Hermann Weyl

Hermann Weyl (born November 9, 1885, Elmshorn, near Hamburg, Germany—died December 8, 1955, Zürich, Switzerland) was a German American mathematician who, through his widely varied contributions in mathematics, served as a link between pure mathematics and theoretical physics, in particular adding enormously to quantum mechanics and the theory of relativity.

As a student at the University of Göttingen (graduated 1908), Weyl came under the influence of David Hilbert. In 1913 he became professor of mathematics at the Technische Hochschule, Zürich, where he was a colleague of Albert Einstein. The outstanding characteristic of Weyl’s work was his ability to unite previously unrelated subjects. In Die Idee der Riemannschen Fläche (1913; The Concept of a Riemann Surface), he created a new branch of mathematics by uniting function theory and geometry and thereby opening up the modern synoptic view of analysis, geometry, and topology.

The outgrowth of a course of lectures on relativity, Weyl’s Raum, Zeit, Materie (1918; “Space, Time, Matter”) reveals his keen interest in philosophy and embodies the bulk of his findings on relativity. He produced the first unified field theory for which Maxwell’s equations of electromagnetic fields and the gravitational field appear as geometric properties of space-time. The influence of these studies on differential geometry is exemplified best by his treatment of the Italian mathematician Tullio Levi-Civita’s concept of parallel displacement of a vector. Weyl freed the concept from dependence on a Riemann metric and thus set the stage for the rapid development of projective differential geometry by Oswald Veblen of the United States and by others. Around this time (and influenced by the work of French mathematician Elié Cartan), Weyl attempted a unified field theory to unite electromagnetism and gravitation, in which he introduced the concept of gauge invariance, which describes how some quantities do not change despite a transformation in the underlying field and which became important in later particle physics. By 1923 Raum, Zeit, Materie had appeared in four more editions.

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Numbers and Mathematics

From 1923 to 1938 Weyl evolved a general theory of continuous groups, using matrix representation. He found that most of the regularities of quantum phenomena on the atomic level can be most simply understood by using group theory. With the findings published in Gruppentheorie und Quantenmechanik (1928; “Group Theory and Quantum Mechanics”), Weyl helped mold modern quantum theory.

Weyl was appointed professor of mathematics at the University of Göttingen in 1930. The Nazi dismissal of many of his colleagues prompted him to leave Germany in 1933 and accept a position at the Institute for Advanced Study, Princeton, New Jersey; he became a U.S. citizen in 1939. After his retirement in 1951, Weyl remained professor emeritus of the institute and divided his time between Princeton and Zürich. Weyl was keenly interested in the aesthetic and philosophical aspects of mathematics and physics, an interest that came to the fore in Symmetry (1952), a profusely illustrated work that examines symmetry in art and nature. He once said, “My work has always tried to unite the truth with the beautiful, but when I had to choose one or the other, I usually chose the beautiful.”