# Hermann Weyl

Hermann Klaus Hugo Weyl ( born November 9, 1885 in Elmshorn, † December 8, 1955 in Zurich ) was a German mathematician, physicist and philosopher, because of its wide field of interest from number theory to theoretical physics and philosophy as one of the last mathematical universalist applies.

## Life

Weyl attended high school Christianeum in Altona. On the recommendation of the Director, who was a cousin of David Hilbert and the talent of the boys impressed Weyl began after graduating from high school in 1904 in Göttingen in Hilbert mathematics and incidentally also to study physics. He also took courses in philosophy with Edmund Husserl, where he met his future wife, Helene. Up to one year in 1905 in Munich, he studied at Göttingen, where he in 1908 David Hilbert with the thesis " Singular integral equations with special reference to the Fourier Integraltheorems " doctorate, habilitation in 1910 and until 1913 taught as a lecturer.

In 1913 he married Helene Joseph from Ribnitz, among other things, later translated many works of the Spanish philosopher José Ortega y Gasset. With her he had two sons. In the same year he was appointed professor for the chair of geometry at the Swiss Federal Institute of Technology Zurich, where he met Albert Einstein, who was developing his theory of general relativity at that time (1916-1918), which Weyl of the intensive study of the mathematical foundations general relativity and their possible extensions (about to take account of electrodynamics and a calibration parameter ) stimulated, but in particular with the underlying differential geometry.

In 1918 he published one of the first textbooks on general relativity (besides textbooks by Max von Laue and Arthur Eddington ), "Space, time, matter ."

Making a call to Göttingen to succeed by Felix Klein, he lashed out. It was not until 1930, after Hilbert's chair was orphaned, he took part in: to become Hilbert's successor, was an honor that he could not refuse him. However, the change from Zurich him fell to Gottingen not easy because he saw the political radicalization and the rise of Nazism in the Weimar Republic with concern as he did in a speech to the Göttingen Mathematical Connection 1930 expression: "Only with some trepidation I find myself out of their [ the traditionally democratic Switzerland ] freer and more relaxed atmosphere back into the yawning, umdüsterte and cramped Germany of today. " Throughout his life he felt obliged to democratic ideals, and in 1933, he was unable, under the rule of the National Socialists to teach Germany, especially since his wife was Jewish. In its from Zurich on October 9, 1933 the sent resignation to the new National Socialist Minister of Education, Bernhard Rust, he wrote: " The fact that I 'm out of place in Göttingen, has paid me very soon, as I in the fall of 1930 after 17 years of service at the Swiss Federal Institute of Technology Zurich moved there as a successor of Hilbert. " through the mediation of Albert Einstein, he took a position at the Institute for Advanced Study in Princeton, where he worked until 1951. At Princeton, died in 1948 his wife, Helene, and he married in 1950 the sculptor Ellen Baer of Zurich, from the Hermann Weyl bust comes, which is in the universities of Princeton, Zurich and Kiel in his memory. He spent his last years mainly in Zurich. In 1955 he received an honorary citizen of his hometown of Elmshorn, and shortly afterwards he died unexpectedly in Zurich due to a heart attack he suffered when sending mail to a mailbox.

Weyl was close friends with Peter Weyl, for example, the Schrödinger. In Zurich he had an affair with Erwin Schrödinger's wife Anny, but the friendship with Schrödinger nothing changed since the Schrödinger in an open relationship survived. Weyl's wife Helene ( called Hella), in turn, had at this time an open relationship with Paul Scherrer.

## Work

Weyl has dealt with many areas of mathematics and wrote several books and over 200 journal articles.

He began as an analyst, according to the interests of the Hilbert school at the beginning of the 20th century ( integral equations, spectral theory ), and his habilitation in 1910 on a singular differential equations and their development in eigenfunctions, which, inter alia, important in mathematical physics (later " spectral theory of self-adjoint operators" mentioned ). 1915 ( Rendicondi Circolo Mathematico di Palermo) he determined the asymptotic distribution of eigenvalues of the Laplace equation and showed that the first term proportional to the volume is what the physicists (including Hendrik Antoon Lorentz ) in the investigation of cavity radiation, the first connections between quantum mechanics classical theory and delivered, had already suspected. Other parameters except the volume do not matter. The general question of whether one can conclude ( the natural oscillations ) on the geometric shape of a region of the spectrum, Mark Kac popularized in his essay " Can one hear the shape of a drum? " (American Mathematical Monthly, 1966).

Less well known is that its Weyl Zurich colleague Erwin Schrödinger not insignificantly supported at its fundamental essay on quantum theoretical wave mechanics, pointing him the way to solve the Schrödinger equation for the hydrogen atom.

In 1913 he published the book "The idea of the Riemann surface ", in which the previously rather heuristically introduced topological methods have been treated more severely and also the modern concept of manifolds was first used systematically.

Since his book on the theory of general relativity Weyl was greatly interested in the connections to physics. He formulated the underlying differential geometry of general and flexible with introduction of an affine connection. In " Space, Time and Matter" and in his essay " Gravitation and Electricity" from 1918, he also leads the first time the concept of a gauge theory, though not initially in its present form, but by a locally variable scale factor. The idea was that if parallel transport of a vector along a closed curve not only the direction is changed (which is expressed by the curvature ), but also the length could vary. He hoped that the electrodynamics in the theory involve. When the electrodynamics comprehensive extension of the theory she was quickly contradictory discarded by Einstein as the experiments. The book "Space, time, matter " also develops systematically the Riccischen tensor calculus and uses the parallel shift ( introduced by Levi -Civita ) of vectors as a fundamental concept.

But Weyl 's also the founder of gauge theories in the modern sense, in a work of 1929, with gauge transformations as phase factors of the quantum mechanical wave functions.

The analysis of Riemann and Helmholtz's ideas about the spatial forms that are possible under "reasonable" physical conditions, Weyl took up in his Spanish lectures " The mathematical analysis of the problem of space " 1920. This led him to applications of group theory, from which probably his employment with continuous groups developed ( Lie groups ).

His most important works ( Mathematical Journal Vol 23, 24, 1925/1926 ) may be seen in the theory of Lie groups whose representation theory he studied, where he also contributes global concepts such manifolds instead of the hitherto predominant local aspects of the Lie - algebra. For example, he said the first time the spinors from the topology of the rotation group. He also suggests a connection here to the methods developed by Frobenius and Schur representation theory of finite matrix groups. Weyl is a general formula (so-called " Weyl character formula " ) for the characters of the irreducible representations of semisimple Lie groups, by examining the investigated already by Cartan and Wilhelm Killing Lie algebras with reflection groups, the Weyl groups.

Another important result of his work is the set of Peter -Weyl ( Mathematische Annalen in 1927, together with his student Peter Fritz ). Are sine and cosine orthogonal function systems with respect to the translation group in one dimension, so there are also those for general compact Lie groups G ( in which an invariant (hair ) measure can be defined as an integral over the group elements). In this feature space, a Hilbert space, the representations of the group G are then given by irreducible representations of the unitary group after the Peter -Weyl theorem.

In " Group Theory and Quantum Mechanics" he gave in 1928 (a little before the books by Bartel Leendert van der Waerden and Eugene Wigner ) a representation of the group-theoretical aspects (and also general mathematical aspects) of quantum mechanics, especially the representation theory of the unitary and orthogonal groups ( the turn to Shur which the symmetric group related ). In the book "The classical groups" of 1939 he extended this to all classical groups and also created the connection to the classical invariant theory, an important part of the algebra of the 19th century.

Together with his son Fritz Joachim Weyl (1915-1977) he published in 1943 the book meromorphic functions and analytic curves, in which the theory of meromorphic functions Nevanlinnasche value distribution is generalized to analytic curves.

Since his studies at Hilbert Weyl was also interested in number theory ( by his own account he spent with the study of Hilbert's financial report during the holidays the happiest months of his life ). For example, he published in Mathematische Annalen in 1916 an essay on analytic number theory " equal distribution of numbers mod 1". In it he showed that the fractional part of the multiples of an irrational number are close not only in the interval [ 0,1], as Kronecker proved, but are equally distributed. So they are easy to use as random numbers.

His philosophical interests, which already appeared in books such as " space, time, matter ," could take on the part of intuitionists on pages Brouwers against the so-called formalists of the Hilbert school party him in the 1920s. The pure intuitionists identify only constructive of inference in finite number of steps on ( and no such objects whose existence is proved using the axiom of choice ), so that would be feasible only with a computer. In later years is Weyl but come to a more balanced view of mathematics between constructive and axiomatic methods. His older conception of the troubled years after the first World War, for example, shown in " About the new foundational crisis of mathematics " (Matt. magazine 1921), his mature philosophy in the book "Philosophy of mathematics and natural sciences."

In the book " symmetry " he gives a popular account of the group concept of snow crystals, ornaments (group of planar translations and reflections / rotations ) to the symmetry of equations under permutation of the roots ( Galois theory ).

## Awards and Honors

- Lobachevsky Prize for Geometry at the University of Kazan in the USSR, 1925.
- In 1928 he gave a plenary lecture at the International Congress of Mathematicians in Bologna ( Continuous groups and their representation by linear transformations ).
- In 1932 he was president of the German Mathematical Society.
- Arnold Reymond Prize May 1954.
- Honorary doctorate from the Swiss Federal Institute of Technology in 1945 and the Universities of Oslo in 1929, Pennsylvania in 1940, Sorbonne (Paris) in 1952, Columbia University, New York in 1954 and the Technical University of Stuttgart in 1929.
- Honorary citizen of the city Elmshorn November 17, 1955.
- A lunar crater is named after Hermann Weyl.
- His 100th birthday of the first international Hermann Weyl Congress was held in his honor in Kiel, attended by, among others Hans Freudenthal, Jean Dieudonné, Erhard disc, Jürgen Ehlers, Julian Schwinger, George Mackey, David Speiser, corner hardware W. Mielke, Friedrich W. Hehl, Gerhard Mack, Bas van Fraassen, Bernard d' Espagnat, Bernulf Kanitscheider, Wolfgang Deppert, John Archibald Wheeler and Andreas Bartels took part with their own original contributions.
- The asteroid " ( 32267 ) Hermannweyl " is named after Hermann Weyl.

## Writings

- The idea of the Riemann surface, Teubner 1997 ( first in 1913, new edition with contributions by Patterson, Hulek, Hildebrandt, Remmert, Schneider, editor: R. Remmert: TEUBNER ARCHIVE mathematics, Suppl 5, 1997)
- Space, time, matter - Lectures on General Relativity, 8th edition, Springer, 1993 ( first in 1918, 5th edition 1922) Online
- The continuum - critical studies on the foundations of analysis, Leipzig, Veit and Comp. , 1918 Online
- Gravitation and Electricity, session reports Preuss. Academy of Wiss. , 1918 ( reprinted in Lorentz, Einstein, Minkowski, The Principle of Relativity ).
- What is matter? - Two Essays on Natural Philosophy, Springer, Berlin 1924
- Philosophy of Mathematics and Natural Sciences, Munich: Oldenbourg Verlag 1927 [= Handbook of Philosophy, ed. by Alfred Baeumler and Manfred Schröter, Part A], 6th edition, Munich: Oldenbourg Verlag 1990
- Group Theory and Quantum Mechanics, University Press, 1977 ( reprint of the 2nd edition in 1931, first in Leipzig, Hirzel 1928)
- The classical groups -their invariants and representations, Princeton University Press 1939, 1946, 1961
- Elementary theory of invariants, Institute for Advanced Study 1936
- Meromorphic functions and analytic curves, Princeton University Press 1943
- Symmetry, Birkhauser 1955, 1981 ( first in 1952, Princeton )
- Selecta Hermann Weyl, Birkhauser Verlag ( selected essays ) 1956
- Algebraic Number Theory, BI university paperback 1966
- Collected Essays, 4 vols, ed K. Chandrasekharan, Springer Verlag 1968
- Riemann's geometric ideas, their impact and their link with group theory, Springer 1988