Hermann Weyl

Hermann Weyl was born in Elmshorn, near Hamburg, in 1885. He studied at Munich and Göttingen, took his doctorate at Göttingen in 1908 under David Hilbert (with a dissertation on singular integral equations), and remained one of Hilbert's closest mathematical descendants for the rest of his life. After a habilitation lectureship at Göttingen and a brief return to Munich, he was called in 1913 to ETH Zürich, where Einstein had just arrived as professor of theoretical physics. The two became close — Weyl's 1918 monograph *Raum, Zeit, Materie* (*Space, Time, Matter*) is one of the great expositions of general relativity, written within three years of Einstein's 1915 field equations — and Weyl held the ETH chair until 1930, when he succeeded Hilbert at Göttingen. He held the Göttingen chair for only three years; in 1933 the Nazi seizure of power and his Jewish wife Helene's situation made staying impossible. He emigrated to Princeton's Institute for Advanced Study, where he remained Einstein's neighbour and colleague until his retirement in 1951.

Weyl's 1918 attempt to unify gravity and electromagnetism is the origin of the word "gauge" in modern physics. He proposed that the local scale of length — the *Eichmaß*, German for gauge — was not absolute but could vary from point to point in spacetime, and that this local scale invariance, together with the metric of general relativity, would generate electromagnetism as a second connection. Einstein flat-out rejected the proposal: a clock transported around a closed loop would, in Weyl's theory, return with a different rate, contradicting the observed constancy of atomic spectral lines. Weyl conceded the point but kept the underlying idea — that a *local* symmetry of the wavefunction could generate a force. In 1929, with quantum mechanics now in hand, he retooled the construction: the local symmetry is a phase rotation ψ → e^{iqΛ(x)/ℏ}ψ of the complex wavefunction, and the gauge field that compensates for it is the electromagnetic four-potential A_μ. The U(1) gauge symmetry of QED, the modern derivation of charge conservation from gauge invariance via Noether's theorem, and the entire gauge-theoretic backbone of the Standard Model trace to that 1929 paper. The Yang-Mills 1954 generalisation to non-abelian groups SU(N) is the same construction with a richer symmetry; Weyl himself recognised the connection to spinors and group representations and laid the mathematical groundwork in his 1928 *Gruppentheorie und Quantenmechanik*.

Weyl's broader mathematical legacy is enormous: foundational work on the representation theory of compact Lie groups (the Weyl character formula, Weyl chamber, Weyl group), the 1918 *Riemann-Hurwitz* calculations for Riemann surfaces, the *Weyl spinor* (the two-component massless spinor used by every particle physicist), Weyl semimetals (named for the 1929 fermion equation he wrote down), and the 1952 popular essay *Symmetry*, still in print and one of the loveliest pieces of writing on mathematics in the twentieth century. He died of a heart attack outside a Zurich post office on 8 December 1955, on his 70th birthday, while mailing a stack of letters thanking colleagues for their birthday wishes.