The first time I met Lilla Vekerdy, she was holding a book.

“What’s that?” I asked.

“A second edition of Galileo’s *Siderius nuncius*. Here,” she added, thrusting the book into my hands. “Take it.”

So began my internship at the Smithsonian Institution’s Dibner Library for the History of Science and Technology.

Many people know the Smithsonian for its museums. The Smithsonian, they know, houses the ruby slippers worn by Dorothy in *The Wizard of Oz*. The Smithsonian houses planes constructed by Orville and Wilbur Wright, the dresses worn by First Ladies on presidential inauguration evenings, a space shuttle, and a *Tyrannosaurus Rex* skeleton. Smithsonian museums line the National Mall in Washington, D.C.—the United States’ front lawn—and march beyond.

Most people don’t know that the Smithsonian has 21 libraries.

Lilla heads the Smithsonian Libraries’ Special-Collections Department. She also directs a library tucked into a corner of the National Museum of American History. I interned at that library—the Dibner—in college. Images of Benjamin Franklin, of inventor Eli Whitney, and of astronomical instruments line the walls. The reading room contains styrofoam cushions on which scholars lay crumbling rare books. Lilla and the library’s technician, Morgan Aronson, find references for researchers, curate exhibitions, and introduce students to science history. They also care for the vault.

The vault. How I’d missed the vault.

The vault contains manuscripts and books from the past ten centuries. We handle the items without gloves, which reduce our fingers’ sensitivities: Interpose gloves between yourself and a book, and you’ll raise your likelihood of ripping a page. A temperature of 65°F inhibits mold from growing. Redrot mars some leather bindings, though, and many crowns—tops of books’ spines—have collapsed. Aging carries hazards.

But what the ages have carried to the Dibner! We^{1} have a survey filled out by Einstein and a first edition of Newton’s *Principia mathematica*. We have Euclid’s *Geometry* in Latin, Arabic, and English, from between 1482 and 1847. We have a note, handwritten by quantum physicist Erwin Schödinger, about why students shouldn’t fear exams.

I returned to the Dibner one day this spring. Lilla and I fetched out manuscripts and books related to quantum physics and thermodynamics. “Hermann Weyl” labeled one folder.

Weyl contributed to physics and mathematics during the early 1900s. I first encountered his name when studying particle physics. The Dibner, we discovered, owns a proof for part of his 1928 book *Gruppentheorie und Quantenmechanik*. Weyl appears to have corrected a typed proof by hand. He’d handwritten also spin matrices.

Electrons have a property called “spin.” Spin resembles a property of yours, your position relative to the Earth’s center. We represent your position with three numbers: your latitude, your longitude, and your distance above the Earth’s surface. We represent electron spin with three blocks of numbers, three matrices. Today’s physicists write the matrices as^{2}

We needn’t write these matrices. We could represent electron spin with different matrices, so long as the matrices obey certain properties. But most physicists choose the above matrices, in my experience. We call our choice “a convention.”

Weyl chose a different convention:

The difference surprised me. Perhaps it shouldn’t have: Conventions change. Approaches to quantum physics change. Weyl’s matrices differ from ours little: Permute our matrices and negate one matrix, and you recover Weyl’s.

But the electron-spin matrices play a role, in quantum physics, like the role played by *T. Rex* in paleontology exhibits: All quantum scientists recognize electron spin. We illustrate with electron spin in examples. Students memorize spin matrices in undergrad classes. Homework problems feature electron spin. Physicists have known of electron spin’s importance for decades. I didn’t expect such a bedrock to have changed its trappings.

How did scientists’ convention change? When did it? Why? Or did the convention not change—did Weyl’s contemporaries use today’s convention, and did Weyl stand out?

I intended to end this article with these questions. I sent a draft to John Preskill, proposing to post soon. But he took up the questions like a knight taking up arms.

Wolfgang Pauli, John emailed, appears to have written the matrices first. (Physicist call the matrices “Pauli matrices.”) A 1927 paper by Pauli contains the notation used today. Paul Dirac copied the notation in a 1928 paper, acknowledging Pauli. Weyl’s book appeared the same year. The following year, Weyl used Pauli’s notation in a paper.

No document we know of, apart from the Dibner proof, contains the Dibner-proof notation. Did the notation change between the proof-writing and publication? Does the Dibner hold the only anomalous electron-spin matrices? What accounts for the anomaly?

If you know, feel free to share. If you visit DC, drop Lilla and Morgan a line. Bring a research project. Bring a class. Bring zeal for the past. You might find yourself holding a time capsule by Galileo.

*With thanks to Lilla and Morgan for their hospitality, time, curiosity, and expertise. With thanks to John for burrowing into the Pauli matrices’ history.*

^{1}I continue to count myself as part of the Dibner community. Part of me refuses to leave.

^{2}I’ll omit factors of

Great post! I reckon that the standard convention comes from considering z as the standard axis of reference for “up” and “down” (x and y being used for the horizontal plane), so that we choose to represent sigma_z as diagonal. On the other hand, if I was to choose from scratch three matrices with the standard properties of Pauli matrices, and going sigma_x, sigma_y, sigma_z (in this order) I would go for Weyl’s choice as a natural choice. So, to me, one choice is more “physically motivated” and the other “mathematically natural”.

Thanks, quantumrules! If I understand correctly, you’re reasoning, “We usually say, ‘x, y, z.’ The x heads that list. So ‘favoring’ x — rather than z, which ends the list — makes sense. Weyl favored x as we favor z. The oddball imaginary matrix elements are relegated to the final entry in Weyl’s list.'” Sounds reasonable.

“We¹ have” — until I read the footnote, I read this “We” as “anyone who is a US citizen”, but it’s true that a Librarian’s “have” is closer.

As to what representation one /should/ use, I imagine Weyl would be quicker than quick to mention unitary equivalence.

Thanks for your thoughts, Peter. I agree that the representations are equivalent. Hence my curiosity about the “choice of convention.”

Great piece, Nicole. The Burndy Library here at the Huntington — the Dibner’s larger cognate, here on the west coast — contains 23 works, including translations, by Weyl. This one — http://catalog.huntington.org/record=b1721602 — was inscribed by Weyl and would be worth scrutinizing to see if there are other notations by him in the copy.

Beyond the Burndy’s 67,000 volumes in the history of sci/tech, there is also a very robust fellowship program: http://www.huntington.org/WebAssets/Templates/content.aspx?id=17896.

Thanks! To readers, especially those in Pasadena and San Marino: I second the invitation to check out the Burndy Library at the Huntington. I also recommend the “Beautiful Science” exhibition that features items from the library collection (http://www.huntington.org/webassets/templates/general.aspx?id=16992). If anyone makes it over and examines the Weyl works, let us know what you find!

The spin of a positron is also hbar/2. The photon results from e = e-, so photon spin is 1

I meant e+ plus e-

The graviton is said to have spin 2. Could it be a ring of positrons and electrons?

If {god forbid} the electron can be broken into six hexons, what would be the spin of a hexon?

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