WEYL, HERMANN

(b. Elmshorn, near Hamburg, Germany, 9 November 1885; d. Zurich, Switzerland, 8 December 1955), mathematics, mathematical physics.

Weyl attended the Gymnasium at Altona and, on the recommendation of the headmaster of his Gymnasium, who was a cousin of Hilbert, decided at the age of eighteen to enter the University of Göttinfgen. Except for one year at Munich he remained at Göttingen, as a student and later as Privatdozent, until 1913, Weyl declined an offer to be his successor at Göttingen but accepted a second offer in 1930, after Hilbert had retired. In 1933 he decided he could no longer remain in Nazi Germany and accepted a position at the Institute for Advanced Study at Princeton, where he worked he worked until his retirement in 1951. In the last years of his life he divided his time between Zurich and Princeton.

Weyl undoubtedly was the most gifted of Hilbert’s students. Hilbert’s thought dominated the first part of his mathematical career; and although later he sharply diverged from his master, particularly on questions related to foundations of mathematics, Weyl always shared his convictions that the value of abstract theories lies in their success in solving classical problems and that the proper way to approach a question is through a deep analysis of the concepts it involves rather than by blind computations.

Weyl arrived at Göttingen during the period when Hilbert was creating the spectral theory of self-adjoint operators, and spectral theory and harmonic analysis were central in his mathematical research throughout his life. Very soon, however, he considerably broadened the range of his inter ests, including areas of mathematics into which Hilbert had never penetrated, such as the theory of Lie groups and the analytic theory of numbers, thereby becoming one of the most universal mathematicians of his generation. He also had an important role in the development of mathematical physics, the field to which his most famous books, Raum, Zeit und Materie (1918), on the theory of relativity, and Gruppentheorie und Quantenmechanik (1928), are devoted.

Weyl’s first important work in spectral theory was his Habilitationsschrift (1910), on singular boundary conditions for second-order linear differential equations. The classical Sturm-Liouville problem consists in determining solutions of a selfadjoint differential equation

in a compact interval 0 ≤ x ≤ l, with p(x)> 0 and q real in that interval, the solutions being subject to boundary conditions

with real numbers w,h; it is known that nontrivial solutions exist only when λ takes one of an increasing sequence (λ_n) of real numbers ≥0 and tending to +∞ (the spectrum of the equation). Weyl investigated the case in which l = +∞; his idea was to give arbitrary complex values to λ. Then, for given real h, there is a unique solution satisfying (2) and(3), provided w is taken as a complex number w(λ,h).When h takes all real values, the points w(λ,h) are on a circle C₁(λ) in the complex plane. Weyl also showed that when l tends to +∞, the circles C₁(λ) (for fixed λ) from a nested family that has a circle or a point as a limit. The distinction between the two cases is independent of the choice of λ, for in the “limit circle” case all solutions of (1) are square-integrable on [o,+∞], whereas in the “limit point” case only one solution (up to a constant factor) has that property. This was actually the first example of the general theory of defects of an unbounded Hermitian operator, which was created later by von Neumann. Weyl also showed how the classical Fourier series development of a function in a series of multiples of the eigenfunctions of the Sturm-Liouville problem was replaced, when l = +∞, by an expression similar to the Fourier integral (the spectrum then being generally a nondiscrete subset of R ); he thus anticipated the later developments of the Carleman integral operators and their applications to differential linear equations of arbitrary order and to elliptic linear partial differential equations.

In 1911 Weyl inaugurated another important chapter of spectral theory, the asymptotic study of the eigenvalues of a self-adjoint compact operator U in Hilbert space H, with special attention to applications to the theory of elasticity. For this purpose he introduced the “maximinimal” method for the direct computation of the nth eigenvalue λ_n of U (former methods gave the value of λ_n only after those of λ₁,λ₂,…,λ_n-1, had been determined). One considers an arbitrary linear subspace F of codimension n–1 in H and the largest value of the scalar product (U · x ǀ x) when x takes all values on the intersection of F and the unit sphere ǀǀxǀǀ = 1 of H; λ_n is the smallest of these largest values when F is allowed to run through all subspaces of codimension 1. This method has a very intuitive geometric interpretation in the theory of quadrics when H is finite-dimensional; it was used with great efficiency in many problems of functional analysis by Weyl himself and later by Richard Courant, who did much to popularize it and greatly extend its range of applications.

Weyl published the famous paper on equidistribution modulo 1, one of the highlights of his career, in 1916. A sequence (x_n) of real numbers is equidistributed modulo 1 if for any interval [α,β] contained in [0,1], the number v(α,βn) of elements x_k such that k ≤ n and x_k = N_k + Y_k’ with α ≤ Y_k ≤ β and N_K a (positive or negative) integer, is such that v(α,βn)/n tends to the length β–α of the interval when n tends to +∞. Led to such questions by his previous work on the Gibbs phenomenon for series of spherical harmonics, Weyl approached the problem by a completely new–and amazingly simple–method. For any sequence (y_n) of real numbers, write M((y_n)) the limit (when it exists) of the arithmetic mean (Y₁ + …+ Y_n)ǀn when n tends to +∞; then to say that (x_n) is equidistributed means that for any function of period 1 coinciding on [0,1] with the characteristic functions of any interval [α,β]. Weyl’s familiarity with harmonic analysis enabled him to conclude (1) that this property was equivalent to the existence of M((f(x_n))) for all Riemann integrable functions of period 1 and (2) that it was enough to check the existence of that limit for the particular functions exp(2π ikx) for any integer K ∊ Z. This simple criterion immediately yields the equidistribution of the sequence (nα) for irrational α (proved independently a little earlier by Weyl, Bohl, and Waclaw Sierpinński by purely arithmetic methods), as well as a quantitative form of the Kronecker theorems on simultaneous Diophantine approximations.

Weyl’s most profound result was the proof of the equidistribution of the sequence (P(n)), where P is a polynomial of arbitrary degree, the leading coefficient of which is irrational; this amounts to showing that

for N tending to +∞. To give an idea of Weyl’s ingenious proof, consider the case when P(n) = αn² + βn with irrational α. One writes

where exp (2πiαrn), I_r being the interval intersection of [0,N] and [-r, N - r] in Z, This yields . One has two majorations, ǀσ_rǀ ≤ N + 1 and ǀσ_rǀ ≤ 1/sin(2παr). For a given , the number of integers r∈ [– N,N] such that 2α r is congruent to a number in the interval [–є,є] has the form 4πєN + o(N) by equidistribution; hence it is ≤ 5є N for large N. Applying to these integers r the first majoration, and the second to the others, one obtains

ǀS_Nǀ² ≤ 5ε N(N + 1) + (2N + 1)/sin(πє) ≤ 6ε N²

for large N, thus proving the theorem. The extension of that idea to polynomials of higher degree d is not done by induction on d, but by a more elaborate device using the equidistribution of a multilinear function of d variable. Weyl’s results, through the improvements made later by I. M. Vinogradov and his school, have remained fundamental tools in the application of the Hardy-Littlewood method in the additive theory of numbers.

Weyl’s versatility is illustrated in a particularly striking way by the fact that immediately after these original advances in number theory (which he obtained in 1914), he spent more than ten years as a geometer–a geometer in the most modern sense of the word, uniting in his methods topology, algebra, analysis, and geometry in a display of dazzling virtuosity and uncommon depth reminiscent of Riemann. His familiarity with geometry and topology had been acquired a few years earlier when, as a young Privatdozent at Göttingen, he had given a course on Riemann’s theory of functions: but instead of following his predecessors in their constant appeal to “intution” for the definition and properties of Riemann surfaces, he set out to give to their theory the same kind of axiomatic and rigorous treatment that Hilbert had given to Euclidean geometry. Using Hilbert’s idea of defining neighborhoods by a system of axioms, and influenced by Brouwer’s clever application of Poincaré’s simplicial methods (which had just been published), he gave the first rigorous definition of a complex manifold of dimension 1 and a thorough treatment (without any appeal to intuition) of all questions regarding orientation, homology, and fundamental groups of these manifolds. Die Idee der Riemannschen Fläiche (1913) immediately became a classic and inspired all later developments of the theory of differential and complex manifolds.

The first geometric problem that Weyl attempted to solve (1915) was directly inspired by Hilbert’s previous work on the rigidity of convex surfaces. Hilbert had shown how the “mixed volumes” considered by Minkowski could be expressed in terms of a second-order elliptic differential linear operator L_H attached to the “Stützfunktion” H of a given convex body; Blaschke had observed that this operator was the one that intervened in the theory of infinitesimal deformation of surfaces, and this knowledge had enabled Hilbert to deduce from his results that such infinitesimal deformations for a convex body could only be Euclidean isometries. Weyl attempted to prove that not only infinitesimal deformations, but finite deformations of a convex surface as well, were necessarily Euclidean isometries. His very original idea, directly inspired by his work on two-dimensional “abstract” Riemannian manifolds, was to prove simultaneously this uniqueness property and an existence statement, namely that any two-dimensional Riemannian compact manifold with everywhere positive curvature was uniquely (up to isometries) imbeddable in Euclidean three-dimensional space. The bold method he proposed for the proof was to proceed by continuity, starting from the fact the (by another result of Hilbert’s) the problem of existence and uniqueness was already solved for the ds² of the sphere, and using a family of ds² depending continuously on a real parameter linking the given ds² to that of the sphere and having all positive curvature. This led him to a “functional differential equation” that he did not completely solve, but later work by L. Nirenberg showed that a complete proof of the theorem could be obtained along these lines.

Interrupted in this work by mobilization into the German army, Weyl did not resume it when he was allowed to return to civilian life in 1916. At Zurich he had worked with Einstein for one year, and he became keenly interested in the general theory of relativity, which had just been published; with his characteristic enthusiasm he devoted most of the next five years to exploring the mathematical framework of the theory. In these investigations Weyl introduced the concept of what is now called a linear connection, linked not to the Lorentz group of orthogonal transformations of a quadratic form of signature (1, 3) but to the enlarged group of similitudes (reproducing the quadratic form only up to a factor); he even thought for a time that this would give him a “unitary theory” encompassing both gravitation and electromagnetism. Although these hopes did not materialize, Weyl’s ideas undoubtedly were the source from which E. Cartan, a few years later, developed his general theory of connections (under the name of “generalized spaces”).

Weyl’s use of tensor calculus in his work on relativity led him to reexamine the basic methods of that calculus and, more generally, of classical invariant theory that had been its forerunner but had fallen into near oblivion after Hilbert’s work of 1890. On the other hand, his semiphilosophical, semimathematical ideas on the general concept of “space” in connection with Einstein’s theory had directed his investigations to generalizations of Helmholtz’s problem of characterizing Euclidean geometry by properties of “free mobility.” From these two directions Weyl was brought into contact with the theory of linear representations of Lie groups; his papers on the subject (1925-1927)certainly represent his masterpiece and must be counted among the most influential ones in twentieth-century mathematics.

In the early 1900’s Frobenius, I. Schur, and A. Young had completely determined the irreducible rational linear representations of the general linear group GL (n,C ) of complex matrices of order n; it was easy to deduce from Schur’s results that all rational linear representations of the special linear group SL (n,C ) (matrices of determinant 1 ) were completely reducibel–that is, direct sums of irreducible representations. Independently, E. Cartan in 1913 had described all irreducible linear representations of the simple complex Lie algebras without paying much attention to the exact relation between these representations and the corresponding ones for the simple groups, beyond exhibiting examples of group representations for each type of Lie algebra representations. Furthermore, Cartan apparently had assumed without proof that any (finite-dimensional) linear representation of a semisimple Lie algebra is completely reducible.

Weyl inaugurated a new approach by deliberately focusing his attention on global groups, the Lie algebras being reduced to the status of technical devices. In 1897 Hurwitz had shown how one may form invariants for the orthogonal or unitary group by substituting, for the usual averaging process on finite groups, integration on the (compact) group with respect to an invariant measure. He also had observed that this yields invariants not only of the special unitary group SU (n) but also of the special linear group SL (n,C ) (the first example of what Weyl later called the “unitarian trick”). Using Hurwitz’s method, I . Schur in 1924 had proved the complete reducibility of all continuous linear representations of SU (n) by showing the existence, on any representation space of that group, of a Hermitian scalar product invariant under the action of SU (n); by using the “unitarian trick” he also was able to prove the complete reducibility of the continuous linear representations of SL (n,C ) and to obtain orthogonality relations for the characters of SU (n), generalizing the well-known Frobenius relations for the characters of a finite group. These relations led to the explicit determination of the characters of SL (n,C ), which Schur had obtained in 1905 by purely algebraic methods.

Starting from these results, Weyl first made the connection between the methods of Schur and those of E. Cartan for the representations of the Lie algebra of SL (n,C ) by pointing out for the first time that the one-to-one correspondence between both types of representations was due to the fact that SU (n) is simply connected. He next extended the same method to the orthogonal and symplectic complex groups, observing, apparently for the first time, the existence of the two-sheeted covering group of the orthogonal group (the “spin” group, for which Cartan had only obtained the linear representations by spinors). Finally, Weyl turned to the global theory of all semisimple complex groups. First he showed that the “unitarian trick” had a validity that was not limited to the classical groups by proving that every semisimple complex Lie algebra ℑ could be considered as obtained by complexification from a well-determined real Lie algebra ℑ_u, which was the Lie algebra of a compact group G_u; E. Cartan had obtained that result through a case-by-case examination of all simple complex Lie groups, whereas Weyl obtained a general proof by using the properties of the roots of the semisimple algebra. This established a one-to-one correspondence between linear representations of ℑ and linear representations of ℑ_u; but to apply Hurwitz’s method, one had to have a compact Lie group having ℑ_u as Lie algebra and being simply connected. This is not necessarily the case for the group G_u, and to surmount that difficulty, one had to prove that the universal covering group of G_u is also compact; the a priori proof that such is the case is one of the deepest and most original parts of Weyl’s paper. It is linked to a remarkable geometric interpretation of the roots of the Lie algebra ℑ_u relative to a maximal commutative subalgebra t, which is the Lie algebra of a maximal torus T of G_u. Each root vanishes on a hyperplane of t, and the connected components of the complement of the union of these hyperplanes in the vector space t are polyhedrons that are now called Weyl chambers; each of these chambers has as boundary a number of “walls” equal to the dimension of t.

Using this description (and some intuitive considerations of topological dimension that he did not bother to make rigorous), Weyl showed simultaneously that the fundamental group of G_u was finite (hence was compact) and that for G_u the maximal torus T played a role similar to that of the group of diagonal matrices in SU (n): every element of G_u is a conjugate of an element of T. Furthermore, he proved that the Weyl chambers are permuted in a simply transitive way by the finite group generated by the reflections with respect to their walls (now called the Weyl group of ℑ or of G_u); this proof gave him not only a new method of recovering Cartan’s “dominat weights” but also the explicit determination of the character of a representation as a function of its dominant weight.

In this determination Weyl had to use the orthogonality relations of the characters of G_u (obtained through an extension of Schur’s method) and a property that would replace Frobenius’ fundamental theorem in the theory of linear representations of finite groups: that all irreducible representations are obtained by “decomposing” the regular representation. Weyl conceived the extraordinarily bold idea (for the time) of obtaining all irreducible representations of a semisimple group by “decomposing” an infinite-dimensional linear representation of G_u. To replace the group algebra introduced by Frobenius, he considered the continuous complexvalued functions on G_u and took as “product” of two such functions f,g what we now call the convolution f * g, defined by (f * g)(t) = ∫ f(st^-1)g(t)dt, integration being relative to an invariant measure. To each continuous function f the operator R(f): g → f * g is then associated; the “decomposition” is obtained by considering the space of continuous functions on G_u as a pre-Hilbert space and by showing that for suitable f (those of the form h * h, where , R(f) is Hermitian and compact, so that the classical Schmidt-Riesz theory of compact operators can be applied. It should be noted that in this substitute for the group algebra formed by the continuous functions on G_u, there is no unit element if G_u is not trivial (in contrast with what happens for finite groups); again it was Weyl who saw the way out of this difficulty by using the “regularizing” property of the convolution to introduce “approximate units” —that is, sequences (u_n) of functions that are such that the convolutions U_n * f tend to f for every continuous function f.

Very few of Weyl’s 150 published books and papers—even those chiefly of an expository character—lack an original idea or afresh viewpoint. The influence of his works and of his teaching was considerable: he proved by his example that an “abstract” approach to mathematics is perfectly compatible with “hard” analysis and, in fact, can be one of the most powerful tools when properly applied.

Weyl had a lifelong interest in philosophy and metaphysics, and his mathematical activity was seldom free from philosophical undertones or afterthoughts. At the height of the controversy over the foundations of mathematics, between the formalist school of Hilbert and the intuitionist school of Brouwer, he actively fought on Brouwer’s side; and if he never observed too scrupulously the taboos of the intuitionists, he was careful in his papers never to use the axiom of choice. Fortunately, he dealt with theories in which he could do so with impunity.

BIBLIOGRAPHY

Weyl’s writings were brought together in his Gesammelte Abhandlungen, K. Chandrasekharan, ed., 4 vols. (Berlin-Heidelberg-New York, 1968). See also Selecta Hermann Weyl (Basel-Stuttgart, 1956).

J. DieudonnÉ