Index photos courtesy ETH-Bibliothek,
Zurich Bildarchiv

Why I'm No Longer a Christian

Who Was Hermann Weyl?

Wheeler's Tribute to Weyl (PDF)

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

email:bill@weylmann.com

Math Tools
Weyl's Spinor and Dirac's Equation
Weyl's Conformal Tensor
Weyl Conformal Gravity
Weyl's 1918 Theory
Weyl's 1918 Theory Revisited
Weyl v. Schrodinger
Why Did Weyl's Theory Fail?
Weyl and the Aharonov-Bohm Effect
The Bianchi Identities in Weyl Space
Weyl's Scale Factor
Weyl's Spin Connection
Weyl and Higgs Theory
Weyl & Schrodinger - Two Geometries
Lorentz Transformation of Weyl Spinors
Riemannian Vectors in Weyl Space
Introduction to Quantum Field Theory
Electron Spin
Clebsch-Gordan Calculator
Bell's Inequality
The Four-Frequency of Light
There Must Be a Magnetic Field!
Non-Metricity and the RC Tensor
Curvature Tensor Components
Kaluza-Klein Theory
A Brief Look at Gaussian Integrals
Particle Chart

Uncommon Valor

Sophie did not forget Jesus!
Hans: "Long live freedom!"

 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.

 This is the End? — Posted Monday, 15 August One of my favorite blogs is written by German physicist Sabine Hossenfelder, a research fellow at the Frankfurt Institute for Advanced Studies, whose articles are posted regularly on her website, Backreaction. Her expertise appears to cover just about every topic in modern physics, especially quantum gravity, but more interestingly she loves to write about both physics and non-physics topics, of which she tends to be very opinionated (her English, like her writing, is far better than mine, although her physicist husband's needs some work). Not too long ago, to supplement her income (she and her husband have two little girls, and like most people they worry about mortgages and the cost of living just like everyone else) she began her Talk to a Physicist consulting service (to my knowledge, unique in its kind). For a very reasonable \$50 fee you can Skype her for 20 minutes and get your questions answered, or you can discuss your own pet physics theories with her, regardless of how ridiculous they may be (those of you who are Sherlock Holmes fans will note the similarity of this to Holmes' "consulting detective" service). One of the more recent subjects Hossenfelder addresses on her website deals with the "End of Physics," a topic that has been around for several years now following the failure of the Large Hadron Collider to get past its celebrated 2012 discovery of the Higgs boson. At 125 GeV, the energy of the Higgs is in the same ballpark of the energies of particles confidently predicted by supersymmetry (SUSY) theory, but which have infamously not been detected at the LHC. A number of notable particle physicists are panicking over this, while others have simply moved the energy goalposts in the hope that the true energies are higher than previously thought. But the LHC, now nearing its design energy of 14 TeV, still hasn't seen any SUSY particles, while the December 2015 "diphoton discovery" was shown last month to have been a statistical phantom. The so-called End of Physics might transpire in the following way: the LHC, despite minor and major adjustments and improvements, might simply not discover any more particles, while proposed, more powerful projects such as the International Linear Collider may do nothing more than add a few decimal places to the Higgs discovery. Physicists would not only be disappointed, but they might also finally realize that with the impossibility of Planck-energy research there would be nothing more for them to do. The Standard Model might get a touch-up here and there, but that would be it. Condensed matter physics would likely take center stage, as would other fields in applied physics, but purely theoretical endeavors like string theory might ultimately find themselves in the pseudoscience bin, to be joined there with their equally unhappy cousin, religion. But a "particle desert" discovery in physics might also herald a new view of the universe. It's possible that we live in a wonderful (if ultimately limited) "goldilocks" universe that's just one of a multiverse of worlds, or we could reside in a digitally simulated universe created by an advanced civilization that itself resides outside ours. One of the characteristics of string theory is that there are some $$10^{500}$$ versions of the theory, and each might represent a separate universe (though, like string theory itself, we'd have no way of experimentally proving this). As for our universe being a computer simulation, there is exactly one way to know if it's true: we find an unresolvable glitch, error or pixelation in the programming somewhere, prompting the simulators to announce their existence to a stunned world. Perhaps the "particle desert" idea represents that glitch. Meanwhile, Harry Cliff is a British particle physicist who spoke recently at a TED talk in Switzerland. He talks about the End of Physics issue in this short video:
 Geometric Algebra Again — Posted Tuesday, 26 July A few years back I was talking about something called geometric algebra or GA. At that time I was discovering that vectors, dot products, cross products, reflections, rotations and all that jazz have a vastly superior formalism that is certainly being used by all intelligent extra-galactic beings in lieu of the rather confusing "Gibbsian" formalism our students are forced to learn in high school and university today. GA is not only powerful but simple as hell, and it has found a home in many computer applications, especially games and animation (though fast, computers simply can't calculate, update and refresh animation and gaming frames in real time using the old vector algebra). This morning I came across a free introductory book on GA written (and uploaded) by author John A. Vince that presents the basics of the algebra as well as its applications to computer animation and games (you can also download the book from my website). The book is short and well-written, with portions in color, but the best part is it's easily understood by high school students. GA is slowly taking over as the standard teaching approach for vectors in university, and by the time I'm dust I hope it's the only approach.
 Veblen and Weyl — Posted Tuesday, 5 July Anyone with the name "Oswald" can't be bad, especially if he's a mathematician. Oswald Veblen (1880 - 1960) was an American mathematician at the Institute for Advanced Study in Princeton. He was much sought after as an academician, and along with Hermann Weyl, Albert Einstein and John von Neumann he was a founding member of the Institute in the period 1932-1933. I haven't read much of his work, but in 1922 he wrote a paper that presented a set of fundamental new identities involving the Riemann-Christoffel curvature tensor $$R_{\, \, \mu\nu\alpha}^{\,\lambda}$$ that touched directly on Hermann Weyl's 1918 theory of the unified gravitational-electromagnetic field. Titled Normal coordinates for the geometry of paths, Veblen presented his paper to the National Academy of Science in April 1922. The paper's tensor notation is in the old style (we can forgive the glaring typo in Equation 8.1), but in it Veblen summarizes the derivation of the identities given by $$R_{\, \, \mu\nu\alpha||\beta}^{\,\lambda} + R_{\, \, \alpha\mu\beta||\nu}^{\,\lambda} + R_{\, \, \beta\alpha\nu||\mu}^{\,\lambda} + R_{\, \, \nu\beta\mu||\alpha}^{\,\lambda} = 0 \tag{1}$$ where the double-bar subscript is my usual notation for covariant differentiation. (I derived the identities by a more straightforward approach, but I never knew about them until I saw Veblen's paper.) Compare this expression with the more famous Bianchi identities, which are given by $$R_{\, \, \mu\nu\alpha||\beta}^{\,\lambda} + R_{\, \, \mu\beta\nu||\alpha}^{\,\lambda} + R_{\, \, \mu\alpha\beta||\nu}^{\,\lambda} = 0 \tag{2}$$ (Notably, the Bianchi and Veblen identities are also valid in non-Riemannian space.) By contracting the indices $$\lambda$$ and $$\beta$$ we can get get rid of the common $$R_{\, \, \mu\nu\alpha||\beta}^{\,\beta}$$ term in (1) and (2) and rewrite the reduced identities as $$\left( R_{\mu\alpha} - R_{\alpha\mu} \right)_{||\nu} + \left( R_{\alpha\nu} - R_{\nu\alpha} \right)_{||\mu} + \left( R_{\nu\mu} - R_{\mu\nu} \right)_{||\alpha} = 0$$ where we have used the identity $$R_{\, \,\mu\alpha\beta}^{\,\mu} = R_{\alpha\beta} - R_{\beta\alpha}$$ (which vanishes in Riemannian space). Because of the antisymmetry, this reduces to the simpler form $$\left( R_{\mu\alpha} - R_{\alpha\mu} \right)_{|\nu} + \left( R_{\alpha\nu} - R_{\nu\alpha} \right)_{|\mu} + \left( R_{\nu\mu} - R_{\mu\nu} \right)_{|\alpha} = 0 \tag{3}$$ where the single subscripted bar represents ordinary partial differentiation. Again, in ordinary Riemannian space we have $$R_{\nu\mu} - R_{\mu\nu} = 0$$, so that (3) is trivially true, but in Weyl's theory we have $$R_{\nu\mu} - R_{\mu\nu} = F_{\mu\nu}$$, where $$F_{\mu\nu}$$ is the antisymmetric electromagnetic tensor. Consequently, in Weyl's theory Veblen's identities become $$F_{\mu\alpha|\nu} + F_{\alpha\nu|\mu} + F_{\nu\mu|\alpha} = 0$$ which expresses the homogeneous set of Maxwell's equations for the electromagnetic field. For example, if we set $$\mu=0, \nu = 1$$ this reduces to the set of Maxwell's equations $$\vec{\nabla} \times \vec{E} = - \frac{1}{c} \frac{\partial \vec{B}}{\partial t}$$ I doubt if Weyl ever considered this application of Veblen's identities to his theory, since by 1922 Weyl had abandoned it in favor of more promising and conventional climes, like group theory and the then-emerging quantum theory. A beautiful summary of Veblen's life and the major role he played in getting Weyl to come to America in late 1933 can be found here.
 More Mathematical Silliness — Posted Sunday, 26 June The great Swiss mathematician Leonard Euler once proved that $$\infty = -1$$, and thus concluded that negative numbers are greater than infinity. Euler's proof is simple, straightforward and difficult to contradict: \begin{eqnarray*} S & = & \sum_{n=0}^\infty 2^n = 1 + 2 + 4 + 8 \ldots \infty \\ & = & 1 + 2(1 + 2 +4 + 8 \ldots \infty ) \\ & = & 1 + 2S \end{eqnarray*} and so $$S = -1$$. As the authors of the fascinating 2013 book Magnificent Mistakes in Mathematics imply, Euler knew this was total nonsense, as Euler was no dummy. The book illustrates once more that treating infinity as a number is a grievous mistake, as the above example shows. It also illustrates that our concept of infinity, either in mathematics or religion, is also highly tenuous. Quantum field theory typically succeeds only when it truncates infinity to a large number that can be later swept under the rug. By comparison, the notion of infinity in religious belief typically succeeds only by relegating it to some nebulous "finite" concept (usually involving time) that no one really comprehends. Mathematically or otherwise, we humans can write down a symbol for infinity ($$\infty$$) or express it in writing as a concept, but we really have no idea what it is. While Hermann Weyl is not mentioned anywhere in the book, that does not necessarily mean he never made any mathematical mistakes, although to date I have yet to find him guilty of any. But again, the book is fascinating, and written at a level anyone can understand. I highly recommend it. By the way, if Herr Euler appears to be winking here, it's only because he suffered from several progressive eye conditions that eventually rendered him almost completely blind. Fortunately, like Stephen Hawking's amyotrophic lateral sclerosis, the conditions did not affect his prolific mind. This causes me to look at myself today — elderly but healthy, with a sound (?) mind and good eyesight, but a complete idiot by comparison. It's just not fair.
 Brexit — Posted Saturday, 25 June "The supreme duty of this nation [America] is to realize her sublime providential mission, and bear the blessed light of the Gospel to all the dark places of the Earth $$\ldots$$" — Rev. A. W. Pitzer, 1890 "And this also," said Marlow suddenly, "has been one of the dark places of the Earth." Joseph Conrad, Heart of Darkness, 1899 In my freshman year at college I took English 101 — a requirement even for chemistry majors back in 1967 — and was disheartened to learn during the professor's first lecture that I would have to write an essay of at least ten pages, due at the end of the semester. For no particular reason, I chose to write on Joseph Conrad's 1899 novella Heart of Darkness, which the professor had mentioned as a possible essay topic. Up to that point, my only essay experience had been as a freshman in high school, when Mr. McNeely tasked his students with the same challenge. I picked Harper Lee's To Kill a Mockingbird, a standard read at the time. The title of my rambling and very inexperienced essay (which I still possess) was "Prejudice," which gained me a "C" grade from McNulty. "No, it's much more than that!" he wrote at the top of my three-page masterpiece. By comparison, my Conrad essay garnered me an "A" grade, though I neglected to preserve the paper. McNeely was my favorite teacher in high school, and he apparently taught me something after all. But I digress. Heart of Darkness ranks as one of the most important literary works ever written, and possibly because of this its lessons have been completely forgotten by mankind. Conrad, who had visited the ironically-named Congo Free State when it was literally owned and operated by Belgium's inhuman King Leopold II, saw firsthand the great tragedy of Christian humanitarianism-turned-profiteering and the attendant wholesale slaughter, maiming and waste it brought upon the Congolese people and the country. Although slavery in the late 1800s had long been banned in Europe, the lure of ivory and rubber was too great to prevent Europeans from enslaving Africans to produce the stuff. Through his agents posted to the continent, Leopold II worked his slaves to the point of death, and when they failed to produce their quota of goods he had their arms and hands chopped off, often with those of their family members cut off as well. Protesting social and political reformers, shocked by horrific stories and photographs coming out of the region, convinced Leopold to forego his barbaric practices. But following Leopold's abdication of the Congo came what was to be called "The Great Forgetting" in Belgium and Europe, the wholesale amnesia of what had been done to the Congolese in terms of forced labor, murder, dismemberment, waste and pillage. So what has all this to do with Britain's leaving the European Union? The main force driving the "leave" referendum involved a British kind of "white fright" — the fear that immigration from other EU nations was taking away jobs and resources that imperiled the standard of living that Brits had come to expect, aided by the perceived pandering of British political leaders whose interests strayed from those of their countrymen. Brexit in many ways parallels what is happening in America today, with one important distinction. The plight of wretched immigrants fleeing Syria and afflicted neighboring Middle Eastern countries is due in large part to American military adventurism in that part of the world, particularly its illegal invasion of Iraq in 2003. The power vacuum created by Bush, Cheney and Bremer resulted in the humanitarian disaster we see today, as evidenced by terrorist groups and armies like ISIS that no one seems to know how to cope with. But America, having created the problem in the first place, does not want to deal with the consequences. Obama's pledge to allow ten thousand Syrian refugees into the United States will almost certainly be overturned by a Republican-dominated Congress whose constituents are mostly stupid, racist, undereducated, fearful white "Christian" gun worshippers whose ignorance of world affairs is unparalleled anywhere else on the planet. And who is leading the charge for this kind of pro-military, anti-immigrant madness but one Donald Trump, who likely has a 50-50 chance of becoming our 45th president. While Britain's voters have told immigrants to "go somewhere else," Trump's battle cry is more like "Exterminate the brutes!" (a reference to Heart of Darkness that has sadly become banal). Already, America's evangelicals are warming up to the policies of Trump, in direct contradiction to the teachings of Jesus. But do not liken this to a "Great Forgetting" moment — America never learned any lessons in the first place.
 Orlando — Posted Sunday, 12 June I had no sooner finished the Great Courses lecture series How the Crusades Changed History and Stephen Platt's 2012 book Autumn in the Heavenly Kingdom: China, the West, and the Epic Story of the Taiping Civil War when a smaller but no less disturbing bloodbath arrived on America's own doorstep late last night in the form of yet another religion-influenced mass murder, this time in Orlando, Florida. The Crusades, you will remember, was a two centuries-long series of attacks fomented by Catholic Popes to wrest Jerusalem from the Muslims. The Templar Knights, holy killing machines, aided by rag-tag hoardes of mostly European peasants (even children) who apparently had nothing else to do, mounted a total of nine crusades into and/or around the Holy Land, bent on restoring Christianity and (no less importantly) retrieving holy relics for veneration. Over the period of roughly 1100 to 1300 CE, the Holy Land was apparently literally awash in the blood of Jesus (vials of it were floating around everywhere), along with pieces of the cross on which he was crucified (there were enough pieces, it has been said, to build a fleet of wooden ships), the original Crown of Thorns, and the bones, hair and scraps of clothing from every saint you could possibly list (oddly enough, the foreskin of the circumcised Savior was somehow never located). But what actually drove the Crusaders, numbering at times around 50,000, was not so much righteous indignation over the Muslims lording it over everyone from God's Very Own City but the promise of plenary indulgences conferred upon the participants by a series of Popes, notably those named Urban and Innocent. Indulgences, which are still recognized by the Catholic Church today, are permanent and absolute ("plenary") affirmations of forgiveness conveyed upon a sinner by God, typically via an intermediary. In Crusader times the Pope was the guy who did the conveying, and believe it or not he got a hefty cut from the wars in the form of money, gold and power. The indulgences extinguished all sins past and future, contingent only upon a Crusader actually fulfilling his holy mission. It just so happens that the fear of eternal damnation and torment in hell was so great among Medieval folk that they actually believed this nonsense (the fear was constantly driven home by the Church, of course). The Crusades are estimated to have resulted in the deaths of approximately 2 million people, combatants and civilians alike. Oddly enough, the Muslims (who managed to maintain their hold on Jerusalem) were far more merciful toward captured Crusaders and Jews and Christians living in the region than the Crusaders themselves, who indiscriminately tortured and slaughtered Muslims, Jews and collaborating Christians alike. Of far less notoriety was the Taiping Rebellion (actually a civil war) that was fought in China against the ruling Manchu Xing dynasty by a large number of mostly poor and indigent Chinese who had converted from Confucianism to Christianity in the late 1840s. The uprising was led by a charismatic Joseph Smith-like character named Hong Xiuquan, a failed civil servant who had studied the Bible under the tutelage of an American Southern Baptist missionary in the 1830s. Xiuquan subsequently experienced a series of revelatory dreams in which he not only conversed with God and Jesus, but was told he was actually the younger brother of Jesus himself! China, which had suffered a series of natural calamities in the 1840s (including a humbling defeat by the British in the Opium War), seems to have been primed for change, and the otherwise absurdly nonsensical stories spun by Xiuquan somehow caught on. At one point, his followers had amassed an army (known as the "God Worshippers") of some 50,000 fighters, and for 14 years they gave the rulers of the Xing Dynasty a real run for their money. But the dynasty won out in the end, and Xiuquan, now calling himself the "Heavenly Emperor," died in 1864, his dream of a Christian China in ruin — but not before a conservatively estimated 20 million people had been slaughtered in that divinely-inspired civil war. By comparison, today we have this nutcase named Omar Mateen killing a paltry 50 people in Orlando (himself included). An alleged devotee of ISIS, Mateen is no doubt now basking in Heaven, surrounded by servants and dark-eyed virgins, or at least that's what ISIS would have you believe. And who the hell is ISIS but yet another band of religiously-motivated crusaders and holy warriors, the likes of which humankind has foisted upon itself time and time again? Let us not forget either that President George W. Bush likened America's bogus, lie-based invasion of Iraq a "crusade." (Then there's the 1953 CIA-led coup in Iran, Vietnam, El Salvador, Honduras, etc., but enough.) "World Without End"? No, this one has an end, and we're rapidly bringing it on ourselves.
 Group Theory — Important, But Dry — Posted Friday, 3 June Some years ago I posted this photo of Hermann Weyl at his desk, thinking that it was taken shortly after his immigration to America in late 1933. Not so — it was taken while he was still in Zürich, Switzerland, sometime in 1927. Still just 41, Weyl had abandoned his earlier theory on gravitational metric invariance and was now infatuated with group theory, though he would remain convinced of the importance of metric conformal invariance for the rest of his days. Group theory was perhaps the true love of Weyl's life. In 1928 he published the first edition of his book Group Theory and Quantum Mechanics, a seminal if somewhat difficult-to-read text today that has easily been surpassed by Anthony Zee's latest book, Group Theory in a Nutshell for Physicists, which I just finished reading (I found only one typo — he misspells "Clebsch" in one of the appendices). Zee, whose books I just love to read, relates a notable story of J.B.S. Haldane is his book. When asked what seemed to be of utmost importance to God when he created everything, Haldane replied "An inordinate fondness for beetles," which are perhaps the most common life form in existence today. But Zee responds instead with "An inordinate fondness for group theory." While I don't believe in the Judeo-Christian-Islamic God, I think Zee hit the nail on the head, though I still see group theory as rather dry and boring. (Photo scanned from the 2009 book Mind and Nature: Selected Writings on Philosophy, Mathematics and Physics by Hermann Weyl with the express permission of the University of Princeton Press. Please respect the copyright when sharing)
 Mr. Cool Again — Posted Wednesday, 1 June Researcher Malcolm Douglas sent me the link to the following presentation, which was delivered in 2007 by Norbert Schappacher of the University of Strasbourg. Entitled Self-Limitation and Holism in Mathematics: The Example of Hermann Weyl and World War I, it addresses purely philosophical issues espoused by Hermann Weyl before and after the Great War. Included are several photos of Weyl I'd never seen before, along with this group photo of the Swiss Mathematical Society meeting in 1917. Interestingly, Weyl's wife Hella is also shown, along with three other women. The shorter man to David Hilbert's right appears to be Marcel Grossmann, the Society's president that year. A lifelong close friend and confidant of Einstein, Grossmann assisted the great physicist in elucidating the tensor formalism of the general theory of relativity, which Einstein published a year earlier. In fact, Grossmann actually tutored Einstein on tensor calculus, which Einstein was certainly unskilled at when he undertook efforts to generalize his theory of special relativity, published in 1905. Also shown in the paper is another view of Weyl on a children's seesaw, which I posted here on 11 January under the heading "Mr. Cool."
 $$1+2+3+4 \ldots$$ — Posted Monday, 30 May My post of 28 April prompted several emails from high school students asking me how on Earth the infinite series $$\sum_{n=1}^\infty n = 1 + 2 + 3 + 4 + \ldots = - \frac{1}{12}$$ could ever converge to a finite (and negative) number, much less be important in some way regarding string theory. The answer is that it does not converge, and is indeed infinite, but there is a "finite" aspect to it that bosonic string physicists have glommed onto that tends to prop up the theory. In my much earlier post of 14 June 2012 I mentioned this same problem and how the Stanford string theorist Leonard Susskind addressed it in Lecture 5 of his wonderful online video series String Theory and M-Theory. Susskind kind of confuses things in his analysis but in the end his answer is correct. The following is essentially what he does in his lecture: Consider the infinite geometric series $$P = \sum_{n=1}^\infty e^{-an}$$, where $$a$$ is some small number that we will ultimately set to zero. This series converges to the simple result $$P(a) = \frac{1}{1-e^{-a}} = \frac{e^a}{e^a - 1}$$ which any high school student can reproduce. Taking the derivative of this with respect to $$a$$, we have $$\frac{dP}{da} = - \sum_{n=1}^\infty n \, e^{-an} = - \frac{e^a}{(e^a - 1)^2}$$ or $$S = \sum_{n=1}^\infty n \, e^{-an} = \frac{e^a}{(e^a - 1)^2}$$ This summation is exactly what we want in the limit that $$a \rightarrow 0$$, but setting $$a=0$$ in the exponential term on the right immediately gives us infinity. What Susskind does instead is expand the numerator and denominator using $$e^a = 1 + a + \frac{1}{2!}a^2 + \frac{1}{3!} a^ 3 \ldots$$ and then employ synthetic division to divide the denominator into the numerator in $$S$$. Susskind's attempt at synthetic division is a complicated mess, but the answer comes out to be $$S = \frac{1}{a^2} \left( 1 - \frac{1}{12} a^2 + \frac{1}{240} a^4 \ldots \right)$$ where the "$$\ldots$$" means orders of $$a^2$$ higher than $$a^4$$. Dividing by $$a^2$$ and then setting $$a = 0$$, we get $$S= \infty - \frac{1}{12}$$ Notice that the answer is still infinity, as expected, but there's a straggling term that physicists have made good use of, including establishing the notion that bosonic string theory is sensible only in 26-dimensional spacetime. What Susskind and others claim is that it all makes sense in the end, because quantum field theory also has a habit of sweeping the infinities under the rug while keeping the finite parts that make the physics work. I have several books on string theory, and you can take my word for it that none of them makes this "$$-1/12$$" business any more sensible. Barton Zwiebach's text A First Course in String Theory is arguably the best introductory book of its kind, and even there the author merely notes that the $$-1/12$$ result may be surprising but that, what the hell, it works. At a much more introductory level, David McMahon's String Theory Demystified analysis simply inserts the $$-1/12$$ term without even bothering to explain where it comes from. (McMahon's book is chock-full of typos and algebraic errors, but you can easily spot them and the book is otherwise a great introduction to the theory.) I'm still not convinced. As Dirac once famously noted, the business of getting infinite results and having to ignore them means something's wrong somewhere. I think he was right.
 Weyl and Family — Posted Monday, 23 May Undated but probably taken sometime in 1942, here's Hermann and Hella Weyl with their son Joachim along with his wife and daughter. The Weyl's first son, Joachim was born in Germany in 1915, with his brother Michael following in 1918. Despite initially knowing little or no English, Joachim studied mathematics at Princeton and received his PhD there in 1939. He and his father collaborated extensively in their research, and Joachim became an influential mathematician in his own right. He died in 1977 at the relatively young age of 62. (From the 2009 book Mind and Nature: Selected Writings on Philosophy, Mathematics and Physics by Hermann Weyl with the express permission of the University of Princeton Press. Please respect the copyright when sharing)
 Math Can't Get No Respect — Posted Sunday, 8 May This story would be funny if it didn't also betray the sad state of paranoia the country's in: a flight was delayed for two hours because a passenger spotted a professor from the University of Pennsylvania writing mysterious equations on his notepad prior to departure and, believing the man to be a terrorist, duly notified the gendarmes. I'd be willing to bet anything that the woman who reported him is a Republican who believes that differential equations should be outlawed, at least in public places. Speaking of mysterious, mathematical logician David Marans sent me this group photo of the mathematicians who participated in the December 1946 Princeton Bicentennial Conference on Problems in Mathematics. Can you spot Hermann Weyl? Well, we couldn't either, but we later learned he's the guy in the fourth row back, fourth from the right. Weyl gave the conference dinner speech that year, but I don't know what the topic was. Probably something I wouldn't understand, anyway — just like the lady on the plane!
 Weyl and Grandson — Posted Monday, 2 May Here's a nice photo of Hermann Weyl with his grandson Peter in 1949. Weyl was probably 63 at the time, four years younger than I am now. God, I feel old. (From the 2009 book Mind and Nature: Selected Writings on Philosophy, Mathematics and Physics by Hermann Weyl with the express permission of the University of Princeton Press. Please respect the copyright when sharing)
 Logic is Fun! (Well, Maybe) — Posted Thursday, 28 April David Marans is Professor of Logic at St. Thomas University in Miami, Florida. I don't know him personally, but he's a fellow Doctors Without Borders (DWB) supporter, professional piano player, world traveler and all-around good guy. We occasionally exchange emails on topics that are usually far over my head. David wrote and occasionally updates his book Logic Gallery, which you can either buy here (a portion of which goes to DWB) or view online for free here. It's literally a gallery of nearly 200 famous logicians, mathematicians and scientists, replete with photos, short articles, quotes, tidbits and links about people like Descartes and even Hermann Weyl. Disclosure: Although I did very well on my chemistry, physics and engineering Graduate Record Exams, my scores in Logic were absolutely pitiful, somewhere below the 40th percentile. Non-mathematical logic simply escapes me, and Logic's poor cousin, Philosophy, finds me equally at a loss. But Logic Gallery is fun to read, and I encourage anyone who is logically impaired like me to check it out.
 Off On Another Tangent — Posted Wednesday, 27 April Turner Classic Movies recently aired the restored version of German director Fritz Lang's 1922 silent masterpiece Dr. Mabuse, der Spieler, starring the great Rudolf Klein-Rogge (Metropolis, 1927). "Spieler" in German generally means "gambler," but here it's more like "puppeteer." Mabuse is a brilliant psychoanalyst whose hypnotic powers can induce people to do his will, usually for monetary, political or sexual gain. At nearly five hours, it requires some effort to get through, but the film is worth it. Midway through the film Mabuse encounters the beautiful Countess Dusy Told, whom he sets out to seduce by first destroying her husband. The philosophy of the monomaniacal Mabuse is summarized in the following scene. The Countess is a good woman, but she's bored with her husband and seeks fun at gambling parties and other "sensational" outlets. But she learns that virtuous love (Liebe) is far nobler than anything the senses can provide, a knowledge she relates to the lustful Mabuse. But his response is "There is no love, only passion! There is no chance, only the will to power!" — Klein-Rogge, whose lifespan (1885-1955) coincided with Hermann Weyl's, is deliciously evil in the film, and its popularity in the Weimar Germany of the 1920s led Lang to direct two successful sequels. However, the Nazis disliked the films, mainly because they portrayed the vain-glorious, all-powerful madman Mabuse in an unfavorable light. Today we would view Mabuse not as a madman but as a neoliberal, a term whose definition has varied widely since its introduction in 1938. Regardless of its original meaning, today's neoliberals are characterized as being disdainful of government and governmental regulation, while praising privatization of all public services in deference to a universal, rule-free market economy. Like Mabuse, neoliberals constantly seek influence and power through competition, viewing cooperation and other mutually-beneficial actions as destructive to free-market principles. Consequently, public services such as social security, welfare and governmental health programs are viewed by neoliberals as social evils whose elimination would result in increased profits to private enterprise. In short, neoliberals, like the good Herr Doktor Mabuse, are in reality evil sociopaths interested only in their own personal gain. And that brings us inescapably to the likes of Republican presidential candidate Donald Trump. Although Trump's sweeping victories in all five state primaries last night mean that he will almost certainly be the Republican nominee, I can now only hope that he is defeated in the general election by the presumptive Democratic nominee, Hillary Clinton. I am not surprised that Trump's victories are due in no small part to white evangelicals, whose earlier hesitant support for Trump has now been replaced with near-overwhelming acceptance. But I am surprised that roughly 50% of Americans can be taken in by this strutting, billionaire egomaniac whose lifestyle and philosophies go against anything remotely appealing to America's "Christians." Perhaps human beings are really nothing more than intelligent but savage apes, grabbing everything they can however they can.
 What's It All About, Alfie? — Posted Tuesday, 26 April The long-wished-for "Theory of Everything" will be so simple that we will be able to summarize it in a single equation on a t-shirt. —Every Living Physicist I already quoted Einstein in my previous post, so I won't repeat it here, but it actually has a more appropriate application regarding what follows below. Dutch theoretical physicist Martinus Veltman was the co-recipient (with fellow Dutch physicist Gerald 't Hooft) of the 1999 Nobel Prize for his work in particle physics. He has written several books, at both the lay and advanced level, and one of his technical but still somewhat readable texts is his 1994 book Diagrammatica: The Path to Feynman Diagrams. In Appendix E Veltman details the full Lagrangian for the Standard Model of physics, consisting of seven parts: Just one of those parts, the weak-force Lagrangian $$\mathcal{L}_w$$, goes like If you happen to be intimately familiar with all seven parts, including the Feynman rules required for their application, you'd be in pretty good shape (don't look at me; I surely am not). But I think you get the picture — although the Standard Model has never failed, it's not quite the theory one would like to display on a t-shirt, much less carry around in one's head. Even worse, the SM doesn't include the gravitational interaction, because gravity just refuses to be put into the box. University of Oxford physicist Vlatko Vedral has a fascinating article in this week's online Aeon magazine that basically asks the question: "With the exception of gravity, the Standard Model describes the underlying physics of all inanimate matter about as well as we could ever hope. Why can't living matter be similarly described?" This is really a very profound question, because although every living thing is composed of inanimate atoms and molecules, self-replicating assemblages of nucleons and elementary particles simply defy mathematical description. In short, physics and mathematics do not explain life; there is no Lagrangian for living systems. If there were, it would surely be infinitely more complicated than the mess displayed above. In a somewhat related Aeon article, British writer Stephen Poole addresses the problem of teleology, which has to do with the meaning or purpose of things. Teleology is really nothing more than the ultimate philosophical question, although it is perhaps the most fundamental issue humankind could ever ponder. Admittedly, physical laws allow for "islands of complexity" in the universe that conserve energy and momentum and do not violate the Second Law of thermodynamics, but why should "complexification" even be allowed in a universe supposedly bound by simple (!) physical and mathematical laws? Religious types might respond to that question with some kind of unsupportable dogma involving God (or gods), but even that does not answer the question, because one could then ask what God's underlying purpose is. To them, the whole issue then becomes a matter of argumentum ad ignorantiam: "We don't really know the answer, but it's as we say, regardless." Believe me, as one gets older these questions become important. My younger son got his PhD in molecular biology, immunology and genetics at UCLA in 2010 and is now a scientist at the Centers for Disease Control and Prevention in Atlanta. Before leaving home, he and I used to have endless discussions on these very topics. We were never able to resolve anything, of course, but oh, God, how I do miss those days now.
 Back to Weyl — Posted Saturday, 23 April Hence, after the development of modern quantum theory, Hermann Weyl interpreted the ideas of gauge invariance and the corresponding mathematical formalism as connected with transplanting the state vector of a quantum-theoretical system. Be this as it may, there seems to be a very suggestive and potentially significant content to [Weyl's 1918 theory]. Physical reasoning led Weyl to a model of differential geometry which is of great theoretical interest and aesthetic appeal. — Adler, Bazin & Schiffer, Introduction to General Relativity, 1975 Physics should be made as simple as possible, but not any simpler. — Einstein Although Weyl's 1918 metric gauge theory is now almost 100 years old it is still regularly resurrected in the literature, usually in the hope that something relevant to modern physics will spin out of it. This is undoubtedly due to the unmistakable beauty of the theory, which today arguably remains the only potentially viable route to a practical non-Riemannian theory of gravity. In the last lecture of Benjamin Schumacher's excellent Great Courses video series Black Holes, Tides and Curved Spacetime: Understanding Gravity, the Kenyon College professor talks about the next revolution in physics, which will undoubtedly be the unification of general relativity and quantum mechanics. He briefly describes string theory, M-theory and loop quantum gravity, all relatively new ideas that are mathematically beautiful and intriguing but totally lacking in any empirical evidence or experimental data to back them up. But he then mentions the very new idea of entropic gravity, which ties together quantum information theory and gravitation via a simple rubber band argument! It's fascinating, and I encourage you to borrow this great (but expensive) video series from your library (as I did). But back to Weyl. In a short and very readable paper released just last month, University of Connecticut physics professor Philip D. Mannheim shows by a clever argument that the gauge freedom of the massless Dirac Lagrangian, metric tensor, Dirac vierbein and electromagnetic potential can be "balanced out," effectively producing a purely Riemannian theory consistent with a local conformal invariance that comes out of the formalism for "free." Symmetry breaking of the Lagrangian can then be used to introduce a mass term. In Mannheim's view, demanding that gravity be conformal at the outset provides the basis for a very natural approach to both gravity and quantum physics. It was Weyl himself who in 1922 introduced the notion of conformal gravity with the conformally-invariant Lagrangian $$\sqrt{-g} \, C_{\mu\nu\alpha\beta} \,C^{\mu\nu\alpha\beta}$$ (where $$C$$ is the Weyl conformal tensor), which itself is the basis for this and much of Mannheim's other work. [Mannheim posted a more detailed version of the paper last year, which made explicit use of the imaginary Weyl potential $$iA_\mu$$ rather than $$A_\mu$$. Adler, Bazin and Schiffer noted in their 1975 book that while this simple substitution automatically explains the quantization conditions for electron orbits in the hydrogen atom, it also implies an imaginary fine structure constant $$\alpha$$. This problem can be tied to the seemingly unphysical variance of vector length in Weyl's original theory, which was convincingly overturned by Einstein himself.]
 Earth Day: Two Versions — Posted Friday, 22 April The planet earth from way up there is beautiful and blue And floating softly through a rainbow But when you touch down, things look different here $$\ldots$$ — Electric Light Orchestra, Mission (A World Record) On April 22, 1970 I was finishing my junior year at California State University Long Beach when I wandered into the Music Center auditorium to hear Jacques Cousteau talk about "Earth Day." As a chemistry major and ocean enthusiast (i.e., the beach), I thought I would attend. I was amazed to find thousands of fellow students there. Today Earth Day celebrates its 46th anniversary, and since that first day there has been a sea change (so to speak) in our knowledge of what human activities are doing to the planet's ecosystems (if you're a Republican you can stop reading now). Cousteau himself could not have imagined that rising atmospheric CO$$_2$$ levels could have such a devastating effect on the oceans he loved, particularly with regard to the effects of acidification and warming on coral reefs. Meanwhile, the amount of plastic and other floating debris (much of it indestructible) in our oceans represents a global disgrace — an alien civilization visiting our planet and scanning the amount of atmospheric, water and soil pollution would be amazed at how stupidly an otherwise supposedly advanced civilization could contaminate its own world. And it doesn't end there — there are roughly 200 million pieces of non-functioning space junk now in orbit around the planet, requiring careful consideration every time a rocket, shuttle or satellite is sent into Earth orbit, lest it get impacted and destroyed, thus becoming yet more space junk. Einstein once remarked that both the universe and human stupidity are infinite, and that the definition of insanity is doing the same thing over and over while expecting a different result. Today we're a lot more knowledgeable regarding the impacts our activities are having on Planet Earth, but our efforts to reverse, minimize or nullify those impacts have been relatively sparse and ineffective. Conversely, the world's humans seem to care a lot less about the impending death of their home planet than they do about the passing of admittedly talented but behaviorally insane and inconsequential cultural icons. For that reason, I am posting this rather more appropriate Earth Day graphic:
 Bottle and Beam — Posted Tuesday, 12 April For those of you who, like me, have wondered just exactly what the half-life of a free neutron is, this month's Scientific American has an interesting article describing two separate attempts to definitively nail down that aspect of this most basic and important particle. Undergraduate physics students learn early on that the half-life of the elementary muon has been very precisely measured to be $$2.1969811(22)\times 10^{-6}$$ s, mainly because the muon is often used in the unfolding of the time dilation effect in special relativity. But the neutron, present in the nuclei of all elements except elementary hydrogen, is almost always assumed to be stable and eternal. Well, it is stable in its bound state, but when isolated the neutron is actually unstable, decaying into a proton, an electron and an anti-electron neutrino: Most textbooks that even deal with neutron decay tend to report that its half-life is around 10 to 15 minutes, an uncertainty that is rather embarrassing to physicists, considering how important and fundamental the neutron is. The authors of the Scientific American article admit this embarrassment and, with each participating in two separate experiments, they have attempted to define the half-life as precisely as possible. The results of these experiments are most interesting. One approach is to put a bunch of neutrons in a bottle, wait for a while, and then count the ones that have not decayed. While the concept is simple and straightforward, the experiment is fraught with difficulties, mainly because neutrons, being electrically neutral, can pass through the bottle's walls. But those difficulties can largely be overcome with some ingenuity. The authors' other approach is to basically monitor a beam of neutrons along a conduit, then count the protons produced in their decay. Protons, being positively charged, cannot easily escape the conduit walls and are easily counted, but again there are difficulties in the experiment, requiring ingenuity in its design. After the scientists performed numerous experiments as carefully as possible, they found to their surprise that the results differed significantly. The difference in the measured half-life of the neutrons was about 9 seconds — the bottle approach gave a half-life of $$879.6\pm0.6$$ seconds, while the beam method gave a figure of $$888.0\pm2.1$$ seconds. This is actually a huge difference but, while the scientists were dismayed, they also considered the possibility that neutron decay, which takes place via the weak nuclear force, may actually involve some exotic (and exciting) new physical phenomenon not yet known to science. Still, of all the basic fundamental constants of Nature, none comes close to that of gravity in terms of measuring frustration. In spite of the fact that it's the most basic of Nature's forces, the gravitational constant $$G$$ has only been measured to the relatively imprecise value of $$6.67408(31)\times10^{-11}$$ N-m$$^2$$-kg$$^{-2}$$ (by comparison, Planck's constant is known to better than 9 decimal places). Even worse, the basic measurement approach for $$G$$ has not changed in over two hundred years — placing two heavy masses suspended by a thin fiber and measuring the tiny amount of rotation that the fiber undergoes when another heavy mass is brought next to one of the masses. For 21st-century physics, this is barely more sophisticated than what you might find in a high school science lab.
 Interestingness — Posted Sunday, 10 April UC Berkeley astronomy professor Alex Filippenko is a phenomenal scientist and lecturer, and his 50-hour, 96-part Great Courses video lecture series is well worth wading through (it took me several months to get through it). The last few lectures are the most enlightening, when he talks about the expansion of the universe, the role gravity played in its formation, and how tiny (1 part per several hundred thousand) density differences in the otherwise perfectly uniform structure of the very early universe ultimately led to the formation of stars and galaxies (and us). Indeed, without gravity (which we still do not fully understand) the universe would have been very boring — a homogenous soup of mostly hydrogen and helium gas, existing forever without doing anything. While I'm not a great fan of the series Through the Wormhole hosted by actor Morgan Freeman, one episode talked about the purpose of the universe and the meaning of it all. Even if you're devoutly religious, you have to admit that the teleology of existence cannot be easily explained. I consider answers like "The purpose of our lives is to become one with God and live in His countenance and love forever" to be naive, meaningless feel-good gibberish, and it's surprising how many people seem to think it's an answer at all. But the question of what exactly's going on here is perfectly justifiable. Contrary to what a lot of fundamentalists believe, the Second Law of Thermodynamics is not violated simply because complex systems come into existence. Those systems effectively borrow energy and entropy from elsewhere, creating "islands" of complexity that would otherwise appear to defy explanation, while overall the entire system strictly obeys $$dS \ge dE/T$$. Over the years I've thought about this many times, and the only answer I've been able to come up with is that, for some reason, the universe likes things to be interesting. But "interestingness" is a very subjective, human concept, and I'd be challenged if I had to quantify it in terms of some teleological purpose. It also gives "agency" to the universe, like a god or super-human computer programmer, and I'm uncomfortable with that. I can accept evolution, natural selection and random change without difficulty, but that hardly addresses the question of "why." Here's a clip from Lecture 93 of Filippenko's talks, showing how super-computer simulation allows us to understand how the observed filamentous structure of the universe probably occurred over the 13.8 billion years following the Big Bang. The "nodules" that also form are the galaxies and super-clusters of galaxies, the individual stars being too small to model. In 2014 a team of scientists at Los Alamos and Stanford completed the largest such effort to date, involving some 1.07 trillion particles in the computer simulation. All of these efforts perfectly match our observations of the universe at the largest scales. Filippenko notes at the end of the clip that the universe is probably not rotating (a good question would be "rotating with respect to what?) As a gift to Einstein on his 70th birthday in 1949, colleague Kurt Gödel found an exact solution to Einstein's gravitational field equations showing that time travel to the past was possible in a rotating universe. Again, interesting!
 Is It Ever Over? — Posted Friday, 8 April Astrophysicists have discovered a galactic black hole that's 17 billion times the mass of the Sun. Located a mere 200 million light-years from Earth, NGC 1600 (New General Catalog item 1600) is a bright but otherwise ordinary galaxy whose heart contains an extraordinarily massive black hole. Its event horizon is larger than our solar system, so an observer falling past the point of no return would not notice notice anything particularly odd except for the appearance of the stars in the sky, which would seem to be collected into a single bright point. At 17 billion $$M_{\odot}$$, the observer would not feel the crushing tidal forces of a much smaller black hole, and her journey to the singularity would be long enough to write a scientific paper about the trip. But could that paper ever be read by others on the outside? Until relatively recently, the answer was no — all information in a black hole was assumed to be destroyed forever. That was Stephen Hawking's position until 2004, when he admitted that the ideas of eternal information preservation espoused by Leonard Susskind and others were correct. Here's a brief video overview of Hawking's current thoughts on the matter:
 Quantum Immortality? — Posted Thursday, 31 March We observe the universe, and in doing so we create the very universe we observe. — Wheeler Frankfurt physicist Sabine Hossenfelder has an article in today's Aeon Magazine on the so-called information paradox that exists between quantum mechanics and Einstein's general theory of relativity. Basically, general relativity says that if a one-kilogram rock is tossed down a black hole, it just increases the hole's mass, while tossing a one-kilogram book or set of videos does exactly the same thing — the rock's "information content" is essentially nil, while the book and videos contain lots of information, but that information is lost forever in the black hole. On the other hand, the principle of quantum unitarity requires that a packet of mixed quantum states remains mixed forever unless acted upon by the unitarity operator $$e^{-iHt/\hbar}$$, meaning that the information content of books, videos and human beings cannot be lost from the universe when they fall into a black hole. Information-wise, a 1-kg rock and a 1-kg book are intrinsically distinct, and their information content cannot be destroyed by a black hole. Hossenfelder's article is interesting, but she notes that the information "conundrum" is based on general relativity's tendency to destroy information. Since we don't have a workable theory of quantum gravity, I believe that statement is either wrong or premature. A number of physicists believe that the information of an in-falling body is somehow preserved in outgoing Hawking radiation, or even transferred to another universe. At the same time, we don't really know how the unitarity operator affects a quantum system when wave function collapse occurs; it is spontaneous (meaning that the operator itself becomes meaningless during collapse) or is it merely rapid, in which case one has to wonder how a simple measurement by a conscious observer can possibly affect the operator. It's really neat how black holes today are increasingly being viewed as the one viable link between quantum physics and gravitation. Hossenfelder points out recent studies indicating that a black hole is essentially a Bose-Einstein condensate consisting of a soup of gravitons, the totality of which serves to preserve information forever. And just exactly where does information originate? I believe it's somehow tied to human consciousness and the myriad, near-inifinite number of observations we make, the decisions and actions we take based on those observations, and how these actions might affect the rest of the universe. Einstein once jokingly asked how the observations of a mouse could possibly induce the very existence of the moon, an inquiry that to him justified the Newtonian determinism of the universe while refuting the notion of quantum indeterminacy. But a mouse is not a sentient being, while humans are (I leave out Republicans, who are neither sentient nor human). By comparison, the late physicist John Archibald Wheeler once suggested that the universe exists so that observers could be created, and that those observers in turn made the universe possible — the ultimate circular logic, to be sure, but who's to say he was wrong? Hossenfelder goes on to posit the possibility that the universe is a gigantic quantum computer, inputting, processing and outputting information on a continuous basis. It's not an original idea, but it seems plausible. Imagine that you're a computer programmer, simulating the flow of water in a waterfall or the motion of several million particles. You may be using a supercomputer to do the calculations, and that computer may take days or months to complete a single time-step computation. But Nature does the same thing instantaneously and seemingly effortlessly. Exactly how does Nature do that? Perhaps that kind of thing is what the universe has been doing all along.
 Weyl and Hilbert — Posted Monday, 28 March Here's an exceedingly rare film of Hermann Weyl with his PhD advisor, mentor and close friend, the great German mathematician David Hilbert. At a conference in Paris in 1900, Hilbert famously proposed 23 mathematical problems that he hoped would be solved in the then-new century. A few of these problems were indeed solved, but only relatively recently. This video clip is taken from the PBS program celebrating the life of the American mathematician Julia Robinson, who spent the bulk of her professional career trying to prove Hilbert's tenth problem. She didn't quite make it, but her work paved the way for others. Hilbert's tenth problem basically asks whether it is possible to prove that, by any finite algorithm, a solution to a given $$n$$-order polynomial expression $$f(x_1,x_2, \ldots x_n) = 0$$ (with integer coefficients) can be expressed in terms of integers. The answer, which was finally proved in 1970, is no. The film was most likely made in or around 1930, when Weyl was appointed the chair of mathematics at the University of Göttingen to replace Hilbert, who retired that year.
 Weyl in Spain — Posted Monday, 21 March Here's a nice photo of Hermann and Hella Weyl taken in the summer of 1922 while they were visiting the ancient Moorish palace at the Alhambra in Granada, Spain. Weyl, already being considered the mathematical heir to his advisor and mentor, the great German mathematician David Hilbert, traveled frequently throughout Europe to lecture, attend conferences and to vacation with his wife, and by the late 1920s he was also visiting America to lecture. He once traveled as far west as Colorado, but the staff at the Einstein Papers Project at Caltech has no record of Weyl ever coming to Pasadena. (From the 2009 book Mind and Nature: Selected Writings on Philosophy, Mathematics and Physics by Hermann Weyl with the express permission of the University of Princeton Press. Please respect the copyright when sharing)
 Dark Energy — Posted Saturday, 12 March The standard form of Einstein's field equations is $$G^{\mu\nu} = R^{\mu\nu} - \frac{1}{2} g^{\mu\nu} R + \Lambda g^{\mu\nu} = \frac{8 \pi G}{c^4} T^{\mu\nu}$$ where the $$\Lambda$$ term is Einstein's famous cosmological constant, thought by many physicists to be entirely or at least partly responsible for the phenomenon known as dark energy, a mysterious force that appears to be causing the observed acceleration of the expansion of the universe. Interestingly, the cosmological constant is a true fudge factor that Einstein included in his field equations to actually halt the expansion and keep it a constant size, an assumption he subsequently called the greatest blunder in his life when Edwin Hubble discovered in 1929 that the universe was truly undergoing expansion. But with the proper sign $$\Lambda$$ acts as an accelerant, causing the universe to expand at an ever-increasing rate. In this month's issue of Scientific American, astrophysicists Adam Riess (co-recipient of the 2011 Nobel Prize in physics for the observed confirmation of the accelerating universe) and Mario Livio write about the "Puzzle of Dark Energy," it being a puzzle because no one really knows what the hell it is, what it's made of or where it comes from, although it appears to account for roughly 70% of all the mass-energy in the universe. The authors suggest three alternatives for dark energy: 1) the cosmological constant, which is believed to be very small but is nevertheless a significant contribution to the Einstein field equations; 2) quintessence, a kind of field that can change with time and position in space that could also result in the ultimate collapse of the universe; and 3) there is no dark energy at all, but simply an aspect of gravity that we don't yet really understand. In any of these alternatives, vacuum fluctuations due to quantum effects (particle creation and annihilation) could also be contributing to what is being observed. [The above graph summarizes the basis for Riess's Nobel Prize discovery. For a larger view, see my post dated 11 November 2013.] The only reason Einstein could stick a cosmological constant into his equations is due to the fact that both sides of the above equation must be have a vanishing divergence; that is, $$G_{\,\,\,||\nu}^{\mu\nu} = 0$$ where the double-bar subscript notation refers to covariant differentiation. In Riemannian geometry (which Einstein's theory is based upon), the metric tensor $$g^{\mu\nu}(x)$$ acts as a pure constant under covariant differentiation, a property that allowed Einstein to introduce $$\Lambda$$ in the first place. But this is not the case in most non-Riemannian geometries; for example, the geometry in Hermann Weyl's 1918 theory includes a metric tensor that does not have this property. On the other hand, the vanishing divergence of Einstein's equations is assumed to mimic the classical conservation condition for mass-energy, but true energy conservation in gravitation has always been an ambiguous issue — it can be shown to be conserved in some coordinate systems but not all. It would then appear that the zero-divergence property of the gravitational field is really not a motivating issue, especially considering the likelihood that quantum fluctuations don't obey anything but Heisenberg's uncertainty principle (which is not a classical conservation principle). Riess and Livio note that the next few years are likely to be a pivotal time for dark energy research, although what is really needed is a consistent theory of quantum gravity, a problem that is now approaching the one-hundred year mark, and one we don't seem to be getting any closer to.
 What We Should All Know About String Theory — Posted Friday, 11 March In June of last year Edward Witten of the Institute for Advanced Study gave a talk at the International Centre for Theoretical Sciences on an elementary approach to quantum gravity. Beginning with a toy theory in one dimension, Witten showed that by considering the metric tensor $$g_{\mu\nu}$$ to be a $$1\times 1$$ matrix the massive Klein-Gordon equation could be derived for both Euclidean and curved space, which represents kind of a first step toward quantum field theory. This one-dimensional approach is analogous to Feynman's diagrams, which consist of 1-D stick figures glued together at vertices, and Witten shows that this is essentially the basis for the unwieldy divergences that plague QFT. But in two dimensions the stick figures become tubes, which merge smoothly together at their vertices and thus avoid the infinities. This is the beginning of string theory. Witten shows, however, that this 2-D approach necessitates a $$2\times 2$$ metric tensor that must be conformally invariant, meaning that a rescaling transformation of the form $$g_{\mu\nu} \rightarrow \lambda(x) g_{\mu\nu}$$ cannot alter any of the underlying physics. Witten refers to these rescaling as Weyl transformations, which formed the basis for Hermann Weyl's 1918 unified field theory. Witten is considered by most physicists today to be the smartest man in the world (although that overused phrase tends to demean the man's vast contributions to both physics and mathematics) and the heir apparent to Einstein. His talk is posted on YouTube under the title "What every physicist should know about string theory." It's suitable for undergraduates, and is well worth watching (but use the full-screen option if you want to actually see the equations) :
 New Superconductor — Posted Tuesday, 8 March Here's an interesting compound: Sn$$_6$$Sb$$_6$$Ba$$_2$$MnCu$$_{13}$$O$$_{26}$$, a metallic ceramic that happens to become superconducting at 110 degrees Celsius, ten degrees higher than the boiling point of water. Are room-temperature superconductors, allowing for the everyday application of lossless power lines and magnetically levitated trains, not too very far in the future? I never took a class in solid-state physics (condensed-matter physics), so I can't offer any intelligible comments on this finding. But thanks anyway to Professor Golden Nyambuya of the National University of Science & Technology Department of Applied Physics for notifying me of this neat discovery.
 An All-Red America? I Sure Hope Not — Posted Tuesday, 8 March As we all know the Republican Party's establishment acolytes are ramping up pressure against front-runner Donald Trump, if only for the purpose of avoiding Trump's outrageous political positions from sticking to the GOP brand. But they generally dislike Ted Cruz, while Marco Rubio is barely hanging on by his political fingernails. John Kasich seems to be the most sane of all the candidates, but Republican voters aren't buying into him. So what's gonna happen? Without question, the GOP establishment is doing what it can to prevent Trump from accumulating the 1,237 delegates he needs to win the Republican nomination. He may indeed be the candidate with the most delegates, but if he doesn't get to 1,237 then all bets are off — the party elites are no longer beholden to him and can nominate anyone they want. That would be the brokered convention everyone's talking about, and while it's unconventional (excuse the pun) it's been done before. In this case, it's almost guaranteed if, by convention time, Hillary Clinton (or maybe Bernie Sanders) is considered all but ensured of winning the general election in November against Trump (or Cruz). If the GOP is successful in preventing Trump from getting the necessary number of delegates, I believe they'll almost certainly nominate former Wisconsin senator and current House Speaker Paul D. Ryan. Yes, he's said on numerous occasions that he would never accept the Speaker position, and yes, he's stated at least once that he would not pursue the presidency at this time, most likely preferring to defer it until 2020 or even 2024, when he would still be only 52. But with the Executive Office and a conservative Supreme Court makeup at risk and with the entire Congress already now in Republican hands, I think Ryan's forced nomination is all but guaranteed. And, like the house speakership, Ryan would almost certainly fall on his sword and accept being drafted come July 18-21, 2016. Ryan's primary objection to accepting the House speakership was based on his stated preference to spend more time with his family, a claim that may well have been legitimate, but a President Ryan would have his family with him in the White House, negating that issue. Also, George W. Bush managed to spend 533 vacation days away from the White House clearing brush on his ranch when he was president, an activity Ryan certainly wouldn't engage in. What's even scarier is that, with a united Republican Party at his back and little serious troublesome political baggage, he would most likely be elected in November.
 The Authoritarians — Posted Wednesday, 2 March When you come to a fork in the road, take it. — Yogi Berra Recently I've tried very hard to minimize my thoughts on politics and religion on this site, as they usually have nothing to do with physics or science, but there are times when I simply have to speak out. Presidential candidate Donald Trump's near sweep of yesterday's Super Tuesday primaries and caucuses did not surprise me, but at the same time how he seems to be leading among poor Americans and evangelicals comes more as a shock than a surprise. What might explain this? Some years ago I read the free online book The Authoritarians by Robert Altemayer, an associate professor of psychology at the University of Manitoba in Canada. The book examines why some people tend to support candidates and programs that would seem to go against their own best interests, along with a number of related issues common to both America and Canada, and Altemayer came up with the answer. In a nutshell, people tend toward authoritarian leaders out of fear. One might wonder why and how fear would cause some people to Not think that spousal abuse is a serious issue; Have a disproportionate craving or obsession for wealth and the security it appears to provide; Have overly conservative economic philosophies that don't benefit them or even have a negative impact; Be overly prejudicial regarding matters of race, religion, ethnicity, gender and sexual orientation; Reject paying for necessary across-the-board social programs in favor of increasing military spending; Reject proposals and laws that would raise taxes on the rich or lower them on the poor. Basically, I think it comes down to this: 1) If you're wealthy, then you don't want things to change because your wealth might decrease; 2) If you're not wealthy, you see a need to maintain the existing system (even if it's rigged) so that you may become wealthy through it; 3) If you're poor or in need, at least you're alive and able to dream about a day when things will get better, and changing things may make your situation even worse; and 4) The Bible says the poor will always be with us, so if you're poor that's just the way it is — you mustn't flout God's design. These are not exactly Altemayer's conclusions, but they're close. Notice that each involves a fear that changing the system may result in adversely affecting one's existing situation — better the devil you know than the one you don't. Nice, all-white crowd, but the mother with the baby really freaks me out. A more recent assessment of (American) authoritarianism was published yesterday by Amanda Taub from Vox, whose lengthy but penetrating article on authoritarianism in many ways resembles the update that Altemayer probably needs to provide for his ten-year-old book. While expanding on many of the issues, observations and conclusions that Altemayer addresses, Taub ominously notes that America today has a three-party system consisting of the Democrats, the GOP establishment and GOP authoritarians, implying that to prevent future fracturing of the GOP vote more and more Republicans will go the authoritarian route. Further, she asserts that this third party of authoritarians will persist and grow even if Trump, Cruz or Rubio are neither nominated nor elected in November. While Taub notes that GOP authoritarians threaten to tear the GOP apart, I see it just the other way round: instead, it bodes ill for the Democratic Party, since approaches such as science, reasoning, rationality, facts, evidence and truth have been useless tactics against the growing segment of Americans that is turning to authoritarianism in these troubled times of worldwide terrorism, climate disruption, environmental degradation, economic inequality and resource depletion. Ultimately, however, I blame increased American religious zealotry for the problem of authoritarianism. You can't think when you're afraid, and religion offers a way to cope with fear that doesn't require any thinking — you simply do what you're told and obey. Just as prayer is an act of desperation when you've exhausted all other options, religion provides that persistent last gasp of hope that the human race is so dependent on for its supposed sanity. It's no wonder then that most evangelicals in this country are Republicans, and it's similarly no surprise that increasing authoritarianism is coming from the Republican Party. I don't see an easy solution.
 Einstein Papers — Posted Tuesday, 1 March Here's a neat site where you can read the papers published by Einstein in the German journal Annalen der Physik (Annals of Physics). Presented by the Max Planck Institute for the History of Science in conjunction with Wiley-VCH, the papers are all in German but the math is readable and there are many notations in English that describe them (the site includes a search function but it's not working yet). Even better is the set of collected works (with English translation supplements) available for free online from the Princeton University Press. Volume 7 covers the years 1918-1921, when Einstein and Hermann Weyl communicated extensively on a range of subjects, including gravitation, cosmology, unified field theory and even philosophy.
 Linger Yet, Thou Art So Beautiful! — Posted Tuesday, 1 March My favorite classical opera was composed by Arrigo Boito, who wrote Mefistofele in 1868. It didn't do well at first (mainly because it was over five hours long), but Boito trimmed it down and it soon became a hit. It's rarely performed today, but in the 1950s and 1960s it was fairly popular. I just finished reading writer Brian Morgan's 2006 book Strange Child of Chaos, a biography of the great bass-baritone Norman Treigle (pronounced Tray-gull). Treigle began performing the part of the opera's devil in 1954 (along with the same part in the 1859 opera Faust by Charles Gounod) and his brilliant singing and acting quickly made him the part's signature player. But what I found most interesting to learn was that Treigle, who was born in New Orleans in 1927, was a devout Southern Baptist who also sang Christian songs and hymns throughout his life, yet was married twice, was something of a ladies' man and smoked four packs of cigarettes and a quart of scotch every day. A lifelong insomniac, he regularly abused sleeping tablets until he died of an accidental overdose in 1975 in London at the age of 47. Nearly six feet tall, he never weighed more than 138 pounds. But his voice was nothing short of miraculous. The book's title is taken from Dr. Faust's line Strano figlio del Caos, which he utters in response to Mefistofele's great "Whistle Aria." Every song in the opera is a gem, and the 1973 New York performance (directed by Julius Rudel and starring the great tenor Plácido Domingo) features the incomparable voice of Monserrat Caballé, perhaps the finest soprano of all time. But Treigle's Mefistofele steals the show. Here is the opera's final aria, All'erta!, in which the damned soul of Faust is saved by grace, accompanied by a deafening chorus of celestial beings, with the defeated Satan taking it all in stride but screaming Trionfa il Signor, ma il reprobo fischia! ("The Lord has triumphed, but the reprobate whistles!"), which he does to the very end.
 Do Not Attempt This At Home On Your Commodore 64 — Posted Tuesday, 16 February Two journals published by the American Physical Society, Physical Review (established in 1893) and Physical Review Letters (established in 1958), are the most prestigious physics journals in the world — every serious physicist wants to get published in them, but only a relative few are accepted. While abstracts of published papers are posted regularly by these journals, the full papers generally are not — unless the papers are of historic significance. Last week, Physical Review Letters posted the entire paper "Observation of Gravitational Waves from a Binary Black Hole Merger," which announced the now-famous detection of gravitational waves (you can download a copy of the paper here or go online, where you can find it just about everywhere now). While every major newspaper on the planet covered the discovery, I think what's nearly as significant as the detection effort itself is the computer programming that simulated the gravitational waves coming off a binary black hole merger. Most students of general relativity rarely get beyond the derivation of the Schwarzschild model, and maybe a few can actually understand the Kerr model of a single rotating black hole. But to take Einstein's equations and apply them to two spinning, co-orbiting black holes is a challenge I cannot even imagine. Yet, the scientists at Caltech and the other participating universities did just that. While detecting a tiny signal and confirming it with a predictive mathematical model is a pretty neat accomplishment in itself, the fact that the project's two detectors saw the exact same pattern separated by a time lapse consistent with the light speed of the passing gravitational wave (the Livingston, Louisiana detector and the Washington State detector in Hanford) just blows my mind: As the black holes converged (and this was 1.3 billion years ago!), the detected gravitational wave pattern was relatively quiet, and the corresponding computer model showed a bit less correlation. But as the holes merged, all three — the computer prediction and the two signals themselves — precisely agreed with one another. That final extended blip at the right end of the graphs is the "chirp" you've been reading about, but the simulated audio version simply doesn't do the science justice. Principal investigator B.P. Abbott from Caltech and his hundreds of colleagues (see the citation list at the end of the paper) are to be congratulated, and I cannot imagine the 2016 Nobel Prize in physics bypassing these guys.
 Quadratic Equations — Posted Tuesday, 16 February Here's a bit of nostalgic silliness. While disposing of a bunch of video files today I found this old episode of Dr. Science I recorded on VHS from the mid-1980s. Airing Saturdays on the old Fox channel, there were only about a dozen episodes, but my kids loved it — my older boy once even dressed up as Dr. Science for a school Halloween pageant. My kids always claimed that Dr. Science and I were spitting images of one another, but of course I was far better looking. Shot on the cheap at the campus of California State University at Dominguez Hills, Dr. Science enjoyed a short-lived vogue for a time. You can still watch the most popular episode The Dr. Science National Science Test on YouTube (originally broadcast on PBS), but the show is now largely forgotten by all except for idiots like me. Some years ago I tried contacting the show's originator, Dan Coffey, by email several times but he never answered. But apparently the Dr. Science radio show is still being broadcast somewhere back east, and there's even a website where you can submit questions and find out useful stuff like why bathroom water and kitchen water taste different to little kids in the middle of the night. (I just posted the episode Dr. Science Meets His Match (from which the above clip was taken) on YouTube.)
 Einstein and Unification — Posted Sunday, 14 February The late physicist Abraham Pais was a friend and colleague of Albert Einstein at the Institute for Advanced Study in Princeton and the author of numerous books. Born in Amsterdam in 1918, he received his PhD in physics from the University of Utrecht in 1941. A Jew, he was forced to hide from the Nazis until nearly the end of the war, when he was captured and sent to a Gestapo prison camp. Somehow he managed to avoid execution until VE Day, when he and the other surviving prisoners were released by the Allies. In Pais' 1982 book Subtle is the Lord: The Science and the Life of Albert Einstein, arguably the finest scientific biography of Einstein ever written, he writes: Then it came to pass that physics veered toward a different course, neither led nor followed by Einstein. First quantum mechanics and then quantum field theory took center stage. New forces had to be introduced. New particles were proposed and discovered. Amid all these developments, Einstein stayed with the unification of gravitation and electromagnetism, the final task he set himself. This insistence brought the ultimate degree of apartness to his life. [It] was not he but others who in the end ushered in the new physics. So it was to remain in the next decade, and the next and the next, until he laid down his pen and died. His work on unification was probably all in vain, but he had to pursue what seemed centrally important to him, and he was never afraid to do so. That was his destiny. Einstein's first "unification" paper appeared in 1922, but as Pais notes in the book he was certainly on the unification track far earlier, perhaps as early as 1917. Back then only two forces of nature were known, gravitation and electromagnetism, and the beauty of Einstein's 1915 general theory of relativity was so profound that many physicists simply knew in their hearts that the two forces could somehow be unified through the same mathematical formalism (similarly, the inverse square laws for both forces are essentially identical, and for many years before Einstein it was believed they must share a common basis). Hermann Weyl's 1918 unification theory was the first unified theory to appear, at least the first that appeared to really work. But, as I've noted here repeatedly, when the theory was shown to be unworkable (by Einstein, no less) he abandoned it and turned to other things. But Einstein either couldn't or wouldn't move on — perhaps his 1915 theory impressed him so much that he was certain it would yield even greater discoveries, and that he would be the one to do it. On 19 June 1945 Einstein, then 66 years of age, submitted a paper to the Annals of Mathematics on a new approach to unification that in retrospect seems to have had the same elements he stuck with until he died 10 years later. I am posting this nearly 71-year-old paper here because it introduces an interesting new aspect to the geometry of spacetime that contains at least the seeds of a formalism that could legitimately be tied to quantum mechanics. In it Einstein introduced a new metric tensor consisting of the sum of the ordinary symmetric metric tensor $$g_{\mu\nu}$$ and the antisymmetric electromagnetic tensor $$F_{\mu\nu}$$, $$\hat{g}_{\mu\nu} = g_{\mu\nu} + i F_{\mu\nu}$$ This new tensor is not symmetric but hermitian, which Einstein might have believed was somehow associated with quantum theory, and he proceeded to derive a set of field equations based on this new quantity. Unfortunately, Einstein's approach necessarily resulted in a non-symmetric connection term $$\Gamma_{\mu\nu}^\alpha$$ that he had to dance around mathematically to get something he deemed physically meaningful (in the end it failed, of course, but to Einstein there were really no failures, only ideas that didn't work). Still, he was sufficiently impressed with the basic idea that the final edition of his 1922 book The Meaning of Relativity, published posthumously in 1956, included an appendix he wrote entitled "Relativistic Theory of the Non-Symmetric Field." Perhaps even more telling is that on his 70th birthday in 1949 Einstein was presented a birthday cake (prepared by his long-time assistant Helen Dukas) decorated in icing with several equations taken from this non-symmetric theory (you can see a bigger photo of the cake on my posting dated 29 July 2015). [For reasons of completeness you might also want to read the sequel to this paper, which Einstein published in early 1946 with colleague Ernst Straus.] Although Einstein's notion of a hermitian metric tensor probably resulted simply from a desire to generalize his 1915 theory any way he could, there is a fundamentally good reason why it might actually have applicability in general relativity. I'm thinking I'll write this up myself, as I need something right now to take my mind off the fact that a leading Republican contender for the presidency (and an avowed devout Christian, natch) wants to carpet-bomb the Middle East and make the sands over there glow.
 Carroll on Gravity Waves and Locality — Posted Friday, 12 February Noted Caltech physicist Sean Carroll has an article in today's Atlantic where he talks about gravitational waves while also musing on the notion of "locality" in general relativity and quantum mechanics. I mentioned earlier Isaac Newton's dissatisfaction with the idea of "action at a distance," of which Carroll writes Newton himself never liked this feature of his own theory. It wasn't primarily the instantaneous speed of gravity, but the fact that the force seemed to propagate through empty space, rather than through a medium like the air. He found this "action at a distance" distasteful, but he left its ultimate solution "to the Consideration of my readers."Newton's inverse square force "law" of gravity does indeed imply instantaneous action but in 1687 Newton had no better understanding of gravity, so he was stuck with it. It's a testament to his genius that he was able to see action at a distance as a flaw that would ultimately be resolved by a better theory. That theory had to wait until 1915, when Einstein demonstrated that gravitation really was a "local" phenomenon that propagated at the finite speed of light — fast, yes, but not instantaneous. However, Carroll goes on to note that quantum mechanics is actually a non-local theory, at least with respect to quantum entanglement, implying that this non-locality really is a form of action at a distance (why the stupid religious fundamentalists over at Conservapedia don't latch onto that aspect of relativity escapes me, but then they're a bunch of idiots anyway). Carroll puts off any discussion on the matter for a later time, but could the infamous incompatibility between general relativity and quantum physics be tied in some way to entanglement? People like Leonard Susskind think so, and it may be no coincidence that black holes today are looking to be the gateway to a unified theory of gravitation and quantum mechanics. Indeed, I think it's highly suggestive that yesterday's announcement of the detection of gravitational radiation involves the violent merger of two black holes some 1.3 billion years ago, a finding that, if indeed true, seems to also make things like time and spacial separation vague if not meaningless concepts. Quantum entanglement simply means that if you make a measurement on one entangled particle the wave function of its partner also collapses, since they're both in the exact same quantum system. The "problem" of non-locality arises only when one considers that the particles might be in different galaxies, separated by billions of light-years, which inescapably invokes our human notions of simultaneity of measurement across space and the erroneous view that one particle must somehow send out an instantaneous signal to the other when it's observed. Yes, it's completely nonintuitive, but we humans aren't nearly as smart as we need to be to understand this stuff. I believe this explains why Hermann Weyl's ideas on the possible irrelevance of spacial scales and distances impressed me so much back in the 1970s. And if the concept of distance is irrelevant, then the concept of time itself must also be irrelevant. A physical theory that does not involve spacetime at all might be the right theory, but who the hell can even imagine what that entails? Gravity is a wave, Rodney, and I'm going to surf that wave. — Dr. Science (Dan Coffey)
 The First — Posted Friday, 12 February Ann Mayes Rutledge (1813-1835) is generally acknowledged as Abraham Lincoln's first (and perhaps only) true love, her life tragically cut short when the young woman died of the "milk sick" at the age of only 22. Lincoln's friends said he never really got over it, and they attributed his lifelong melancholy not to the travails of the Civil War or his battles over wife Mary Todd Lincoln's spending but to the death of someone near and dear to him 30 years earlier. Lincoln's love life is a fascinating story of fits and starts, replete with tales of indecision, awkwardness, shyness and even hints of homosexuality. A year after Ann's death Lincoln found himself engaged to Mary Owens, a woman he barely knew and clearly did not even like. It was apparently a set-up situation through a mutual female friend, who introduced Owens to Lincoln in 1833. For whatever reason Lincoln agreed to marry her, but after an extended, long-distance mail correspondence with the woman Lincoln practically begged to be released from his promise. When he saw her in the flesh again some years later, she had grown so fat and unattractive that Lincoln remarked in an 1836 letter to a friend "I knew she was over-size, but she now appeared a fair match for [Shakespeare's] Falstaff $$\ldots$$ when I beheld her, I could not for my life avoid thinking of my mother; and this, not from withered features, for her skin was too full of fat to permit its contracting into wrinkles, but from her want of teeth, weather-beaten appearance in general, and from a kind of notion that ran in my head that nothing could have commenced at the age of infancy and reached its present bulk in less than thirty-five or forty years; and, in short, I was not at all pleased with her." (I nearly always die laughing when I read this). A similar situation presented itself in the period 1840-1842, when Lincoln met and then married Mary Ann Todd, though Lincoln's apprehensions were less pronounced, if only because this Mary was from a wealthy family and she was even better educated than he at the time. Still, another close friend described Lincoln's relationship with Mary Todd as "a burning, scorching hell, as terrible as death and as gloomy as the grave." But, as they said then and still say today, "Lincoln soldiered on." Though somewhat uncomfortable around women, Lincoln had a great sense of humor and was a famous teller of off-color jokes and stories, some of which would raise eyebrows even today, though I find it difficult to imagine Lincoln and Mary having much of a sex life (Abe: Brace yerself, Mary! Mary: Oh, Abe!) I mention this story because of an article I read today by Ellen McCarthy, a writer for the Washington Post, and it struck a nerve. It deals with that first real brush of adoration we all experience, usually at a post-adolescent but still impressionable age. For me it was in Mrs. Farrell's French II class in my sophomore year in high school (and no, it wasn't Mrs. Farrell herself, who I spoke of in my 1 March 2013 post, and it wasn't Shirley or Bonnie or Joanne). Still, 51 years later and counting, I still think about that girl classmate with the golden hair. "I think it's not just about the other person. It's about who we were at that time. We're relishing the image of ourselves. They give us license to be the person we were once again — young and vibrant and beautiful." — Jefferson Singer I think Connecticut College psychologist Singer's analysis is correct, although I can never recall a time when I was "young and vibrant and beautiful" — maybe just young. And, for whatever reason, when I read the article my mind went straight to Lincoln and Ann Rutledge. Ann has a new headstone now, but the original was inscribed with this poem by Edgar Lee Masters: Out of me unworthy and unknown The vibrations of deathless music: "With malice toward none, with charity toward all." Out of me the forgiveness of millions toward millions, And the beneficent face of a nation Shining with justice and truth. I am Ann Rutledge who sleep beneath these weeds. Beloved in life of Abraham Lincoln, Wedded to him, not through union, But through separation. Bloom forever, O Republic, From the dust of my bosom! Memory, especially long-term memory, can be both a blessing and a curse. In the immortal words of yet another poet, England's great Walter de la Mare: Here lies a most beautiful lady, Light of step and heart was she; I think she was the most beautiful lady That ever was in the West Country. But beauty vanishes, beauty passes; However rare — rare it be; And when I crumble, who will remember This lady of the West Country?
 Joseph Weber — Posted Thursday, 11 February In my 5 February post I mentioned an article written by UC Irvine astronomer Virginia Trimble (please see this link to the 1962 article "Behind a lovely face, a 180 I.Q.") She just happens to be the widow of Joseph Weber, largely recognized as the father of gravitational wave detection. He died in 2000, and I like to think he would have been tickled by today's news regarding the confirmed detection of gravitational waves. Weber's approach was to construct large cylinders of aluminum outfitted with sensitive strain gauges, a design that he hoped would be able to pick up the faint signal of a passing gravitational wave. That design failed, if only because today's announcement indicated that a gravitational wave would produce a deformation in spacetime smaller in size than the diameter of a proton. The LIGO experiment that successfully detected gravitational waves relied on interferometry, an insanely-sensitive approach equivalent to that used in the 1880s by Michelson and Morley to dispel the notion of the "luminiferous aether." That was also a ground-breaking discovery, as it conclusively demonstrated that the velocity of light is independent of the velocity of the observer. Einstein's 1905 theory of special relativity relied on that independence, and it quickly became a cornerstone of modern physics. I will now reiterate a comment I have made numerous times on this site: fundamentalist religion rejects all of Einstein's theories precisely because those theories contradict the notion of "action at a distance," which Christian theologians have relied upon for 2,000 years to enable the instantaneous action of God's miracles. The concept of action at a distance resulted when Isaac Newton developed his theory of gravitation, but he was reluctant to accept it because he could not understand how a physical phenomenon could occur instantaneously. I again direct my readers to the Christian website Conservapedia, a highly influential website created by conservative writer and author Phyllis Schlafly and her son Andrew that purports to overturn science (especially evolution and relativity) in favor of fundamentalist religion.
 Gravitational Wave Announcement — Posted Thursday, 11 February What happens when two black holes having masses of 29 and 36 times that of our Sun rotate about one another? They blow off energy in the form of gravitational waves en route to eventually merging into a single black hole of 62 solar masses, the mass difference having been converted into gravitational radiation. At least that's the scenario about to be reported by the LIGO (laser interferometer gravitational-wave observatory) team. Official announcement of the LIGO finding is imminent, and if it bears out then science will have a truly major discovery, one that goes far beyond mere confirmation of the existence of gravitational waves, which physicists have been pretty sure of since Einstein predicted them in 1916. Indirect (but highly precise) evidence of energy loss due to gravitational radiation in a binary pulsar system led to the 1993 Nobel Prize in physics. If the current discovery holds up, it will likely win another prize. We'll see. Update: It's been confirmed (see also today's New York Times, which has a great video explaining the discovery). When all the shouting dies down we'll find out exactly what they discovered. A great day for science!
 Gravitational Waves Discovered? — Posted Tuesday, 9 February "Tidal" effect of a passing gravitational wave. From Adler, Bazin and Schiffer, 1975. Southern California has been laboring under the apparent delusion that the "Mother of All El Niños" would be drenching us non-stop with record rains, and I admit that I was also caught up in the hope that the four-year drought here might finally be broken. But the facts appear to belie the hype, much to SoCal's great disappointment. In recent weeks I've blogged repeatedly about the tanned, wide-stanced, action-figure television weather forecasters who constantly praise SoCal's warm weather and cloudless, sunny clear skies while surreptiously avoiding the uncomfortable certainty that we're headed into a fifth year of record-setting drought. For this reason I'm hesitant to embrace recent reports that gravitational waves have finally been discovered, with a five-sigma reliability to boot. While even noted mathematical physicist John Baez is hopeful that Einstein's as-yet unproven prediction has finally been proved, I remain unconvinced. In 1922 Hermann Weyl examined the mathematics behind gravitational waves in the context of the weak-field limit ($$g_{\mu\nu} \rightarrow \eta_{\mu\nu} + \gamma_{\mu\nu}$$) of Einstein's gravitational field equations (others, including Einstein, also examined the effect). Because the field equations are highly non-linear, only an approximate solution is possible, which indicates that gravitational waves are essentially akin to electromagnetic waves propagating at the speed of light. Anyway, the weak-field limit predicts that separated test particles will experience a very slight physical displacement when a gravitational wave passes by. This displacement is exceedingly tiny, yet recent LIGO results indicate that such a displacement has indeed been detected. The result will be published this Thursday in the journal Science. The experimental search for gravitational waves has been ongoing for nearly fifty years. I hope this settles it.
 Weyl and Noether — Posted Tuesday, 9 February Here's the photo I alluded to earlier this year. It's summer, 1932, at the Gasthof Vollbrecht in the little town of Nikolausberg, a little outside of Göttingen, Germany, where Hermann Weyl found himself the chair of mathematics at the university following the retirement of his mentor, the great German mathematician David Hilbert. To the immediate left we have the very young German mathematician Ernst Witt, beside him the Swiss mathematician Paul Bernays, and then Hella and Hermann Weyl with their 17-year-old older son Joachim, who also became a noted mathematician. The chunky woman is none other than Emmy Noether (pronounced nuhr'tah), arguably the greatest female mathematician who ever lived. The attractive Mädchen on the far right is "E. Bannow", whose identity I have been unable to determine. [On 12 February I received an email from Malcolm Douglas informing me that the young lady is Erna Bannow, Ernst Witt's first doctoral student. Born in 1911, she got her PhD under Witt in 1939, and the two married in 1940. While flanking the group in the photo, I can't help wondering if they already had designs on one another! I also learned that Erna was the first to sign a 14-page petition to the German authorities to allow Noether to remain as a teacher: The last sentence reads "This is the reason why we would appreciate it if Frau Prof. Noether, [whose reputation] stands alone in Germany, was given the opportunity to continue to exercise her activities as a teacher." The petition failed, and Noether was banished from teaching in Germany.] Weyl and Noether both emigrated to the United States in 1933 to escape Hitler, although Noether dallied a bit before leaving. As noted, Noether was peremptorily fired from teaching due to her being a Jew, and she was reduced to teaching informally from her apartment until she finally left for America. Even then she could only find a half-salaried position at Bryn Mawr University in Pennsylvania, where she died unexpectedly from a malignant ovarian tumor in 1935 at the age of only 53. Einstein and Weyl presided over her funeral, where Weyl nobly accredited her mathematical genius as far superior to his. (From the 2009 book Mind and Nature: Selected Writings on Philosophy, Mathematics and Physics by Hermann Weyl with the express permission of the University of Princeton Press. Please respect the copyright when sharing)
 Turtles All The Way Down? — Posted Friday, 5 February When I first heard the following story I thought it was attributed to Richard Feynman, but Stephen Hawking notes otherwise: A well-known scientist (some say it was Bertrand Russell) once gave a public lecture on astronomy. He described how the Earth orbits around the Sun and how the Sun, in turn, orbits around the center of a vast collection of stars called our galaxy. At the end of the lecture, a little old lady at the back of the room got up and said: "What you have told us is rubbish! The world is really a flat plate supported on the back of a giant tortoise." Upon hearing this, the scientist flashed a superior smile and replied, "Well, then what is the tortoise standing on?" "You're very clever, young man, very clever," said the old lady. "But it's turtles all the way down!" This story is mentioned by University of Irvine astronomer Virginia Trimble in an article she wrote in 2013 on physicist Julian Schwinger (shown) entitled Gravity Before Einstein and Schwinger Before Gravity. Schwinger was the co-recipient of the 1965 Nobel Prize in physics (for his contributions to the theory of quantum electrodynamics), along with Feynman and Sin-Itiro Tomonaga, and Trimble's essay notes on more than one occasion the man's eccentricities, the most noteworthy, to me, being Schwinger's fascination with Green's functions — and anyone who has studied Green's functions in detail knows they are to be avoided like the plague. Schwinger's work was perhaps the most erudite, formal, well-written and mathematically thought-out of any physicist, but as Trimble notes his work was also almost unintelligible, even to his colleagues. I once owned a book, which I donated to Caltech, entitled Selected Papers on Quantum Electrodynamics, and the papers by Schwinger were indeed all erudite, formal and well-written, as well as completely unintelligible. I stumbled upon Trimble's article because I was looking into Schwinger's ideas on gravity and also considering buying the book Fields of Color: The Theory That Escaped Einstein by retired Harvard physicist Rodney A. Brooks. Brooks was a student of Schwinger, and the main thing he came away with was Schwinger's belief that all of reality (or at least most of it) is described by either classical or quantum fields, with particles representing merely highly localized fluctuations of those fields. The "everything is a field" concept might indeed actually underlie all of reality, but our brains are too puny to conceptualize anything other than particles and the matter they're made of. (After watching several of Brooks' YouTube video lectures and reading reviews of his otherwise excellent book, I opted not to buy it because it's too elementary, though you might find it useful.) Here's a neat photo. It's Schwinger's gravestone in Cambridge, Massachusetts, emblazoned with the reduced fine structure constant, which he used on many occasions in his quantum field theory calculations: By the way, Schwinger came from a Jewish family, and it still astonishes me that so many of our greatest scientific thinkers were and are Jewish. In 2013 the noted astrophysicist Neil deGrasse Tyson was featured in a short YouTube video talk in which he notes how, in spite of the many profound physical and mathematical discoveries that were made by the early Arabs, in or around 800-1100 CE Islamic theologians came to believe that physics and mathematics were actually the work of Satan. With few exceptions, Tyson notes, progress in science and mathematics was then halted in Islam-ruled countries with the result that today, some 900 years later, there are no longer any significant discoveries being made by Muslims (I'm sure I'll get some angry emails in response to this). Tyson's talk is only about 10 minutes long —
 MOND — Posted Monday, 1 February Frankfurt theoretical physicist Sabine Hossenfelder has an interesting article in this week's online Aeon magazine that talks about the dark matter problem. As for what that problem is and why it's so important to physicists today can be gotten from the article itself, but it raises the potentially greater problem of what gravity is and how we've looked at it over the last 100 years since Einstein brought it into its modern, post-Newtonian form. Hossenfelder notes that the failure to date for experimentalists to irrefutably detect a single dark matter particle may be due to the fact that it's not a particle problem but a gravity problem. She cites a recent paper by Lasha Berezhiani and Justin Khoury of the University of Pennsylvania pointing to the possibility that dark matter is essentially a superfluid that exists thanks to the extreme coldness of interstellar space. The lengthy (44 pages) paper infers that it's undetectable because our Solar System is basically too warm for its gravitational effects to be noticed in our neighborhood, although its effects are noticeable in the dynamics of other galaxies. In this respect, Hossenfelder claims that the superfluidity of dark matter allows it to mimic gravitational effects, thus making the Berezhiani/Khoury idea a kind of modified Newtonian dynamics (MOND) theory. I've skimmed over the latter's 2015 paper, and while I haven't fully digested it yet I am confused — dark matter either exists or it doesn't, as Hossenfelder also notes, but if it does then I don't see it as being any kind of MOND theory at all. In 2009, John Moffat of the Perimeter Institute for Theoretical Physics in Canada published Reinventing Gravity: A Physicist Goes Beyond Einstein, in which he posits a true MOND theory that makes certain adjustments to Einstein's 1915 gravity theory to fit observations. However, like other theories that impose various tensor, vector, spinor and scalar components into a suitable action Lagrangian density for gravity, Moffat's theory fits the observational data well but only at the cost of introducing a lot of parameters that reduces it to an exercise in mere curve fitting. And while I haven't completely read the Berezhiani/Khoury paper, I suspect it might be something along the same lines. Conversely, I still prefer the approach first proposed some years ago by Mannheim and Kazanas, which to me represents a true MOND theory, although it utilizes Hermann Weyl's conformal tensor as the starting point. You might also want to look at this paper by the same folks (to see the entire paper, click on 'print this article', which will bring up the entire paper. I dare you to check the calculations in Appendix A, which are undoubtedly the work of some poor grad student). I hope time will tell who's right in the end.
 Pure Math is a Bitch — Posted Wednesday, 27 January I occasionally get questions about some aspect of Hermann Weyl's mathematics. Although Weyl's favorite topic was group theory, a subject that has deep and profound applications to modern physics, much of his work involved pure mathematics. Unfortunately, I understand almost none of it. While I took numerous courses in graduate-level mathematics at university, it was all what might be called "physics math," often called applied math. I did take one class in pure mathematics dealing with the calculus of variations, but it was taught by a pure mathematician using a math language that I felt very uncomfortable with — lots of stuff about unions, intersections and categories, along with symbols I had never seen before, much less used. So while I can follow Weyl's physics math pretty well, his pure math has always been undecipherable to me. And that's why I never talk about it here! (And it's not just because I'm an idiot — Paul Dirac, arguably the greatest physicist who ever lived, never understood Weyl's math, either. To see the story behind this, see the first page of my paper Weyl's Spinor and Dirac's Equation.) Before winning the Nobel Prize in physics with T. D. Lee in 1957 for the discovery of non-conservation of parity, C. N. Yang worked with Robert Mills on nonabelian extensions of Weyl's 1918 gauge theory, which in 1929 was found to underlie all modern quantum theories. Today, Yang-Mills theory underlies the Standard Model's explanation of electroweak unification and quantum chromodynamics, although in the early 1950s Yang had few notions as to how successful it would turn out to be, and he had no idea that Weyl had started it all. Anyway, Yang made a statement back in 1980 that endeared him to me forever: "If you try to understand fibre bundles [an abstract view of modern gauge theory] by reading mathematics, you would probably not succeed, because modern mathematics is extremely difficult to read, and I believe that there exist only two kinds of modern mathematics books: one which you cannot read beyond the first page, and one which you cannot read beyond the first sentence."Yang is 93 years old now, and lives in China. Interestingly, after Weyl retired from the Institute for Advanced Study in Princeton he sold his house to Yang, where the latter lived from 1957 to 1966. It's located at 284 Mercer Street (a few blocks from the house Einstein lived in), and you can see a larger photo of the house in my post dated September 17, 2011.
 On the Beach, Again — Posted Wednesday, 27 January Another beach photo, this time taken while on a trip to Skagen, a tourist town at the northern-most tip of Denmark. It was probably snapped in the summer of 1951 when Weyl was 65, about the time of his second marriage to the noted artist and sculptor Ellen Bär and around the time of his retirement from the Institute for Advanced Study in Princeton. Widowed in 1948, Weyl and Ellen split their time between Princeton and Zürich until his unexpected death in December 1955. Ellen was born in Switzerland in 1902 and passed away in 1998 at the age of 96. (From the 2009 book Mind and Nature: Selected Writings on Philosophy, Mathematics and Physics by Hermann Weyl with the express permission of the University of Princeton Press. Please respect the copyright when sharing)
 Black Planet, or Pluto's Revenge — Posted Friday, 22 January Why do astrophysicists believe black holes exist, even though they've never been directly observed? Well, imagine that our Sun somehow collapsed into a black hole. The planets in our solar system would continue to revolve about the Sun as usual, only over time they'd get much colder, to be sure. An observer from a distant stellar system who happened to be tracking Jupiter would see something strange: a large planet orbiting a point in space where nothing appeared to exist. That observer would almost certainly deduce that the central object was a black hole. And that's essentially how scientists know that black holes exist, because they've observed exactly this behavior with extrasolar planets, stars and nebulae. Researchers Konstantin Batygin and Mike Brown at Caltech recently observed similar behavior with observable objects in the Kuiper Belt beyond the orbit of Neptune. The slight variations in the orbits of these objects (mostly large, visible icy debris) intrigued the scientists, and after carefully accounting for the perturbative gravitational effects of the Jupiter, Saturn, Uranus and Neptune they deduced that there's another planet lurking out there that's roughly 10 times the mass of Earth and from two to four times its diameter. So why hasn't it been detected before? Batygin and Brown say it's due to the fact the the orbital eccentricity of the planet is so great that its orbit takes 10,000 to 20,000 years to complete one revolution around the Sun, far longer than astronomers have been around. Provisionally dubbed "Planet Nine," the scientists are pretty positive that it's really out there. Time, and further studies, should decide the planet's existence once and for all. An important fact that the linked article does not mention is that more than 200 years ago astronomers observed a similar discrepancy in the orbit of Mercury, the planet closest to the Sun. After laboriously calculating the perturbative gravitational effects of all the other known planets (this much have been a real bitch, as all calculations had to be done by hand), they could not account for the slight observable discrepancy in Mercury's orbit using Newtonian physics. Despite repeated observations and calculations, the discrepancy persisted, leading scientists of the day to conclude that either Newtonian physics was wrong or there was another planet orbiting the Sun that had never been observed. Newton's law of gravitation at the time was sacrosanct, so astronomers began to look for a new orbital body they called "Vulcan." They calculated that if Vulcan orbited the Sun in a manner exactly opposite to Earth's orbit, it would forever lie behind the Sun and thus be unobservable (this exact problem came up on a physics final I took as a graduate student many years ago). It wasn't until Einstein came along in 1915 that the discrepancy in Mercury's orbit was explained, thus killing off the Vulcan theory. In a calculation he made in 1916, Einstein discovered that his general theory of relativity exactly explained the discrepancy. In his theory planets do not orbit the Sun in perfect ellipses, as Newton's physics predicted, but in ellipses that precess over time. Shortly after he made this discovery, Einstein wrote to a friend that he was so deliriously happy that he was unable to sleep for several days. Nevertheless, lay people at the time did not appreciate the enormity of Einstein's discovery. But in 1919 another of Einstein's predictions, the deflection of starlight by gravitating masses, was definitively observed, making him a scientific superstar overnight. Upon hearing of the Caltech finding, my son Kris immediately dashed off this little cartoon, which provides a scarier aspect to Pluto's being downgraded from planet to run-of-the-mill planetoid:
 Palin and Riefenstahl — Posted Thursday, 21 January I posted this photo on my site back in 2010, but I still think it's appropriate considering how the political zombie Sarah Palin has risen yet again from the grave of irrelevance to plague America, this time thanks to her public support of GOP presidential candidate Donald "Two Corinthians" Trump. "Momma, when I am prezdint I wanna nucular war this big!" (Odd how Text Edit keeps wanting to spell it "nuclear." Must be one of them Democrat Party word processors.) No, Ms. Palin did not blame President Obama for Trig's Down syndrome, but she is blaming Obama for older son Track's recent arrest for domestic abuse. Similarly, she didn't blame Obama for her and hubby Todd's 2008 legal problems, presumably only because they occurred before Obama's election that year, and even Sarah has some concept of and appreciation for physical and temporal causality. Meanwhile, Sarah's daughter Bristol (the noted Christian abstinence spokesperson) recently had yet another child out of wedlock. Well, as King Claudius put it in Hamlet, "When sorrows come, they come not single spies, but in battalions." But I believe we can legitimately blame Obama for Sarah's recent resurrection from the dead if only because the GOP, having now gone full-blown fucking insane, would not even consider running the likes of Donald Trump, Ted Cruz, Marco Rubio, Bobby Jindal or Mike Huckabee as 2016 presidential candidates had not Obama thoroughly messed with their minds. Even worse, it's becoming ever clearer that the GOP will have to contend with Trump being their nominee, meaning they'll also have to somehow completely ignore everything Jesus supposedly taught them in the Gospel of Matthew regarding greed, pride and hypocrisy. We can blame Obama for that, too. While Obama's to blame for everything wrong in the observable universe (and in several nearby parallel universes), the media somehow neglected to adequately cover candidates Cruz, Huckabee and Jindal in their appearance at one of those "Freedom and Faith" rallies that the Republicans are so fond of holding. Watch how noted Colorado pastor Kevin Swanson promotes his belief in the actual ritual killing of sinners with accomplices guests Cruz, Huckabee and Jindal in tow, who show no signs of discomfort over Swanson's insane rantings: Somehow this reminds me of the 1935 film Triumph des Willens (Triumph of the Will) by Leni Riefenstahl, the gifted and beautiful German film director and Nazi propagandist whose work could legitimately be compared to what the equally beautiful but brainless Sarah Palin is doing for die Partei today. And yet, nearly 50% of Americans today are Republican or Republican-leaning. Hell, they would nominate Heinrich Himmler if it meant defeating Clinton or Sanders. God help us.
 Being Number Two Ain't So Bad — Posted Friday, 15 January Japanese astrophysicists believe they've found the second largest black hole in our galaxy. They deduced the presence of a black hole from the range of observed velocities in the gas cloud known as CO-0.40-0.22, estimating the hole's mass to be that of about 100,000 Suns (100,000 $$M_ \odot$$). The cloud is located only about 200 light-years from the galactic center, almost a literal stone's throw from the biggest known black hole in the Milky Way (called Sagittarius A*), a monster of some 4 million $$M_\odot$$. Number Two is not in the cloud itself, but was deduced from observed extreme Doppler shifts of the emission spectra of gaseous hydrogen cyanide molecules. The data are consistent with a nearby black hole attracting the gases and then flinging them past the hole at high velocity in accordance with standard orbital "slingshot" physics. Interest in black holes has grown substantially over the past few decades. Originally dismissed by Einstein and others in the 1920s, by the 1960s it was apparent that no known physical or nuclear force could prevent their formation given enough "starting" mass. More recent observations appear to show that a massive black hole lies at the center of all galaxies, and there seems to be a very fundamental connection between black holes and the evolution of their host galaxies from nebulous gas clouds. Recent studies also indicate that there is a strong correlation between the mass of a galaxy and its central black hole, and just last month a group of astrophysicists calculated that black holes may have a theoretical limit of about 50 billion $$M_\odot$$. Earlier I posted a video link to a lecture Leonard Susskind gave recently on the connection of black holes with quantum physics, entropy and information theory. It is entirely possible that the birth, evolution and fate of the universe are tied to black holes in some fundamental way we don't yet understand. In addition, according to Susskind and others, the preservation of the principle of quantum unitarity (specifically, that information cannot be destroyed) may depend on black holes in one universe serving as portals to other universes. Fascinating.
 Waiting for Einstein — Posted Thursday, 14 January During a visit to the optometrist yesterday I took along my copy of Anthony Zee's Einstein Gravity in a Nutshell (unquestionably the best book of its kind), intending to read the chapter on Kaluza-Klein theory while waiting for the pupil-dilation drug to take effect. I was delighted to discover that the optometrist was a physics and math major in college, and he immediately recognized the book I was reading. A nice discussion followed, and we both noted that finding a kindred spirit in such an accidental manner is a rare but welcome occurrence. Anyway, in the chapter Zee discusses how Einstein reacted to the work of Hermann Weyl and Theodor Kaluza (shown) regarding their ideas on the unification of gravity and electromagnetism, and he quotes from some of the Einstein letters that Kaluza's son had preserved. April 21, 1919: Einstein writes "The idea [of unifying electromagnetism with gravity] has also frequently and persistently haunted me. The idea, however, that this can be achieved through a five dimensional cylinder-world has never occurred to me and would seem to be altogether new. I like your idea at first sight very much. From a physical point of view it appears to me more promising than the mathematically so penetrating ansatz of Weyl, because it concerns itself with the electric field and not with the, in my opinion, physically meaningless four-potential." (Zee notes that Einstein was completely wrong in this last sentence, as the four-potential is of far more fundamental importance than the electric and magnetic fields.) April 28, 1919: Einstein starts with "I have read through your paper and find it really interesting" but then adds "the arguments $$\dots$$ do not appear convincing enough." Zee goes on to note that here Einstein starts hedging, as he will ultimately (and unnecessarily) delay the publication of Kaluza's paper for two years. May 29, 1919: "It is true that I made a blunder with [some remark Einstein made in a previous letter] $$\dots$$ I see that you thought the matter over quite carefully. I have great respect for the beauty and boldness of your idea. But you will understand that I cannot take side with it as originally planned given the present factual doubts." More delays. October 14, 1921: Einstein again writes to Kaluza, saying "I am having second thoughts about having restrained you from publishing your idea on a unification of gravitation and electricity two years ago. Your approach seems in any case to have more to it than the one by H. Weyl. If you wish I shall present your paper to the academy after all, provided you send it to me." At last Einstein condescends to advise publication, but what I find remarkable is that Einstein seems to have misplaced Kaluza's original paper! Considering the gravity (no pun intended) of Kaluza's highly original and influential idea (today it's the basis of string theory and all higher-dimensional field theories), it's amazing that Kaluza, a fellow German, didn't tell Einstein to just go to hell and find someone else to promote his paper. And I find the "If you wish" part of Einstein's remark in his last letter to be particularly infuriating. What I find also remarkable is that Kaluza was born on the same day as Weyl, 9 November 1885, although he died a year earlier in 1954.
 Mr. Cool — Posted Monday, 11 January A few pictures. The first has Hermann Weyl on a seesaw at a Gasthaus in Nikolausberg, near Göttingen, Germany in 1932. In the second photo, Weyl and his wife Hella are at a beach in New Jersey in 1937, some four years after their emigration to America from the Nazis. I find it odd that anyone would visit the seashore dressed in a suit and carrying a briefcase, but times were different then (I can still remember neighbors mowing their lawns in the 1950s wearing slacks and dress shoes à la Ward Cleaver. Just what the hell was wrong with people in those days?!) Weyl was a true German intellectual in the classical sense, and he loved teaching in Switzerland (1913-1930) and even in Göttingen (1930-1933), where he took over the mathematics chair when the great German mathematician David Hilbert retired. When offered a position at the then-new Institute for Advanced Study (IAS) in Princeton, Weyl initially accepted. But he spoke little English, and he detested the thought of leaving the cultured academic life he had known for a country he had visited several times but still felt uncomfortable in. At the same time, his wife was Jewish, and Weyl knew things would go downhill very rapidly in Nazified Germany. Still, he accepted and rejected repeated offers from the IAS over a number of months in 1933, at one point becoming essentially mentally and physically disabled from the stress of having to make a decision. After much pleading from colleagues and the IAS, Weyl at last accepted and came to America in November 1933. (Photos scanned from the 2009 book Mind and Nature: Selected Writings on Philosophy, Mathematics and Physics by Hermann Weyl with the express permission of the University of Princeton Press. Please respect the copyright when sharing)
 Rambling Sunday Thoughts — Posted Sunday, 10 January People have written me to ask if I really believe in the multiverse, entangled black holes, holographic universes and all that stuff. I tell them that I try to keep up with current theories in modern physics, but at times the sheer volume of bizarre ideas that are out there today (just look at the number of papers that are being posted on arXiv.org) makes me wonder if things are getting out of hand. Twenty years ago the pinnacle of crazy, math-dense theories was considered to be M-theory, with its 11 spacetime dimensions, while until recently supersymmetry (with its bosonic-fermionic, partner-particle idea) was not only popular, but considered perhaps even true. Out of these came superstring theory and supergravity, and all the while modern physics was looking as if the mathematicians had not only taken over all of physics, but had gone mad as well. With the Large Hadron Collider scheduled for renewed power-up in March, scientists will have access to unprecedented energies (approaching 14 TeV) for particle research. While the LHC produced nearly unequivocal evidence for the Higgs boson in 2012, a series of negative results nearly hammered the last nail in supersymmetry's coffin. The collider's new power will hopefully send SUSY completely back to the mathematicians, who may then choose to make a religion out of it, but variants of string theory are likely to survive, especially if evidence for extra dimensions is found. And while I have absolutely no justification to say it, I think it's also possible that the LHC will provide nothing more than additional confirmation of the Standard Model. Beyond that may lie a virtual desert, devoid of any exciting new particles, forces and fields. If then even more powerful machines are built that produce the same non-results, would physics be essentially "finished" at that point, with nothing left for scientists to do but tack on more decimal places to the Standard Model's predictions? It's not out of the question. At the other end, perhaps at its deepest level physical reality lies at the Planck scale, where the so-called reduced Planck energy, roughly $$10^{19}$$ GeV, is needed to fully resolve what the hell's going on. But that energy is unimaginably greater than anything the LHC could ever produce (Susskind sees Planck-scale accelerators having the dimensions of the observable universe), meaning that humans will never be able to do Planck-scale physics. So, discounting a few new discoveries that the LHC may provide, there really is an ultimate limit to all this. Both string theory and loop quantum gravity (LQG) posit the existence of Planck-size vibrating strings and interconnected networks of spacetime, but again it is doubtful these ideas can ever be tested experimentally. In my opinion, LQG is more beautiful than strings, as it does away with the concept of time altogether, with space becoming granular or foamy, with quantized areas and volumes making up the resulting 3D network mesh of space. There used to be a draft version of Carlo Rovelli's 2014 book Covariant Loop Quantum Gravity available on the Internet, but it's gone now. It's hardly elementary, but the beautiful ideas it presents make me think that if I were the Creator, I might have resorted to that kind of thing. Ask your local public library if they have an interlibrary loan program that can get it for you, and check it out. And speaking of Creators, my mind goes back again to the notion of the simulation hypothesis, which is making more and more sense to me as I get older and even stupider. Unlike the "life is just a dream" idiom, it posits that in the distant future humans (assuming they're still around) will possess computer capabilities far in excess of anything we can imagine today (just extrapolate Moore's Law out a few hundred or thousand years and you'll understand what I'm talking about). Advanced humans (or what Oxford University's Nick Bostrom calls "post-humans") will be able to simulate nearly anything using computers, even computerized life forms having sentience and free will, all residing on a hard drive possessing incomprehensible capacity and speed. Bostrom believes that our own existence may in fact be a computer simulation which, if nothing else, would neatly explain the problem of theodicy. Those of you interested in this seeming lunacy fascinating idea might want to read Daniel F. Galouye's 1964 science fiction book Simulacron-3, or watch the excellent 1999 film The Thirteenth Floor based on the book. Other films, notably Inception, Dark City and World on a Wire deal with the same topic (Tom Cruise's Vanilla Sky also uses the idea, but, sorry to say, the film really stinks). From The Thirteenth Floor © Columbia Pictures and Centropolis Film Productions, 1999 Something we never could have expected?! Hell, that's the first thing I'd suspect. Perhaps the simulations go down forever, and go up forever. Perhaps their ends even link up, so that there are no simulators to begin with, and the universe we know creates itself out of nothing.
 Susskind Again — Posted Friday, 8 January Now approaching his 76th year, renowned Stanford University physics professor Leonard Susskind just keeps on going. Here he is in 2013, lecturing to a group of mostly grad students on the transition of scientific reality from the 20th century (general relativity, quantum mechanics, the advent of accelerators, a little thermodynamics and even less information theory) to the 21st century (quantum entanglement, quantum information, black holes, entropy and how they might all relate to general relativity). The talk deals with quantum gravity and why is is so difficult to find a workable theory, but Susskind never really gets around to it, having to present a bunch of prerequisite material first which takes up most of the 102-minute lecture. The sound quality is uncharacteristically bad for a Susskind video lecture, and it's not helped by the noise generated by fidgeting students who constantly tap with their pencils, open and close doors and scrooch around in their desks, usually right next to the microphone. Worse, the position of the video camera is static, making the blackboard difficult to make out at times. On the plus side, you've got to admire an elderly man lecturing authoritatively in a pair of shorts, a tee shirt and running shoes. One of the reasons for the transition to 21st century physics, Susskind explains, was the limitations imposed due to the increasing energy of particle accelerators, despite the fact that the LHC now has a resolution capability of some 14 TeV. The 20th-century concept of building bigger and more powerful machines was in fact limited by the tendency of such machines to produce new particles from the very energy put into them to see smaller and smaller things. Today, Susskind notes, much higher energies will not provide better resolution, but the creation of mini black holes instead. It is at this point that Susskind begins to talk about entanglement and its close relationship with entropy, black holes and relativity in general which, to be frank, he does a much better job at in some of his other videos. Susskind does give the listener the hope that a consistent theory of quantum gravity will spin out of all this, and perhaps even redefine our concepts of reality and existence. I'm trying to imagine Einstein, Hermann Weyl or Paul Dirac lecturing in this fashion. Einstein was wont to wear sockless shoes, a rumpled pair of pants and a floppy sweatshirt in his later years, but earlier he and all other scientists I can think of were always immaculately dressed. Well, times have changed, and I think for the better.
 Weyl and Sons — Posted Wednesday, 6 January I know I've mentioned this before, but Hermann Weyl had two sons, Fritz Joaquim Weyl (1915 - 1977) and Michael Weyl (1917 - 2011). Fritz became a renowned mathematician like his father, while Michael was a high-ranking employee of the U.S. State Department who worked in various capacities, including interrogator of German prisoners during World War II. Both came to the United States when Hermann and wife Hella fled Nazi Germany in late 1933. Interestingly, Fritz graduated Pennsylvania's Swarthmore College in 1935 with a BA in mathematics, although when he matriculated he spoke only German! Both he and his brother graduated Princeton University in 1937, Fritz with a MA in math (earning his PhD there in 1939) and Michael with degrees in German literature and art history. Here's a photo of Michael and his father taken in 1937, around the time he graduated: And here are two photos of older brother Fritz, the first taken in 1932 at the age of 17 with his father and mother and another in 1970, at the age of 55: (The arm of the woman clipped off on the right of the first photo is that of Emmy Noether, a close friend of the Weyl family and generally considered to be the most renowned woman mathematician of all time; I'll put up the entire photo at a later date.) While Michael lived to the ripe old age of 93, his brother died at the age of only 62 (I don't know the cause, but it was unexpected). Lastly, here's a beautiful photo of Weyl with his grandsons Peter and Thomas taken around 1954 with the Matterhorn providing a nice backdrop: (Photos scanned from the 2009 book Mind and Nature: Selected Writings on Philosophy, Mathematics and Physics by Hermann Weyl with the express permission of the University of Princeton Press. Please respect the copyright when sharing)
 Weyl and God — Posted Monday, 4 January The mathematical lawfulness of Nature is the revelation of divine reason. — Hermann Weyl This quote from Weyl's 1932 essay "The Open World" predates by nearly 30 years a similar (though more secular) observation made by Nobel laureate Eugene Wigner entitled The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Both papers reflect on the unusual relationship that mathematics shares with Nature's laws but, more importantly, on the profound connection that appears to exist between those laws and their mathematical description. There's really no reason why this should be the case; it's one thing to make a physical observation and subsequently tie it to a suitable mathematical expression, but to discover that this expression then holds in a completely different physical application makes one wonder exactly which is more fundamental — the physics or the mathematics. The noted MIT physicist Max Tegmark has gone so far as to assert that Nature is nothing but an enormous mathematical expression. In all my readings of Weyl, I've never been able to precisely define his attitude toward God, though he wrote regularly about the divine nature of things. The same applies to Paul Dirac, who once noted that God is a brilliant mathematician, though Dirac was probably no more than an informed agnostic on the subject of Nature and its connection with religious faith. (Fun fact: Dirac married Eugene Wigner's sister in 1937. When presenting his wife to someone, he would often introduce her by saying "This is Wigner's sister.") I myself am a confirmed agnostic who nevertheless sees a profound connection between Nature and a Higher Something (although I could be wrong). The problem is, we have no idea what that Higher Something might be, if it even exists. The Bible is too full of irreconcilable contradictions and silly nonsensical stories to have any real validity on the subject, and while I haven't studied the Koran or any of the Eastern religions to any great extent I suspect the same conclusion holds with them as well (I detest the idea of fear-based faith, which I see all around us). Consequently I believe that the so-called God of Einstein and Spinoza is the most correct — a Higher Something that simply created everything, set things in motion, then walked away to let us hapless humans try to understand it all. While consolidating some files on my old (and dying) Mac Book Pro, I came across this photo of a contemplative Weyl with the title "Zürich, late 1955." I can't remember now where it came from. I have a few others, but I need to check out the copyright issue before posting them. Update: Okay, this photo and others are from the 2009 book Mind and Nature: Selected Writings on Philosophy, Mathematics and Physics by Hermann Weyl. I have obtained permission to post these copyrighted materials from the University of Princeton Press via the Copyright Clearance Center and will put them up here when I have the time.
 Uuo 118 and Friends — Posted Monday, 4 January Uranium has the highest atomic number of all naturally occurring elements. With 92 protons and 92 electrons, it can exhibit a varying number of neutrons in the nucleus that characterize its more common isotopes. (Plutonium, with 94 protons, is also primordial but is no longer found in Nature.) Elements with higher atomic weights are all synthetic, being produced in accelerators by bombardment of heavy elements with other elements and particles, resulting in unstable nuclei that tend to decay quickly. We thus have elements like einsteinium and fermium, along with later elements with strange names like ununnilium (atomic number 110) and ununbium (atomic number 112). A team of scientists from Japan, Russia and the United States now reports the creation of four new synthetic elements that stretch the atomic number up from 113 to 118, thus completing the seventh row of the periodic table. These species all decay almost instantaneously, exhibiting vanishingly small half lives because of the enormous numbers of protons and neutrons that are packed into their nuclei. The strong nuclear force may be very strong, but it can't hold these things together for very long. As a former chemistry major and chemist, I can't for the life of me understand what practical use these experiments represent. Perhaps, like pure mathematics, pure chemistry does not concern itself with applications, just the pursuit of knowledge, which I suppose is okay, but then I could never appreciate pure mathematics, either. An article in this week's New Scientist describes the findings, along with the scientists' hopes that an "island of stability" will be found within Group 18 in the periodic table (where the "noble gases" helium, neon, argon and krypton are found). However, Element 118 found by the scientists also falls into Group 18, but it doesn't appear to be particularly stable. Provisionally named ununoctium (Uuo), it might actually be what could be called a "noble solid," since quantum theory predicts it would be a solid, not a gas. In addition, the outer electrons of Uuo are moving close to the speed of light, making it truly a "relativistic element" whose physical and chemical properties should be very interesting. No more than a relative (!) handful of Uuo atoms has been produced so far, however, making it highly unlikely that Uuo office paperweights will be showing up any time soon. The U.S. military will also have to wait for a Uuo replacement for its armor-piercing projectiles (which currently use depleted uranium).
 Two Places at the Same Time? — Posted Friday, 1 January Stanford University's Mark Kasevich and colleagues report the creation of a macroscopic object that exists in two different places at the same time. The object is a small but observable "cloud" of 10,000 rubidium atoms cooled to a temperature near absolute zero. Because the cloud is a Bose-Einstein condensate, each of the atoms exists in the exact same quantum state. A laser beam is then used to push this cloud up an evacuated 10-meter chamber, while also splitting the cloud into two distinct quantum states. At the top of the chamber the cloud represents a 50-50 superposition of these two states, but the atoms in these states are now separated by 54 centimeters. Falling back down, the laser beam converts the cloud back to the original, single quantum state. The differing arrival times of the atoms demonstrates that the cloud did indeed exist in two places at once at the top of the chamber. Because of the extreme sensitivity of the apparatus to extraneous interferences (including the rotation of the Earth), the results are subject to experimental confirmation, but Kasevich asserts the finding is solid. I want to see the complete paper in the journal Nature before I can even begin to believe this. The 28 December 2015 edition of the magazine New Science has additional information on the experiment.