Distribution of eigenvalues In 1911 Weyl published
Über die asymptotische Verteilung der Eigenwerte (
On the asymptotic distribution of eigenvalues) in which he proved that the
eigenvalues of the
Laplacian in a compact domain are distributed according to the so-called
Weyl law. In 1912 he suggested a new proof, based on variational principles. Weyl returned to this topic several times, considered elasticity system and formulated the
Weyl conjecture. These works started an important domain—
asymptotic distribution of eigenvalues—of modern analysis.
Geometric foundations of manifolds and physics In 1913, Weyl published
Die Idee der Riemannschen Fläche (
The Concept of a Riemann Surface), which gave a unified treatment of
Riemann surfaces. In it Weyl utilized
point set topology, in order to make Riemann surface theory more rigorous, a model followed in later work on
manifolds. He absorbed
L. E. J. Brouwer's early work in topology for this purpose. Weyl, as a major figure in the Göttingen school, was fully apprised of Einstein's work from its early days. He tracked the development of
relativity physics in his
Raum, Zeit, Materie (
Space, Time, Matter) from 1918, which reached its 4th edition in 1922. In 1918, he introduced the notion of
gauge, and gave the first example of what is now known as a
gauge theory. Weyl's gauge theory was an unsuccessful attempt to model the
electromagnetic field and the
gravitational field as geometrical properties of
spacetime. The Weyl tensor in
Riemannian geometry is of major importance in understanding the nature of
conformal geometry. His overall approach in physics was based on the
phenomenological philosophy of
Edmund Husserl, specifically Husserl's 1913
Ideen zu einer reinen Phänomenologie und phänomenologischen Philosophie. Erstes Buch: Allgemeine Einführung in die reine Phänomenologie (Ideas of a Pure Phenomenology and Phenomenological Philosophy. First Book: General Introduction). Husserl had reacted strongly to
Gottlob Frege's criticism of his first work on the philosophy of arithmetic and was investigating the sense of mathematical and other structures, which Frege had distinguished from empirical reference.
Topological groups, Lie groups and representation theory From 1923 to 1938, Weyl developed the theory of
compact groups, in terms of
matrix representations. In the
compact Lie group case he proved a fundamental
character formula. These results are foundational in understanding the symmetry structure of
quantum mechanics, which he put on a group-theoretic basis. This included
spinors. Together with the
mathematical formulation of quantum mechanics, in large measure due to
John von Neumann, this gave the treatment familiar since about 1930. Non-compact groups and their representations, particularly the
Heisenberg group, were also streamlined in that specific context, in his 1927
Weyl quantization, the best extant bridge between classical and quantum physics to date. From this time, and certainly much helped by Weyl's expositions,
Lie groups and
Lie algebras became a mainstream part both of
pure mathematics and
theoretical physics. His book
The Classical Groups reconsidered
invariant theory. It covered
symmetric groups,
general linear groups,
orthogonal groups, and
symplectic groups and results on their
invariants and
representations.
Harmonic analysis and analytic number theory Weyl also showed how to use
exponential sums in
diophantine approximation, with his criterion for
uniform distribution mod 1, which was a fundamental step in
analytic number theory. This work applied to the
Riemann zeta function, as well as
additive number theory. It was developed by many others.
Foundations of mathematics In
The Continuum Weyl developed the logic of
predicative analysis using the lower levels of
Bertrand Russell's
ramified theory of types. He was able to develop most of classical
calculus, while using neither the
axiom of choice nor
proof by contradiction, and avoiding
Georg Cantor's
infinite sets. Weyl appealed in this period to the radical
constructivism of the German romantic, subjective idealist
Fichte. Shortly after publishing
The Continuum Weyl briefly shifted his position wholly to the
intuitionism of Brouwer. In
The Continuum, the constructible points exist as discrete entities. Weyl wanted a
continuum that was not an aggregate of points. He wrote a controversial article proclaiming, for himself and
L. E. J. Brouwer, a "revolution." This article was far more influential in propagating intuitionistic views than the original works of Brouwer himself.
George Pólya and Weyl, during a mathematicians' gathering in Zürich (9 February 1918), made a bet concerning the future direction of mathematics. Weyl predicted that in the subsequent 20 years, mathematicians would come to realize the total vagueness of notions such as
real numbers,
sets, and
countability, and moreover, that asking about the
truth or falsity of the
least upper bound property of the real numbers was as meaningful as asking about truth of the basic assertions of
Hegel on the philosophy of nature. Any answer to such a question would be unverifiable, unrelated to experience, and therefore senseless. However, within a few years Weyl decided that Brouwer's intuitionism did put too great restrictions on mathematics, as critics had always said. The "Crisis" article had disturbed Weyl's
formalist teacher Hilbert, but later in the 1920s Weyl partially reconciled his position with that of Hilbert. After about 1928 Weyl had apparently decided that mathematical intuitionism was not compatible with his enthusiasm for the
phenomenological philosophy of
Husserl, as he had apparently earlier thought. In the last decades of his life Weyl emphasized mathematics as "symbolic construction" and moved to a position closer not only to Hilbert but to that of
Ernst Cassirer. Weyl however rarely refers to Cassirer, and wrote only brief articles and passages articulating this position. By 1949, Weyl was thoroughly disillusioned with the ultimate value of intuitionism, and wrote: "Mathematics with Brouwer gains its highest intuitive clarity. He succeeds in developing the beginnings of analysis in a natural manner, all the time preserving the contact with intuition much more closely than had been done before. It cannot be denied, however, that in advancing to higher and more general theories the inapplicability of the simple laws of
classical logic eventually results in an almost unbearable awkwardness. And the mathematician watches with pain the greater part of his towering edifice which he believed to be built of concrete blocks dissolve into mist before his eyes." As John L Bell puts it: "It seems to me a great pity that Weyl did not live to see the emergence in the 1970s of smooth infinitesimal analysis, a mathematical framework within which his vision of a true continuum, not “synthesized” from discrete elements, is realized. Although the underlying logic of smooth infinitesimal analysis is intuitionistic — the
law of excluded middle not being generally affirmable — mathematics developed within avoids the “unbearable awkwardness” to which Weyl refers above."
Weyl equation In 1929, Weyl proposed an equation, known as the
Weyl equation, for use in a replacement to the
Dirac equation. This equation describes massless
fermions. A normal Dirac fermion could be split into two Weyl fermions or formed from two Weyl fermions.
Neutrinos were once thought to be Weyl fermions, but they are now known to have mass. Weyl fermions are sought after for electronics applications.
Quasiparticles that behave as Weyl fermions were discovered in 2015, in a form of crystals known as
Weyl semimetals, a type of topological material. ==Quotes==