Scientists and mathematicians #3…
Some theoretical physicists play loose with mathematics. Feynman, for example, never worried about the convergence of the series needed to calculate things in QED (that’s quantum electrodynamics); in fact, he designed a technique, now called Feynman diagrams, to allow him to throw away infinities easily and just hoped things would work out. That’s called renormalization. The renormalization of quantum field theories became a big deal. The electro-weak theory of Weinberg, Glashow, and Salam, designed to explain and combine the electromagnetic and the weak forces, for example, was never fully accepted until ‘t Hooft proved it could be “renormalized.”
Dirac’s bras and kets were just vectors and linear functionals on a Hilbert space, but I doubt he worried too much about functional analysis, the study of infinite-dimensional vector spaces and their linear operators. Even Maxwell’s brilliant synthesis of all classical electromagnetic phenomena—gamma rays, x-rays, visible light, electricity, and magnetism—had to wait years until vector calculus was invented by Gibbs before its true beauty could be seen. A classical vector field is determined by its curl and divergence, and that’s exactly what Maxwell’s equations say about the electric and magnetic fields.
Sometimes physics gets ahead of mathematics. Sometimes it’s the reverse. The key to quantum chromodynamics isn’t Gell-Mann’s Eightfold Way. The representations for the special unitary group SU(3) he used to organize hadrons into composites of quarks already existed. His contribution was to recognize that the representations could organize the hadronic particle zoo. Similarly, I always thought that algebraic topology was an esoteric branch of mathematics, and yet it has found multiple uses in particle physics.
On the flip side, Ed Witten’s treatment of string theory (part of quantum field theory) has led into many breakthroughs in the theory of knots, an unusually esoteric mathematical subject, so much so that Witten received the Fields Medal, the prestigious mathematics equivalent of a Nobel prize (the story about why Nobel didn’t want to give a prize to mathematicians seems apocryphal). Much earlier, Einstein’s General Theory of Relativity wouldn’t have gone anywhere without tensor calculus—his friend Grossman, a mathematician, even helped him with the math.
When physicists and mathematicians are in front of an audience at the blackboard (whiteboard or overhead projector today), they often show their humorous side, and the humor can cover a wide spectrum in function of culture and period. One story has Dirac dozing off in a seminar. He awakes when the presenter finishes and comments, “The sign is different in your preprint.” (A preprint is like a writer’s first draft. It often becomes a scientific paper later. Dirac might have called it a draft, in fact.) The presenter responded, “I must have made a sign mistake someplace.” And Dirac responds with dry British humor, “In an odd number of places.”
Feynman was giving a lecture in front of a less specialized audience comprised of professors, grad and undergrad students, and rubberneckers wanting to see the famous man. Like Bernie Sanders, he grew up in Brooklyn. His accent was more pronounced than Bernie’s, though. That probably explains why it took the audience so much time to figure out that the seagulls he was referring to were only the inverted circumflexes over some of his symbols. (This story isn’t in Surely You’re Joking, Mr. Feynaman, but that’s a good place to find a few more.)
Max Zorn in his emeritus professor years went to warmer climates, but not as far south as Dirac. The creator of the lemma that bears his name—not really a lemma but an axiom, and equivalent to set theory’s Axiom of Choice (it was referenced in Fox’s The Simpsons, of all places)—went to the University of Maryland while Dirac went to Florida State. Old professors tend to nap in seminars (the competition between Dirac and Wheeler was mentioned in a previous post). You can’t blame them much—some seminars are so boring you want to slash your wrists, or those of the presenter, so a quick nap seems the safer and less violent way to go.
Zorn, like Dirac, woke up when the presenter finished. The latter had just defined a class of operators on Banach spaces (the Hilbert space of quantum mechanics is a special type of Banach space, so they’re studied in functional analysis). Zorn pipes up, “Give me an example of such an operator.” No one could. Zorn proceeded to show by reductio ad absurdum that no such operator could exist. The presenter had defined an empty set and derived marvelous things about operators in that set! (An empty set has no elements.)
People often forget that scientists and mathematicians are human beings. Movies like A Beautiful Mind, The Theory of Everything, and The Imitation Game have done a good job of showing that. But the movies aren’t needed by scientists and mathematicians who see that all the time en carne propia, as my Latino friends would say (that can be loosely translated as “with their own eyes”).
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Alien invasions are almost cliché. Mine isn’t. In More than Human: The Mensa Contagion, human beings are forever changed. So, what do they do? Go to Mars! The first part of the story stars a mutating virus. Is it a sly way for ETs to take over Earth? The second part features a Mars colony. Don’t look for potato-growing farmers, but you’ll meet some real ETs that human beings have grown. If you like your sci-fi stew with a new, zesty flavor, this novel will entertain you. If you don’t, go grow some potatoes. Available in all ebook formats.
In libris libertas…
