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Who Was Hermann Weyl?

Wheeler's Tribute to Weyl (PDF)

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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?
Did Weyl Screw Up?
Weyl and the Aharonov-Bohm Effect
The Bianchi Identities in Weyl Space
Conformal, Parameter-Free Riemannian Gravity
Gravity Wave Tutorial
Conformal Kerr-de Sitter Gravity
A Child's Guide to Spinors
Levi-Civita Rhymes with Lolita
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
A Children's Primer on Quantum Entanglement
Veblen and Weyl
Graphing the Lorentz Transformation
FLRW in de Sitter Spacetime
FLRW with a Constant Curvature Scalar
Is There a Flaw in the FLRW Metric?
On the Mannheim-Kazanas Spacetime
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
The Divergence Myth in Gauss-Bonnet Gravity
Schrodinger Geometry
A Brief Look at Gaussian Integrals
Differential Forms for Physics Students
Particle Chart
Einstein's 1931 Pasadena Home Today

Uncommon Valor

She did not forget Jesus!
"Long live freedom!"

Visitors since 11-4-2004:


      God exists since mathematics is consistent, and the Devil
       exists since its consistency cannot be proved.

       Symmetry, as wide or narrow as you may define its meaning,
       is one idea by which man through the ages has tried to
       comprehend and create order, beauty, and perfection.

    Hermann Weyl, 1885-1955


Weyl, ca. 1930 (Göttingen). Courtesy of the Archives of the Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach Photo Collection.

Hermann Weyl was born in the little town of Elmshorn (near Hamburg), Germany on November 9, 1885. The son of Ludwig and Anna Weyl, he showed an unusual aptitude for math and science as a youth. From 1904 to 1908 he studied mathematics and physics in Munich and Göttingen, obtaining his PhD (at the age of 22) in 1908 under the supervision of the great German mathematician David Hilbert. His doctoral dissertation explored singular integral equations with special consideration of Fourier integral theorems.  Following a teaching post in Göttingen, he took a professorship at the Eidgenössiche Technische Hochschule (Swiss Federal Technical University) in Zürich, Switzerland, where he was appointed the chair of mathematics in 1913. Just 27 years old, Weyl was now a senior colleague of Einstein who, already a professor of physics at the school, was working feverishly on his greatest achievement – the General Theory of Relativity.    

In 1930, Weyl accepted the mathematics chair at Göttingen as a replacement for Hilbert, who retired that year. With the rise of the German Nazi regime in January 1933, Weyl left Göttingen and accepted an academic position at the Institute for Advanced Study in Princeton, New Jersey, where he worked with Einstein and others until his retirement in 1952. In his last years he returned to Zürich, where he died on December 8, 1955 at the age of seventy.

In addition to pioneering work on matrix representations of continuous groups, analytic number theory, Riemann surfaces and topology, Weyl developed a gauge theory of the 4-dimensional spacetime metric
gμν with which he sought to unify Einstein's theory of general relativity with Maxwell's equations of the electromagnetic field, thereby establishing one of the earliest (if not the first) unified field theories. In 1929, Weyl demonstrated that his gauge principle tied the quantum wave function Ψ  to the electromagnetic field by a gauge transformation of the electromagnetic 4-potential. Gauge invariance thus mandates electrodynamics, and is the underlying principle behind the conservation of electric charge!  He also showed that Dirac's relativistic electron equation decouples into two expressions for massless spin-1/2 particles that violate the conservation of parity, thereby paving the way for the mathematical description (and eventual discovery) of neutrinos. These discoveries are all the more remarkable in light of the fact that Weyl was first and foremost a mathematician and not a physicist. 

The great Austrian physicist and Nobel laureate Wolfgang Pauli admonished Weyl for having strayed from mathematics into the realm of quantum physics with this remarkable correspondence:

Before me lies the April edition of Proc. Nat. Acad. [the journal that pre-published Weyl's 1929 paper]. Not only does it contain an article from you under "Physics" but shows that you are now in a "Physical Laboratory": from what I hear you have even been given a chair in "Physics" in America. I admire your courage; since the conclusion is inevitable that you wish to be judged, not for your successes in pure mathematics, but for your true but unhappy love for physics.

Soon afterward, however, Pauli apologized to Weyl when he came to fully comprehend the significance of Weyl's discovery:

In contrast to the nasty things I said ... Here I admit your ability in physics. Your earlier theory [Weyl's 1918 effort] with g'μν = λ(x)gμν was pure mathematics and unphysical. Einstein was justified in criticising and scolding. Now your hour of revenge has arrived.

During his lifetime, Weyl published a large number of books and papers on space, time, matter, philosophy, logic, and the history of mathematics. His books include:      
        Was ist Materie? (What is Matter?), 1924
        Gruppentheorie und Quantenmechanik (Group Theory and Quantum Mechanics), 1928. 
        Elementary Theory of Invariants
, 1935 
        Philosophy of Mathematics and Natural Science, 1949.
        Raum-Zeit-Materie (Space-Time-Matter), 1952
        Symmetry, 1952
        The Concept of a Riemannian Surface, 1955

Weyl's gauge theory of the gravitational and electromagnetic field and/or closely related mathematical material can be found in Space-Time-Matter and in

The Mathematical Theory of Relativity (A.S. Eddington, Chelsea Publishing, 1923)
Tensor Calculus (J.L. Synge and A. Schild, Dover Publications, 1949)
Space-Time Structure
(E. Schrödinger, Cambridge University Press, 1950)  [A wonderful little book!]
The Principle of Relativity
(Dover Publications, 1952) 
The Theory of Relativity
(W. Pauli, Dover Publications, 1958)
Introduction to General Relativity
(Adler et al., McGraw-Hill, 2nd. Ed., 1975)
Subtle is the Lord
(A. Pais, Oxford University Press, 1982)  
An Elementary Primer for Gauge Theory
(K. Moriyasu, World Scientific Publishing, 1983)
The Dawning of Gauge Theory
(L. O'Raifeartaigh, Princeton Univ. Press, 1997)  [Another neat book!]
Gravitation and Gauge Symmetries (M. Blagojević, IOP Publishing, 2002)