Hermann Weyl

How to use this course

Hermann Weyl worked in more fields than almost any mathematician of his century: analysis, number theory, Riemann surfaces, differential geometry, relativity, Lie groups, quantum mechanics, logic and philosophy. The parts are connected. Again and again he found that the right way to understand a problem was to ask what symmetry it has, and how that symmetry acts. The word “Weyl” now appears in the names of a law, a criterion, a tensor, a group, a character formula, a spinor equation, a quantization and a lemma.

Each technical lesson opens with an italic paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. The lessons follow the order in which Weyl took up each subject, from spectral theory (1908–1912) to the foundations of mathematics, which occupied him from 1918 onward.

Timeline

Elmshorn, Altona, Göttingen, 1885–1904

Hermann Klaus Hugo Weyl was born on 9 November 1885 in Elmshorn, a small town in Schleswig-Holstein north-west of Hamburg. His father Ludwig was the director of a bank; his mother was Anna Weyl, born Dieck. He went to the Gymnasium Christianeum in Altona, then a separate city beside Hamburg. Its headmaster was a cousin of David Hilbert, and it was through this connection that Weyl, a strong student in mathematics and ancient languages, decided to study at Göttingen.

He arrived in 1904, at eighteen, and was overwhelmed. Hilbert was then at the height of his powers, with Felix Klein and Hermann Minkowski beside him. Weyl later wrote that he resolved, on reading Hilbert’s report on algebraic number fields during his first vacation, to study everything this man had written. He spent a period at the University of Munich, where he heard lectures in physics, before returning to Göttingen to stay.

Hilbert’s student, 1904–1913

Hilbert was then building the theory of integral equations and what would become Hilbert space. Weyl’s doctoral thesis of 1908, Singuläre Integralgleichungen mit besonderer Berücksichtigung des Fourierschen Integraltheorems, extended Hilbert’s theory to kernels on infinite intervals, where the spectrum need not be discrete. His habilitation of 1910 treated Sturm–Liouville operators \(-(pu')'+qu\) on half-lines and found the limit-point and limit-circle alternative, still the starting point of the spectral theory of singular differential operators. He became Privatdozent at Göttingen.

In 1911 he published the asymptotic law for eigenvalues that bears his name (Lesson 04), answering a question raised by physicists studying black-body radiation. In the same year, with Andreas Speiser, he helped Hilbert edit the collected works of Minkowski, who had died in 1909. In 1911–12 he lectured on Riemann surfaces, and turned the course into a book.

Die Idee der Riemannschen Fläche (1913) gave Bernhard Riemann’s intuitive surfaces their first rigorous definition, as abstract two-dimensional manifolds (Lesson 06). Klein had taught the subject geometrically; Weyl’s book made it a part of topology and analysis. The same year he accepted a chair at the Eidgenössische Technische Hochschule (ETH) in Zürich, and in September married Helene Joseph, a student of philosophy and of Edmund Husserl. They had two sons, Fritz Joachim (born 1915) and Michael (born 1917).

Zürich, war and relativity, 1913–1923

At the ETH Weyl overlapped for one year with Albert Einstein, who was completing the road to general relativity before leaving for Berlin in 1914. In 1915 Weyl, still a German citizen, was called up for military service. The Swiss government asked for his release, and in 1916 he returned to Zürich, where he stayed for the next fourteen years.

The publication of general relativity in 1915–16 turned Weyl toward geometry and physics. His 1917 lectures at the ETH became Raum, Zeit, Materie (1918), the first great textbook of the subject, which ran through five editions by 1923. In 1918 he generalized Riemannian geometry so that lengths, as well as directions, depend on the path along which they are compared. The result was a unified theory of gravitation and electromagnetism, and the first gauge theory (Lesson 07). Einstein admired its elegance and rejected it on physical grounds.

The same year Weyl published Das Kontinuum, a critique of the set-theoretic foundations of analysis, and in 1921 he joined L. E. J. Brouwer in calling for a new, constructive mathematics (Lesson 10). This rebellion against his teacher pained Hilbert, who answered by launching his formalist programme. Zürich also brought him a close colleague in physics: Erwin Schrödinger, at the University of Zürich from 1921, who discovered wave mechanics there in 1926.

Spectra: the Weyl law

For the general reader. A drum of a given shape can vibrate only at certain frequencies. Can you hear its area? Weyl proved in 1911 that you can: counting the frequencies up to a large limit gives the area, whatever the shape. Physicists needed exactly this fact to make sense of the radiation inside a hot oven, and it has since become a central theme of analysis and quantum chaos.

Let \(\Omega\subset\mathbb R^d\) be a bounded domain with piecewise smooth boundary. The Dirichlet problem \(-\Delta u=\lambda u\) in \(\Omega\), \(u=0\) on \(\partial\Omega\), has eigenvalues \(0<\lambda_1\le\lambda_2\le\cdots\to\infty\). Let \(N(\lambda)=\#\{k:\lambda_k\le\lambda\}\).

Proof sketch

Boxes. For \(\Omega=\prod_i(0,a_i)\) the eigenfunctions are \(\prod_i\sin(\pi n_ix_i/a_i)\), with eigenvalues \(\pi^2\sum n_i^2/a_i^2\), \(n_i\ge1\). So \(N(\lambda)\) counts lattice points in the positive part of an ellipsoid with semi-axes \(a_i\sqrt\lambda/\pi\), whose volume is \(2^{-d}\omega_d\prod_i(a_i\sqrt\lambda/\pi)=\omega_d\operatorname{vol}(\Omega)\lambda^{d/2}/(2\pi)^d\). The lattice-point count equals the volume up to a surface error of lower order.

General domains. By the minimax principle,

with \(V\) ranging over subspaces of \(H^1_0(\Omega)\). Enlarging the domain enlarges the space of test functions and lowers the eigenvalues (domain monotonicity). Weyl filled \(\Omega\) from inside and covered it from outside by finite unions of small cubes, using Dirichlet conditions inside and Neumann conditions outside. Both counts have the same asymptotics in the limit of fine cubes, which squeezes \(N(\lambda)\).

Weyl conjectured in 1913 a second term, \(-\frac{\omega_{d-1}}{4(2\pi)^{d-1}}|\partial\Omega|\,\lambda^{(d-1)/2}\) for Dirichlet conditions. Victor Ivrii proved it in 1980 under a condition on billiard trajectories that holds for generic domains.

Equidistribution modulo one

For the general reader. Take the multiples of \(\sqrt2\), keep only their fractional parts, and mark them on a ruler of length one. The marks spread out evenly, so each interval eventually gets its fair share. Weyl proved this and much more in 1916, with a test that reduces the question to sums of waves. His method became a basic tool of analytic number theory.

A sequence \((x_n)\) of reals is equidistributed modulo 1 if, for every \(0\le a<b\le1\), the proportion of \(n\le N\) with \(\{x_n\}\in[a,b)\) tends to \(b-a\). Write \(e(x)=e^{2\pi ix}\).

Proof sketch. Equidistribution says \(\frac1N\sum f(\{x_n\})\to\int_0^1f\) for indicator functions of intervals, and then, by approximation from above and below, for every continuous 1-periodic \(f\). The criterion says it for the characters \(e(hx)\). Trigonometric polynomials are uniformly dense in continuous periodic functions (Fejér, or Weierstrass), which gives the equivalence for continuous \(f\); continuous functions squeeze indicator functions.

The key is Weyl differencing. Expanding \(|S|^2\) for \(S=\sum_{n=0}^{N-1}e(f(n))\) and grouping terms by \(h=m-n\) gives

Differencing lowers the degree of a polynomial by one, so induction reduces degree \(k\) to degree 1, a geometric sum. Johannes van der Corput turned the argument into a clean principle in 1931: if \((x_{n+h}-x_n)_n\) is equidistributed for every \(h\ge1\), then so is \((x_n)\).

The idea of a Riemann surface

For the general reader. The square root of a complex number has two values, and following one continuously around zero turns it into the other. Riemann’s answer was to replace the plane by a surface with two sheets, on which the square root is a genuine function. Riemann described such surfaces by intuition. Weyl, in 1913, defined them precisely, and in doing so wrote down for the first time the modern idea of a manifold.

Weyl formulated the topology through systems of neighbourhoods, a year before Felix Hausdorff’s axioms for topological spaces, and defined the surface first and the analytic functions on it second. The surface is no longer spread over the plane; the plane is only a local picture. Holomorphic and meromorphic functions, differentials and their periods are then defined chart by chart.

The existence problem

Does every Riemann surface carry non-constant meromorphic functions? Riemann answered yes by the Dirichlet principle: minimize the energy \(\int|\nabla u|^2\) among functions with prescribed singularities, and the minimizer is harmonic. Weierstrass objected that a minimum need not exist; Hilbert rescued the principle in 1900. Weyl built his book on Hilbert’s rescued principle, constructing harmonic differentials with given periods and singularities, and from them the Riemann–Roch theorem and uniformization.

Weyl proved this in his 1940 paper on the method of orthogonal projection, which made the Dirichlet principle a short argument in Hilbert space: project onto the orthogonal complement of exact forms, then use the lemma to show that the weak solution is a genuine harmonic form. It is the prototype of elliptic regularity and of Hodge theory.

Gauge invariance, 1918 and 1929

For the general reader. In 1918 Weyl proposed that the length of a measuring rod might change as it is carried around a loop, and that this change is electromagnetism. Einstein showed that nature does not work that way. Eleven years later, after quantum mechanics, Weyl reused the same idea with the length replaced by the phase of a quantum wave. The result, the gauge principle, is now the basis of all theories of fundamental forces.

1918: scale gauge

In Riemannian geometry a vector can change direction under parallel transport, but its length is preserved. Weyl allowed the length \(\ell\) to change too, along a curve with tangent \(dx^\mu\), by

for a 1-form \(\varphi_\mu\). The metric is fixed only up to a local rescaling (“Eichung”, gauge) \(g_{\mu\nu}\mapsto e^{2\lambda(x)}g_{\mu\nu}\), under which \(\varphi_\mu\mapsto\varphi_\mu-\partial_\mu\lambda\). The length change around a closed loop is governed by \(f_{\mu\nu}=\partial_\mu\varphi_\nu-\partial_\nu\varphi_\mu\), which is invariant. Weyl identified \(\varphi_\mu\) with the electromagnetic potential and \(f_{\mu\nu}\) with the field: gravitation and electromagnetism as one geometry.

Einstein objected at once. If lengths depended on history, so would the rates of clocks, and two identical atoms with different pasts would emit light of different frequencies. Sharp spectral lines show that they do not.

1929: phase gauge

After Vladimir Fock (1926) and Fritz London (1927) observed that the potential enters quantum mechanics through the phase of the wave function, Weyl wrote Elektron und Gravitation (1929). Demand that physics be unchanged under a local phase change \(\psi\mapsto e^{i\alpha(x)}\psi\). Ordinary derivatives fail, since \(\partial_\mu(e^{i\alpha}\psi)=e^{i\alpha}(\partial_\mu\psi+i\partial_\mu\alpha\,\psi)\). Introduce a field \(A_\mu\) and the covariant derivative

The same paper introduced two-component spinors in general relativity, with tetrads (vierbeins) to carry them, and the massless Weyl equation (Lesson 09). In 1954 Chen Ning Yang and Robert Mills replaced \(\mathrm U(1)\) by \(\mathrm{SU}(2)\); the Standard Model is a gauge theory in Weyl’s sense with group \(\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm U(1)\).

The conformal tensor

Weyl’s 1918 geometry led him to the part of the Riemann curvature that is unchanged by rescaling the metric. In \(n\ge3\) dimensions the Weyl tensor \(C_{abcd}\) is the trace-free part of \(R_{abcd}\). It is conformally invariant (as \(C^a{}_{bcd}\)), and for \(n\ge4\) it vanishes if and only if the metric is locally conformally flat. In general relativity it describes tidal forces and gravitational waves in vacuum, where the Ricci tensor vanishes.

Compact groups and their characters

For the general reader. A symmetry group can act on many spaces at once, and these actions break into indivisible pieces called irreducible representations. In 1925 and 1926 Weyl found all the pieces for the continuous symmetry groups of physics and geometry, with a single formula for their characters and dimensions.

Averaging and the unitarian trick

A compact group \(G\) has a unique invariant probability measure \(dg\) (Haar measure; for Lie groups Adolf Hurwitz had constructed it in 1897). If \(V\) is a finite-dimensional representation, averaging any inner product gives an invariant one, \(\langle u,v\rangle_G=\int_G\langle gu,gv\rangle\,dg\). The orthogonal complement of an invariant subspace is then invariant, so every representation is a direct sum of irreducibles. Weyl’s unitarian trick transfers this to non-compact groups: the holomorphic representations of \(\mathrm{SL}_n(\mathbb C)\) are the same as the representations of its compact form \(\mathrm{SU}(n)\), because \(\mathrm{SU}(n)\) is Zariski dense. So complete reducibility holds for complex semisimple groups too.

The character formula

Let \(G\) be compact and connected with maximal torus \(T\), roots \(\Phi\) with a choice of positive roots \(\Phi^+\), half-sum \(\rho=\frac12\sum_{\alpha>0}\alpha\), and Weyl group \(W=N(T)/T\), a finite group generated by reflections. Every element of \(G\) is conjugate into \(T\), so a character is determined by its restriction to \(T\).

Proof idea. The Weyl integration formula reduces integrals of class functions over \(G\) to integrals over \(T\) with density \(\frac1{|W|}|\Delta|^2\), where \(\Delta=\sum_W\varepsilon(w)e^{w\rho}\) is the Weyl denominator. So \(\chi\Delta\) is a \(W\)-antisymmetric function on \(T\) whose square norm is \(|W|\) times that of \(\chi\). Irreducibility means \(\|\chi\|^2=1\); an antisymmetric integer combination of characters of \(T\) with norm \(|W|\) must be \(\pm\) a single alternating sum \(\sum\varepsilon(w)e^{w\mu}\). The highest term identifies \(\mu=\lambda+\rho\).

For \(G=\mathrm U(1)\) this is the theory of Fourier series; Peter–Weyl is Fourier analysis on every compact group. Weyl’s book The Classical Groups (1939) treated the invariant theory and representations of \(\mathrm{GL}_n\), \(\mathrm O_n\) and \(\mathrm{Sp}_{2n}\) together, including the duality between \(\mathrm{GL}_n\) and the symmetric group acting on tensors, now called Schur–Weyl duality.

Groups and quantum mechanics

For the general reader. When quantum mechanics appeared in 1925–26, Weyl saw at once that its rules were statements about symmetry groups. His 1928 book taught a generation of physicists, some of them reluctantly, that atomic spectra, spin and the exclusion principle are representation theory. Along the way he found a way to turn classical quantities into quantum operators, and an equation for massless particles that turned out to describe neutrinos.

The Weyl relations

The Heisenberg relation \([Q,P]=i\hbar\) cannot hold for bounded operators. Weyl (1927) replaced it by relations between the unitary groups \(U(s)=e^{isQ}\), \(V(t)=e^{itP}\):

This settled the equivalence of Heisenberg’s matrix mechanics and Schrödinger’s wave mechanics. In modern terms, the Weyl relations define the Heisenberg group, and the theorem classifies its irreducible unitary representations with a given central character.

Weyl quantization

Weyl proposed assigning to a classical observable \(a(q,p)=\iint\hat a(s,t)\,e^{i(sq+tp)}\,ds\,dt\) the operator

This symmetric ordering is the basis of the Wigner function and of the pseudodifferential calculus.

The Weyl equation

In Elektron und Gravitation (1929) Weyl wrote the equation for a massless two-component spinor field,

with \(\boldsymbol\sigma\) the Pauli matrices. Wolfgang Pauli rejected it because it is not invariant under reflection. After parity violation was discovered in 1956–57, the Weyl equation became the description of neutrinos in the Standard Model (before neutrino masses were found). In 2015 Weyl fermions were observed as quasiparticles in crystals called Weyl semimetals.

The continuum and the foundations crisis

For the general reader. Is the real line made of points, each picked out by a definition, or is it a whole that cannot be assembled from parts? Weyl thought classical analysis had built on sand, and in 1918 tried to rebuild as much of it as possible using only definitions that do not presuppose the totality they define. For a few years he went further, with Brouwer. He later returned to a middle ground, but the questions he raised shaped mathematical logic.

Predicative analysis

A definition is impredicative if it defines an object by quantifying over a collection that contains the object. The classical least upper bound of a bounded set \(S\) of reals is defined, via Dedekind cuts, as the set of rationals \(q\) such that \(q<s\) for some \(s\in S\). When \(S\) is itself defined by quantifying over all reals, the result is a real defined by reference to all reals, itself included.

Solomon Feferman later showed that Weyl’s system is a conservative extension of Peano arithmetic and that most of the analysis used in physics can be carried out in it.

With Brouwer, and back

In 1921 Weyl’s essay on the “new foundational crisis” of mathematics declared that Brouwer’s intuitionism was the revolution. Hilbert replied by founding proof theory, intended to secure classical mathematics by finitary consistency proofs. Weyl gradually returned to a pragmatic position, judging that intuitionistic mathematics was too poor to support physics. In 1918 he had bet George Pólya that within twenty years leading mathematicians would come to regard statements such as the least upper bound principle for arbitrary bounded sets as false or meaningless. When the twenty years had passed, the Zürich colleagues asked to judge the bet decided in Pólya’s favour.

Zürich, Göttingen, Princeton, 1923–1951

In Zürich in the 1920s Weyl produced his deepest work in pure mathematics, the representation theory of Lie groups (1925–27), and he was quick to bring it to the new quantum mechanics. He supervised a series of doctoral students at the ETH, among them Alexander Weinstein and the future president of MIT, Julius Adams Stratton. His 1927 Lobachevsky Prize from the University of Kazan recognized his geometric work. He spent 1928–29 as a visiting professor of mathematical physics at Princeton and gave a plenary address at the International Congress of Mathematicians in Bologna in 1928.

In 1930 he accepted the call to succeed Hilbert at Göttingen. The institution he joined was at its peak, with Emmy Noether, Richard Courant, Edmund Landau and a new Mathematical Institute building. It lasted three years. After the Nazis came to power in 1933, the dismissals destroyed the department. Weyl, who was not Jewish, was at risk because his wife was. After much hesitation he accepted a permanent position at the newly founded Institute for Advanced Study in Princeton, alongside Einstein, John von Neumann, Oswald Veblen and James Alexander. Two of his last Göttingen doctoral students were Gerhard Gentzen (1933), whose work was guided by Paul Bernays, and Saunders Mac Lane (1934).

At Princeton Weyl spent much energy helping refugee scholars find positions. In 1935 he gave the memorial address for Emmy Noether at Bryn Mawr. He became an American citizen in 1939, the year of The Classical Groups. Algebraic Theory of Numbers followed in 1940, and Philosophy of Mathematics and Natural Science, an expanded version of a 1926 work, in 1949. He was elected a Fellow of the Royal Society in 1936 and to the US National Academy of Sciences in 1940, and gave the Gibbs Lecture of the American Mathematical Society in 1948.

Last years and legacy, 1948–1955

Helene Weyl died in Princeton in September 1948. In 1950 Weyl married the sculptor Ellen Bär, born Lohnstein, of Zürich. He retired from the Institute in 1951 and afterward divided his time between Princeton and Zürich. His Princeton lectures on symmetry, given on his retirement and published as Symmetry (1952), remain one of the finest books on mathematics for a general audience. In 1955 the English edition of his Riemann surface book appeared, rewritten in modern language. On 8 December 1955, a month after his seventieth birthday, he collapsed in a Zürich street while walking home and died. His ashes were interred in Princeton Cemetery in 1999.

Legacy

• Physics: the gauge principle underlies the Standard Model; Weyl spinors are the basic fermions of the theory; the Weyl tensor is central in general relativity; Weyl quantization and the Weyl relations are basic in quantum mechanics.
• Representation theory: the character formula, Weyl group, Weyl chambers, and Peter–Weyl theorem are the foundation of the theory of Lie groups and of harmonic analysis on compact groups.
• Analysis and number theory: the Weyl law opened spectral asymptotics and semiclassical analysis; equidistribution and Weyl sums are standard tools from the circle method to ergodic theory; Weyl’s lemma began elliptic regularity.
• Geometry and logic: the abstract manifold, Weyl geometry and conformal geometry; predicative analysis and the reverse-mathematics programme it anticipated.
• Writing: Raum, Zeit, Materie, Gruppentheorie und Quantenmechanik, The Classical Groups and Symmetry remain in print. His collected papers were edited by Komaravolu Chandrasekharan (1968).
• Honours: the lunar crater Weyl is named after him.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Hermann Klaus Hugo Weyl, MacTutor History of Mathematics.
2. Hermann Weyl, Wikipedia.
3. Weyl, Hermann, Historisches Lexikon der Schweiz.
4. Hermann Weyl, Bhāvanā.
5. M. H. A. Newman, Hermann Weyl, 1885–1955, Biographical Memoirs of Fellows of the Royal Society 3 (1957), 305–328.
6. C. Chevalley and A. Weil, Hermann Weyl (1885–1955), L’Enseignement Mathématique 3 (1957), 157–187.
7. M. Atiyah, Hermann Weyl, 1885–1955, Biographical Memoirs of the National Academy of Sciences 82 (2002).
8. K. Chandrasekharan (ed.), Hermann Weyl 1885–1985: Centenary Lectures, Springer, 1986.

Mathematics and physics

1. H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen, Math. Ann. 71 (1912), 441–479.
2. H. Weyl, Die Idee der Riemannschen Fläche, Teubner, 1913; English edition The Concept of a Riemann Surface, 1955.
3. H. Weyl, Über die Gleichverteilung von Zahlen mod. Eins, Math. Ann. 77 (1916), 313–352.
4. H. Weyl, Raum, Zeit, Materie, Springer, 1918 (5th ed. 1923); Das Kontinuum, Veit, 1918.
5. H. Weyl, Gravitation und Elektrizität, Sitzungsber. Preuss. Akad. Wiss. (1918), 465–480.
6. H. Weyl, Theorie der Darstellung kontinuierlicher halbeinfacher Gruppen durch lineare Transformationen I–III, Math. Z. 23–24 (1925–26).
7. F. Peter and H. Weyl, Die Vollständigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe, Math. Ann. 97 (1927), 737–755.
8. H. Weyl, Quantenmechanik und Gruppentheorie, Z. Physik 46 (1927), 1–46; Gruppentheorie und Quantenmechanik, Hirzel, 1928.
9. H. Weyl, Elektron und Gravitation I, Z. Physik 56 (1929), 330–352.
10. H. Weyl, The Classical Groups, Princeton, 1939; Symmetry, Princeton, 1952.
11. L. O’Raifeartaigh, The Dawning of Gauge Theory, Princeton, 1997 (translations of Weyl’s 1918 and 1929 papers).
12. W. Arendt, R. Nittka, W. Peter, F. Steiner, Weyl’s law: spectral properties of the Laplacian in mathematics and physics, arXiv:0908.3207.
13. S. Feferman, Weyl vindicated: Das Kontinuum seventy years later, in In the Light of Logic, Oxford, 1998.
14. E. Scholz (ed.), Hermann Weyl’s Raum–Zeit–Materie and a General Introduction to his Scientific Work, Birkhäuser, 2001.