous posts I've talked about science writer Michael Shermer and his thoughts on patternicity and agenticity. Humans are hardwired to see patterns in nature, Shermer notes, probably as a survival mechanism, even when there are no patterns. The tendency for humans to see unusual or improbable patterns and associate them with an agent (like God) is also probably hardwired into our brains. But, as Hand points out, given an infinity of alternatives and enormous spans of time for which these alternatives to play out in, and there is absolutely nothing unusual about highly improbable events.

On the most recent episode of Cosmos, astrophysicist and host Neil deGrasse Tyson talked about evolution and intelligent design. He raised a favorite topic of the IDers, namely the human eye and the notion that it's too complex to have come into existence via evolution — it simply had to have an Intelligent Designer behind its construction. But Tyson notes that innumerable numbers of ocean-borne molecules (simple and complex), coupled with billions of years of random interactions, made the eye and even life itself not only probable but a certainty.

We do not know if there's a Great Designer behind all things, and we'll likely not know until we die. In the meantime, I would suggest that we put our biases aside, along with our fears of death and the unknown, and try to put this universe and all its inconceivable wonders into a context that leaves us simply in awe, and not afraid.

PS: I can't remember who said "It's difficult to convince someone of a factual truth when their faith or their salary depends on their not believing it." That's what we mean by bias. Here's a cute cartoon demonstrating the concept:

Gravity Waves Detected? — Posted Monday, March 17 2014

Years ago it was noted that the observed cosmic microwave background (CMB) temperature of the universe appears to be far too uniform. If the Big Bang was literally the beginning of everything (including time and space), the expansion of the universe, however rapid it might have been, would be expected to produce some non-uniformity in the observed CMB temperature pattern (it averages only about 2.7 Kelvin, close to absolute zero). To explain this uniformity, Alan Guth proposed the theory of inflation, which conjectured that within the first \(10^{-35}\) second or so after the Big Bang the universe experienced a brief but extremely rapid expansion, so rapid that any non-uniformities would have been effectively smoothed out. Inflation has since become the leading theory of how the universe got to be so uniform, although direct experimental evidence has been lacking.

If inflation (and general relativity) is correct, then gravitational waves resulting from the Big Bang would have been produced copiously and with unimaginably small (and immeasurable) wavelengths, but within a short time inflation would have stretched the waves out to more reasonable size, making them (at least indirectly) observable. The results of the latest research appears to show that this is correct.

The preliminary papers are out on arXiv.org (links here). I haven't read them yet, but notables like Lawrence Krauss, Alan Guth and Andrei Linde are saying that if the data holds up this will be the biggest thing in cosmology in the last 30 years, and certain to earn at least one Nobel Prize.
Weyl as Art — Posted Thursday, March 13 2014
This is interesting — a beautiful interpretation of Hermann Weyl by artist Sarah Kaiser, which she created for the cover of the June 3, 2010 issue of Nature magazine.
Multiverse — Posted Saturday, March 8 2014
After reading an interview with Wesleyan University professor of religion Mary-Jane Rubenstein on the Religion Dispatches website, I picked up her new book Worlds Without End: The Many Lives of the Multiverse, in which she tries to come to terms with the commonalities and differences between science and religion as envisioned by her in the multiverse theory, which is currently enjoying a vogue with the lay public right now.

Imagine flipping a coin five times and getting heads with each flip. No big deal, you say, it's common enough. If you flipped it ten times and got heads on each flip, you'd probably think it was kind of remarkable, but still no big deal. However, if you flipped it twenty-five times and got heads each time, you'd likely think the coin was flawed, or rigged in some way, perhaps heavily weighted on the tails side. But you could still entertain the possibility that it was still only a statistical fluke, albeit a very unusual one. The odds of getting heads 25 times in a row on an honest coin are only about 0.000003%, but that's still far from zero. You say to yourself, hey, it could happen.

You now flip the coin \(10^{100}\) times (you couldn't live long enough to do it for real, but a computer might simulate the flips in a few million years or so). If the results were all heads, and you were absolutely positive that the coin (or the computer) was not flawed or rigged in any way, and you had absolutely eliminated the possibility of an outside physical force or other influence acting on the coin (electromagnetic induction, gusts of wind, losing your mind, etc.), then you would almost certainly ascribe the result to some supernatural cause or entity.

In a nutshell, that's the situation between religion and the multiverse theory today. A religious person would say that it's absolutely impossible for the universe to exhibit the multitude of precise and apparently inviolable physical laws and biological processes we observe by chance alone. But a physicist or mathematician would say that the odds, while inconceivably small, are not zero. And if you have a theory in which the number of possible universes is infinite, or if the amount of time available is essentially unlimited, then the universe we live in is not just a fluke, they say, but a statistical certainty. We just happen to be living in it. If we weren't, we wouldn't be wondering about all this. That is what Stephen Hawking meant when he claimed that the universe does not need God (which isn't the same as saying that God doesn't exist).

Noted science writer and author Michael Shermer (who lives a few blocks from me in nearby Altadena) has asserted the idea that humans are endowed with two attributes that together conspire to create and enforce religious or superstitious belief. The first is a survival instinct that he calls patternicity. Humans tend to see patterns in things, even when there are no actual patterns. To one person, a random collection of data points is just a meaningless scatter plot, while another person might see a definite or meaningful pattern or trend in the data. A more basic form of patternicity that humans developed long ago involves imagining dangerous animals (or rival tribe members) lurking in bushes and forests. If you see a nearby shrub shaking unexpectedly, most likely it's just the wind blowing the leaves around. But why take a chance? You run like hell! This instinct gave rise to the concept of false positives and false negatives — better to run from both if you want to survive. If it's just the wind, then all you've lost is a little energy. If it's a hungry Smilodon or cave bear, then you've improved your survival chances by hightailing it out of there. And it worked great for thousands, maybe even millions of years of human existence.

The second attribute is what Shermer calls agenticity, which is the tendency to ascribe supernatural intent or force to things we do not understand or are afraid of. This also provided a sense of comfort and security and control over the natural world to early humans, which for much of our existence was totally beyond control. Agenticity also provided humans with a sense of purpose and meaning; belief in supernatural forces gave rise to acknowledgment over their power and influence, which then led to ritualized worship, obedience and even sacrifice, actions that provided a measure of purpose in human affairs. Ritual practices also made supernatural belief more "believable," even sensible. That is why atheists, agnostics and "nones" even today are shunned, if not persecuted outright by believers — non-belief on anyone's part serves to induce doubt in the believers, and they don't like that. In addition, just the act of tolerance on the part of believers induces the fear that their god will be angry with them for not getting rid of the non-believers. The Old Testament books of Exodus, Numbers and Deuteronomy are classic examples of extreme religious intolerance, while today American fundamentalist Christians fear God's wrath if they do not expunge the country from the evil-doers in their midst.

But what the multiverse scientists cannot adequately explain is why physical and mathematical laws are so beautiful. And "beautiful" here is not simply a subjective description, but an undeniable attribute of our universe that chance alone would seem to have no business in creating. I suppose it's fair to say that if there is truly an infinite number of possible universes, then there is also an infinite number (or at least a large number) of worlds in which non-subjective "beautiful" physical laws and their underlying mathematical symmetries exist. But to date, I haven't been able to accept that.

If there's anything interesting in Rubenstein's book worth sharing, I'll come back to it.
Hidden Variables — Posted Thursday, March 6 2014
I'm laying hardwood flooring and, having removed all the carpeting, I carefully covered all the tack strips so I wouldn't step on them (those little nails are sharp). Well, I didn't step on any, at least until I took my shoes off for bed. I then immediately stepped on one I somehow forgot to cover. Now aching from a tetanus shot and a pierced foot (not to mention a sore back, because at 65 I'm too damned old to do this anymore), I have little to do but lay here and talk about gauge theory, one of my favorite topics.

The Weyl gauge in electrodynamics is an exceptionally simple prescription that gives the electric potential \(\Phi\) rather directly; in fact, it's just \(\Phi = 0\). Why this prescription is attributed to Weyl escapes me, but in certain situations it's obviously very handy. But even then, the electric field \(E\) itself remains largely undetermined.

It has always amazed me that electric and magnetic fields are so easily detected and measured (an EMF meter, which is actually a kind of antenna that detects and quantifies deflections, like the jump of the needle in a voltmeter, is one such device), while the underlying four-potential \(A^\mu = (\Phi, \vec{A})\) that \(E\) and \(B\) are made from is essentially undetectable and immeasurable. There is no device that can tell us in a straightforward manner that a "bare" electric potential \(\Phi\) is nearby, or that a non-zero vector potential \(\vec{A}\) is lurking about. And, for that matter, no device can measure their intensity. The reason for this has to do with the gauge freedom of the four-potential. As is well-known, Maxwell's equation are unchanged under the pair of gauge transformations $$ \vec{A} \rightarrow \vec{A} - \vec{\nabla} \lambda, \quad \Phi \rightarrow \Phi + \frac{1}{c} \frac{\partial \lambda}{\partial t}, $$ where \(\lambda(x,t)\) is a completely arbitrary scalar function of space and time. Consequently, there is no such thing as the four-potential \(A^\mu\) because \(\lambda\) can have any value.

Nevertheless, the gauge parameter \(\lambda\) can be specified in such a way that makes Maxwell''s equations easier to solve. (In fact, the potentials were originally viewed as just a mathematical convenience with no physical validity, while gauge invariance was seen as a mere happenstance of the formalism.) Recall that the electric and magnetic fields can be expressed in closed form by considering the homogeneous set of Maxwell's equations $$ \vec{\nabla} \times \vec{E} + \frac{1}{c} \frac{\partial \vec{B}}{\partial t} = 0, \quad \vec{\nabla} \cdot \vec{B} = 0, $$ which can be solved using simple vector identities to give $$ \vec{E} = -\vec{\nabla} \Phi - \frac{1}{c} \frac{\partial \vec{B}}{\partial t}, \quad \vec{B} = \vec{\nabla} \times \vec{A} $$ As is easily shown, these physical quantities do not change under the transformations given above. But what about the inhomogeneous Maxwell's equations, which specify the sources? They are $$ \vec{\nabla} \cdot \vec{E} = 4\pi \rho, \quad \vec{\nabla} \times \vec{B} - \frac{1}{c} \frac{\partial \vec{E}}{\partial t} = 4\pi \vec{j} $$ Using the identities for \(E\) and \(B\) above, these go over to $$ \nabla^2 \Phi + \frac{1}{c} \frac{\partial (\vec{\nabla}\cdot\vec{A})}{\partial t} = -4\pi \rho $$ and $$ \nabla^2 \vec{A} - \frac{1}{c^2} \frac{\partial^2 \vec{A}}{\partial t^2} - \vec{\nabla} \left( \frac{1}{c}\frac{\partial \Phi}{\partial t} + \vec{\nabla}\cdot\vec{A} \right) = - 4\pi \vec{j} $$ As many have noted, these last two expressions are ugly as hell, not to mention the fact that they're inextricably coupled in \(\Phi\) and \(\vec{A}\). But they are both invariant with regard to a gauge transformation, and we can use that fact to simplify them.

Consider the scalar quantity $$ S = \frac{1}{c}\frac{\partial \Phi}{\partial t} + \vec{\nabla}\cdot\vec{A} $$ A gauge transformation then gives $$ S \rightarrow S + \frac{1}{c^2} \frac{\partial ^2 \lambda}{\partial t^2} - \nabla^2 \lambda $$ Thus, \(S\) can be made gauge invariant if the gauge parameter satisfies the wave equation of light, which is $$ \Box^2 \lambda = \frac{1}{c^2} \frac{\partial ^2 \lambda}{\partial t^2} - \nabla^2 \lambda = 0 $$ Selecting a gauge parameter \(\lambda\) whose d'Alembertian \(\Box^2 \lambda\) vanishes (which we can always do, since it's arbitrary) thus turns \(S\) into an arbitrary gauge invariant scalar, and for simplicity we may as well set it to zero: $$ \frac{1}{c}\frac{\partial \Phi}{\partial t} + \vec{\nabla}\cdot\vec{A} = \partial_\mu A^\mu = 0 $$ This is called the Lorenz gauge, and its primary value is that it uncouples the above equations in \(\Phi\) and \(\vec{A}\) to give the beautifully symmetric expressions $$ \frac{1}{c^2} \frac{\partial^2 \Phi}{\partial t^2} - \nabla^2 \Phi = 4\pi\rho, \quad \frac{1}{c^2} \frac{\partial^2 \vec{A}}{\partial t^2} - \nabla^2 \vec{A} = 4\pi \vec{j} $$ or, in lovely covariant language, $$ \Box^2 A^\mu = 4\pi j^\mu $$ where \(j^\mu = (\rho, \vec{j})\). In principle, if you can solve one of these equations, then you can solve the other using the same approach.

I've always had problems with the Lorenz gauge. For one thing, I always confused Ludwig Lorenz with the far more famous Hendrik Lorentz of Lorentz contraction fame. They're not the same guy. For another, setting \(S = 0\) requires that the gauge parameter \(\lambda\) satisfy the wave equation, but the opposite is not necessarily true. The only sure way of justifying the Lorenz gauge is by appealing to a rather nasty theorem in vector calculus called Helmholtz's theorem, which states that any well-behaved vector function can always be expressed as the sum of a divergence and a curl. For the vector \(\vec{A}\), the curl is already specified in terms of the magnetic field via \(\vec{B} = \vec{\nabla} \times \vec{A}\). But the divergence of \(\vec{A}\) is undetermined, so we can select any arbitrary value for it. That's what I was taught, but I still don't get it. I know it should have something to do with gauge transformations, but I'll be darned if I know how.

At any rate, to me the four-potential is something like God — it never makes its existence known, and is a total and profound mystery, yet it's somehow there, and can be deduced mathematically and by physical reasoning. I cover some of these thoughts in my elementary write-up on the Aharonov-Bohm effect, which explains how the theoretical existence of the four-potential was finally demonstrated by a very clever (and beautiful) quantum-mechanical experiment.

By the way, noted UC Berkeley physics professor J.D. Jackson has written a lengthy paper detailing many useful types of gauge transformations. But be warned — Jackson is also the author of many a grad student's greatest nightmare, the seemingly impenetrable textbook Classical Electrodynamics.
Quote of the Week — Posted Monday, March 3 2014
US Secretary of State John Kerry, on Russian warmongering in Ukraine:
"You just don't invade another country on phony pretext in order to assert your interests."
And that's from 2004 Democratic presidential candidate John Kerry. I can only wonder if he was intentionally including a "dog whistle" message in that statement, or if he's even aware of how monstrously hypocritical it sounds.

And speaking of that erstwhile "Empire of Evil" or "Axis of Evil" country, I can remember George W. Bush (easily the stupidest and most insanely corrupt President we ever spawned) talking about Vladimir Putin not that long ago, when Bush "looked into his eyes and saw his soul." Do you remember that? Well, it seems the country's conservatives don't. But they're all over President Obama for being a total wuss for not getting tough with Putin.

Fratboy George W. Bush in happier, cockier days. He's still an asshole.

Yeah, let's shoot off those nuclear-tipped ICBMs and get it the fuck over with.
Schrödinger Again — Posted Saturday, February 22 2014
Most people know Erwin Schrödinger as the father of wave mechanics and the co-recipient (with Dirac) of the 1933 Nobel Prize in physics. But he was also interested in numerous other scientific areas, including biology, genetics, general relativity and color measurement (he was also a noted womanizer, but that ain't scientific). In the 1940s his interests turned to a fundamental topic in differential geometry, that of affine connections.

I just posted an online paper (suitable for undergraduates) concerning one particularly simple connection that Schrödinger presented in his short but illuminating 1950 book Space-Time Structure. I bought the book back around 1978 and still turn to it on occasion.

One warning — I kind of dump on Hermann Weyl in this paper, as I believe Schrödinger's connection makes more sense than Weyl's. But whatever.
Einstein's Cake — Posted Monday, February 17 2014
I'm getting a renewed interest in (and even appreciation of) the latter work of Erwin Schrödinger in what he referred somewhat extravagantly to as "the final laws" of gravity and electromagnetism, which he developed in the years immediately following the end of World War II. This work in many ways paralleled that of Einstein, whose interest in a unified theory of gravitation and electromagnetism continued unabated from around 1925 until his death in 1955.

By 1939 Schrödinger had moved to Dublin, Ireland from his native Austria, following a long bout of political persecution from Nazi Germany, which had annexed Austria two years prior to the war. (He was not a Jew, but his progressive ideas were nevertheless annoying to the Germans. But as a co-recipient of the 1933 Nobel Prize in physics his fame fortunately outweighed his infamy in Nazi eyes, so his life was never in danger.)

Schrödinger helped establish the Institute for Advanced Study in Dublin and became a naturalized citizen there in 1948. Following his retirement in 1955 he moved back to Austria, where he died in 1961. During his years in Ireland he wrote numerous papers on unified field theory (while simultaneously siring several illegitimate children with two Irish women), one series of which was titled The Final Affine Field Laws.

Like Einstein, Schrödinger had decided that the symmetry of the affine connection \(\Gamma_{\mu\nu}^\lambda\) in the lower two indices should be abandoned in order to derive a workable theory. This idea had been considered by many physicists, even as far back as 1918, but it introduces many problems. Any asymmetry in the connection of course goes unnoticed in the equations of the geodesics $$ \frac{d^2x^\lambda}{ds^2} + \Gamma^\lambda_{\mu\nu} \frac{dx^\mu}{ds} \frac{dx^\nu}{ds} = 0, $$ but quantities like the Riemann-Christoffel and Ricci tensors become unwieldy, and the field equations associated with these more general tensors don't seem to produce any useful physics. Furthermore, the mathematical notation itself is messy, as one must continually work to keep the symmetric and asymmetric pieces distinct from one another throughout the derivations. In my humble opinion, it's all an exercise in futility, and I suspect Einstein and Schrödinger both feared this was indeed the case.

But there are many physicists today who remain undaunted by these difficulties. Notable is physics professor Nikodem Poplawksi of the University of New Haven in Connecticut (previously with Indiana University), who has authored many papers involving general affine and asymmetric connections. His work (some of which has been featured on television programs) ranges from conventional to truly interesting, even profound to crackpot. But I don't think there are many who have investigated the connection and its possible relation to gravitation and electromagnetism as much as he has. His many papers, most of which are available on arXiv.org, have the benefit of being accessible to the motivated undergraduate, and I encourage the interested student to look into the ideas of this young New Haven professor.

Near the end of his life, when he had completed his failed work on unified field theory, Einstein's personal secretary Helen Dukas had a cake baked for the great scientist in honor of his questionable "achievement." The cake was decorated with the field equations themselves, in red icing. I have a neat photo of the cake laying around here somewhere on my hard drive, and if I can locate the damned thing I'll post it here. In the meantime, you can pop over to the jstor.org academic publishing website where you can read Schrödinger's own papers (in English) on the final affine laws yourself (requires a free subscription).
And the Survey SAYS — Posted Sunday, February 16 2014
In Arthur Conan Doyle's A Study in Scarlet, Sherlock Holmes admits to an astonished John Watson that he is not aware that the Earth orbits the Sun, nor does he care:
"What the deuce is it to me?” he interrupted impatiently; “you say that we go round the Sun. If we went round the Moon it would not make a pennyworth of difference to me or to my work."
By way of explanation, Holmes reveals that he is careful not to commit such "useless" information to his brain, lest it interfere with more important things, like the science of deduction.

Doyle's novel was written in England in 1886, yet even then it would have been impossible to find someone who thought that the Sun revolves around the Earth. But here in 2014 America such a finding would not be unusual at all. A recent survey conducted by the National Science Foundation showed that 26% of Americans actually believe just that. Similarly, only 39% of Americans believe in the Big Bang, and only 48% express a belief in evolution. And half of Americans believe that antibiotics are effective against viruses. Not surprisingly, Europeans and Asians fared much better in the survey.

Any astrophysicist will tell you that the Earth and Sun actually revolve about a common center of mass located very close to the Sun's core. I'll bet nearly 100% of Americans polled would not know that fact, but that is of course quite excusable. But to have 26% of Americans think that the Sun goes round the Earth confirms my theory that a quarter of us are certifiably insane. And they're called Republicans.
Making the Necessary Adjustments — Posted Saturday, February 15 2014
The first-ever winner of the Hermann Weyl Mathematics Prize (2002), Edward Frenkel is a Russian-born professor of mathematics at UC Berkeley whose short NY Times article this Sunday touches on the subjectivity and objectivity of mathematics. Like Einstein's colleague Kurt Gödel, Frenkel asks whether mathematics simply "is," and is therefore subject only to discovery and analysis by humans, or if it's purely an invention, in which case it is subjective to some extent.

Frenkel also addresses a favorite topic of mine, which is the question of what the ultimate reality might be. His article references a recent paper by physicists Silas Beane, Zohreh Davoudi and Martin Savage, which considers the possibility that our universe is actually a computer simulation (I posted the URL for this article two years ago, but you can also link to it from Frenkel's article). Frenkel suggests that if we are indeed living in a simulation then our mathematics might not be inherently absolute but simply a kind of artificial version handed down to us from our simulators, who decided early on that \(1+1 = 2\) and not \(3\), for example.

The Beane et al. paper actually addresses the possibility that the proposed computer simulators' mathematics is not an invention at all, but a fixed logic like ours from which their simulation is based. What makes the paper interesting is the authors' supposition that, either due to oversight or the limitations of their technology, the simulation is "flawed" at some level, making it possible for us otherwise unwitting humans to discover that we've been had. I once suggested that far more powerful Large Hadron Collider-like machines might someday reveal such flaws — for example, we may not discover a wealth of new physics, particles and forces at all, but a barren desert representing the "pixel" limits of the simulators' impressive but ultimately constrained technology.

For some reason, the article reminded me of an Amazing Stories episode from long ago (or something like it*), which featured a man who suddenly realizes that nearly everything he knows is wrong. In the end (if I remember it correctly) he has to be re-educated by his 4-year-old daughter, who reads to him from a child's reading primer. She shows him a picture of a cake, which is labeled "dinosaur" in the book. Needless to say, the man realizes he has a lot to learn (or relearn).

If we are ever granted access to the true reality behind our existence, I wonder if it will be like that. But perhaps it will be like this:

From The Thirteenth Floor (1999). Simulant (Vincent D'Onofrio) meets simulator (Craig Bierko) with unpleasant results.

* A friend has since informed me that the episode was "Wordplay", from the newer (1985) Twilight Zone series.
Schrödinger on Weyl — Posted Friday, February 7 2014
In 1922, Schrödinger submitted a paper to Zeitschrift für Physik that apparently represents the first attempt to tie Hermann Weyl's 1918 gauge theory to quantum mechanics. Schrödinger's On a Remarkable Property of the Quantum-Orbits of a Single Electron (Zeit. f. Phys. 12 1922, 13) notes that Weyl's proposed metric $$ \hat{g}_{\mu\nu} = e^{-k\int \phi_\mu dx^\mu} g_{\mu\nu} $$ explains the energy spectrum of an electron in the hydrogen atom if the term in the exponential is an integral multiple of \(ie/\hbar c\). Schrödinger adds that
It is difficult to believe that this result is merely an accidental mathematical consequence of the quantum conditions, and has no deeper physical meaning.
At the same time, he is hesistant (or unable) to expound on the role that Weyl's theory might actually play in quantum theory, which was then still in its infancy. Perhaps Schrödinger's observation that the factor was pure imaginary bothered him, since Einstein's and Weyl's theories were, after all, classical theories.

Two pages from Schrödinger's notebook (mid-1925) — the genesis of his wave function concept

Note that Schrödinger wrote this paper three years before his own seminal announcement of the wave equation, which for the first time fully explained the strange quantum behavior of hydrogenic electrons that Bohr originally reported on in 1913. Indeed, the then still-emerging quantum theory had not advanced appreciably beyond Bohr's work, and the paper demonstrates the kind of brilliant thinking that was to characterize Schrödinger's contributions to physics, which were finally rewarded when he shared the 1933 Nobel Prize with Dirac.

One must also remember that Weyl was essentially trying to eliminate the subjective concept of scale in his theory. This has since led to conformal (scale- or length-independent) cosmological theories, which may or may not have anything to do with the problems of dark matter and dark energy. Certainly, when the Universe was born out of the Big Bang, the concept of scale or length had little if any physical meaning, since spacetime "outside" the Big Bang did not even exist!

I have been unable to locate an English translation of Schrödinger's paper and, loath as I am to translate the entire (rather lengthy) article from my original German copy, am presenting here the abbreviated version reproduced in the late Lochlainn O'Raifeartaigh's indispensable 1997 book The Dawning of Gauge Theory.

The Alpbach, Austria graves of Erwin and Anny Schrödinger, with perhaps
the most profound physics equation of all time (and yes, it beats \(E=mc^2\))  

Ham on Nye — Posted Wednesday, February 5 2014

This graphic is wrong. The Ark instead had room for all 9,000,000 of Earth's species. (Yeah, right.)

Paris, France has the Musée de l'Homme. Petersburg, Kentucky has the Creation Museum.

I had to force myself to watch last night's Evolution v. Creationism debate between Bill Nye and Ken Ham (CEO of Answers in Genesis), mainly because I believed that the pro-science Nye never had a chance from the get-go. The debate was televised live from Ham's Creation Museum, that palace of idiocy where full-sized models of dinosaurs can be seen equpped with saddles, being joyfully ridden by the contemporaries of Adam and Eve. How Nye could even set foot in the place is beyond me.

Ham preaches that before the Fall of Man™ all dinosaurs were happy, herbivorous and friendly, the teeth of Spinosaurus and T. rex being huge, serrated and sharp only because they needed them to crack open the tasty coconuts they relished. Mosquitoes, chiggers and ticks all sucked tree sap, while giant snakes stalked juicy pomegranates (and the occasional apple). According to Ham, it was only after Adam and Eve were kicked out of the Garden of Eden that these normally docile and tame reptiles became blood-thirsty monsters (because, you see, sin had entered the world). And, adds Ham, this happened only about 6,000 years ago, a short time after God created the Universe.

That a man as sensible as Bill Nye the Science Guy would ever consent to "debate" this kind of insanity staggers the imagination. And so, as could be expected, Nye's appeals to reason, evidence and fact were no match for Ham's biblical dogma, which requires no reasoning, evidence, facts or intelligence whatsoever. At times Nye appeared to be debating with a piece of furniture.

Ham was of course the winner, because evidentialism stands no chance against blind faith and stupidity:
Can God create a prime number between 7 and 11? Of course, God can do anything!
Can God make a new element between sodium and magnesium in the periodic table? Sure!
Can God do all this and more without even existing? Yes, because He's all powerful! Er ...
The only reason that idiots like Ken Ham can continue to delude and fleece the stupid sheep who believe his nonsense is because America remains a largely science-illiterate country. This is especially true in Kentucky, where the State Motto is "I'm With Stupid."

Go to, I'll no more on't, it hath made me mad. — Hamlet

100 Years of Charlie Chaplin — Posted Sunday, February 2 2014
One hundred years ago today, 24-year-old Charles Spencer Chaplin, already a veteran of the English stage, made his film debut in 1914's Making a Living. For us today it's a forgettable, grainy Mack Sennett one-reeler starring a character looking nothing like Chaplin's familiar little tramp, but on this date in 1914 silent film audiences were literally rolling in the aisles.

A rare color autochrome photo of Chaplin, taken in 1918

I have an extensive collection of silent films, but relatively few featuring Chaplin. While I admit that the short films he made in his Essanay, Mutual and First National periods were inventive, I never found them very funny. His later feature films, silent and otherwise, were much better, but I like them because their combination of humor, pathos, politics and cultural satire is much more interesting. In my opinion, Chaplin's City Lights (1931) is his best work. It premiered on January 30, 1931 at the new Los Angeles Theatre at 615 S. Broadway in downtown Los Angeles, and Chaplin's personal guest was none other than Albert Einstein.

The heartbreaking final scene with The Tramp and the blind flower girl
(Virginia Cherrill), who finally realizes who her benefactor really was

Chaplin's leftist political leanings ultimately resulted in his expulsion from America. In 1942, at age 53, he married writer Eugene O'Neill's 18-year-old daughter Oona and retired to Switzerland.
Decay Rates and Religion — Posted Wednesday, January 29 2014
A religious friend of mine recently wrote to me regarding an episode of Through the Wormhole, a popular Discovery Channel series hosted by Morgan Freeman. The episode featured an apparent correlation between solar flare activity and the observed decay rate of certain radioactive elements. My friend (who is fairly science literate) asked me what I thought about this, and whether it meant that radioactive decay rates could have been much higher in the early Solar System. She's not a young-Earth creationist (YEC), but thought that maybe this information might be useful to YECs to justify a 6,000-year-old Earth, in agreement with the Old Testament's version of creation.

This is a very old story that has been around for a number of years. I haven't watched it, but the particular Wormhole episode can be viewed for free here. It deals with the work of Purdue University's Jere Jenkins and Ephraim Fischbach, who in 2006 reported an apparent effect of solar flares on the decay rate of an unstable manganese isotope. Solar flares pump out enormous quantities of electrons, protons and x-rays but the flux is generally thought to be insufficient to promote electron capture of elements here on Earth, so decay rates should not be appreciably effected.

It should be noted that many more experimental studies have reported no effect. A recent study (2013) was conducted by E. Bellottia and colleagues at the University of Milan, who investigated the possible effect of large solar flares on radioactive cesium and other elements. They reported no statistically significant change in the decay rates. A cursory review of related studies posted on arXiv.org and elsewhere report similar null results.

Decay rate residuals from Bellottia et al. Looks pretty solid to me.

Recall that three years ago researchers in Italy reported that they had observed superluminal neutrinos streaming into their detector arising from proton beam studies at CERN. This gave conservative relativity-haters here in the United States hope that Einstein's theory of relativity is wrong, and that God's magic action-at-a-distance process is in fact what's really going on. As is well known, the researchers subsequently discovered that a faulty data cable had skewed the results, and so the neutrino is back to its old sub-light speed again.

(By the way, the cited Conservapedia article actually asserts that President Obama has used relativity theory to justify abortion!)

But it gets better. The good folks at Conservapedia are now busy rewriting the Bible. It seems that Jesus was a tad bit too easy on the poor and downtrodden for their tastes, and they want to portray Christ as the free-market, entrepreneurial wheeler-dealer He really was. One example: the woman taken in adultery in John 7:53-8:11, whom Jesus forgave. Well, she's outta here. Another example: the Old Testament Hebrew writers really meant bethulah (virgin) and not almah (young woman), so the later Greek mistranslation parthenos (virgin) is correct after all, and Isaiah 7:14 really is a magical prophecy. Yeah, what the hell did those Hebrews know, anyway!
On why English should be the official language of Texas: "If English was good enough for Jesus Christ, it ought to be good enough for the children of Texas!" — Miriam A. Ferguson
Mathematical Justice? — Posted Wednesday, January 29 2014
Here's an interesting paper from last year by Alexander Afriat of the Université de Bretagne Occidentale entitled How Weyl Stumbled Across Electricity While Pursuing Mathematical Justice (see also this paper). The "justice" that Afriat talks about has to do with the equality of direction and distance in Riemannian geometry — vectors can have any direction they want, but their lengths are required to be fixed. This is in direct conflict with quantum mechanics, where the direction of a state vector \(|\psi\rangle\) can be anything, likewise its length; multiplying the state vector by any real or complex number doesn't change the vector at all — the length of the vector is in fact essentially meaningless.

But I think Afriat has confused mathematical justice with mathematical symmetry, which is what I believe Weyl was actually interested in. All of our physics appears to arise from mathematical symmetry, which is essentially the invariance of our theories with respect to coordinate change, linear and rotational translation, time translation and quantum-mechanical gauge or phase translation. These symmetries also give us the conservation laws, like those for energy, linear and angular momentum and electrical charge. To me, these symmetries are the most sublime and beautiful evidence we have that there is a Great Intelligence behind everything. And what is that Great Intelligence, you ask? I haven't the faintest idea.

At any rate, Afriat's paper provides lots of neat quotes from Weyl that appear to support the contention that Weyl was indeed on a kind of philosophical or spiritual quest for the truth. I just wouldn't call it "justice."
Hawking on Black Holes and the Weather — Posted Saturday, January 25 2014
Is there no end to mischief from this man? Just kidding.

Today's New Scientist has an article on a new theory Stephen Hawking has proposed regarding the fate of information during black hole evaporation. Quantum unitarity, a cornerstone of quantum theory, says that any information falling into a black hole cannot be lost, nor can it be lost when the hole evaporates to nothing via Hawking radiation. Somehow, the information is still there.

General relativity says that a person falling into a black hole would not notice anything unusual as she passes the event horizon (assuming the hole is massive enough to prevent her spagettification); she'd go right through and on toward the singularity, where God-knows-what happens to her (but probably instant death). Because of time dilation, observers on the outside of the hole would see her apparently slowing to a stop just above the horizon, while light reflecting off her would be red-shifted to the point that she'd become invisible. I'd say that these two points of view qualify as the most extreme anyone could ever imagine — kind of like Schrödinger's alive-and-dead cat.

Anyway, a lot of phyicists are unhappy with the information-preserving aspects of black hole unitarity, and have suggested that an almost literal firewall exists at the event horizon that would destroy anyone falling in. The outside observer might still see red-shifted radiation, but the poor woman would now know she's burning up. Hawking has proposed a kind of happy medium in which the event horizon is sort of virtual; and while some information loss occurs, it's akin to the loss of weather information, which may be accurate today but fades into chaotic gibberish after a few days in terms of predictability.

The noted Stanford physicist Leonard Susskind once tried to demonstrate information conservation using a sink full of water, into which he let red ink fall from an eyedropper with a Morse-code sequence of drops. After an hour or so, the ink diffuses into the water, producing a uniform pink tint. But, Susskind asserts, the Morse code is still there, and if one knew the precise position and motion of all the water molecules and the details of the ink drops from the start, the associated information could in principle be recovered.

I'm not sure that's entirely true. What I do know is that nothing's dropping out of the sky here in perpetually sunny California, which is currently in the worst drought it has ever experienced.
Physics as Religion — Posted Tuesday, January 21 2014
The journal New Scientist has an article about how scientists' ideas about the multiverse might be crossing over into fantasy and even religion, given the fact that many if not most of these ideas can never be proved. The article concentrates on the recent book by noted MIT physicist Max Tegmark entitled Our Mathematical Universe, in which the author posits the possibility of a heirarchy of multiverses that goes from actual parts of space too far away to ever be seen to an infinite number of hypothetical mathematical universes that can exist only in the imagination.

I bought Tegmark's book last week and am still reading it. So far I like it a lot, but haven't decided if his "Level 4" multiverse is any different from a purely religious notion or mere wishful thinking. In a way, it's akin to believing that Schrödinger's wave equation has an actual physical existence, and is not just a useful mathematical tool that describes the physical world. This is not inherently different from merely believing in an infinite, complex Hilbert space that holds all possible outcomes from a given experiment to believing a Hilbert space that is an actual physical thing, an object that one might bump one's head onto while exploring the universe.

Dirac once remarked that if God exists, then He's a mathematician. I agree, but that's much different from saying that God is mathematics, which is kind of what Tegmark is implying.

Still, Tegmark might be onto something. I am reminded of a remark that Nobel physicist Eugene Wigner made around 1963, in which he noted the "unreasonable effectiveness" of mathematics in describing the world as it is. I am also reminded of fellow Nobelist Murray Gell-Mann's statement that in quantum physics "Whatever is not expressly forbidden is compulsory." Both of these remarks have proven to be true. But it's a huge leap to go from there to believing that the 11 dimensions of M-theory and the infinite number of dimensions in Tegmark's ideas are somehow reasonable, if not out and out real.

But as the New Scientist article implies, it is misleading to believe that just because something can be imagined it might actually exist or be true. Famed mathematician and philosopher Bertrand Russell's orbiting teapot is a good example. He fancifully imagined a teapot orbiting the Sun between the orbits of Earth and Mars which, because of its small size, could never be observed by any telescope. Russell also sadly noted that, for many people, the hypothetical existence of something like an invisible orbiting teapot becomes actual fact simply because its non-existence cannot be proved. And if belief in an invisible orbiting teapot provides some kind of comfort to people, then it becomes religion. To me, what is, is, and what is not, is not. It is the business of science to find out what is, to offer explanatons as to how it comes about, and to test those explanations. But while Tegmark's ideas are a little too close to religion in my book, it's still fun to imagine.
Calling George Jetson — Posted Sunday, January 5 2014
I've lost count of how many times I saw George Pal's 1960 film The Time Machine as a kid at the now-defunct Lyric Theatre in Monrovia, California, but the movie still intriques me. And how many times I wished I had the model Rod Taylor sent off into the future!

On occasion some brave academic or two will try to address the issue of time travel to the past in a serious way. Caltech physicist Kip Thorne notably touched on the matter but in general he advised his doctoral students to avoid it, lest it damage their career potential. But R.J. Nemiroff and T. Wilson of Michigan Technological University figured that, if they can't build a time machine themselves, they would at least try to catch time travelers from the future in the act.

In their 11-page Searching the Internet for evidence of time travelers, Nemiroff and Wilson describe an innovative and clever way of ferreting out visitors from the future by mining the Internet for tell-tale clues of their visitations. You see, time travelers are a shy lot, and they invariably cover their tracks, perhaps out of fear that they might change history and destroy their own future existence. That, or they simply don't exist.

But the authors wisely cover their respective academic rear ends, as they note the various logical and physical improbabilities (and impossibilities) associated with backward time travel. Focusing on the keywords "comet_ison" and "pope_francis" (terms that did not exist in the authors' 2006-2012 search window), their detailed Internet seach, which included investigations of Face Book and Twitter sites, sadly did not turn up anything interesting. Time travelers are either pretty damned wary or, as I suggested, they don't exist.

Probably the best and most comprehensive resource on time travel is the 1993 book Time Machines: Time Travel in Physics, Metaphysics and Science Fiction by electrical engineer and University of New Hampshire professor Paul J. Nahin. I still turn to it whenever a time-travel idea pops into my head, and I invariably discover that Nahin has already covered it.

An endlessly fascinating subject.
Forever Mine, or More Rambling Thoughts — Posted Sunday, January 5 2014
In Siberia, archaeologists have discovered two Middle Bronze Age skeletons, male and female, apparently embracing one another and holding hands. The 3,500-year-old find may indicate the prevalence of nuclear-family sentiments, a cultural distinction common today but previously considered doubtful for couples (husband and wife, lovers) in ancient times.

Fossilized remains of individual Neanderthal graves dating back more than 50,000 years have yielded evidence of flowers, beads and other materials associated with deliberate funereal practices, but never couples.

More skeletons: dramatized film versions of Victor Hugo's The Hunchback of Notre Dame typically have Quasimodo winning Esmerelda, or pitifully watching her ride off happily with her lover, Pierre Gringoire, but the novel has a much different ending. True, Claude Frollo gets tossed off the cathedral tower by an enraged Quasimodo, but he is too late to save Esmerelda, who is hanged in the courtyard below. Griefstricken, the hunchback simply runs off and disappears from the story. Except for years later, at the very end of the novel, where Hugo takes the reader on a grisly tour of an ancient underground burial vault in Paris:
[A]mong these hideous carcasses were found two skeletons in a singular posture. One of these skeletons, which was that of a woman, had still upon it some fragments of a dress that had once been white; and about the neck was a necklace of seeds of adrezarach, and a little silk bag braided with green beads, which was open and empty. These things were of so little value that the hangman no doubt had not thought it worth his while to take them. The other, by which this first was closely embraced, was the skeleton of a man. It was remarked that the spine was crooked, the head depressed between the shoulders, and one leg shorter than the other. There was however no rupture of the vertebrae of the neck, and it was evident that the person to whom it belonged had not been hanged. He must have come hither and died in the place. When those who found this skeleton attempted to disengage it from that which it held in its grasp, it crumbled to dust.
[The 1939 film with Charles Laughton as Quasimodo is by far the best version. The music was composed by Alfred Newman, whose score was nominated for an Oscar by the Academy of Motion Picture Arts and Sciences (he eventually won a total of nine Oscars for his film work). Here is the heartbreaking but rejoiceful final track from the film, which I cannot listen to without tearing up.]

Alfred Newman, Love Theme/Hallelujah from The Hunchback of Notre Dame, 1939.
Copyright 2009, Marco Polo Music, Inc.
Something is Wrong with the Standard Model ... of Economics — Posted Wednesday, January 1 2014
The major subject of my doctoral dissertation at MIT was inequality, its evolution over time, and its consequences for macroeconomic behavior and especially growth. I took some of the standard assumptions (of what is called the neoclassical model) and showed that under those assumptions there should be a convergence to equality among individuals. It was clear that something was wrong with the standard model, just as it was clear to me, having grown up in Gary [Indiana], that something was wrong with a standard model that said the economy was efficient and there was no unemployment or discrimination. It was the realization that the standard model didn’t describe well the world we lived in that set me off on a quest for alternative models in which market imperfections, and especially imperfections of information and “irrationalities,” would play such an important role.
Thus writes Columbia University's Joseph E. Stiglitz, the 2001 recipient of the Nobel Prize in Economics, in the introduction to his 2013 best-selling book The Price of Inequality: How Today's Divided Society Endangers Our Future. Stiglitz received his PhD in economics from MIT, but it is interesting to note that he began his undergraduate studies as a physics major at Amhurst College. As he also points out in the book's introduction, as a child he fell in love with the beauty of mathematical equations and their unerringly accurate application to the physical world. But his parents were social progressives, and under their influence the Jewish Stiglitz' interests ultimately veered toward economics. Since receiving his doctorate in 1967, his highly influential work has received far too many awards to mention here, so I will only add that Stiglitz has simultaneously been a supporter of and advisor and strong vocal opponent to President Obama. Today he is universally loved by progressives and loathed by the supply-siders.

It is also interesting to note that Stiglitz' mention of the Standard Model of economics contrasts strongly with the Standard Model of physics. While it is safe to assume that everyone considers themselves to be experts in economics (at least as far as their own finances are concerned), only a tiny fraction of the population is even remotely aware of the Standard Model in physics, despite its huge, media-lauded success in predicting the Higgs boson, which was finally discovered in 2012. It is probably also safe to say that most Americans would consider the economic system of their country to be sound at least in theory. This would explain why, despite having nearly destroyed the country in 2008, economic advisors like Alan Greenspan, Hank Paulson, Larry Summers, Tim Geithner and Ben Bernanke and their ilk remain largely admired, and have yet to be strung up or even run out of town. (By becoming Bush 43's Treasury Secretary, Paulson saved an estimated $250 million in capital gains taxes from his prior position as CEO of Goldman Sachs, perhaps the single most corrupt corporation in US history. For details, read Matthew 7:18.)

By comparison, the SM of physics has never experienced a single failure. All of its predictions have been unerringly correct, in some instances with fantastic accuracy (for example, the model's calculated value for the gyromagnetic ratio of the electron is accurate to within 12 decimal places of the experimental result). So what explains the preference and confidence that people have for economics over physics?

Americans tend to dismiss much of science as just "theories," but economics is really just a lot of unsubstantiated theories based on supply and demand and the dynamics of the market. Worse, economics makes predictions that involve the preferences and emotions of consumers, which are hardly understandable in any accurate sense. Similarly, many people complain that science and math are just too hard to understand, yet the average Susie Housecoat and Joe Sixpack (to quote Montgomery Burns) can balance their checkbooks but can't make heads or tails out of even basic economics.

Still other people may say that economics is more familiar and hits "closer to home" than physics, in the sense that the cost of goods, mortgage interest, inflation and employment rates are all tied to economics. That's true, but the computers, cell phones, MRI scanners, GPS systems and innumerable other technologies that people use every day are all due to modern science and are just as familiar. But hasn't science also given us thermonuclear weapons and biological and chemical warfare agents? That's also true, but these resulted primarily from political misuse. Most of the Manhattan Project physicists who worked to develop the atom bomb at Los Alamos had major misgivings about the destruction of Hiroshima and Nagasaki and the deaths they produced, and some even left science altogether.

In his book, Stiglitz recounts how Alan Greenspan and fellow right-wing economist Friedrich Hayek worked side-by-side with the war criminal and mass murderer Generalissimo Augusto Pinochet to turn a democratic Chile into a supply-side money machine for Chilean elitists and American investors (the story is also told in Thom Hartmann's 2013 book The Crash of 2016). Sorry, but I don't see Greenspan having anywhere near the same moral sense as J.R. Oppenheimer.

The Crash of 2008 resulted in unimaginable misery for many Americans yet, as Stiglitz notes with some irony, no one at the top of the money chain has been held accountable. Even now, the pain felt by that economic disaster is being numbed over by rising housing prices and marginal increases in the employment rate (unless you're young, old or a minority). Hartmann describes this as the "Great Forgetting," a tendancy of Americans to forget the economic and cultural lessons of the past (assuming they ever learned them in the first place).

But, as Stiglitz is also quick to note, American capitalism over the past 200 years has been a great blessing to the rich and poor alike. It's only that in the past three decades that capitalism's Standard Model has shown that it is fatally flawed — inequality was always present in the model, but today it is rampant. The stock markets are currently soaring to record heights, yet 99% of Americans are not benefiting. Feudalism in America is very much a possibility today, but most Americans still believe they'll somehow get rich, so it won't matter.

So where does this leave the Standard Model of physics? Most people still couldn't care less.
Entangled Systems — Posted Wednesday, January 1 2014
Although Hermann Weyl's 1918 theory of conformal invariance failed as a model for the unification of gravity and electromagnetism, it was a phenomenal success when it was applied ten years later to quantum physics. Even today, it seems remarkable that a theory that is invariant with respect to the simple phase transformation \(|\psi\rangle \rightarrow e^{i \theta} |\psi\rangle\), where \(\theta(x)\) is an arbitary function of the spacetime coordinates, could explain the conservation of electric charge. Renamed gauge invariance, Weyl's idea is a cornerstone of modern quantum theory.

But there is another way of dealing with quantum state vectors that does not involve phase arbitrariness, and that is the density operator approach. If the state vector \(|\psi\rangle\) contains everything we are allowed to know about the quantum state \(\Psi\), then surely the slightly more complicated dyad operator \(|\psi\rangle\langle\psi|\) contains the exact same information. Indeed, it not only contains the same information and is phase invariant, but it also provides a means for understanding random collections of quantum states (or mixed states), a topic usually skipped in undergraduate courses. The density operator formalism also opens the door to the branch of quantum physics known as quantum information theory, in which the mystery of quantum entanglement is explained.

Although most of his texts are highly technical and written in German, University of Konstanz professor of physics Jürgen Audretsch's 2007 book Entangled Systems: New Directions in Quantum Physics is written in English at an undergraduate level that is accessible to students of physics, mathematics, chemistry and even computer science. While the book presupposes a beginner's understanding of basic quantum mechanics, it's the most accessible introduction to quantum entanglement, information and entropy I've ever seen (even better than Leonard Susskind's video lectures). The book is a great self-teaching tool, and includes many exercises. (Amazon wants seventy bucks for the book, but you can read a good portion of it for free over at Google Books.)

I'm particularly impressed because for years now I've ruminated on the idea that physical reality is fundamentally rooted in the creation, propagation and annihilation of information and that, for whatever reason, the universe itself is somehow tied to the notion of "interestingness." For many years, physicists have asked the question "Why is there something rather than nothing in the universe?" The answer might simply be "Because something is more interesting than nothing."

[On the downside (at least for me), Audretsch's book addresses some topics that I wanted to use in a book of my own. How can I write anything when other people keep beating me to it?]

Anyway, Happy New Year!
Is We Evolving? — Posted Tuesday, December 31 2013
To me, the most beautiful and profound aspect of physical law is that Nature invariably strives to be as efficient as possible. This is succinctly demonstrated by the fact that the mathematical quantity known as the action, which Nature is somehow intimately familiar with, is invariably extremalized (and usually minimized) in all physical interactions, from the very small (quantum physics) to the very large (gravitation). Since action is always expressed in units of momentum-displacement (\(p \cdot x\)) or energy-time (\(E \cdot t\)), Nature evidently likes to do things using the the least momentum along the shortest path, or the least energy in the least time. Indeed, the principle of least action as first developed in the 18th century was viewed as the best scientific evidence for the existence of God.

When biologist Charles Darwin visited the Galápagos Islands in the 1830s he noted that there was a diverse variety of birds (notably finches) whose food preferences depended to a great extent on the size and shape of their beaks. Those who fed primarily on small seeds had small beaks, while large-beaked birds ate larger, harder seeds. Some birds, which fed mostly on hard-to-reach seeds (like those in cactus), had long, narrow beaks. While not a physicist or mathematician, Darwin saw this diversity as evidence of an evolutionary tendency that trends toward efficiency — for example, birds who feed on small seeds do not carry around heavy beaks, as this would be a waste of energy. I doubt very much if Darwin was ever aware of the principle of least action, but if he was he would probably have viewed evolution as a good example of it. True, Darwin likely saw evolution as a slow process, taking many generations of animals over huge time periods, but given the fact that environmental stressors such as climate change, disease, inter-species competition and predator population change all vary slowly with time, Darwin probably considered evolution to be a very slow but efficient process overall.

[Side story — in 1971 I took an undergraduate class in biochemistry, and the professor calculated the energy efficiency of the electron-transport mechanism at 67%. He compared this with the efficiency of a car engine, which at best is only around 25%. Nature wins this contest, hands down.]

But for many people at the time (and even today), Darwin's On the Origin of Species, written in 1859, went too far. It implied that humans themselves had physically evolved over time, in seeming contradiction to the Bible. Most people in Darwin's time believed that God had created man roughly 6,000 years earlier, and that man and all the other creatures were created by God in the exact same form that we see them today. Worse, Darwin hypothesized that humans had evolved from earlier primate forms related to monkeys and apes. Human ego rejected the notion that people were descended from monkeys — "If monkeys had evolved into people", the saying went, "then why are there still monkeys walking around?"

Although we now know from sound genetic evidence that the genetic lines of apes/monkeys and man diverged some 5 million years ago (which is why monkeys are still around) and that physical evolution can take place startlingly fast, it should not be surprising that the theory of evolution is still being questioned today. After all, Darwin's work is rarely actually read by anyone (much less studied), and biblical genealogy from Adam and Eve on down to Jesus supports the notion that only about 6,000 years have elapsed since time began. To a person of strong Christian, Jewish or Muslim faith, evolution is not only puzzling but contradictory to religious belief as well. But what is surprising, however, is that in spite of new, ongoing and profound fossil and genetic discoveries, evolution is being increasingly rejected by people of faith in America today.

Some 1,983 American adults from all 50 states were polled in a new study conducted by the Pew Research Center regarding their views on human evolution. Although 60% of Americans indicated they believe human evolution has occurred, fully 33% believed that humans have remained physically the same since they were first supposedly created by God. While marginally more Americans believe in evolution today, the percentage of white, evangelical Protestants who reject evolution has increased. Indeed, nearly two-thirds of white evangelicals polled do not believe in evolution:

I don't know about you, but this scares the crap out of me.

When I was teaching I used to tell my students that there are no laws in science, only theories. Electromagnetism, quantum mechanics, gravitation, chemistry, aerodynamics and germ theory are and will always be just theories, no matter how much evidence is compiled to support them — that's what the scientific method is all about. People universally believe these theories, but 33% hold that evolution is "just a theory." And I suspect that a sizable subset of that 33% is also wondering why monkeys are still walking around if there's anything at all to Darwin's ideas.

Study after study now confirm that the country is becoming ever more politically and culturally polarized. I keep asking myself how otherwise intelligent people can reject facts, reason and empirical evidence in favor of dogmatic allegiance to illogical, self-contradictory religious nonsense — people who can drive a car, hold a job, program a DVR, balance a checkbook and even teach at university, but choose willful ignorance over a life of rational thought. I keep asking myself what they're afraid of, what is it that they find so frightening that they would abandon the thinking, reasoning brain that God gave them.

I don't have any answers, and I have no idea what's going on around here. I can only hope that 2014 turns out better than 2013.

Those who can make you believe in absurdities can make you commit atrocities. — Voltaire