Hermann Weyl

Über die Gleichverteilung von Zahlen mod. Eins

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Quantenmechanik und Gruppentheorie

ZusammenfassungEinleitung und Zusammenfassung. — I. Teil. Bedeutung der Repräsentation von physikalischen Größen durch Hermitesche Formen. § 1. Mathematische Grundbegriffe, die Hermiteschen Formen betreffend. § 2. Der physikalische Begriff des reinen Falles. § 3. Die physikalische Bedeutung der repräsentierenden Hermiteschen Form. § 4. Statistik der Gemenge. — II. Teil: Kinematik als Gruppe. § 5. Über Gruppen und ihre unitären Darstellungen. § 6. Übertragung auf kontinuierliche Gruppen. § 7. Ersatz der kanonischen Variablen durch die Gruppe. Das Elektron. § 8. Übergang zu Schrödingers Wellentheorie. — III. Teil. Das dynamische Problem. § 9. Das Gesetz der zeitlichen Veränderung. Die Zeitgesamtheit. § 10. Kinetische Energie und Coulombsche Kraft in der relativistischen Quantenmechanik. — Mathematischer Anhang.

Gravitation und Elektrizität

Nach Riemann2) beruht die Geometrie auf den beiden folgenden Tatsachen: 1. Der Raum ist ein dreidimensionales Kontinuum, die Mannigfaltigkeit seiner Punkte last sich also in stetiger Weise durch die Wertsysteme dreier Koordinaten x 1 x 2 x 3 zur Darstellung bringen; 2. (Pythagoreischer Lehrsatz) Das Quadrat des Abstandes ds 2 zweier unendlich benachbarter Punkte

Raum, Zeit, Materie : Vorlesungen über allgemeine Relativitätstheorie

Inhaltsubersicht: Einleitung.- Der Euklidische Raum: seine mathematische Formalisierung und seine Rolle in der Physik.- Das metrische Kontinuum.- Relativitat von Raum und Zeit.- Allgemeine Relativitatstheorie.- Literatur.- Sachverzeichnis.- Anmerkungen und Erganzungen des Herausgebers.- Literaturerganzungen.

David Hilbert and his mathematical work

A great master of mathematics passed away when David Hilbert died in Gottingen on February the 14th, 1943, at the age of eighty-one. In retrospect it seems to us that the era of mathematics upon which he impressed the seal of his spirit and which is now sinking below the horizon achieved a more perfect balance than prevailed before and after, between the mastering of single concrete problems and the formation of general abstract concepts. Hilbert’s own work contributed not a little to bringing about this happy equilibrium, and the direction in which we have since proceeded can in many instances be traced back to his impulses. No mathematician of equal stature has risen from our generation.

Ramifications, old and new, of the eigenvalue problem

Since this is a lecture dedicated to the memory of Josiah Willard Gibbs let me start with that purely mathematical discovery which Gibbs contributed to the theory of Fourier series. Fourier series have to do with the eigenvalues and eigenfunctions of the oldest, simplest, and most important of all spectrum problems, that of the vibrating string. In preparing this lecture, the speaker has assumed that he is expected to talk on a subject in which he had some first-hand experience through his own work. And glancing back over the years he found that the one topic to which he has returned again and again is the problem of eigenvalues and eigenfunctions in its various ramifications. I t so happens that right a t the beginning of my mathematical career I wrote two papers on what we now call the Gibbs phenomenon.

How Far Can One Get With a Linear Field Theory of Gravitation in Flat Space-Time?

Introduction and Summary. G. D. Birkhoff’s atempt to establish a linear field theory of gravitation within the frame of special relativity makes it desirable to probe the potentialities and limitations of such a theory in more general terms. In thus continuing a discussion begun in another place I find that the differential operators at one’s disposal form a 5 dimensional linear manifold. But the requirement that the field equations imply the law of conservation of energy and momentum in the simple form ∂T k i /∂xk = 0 limit these ∞ possibilities to ∞, which, however, reduce to two cases, a regular one (L) and a singular one (L′). The regular case (L) is nothing but Einstein’s theory of weak fields. Resembling very closely Maxwell’s theory of the electromagnetic field, it satisfies a principle of gauge involving 4 arbitrary functions, and although its gravitational field exerts no force on matter, it is well suited to illustrate the role of energy and momentum, charge and mass in the interplay bwtween matter and field. It might also help, though this is much more problematic, in pointing the way to a more satisfactory unification of gravitation and electricity than we at present possess. Birkhoff follows the opposite way: by avoiding rather than adopting the ∞ special operators mentioned above, his ”‘dualistic”’ theory B) destroys the bond between mechanical and field equations, which is such a decisive feature in Einstein’s theory.

David Hilbert 1862-1943

David Hilbert, upon whom the world looked during the last decades as the greatest of the living mathematicians, died in Gottingen, Germany, on 14 February 1943. At the age of eighty-one he succumbed to a compound fracture of the thigh brought about by a domestic accident. Hilbert was born on 23 January 1862, in the city of Konigsberg in East Prussia. He was descended from a family which had long been settled there and had brought forth a series of physicians and judges. During his entire life he preserved uncorrupted the Baltic accent of his home. For a long time Hilbert remained faithfully attached to the town of his forbears, and well deserved its honorary citizenship which was bestowed upon him in his later years.

Über die kombinatorische und kontinuumsmässige Definition der Überschneidungszahl zweier geschlossener Kurven auf einer Fläche

SummaryThe characteristic of two closed curves on an oriented surface is, roughly speaking, the algebraic sum of their intersections, counting a crossing from left to right as +1, a crossing in the opposite sense as −1. To make this definition precise, combinatorial topology replaces the surface by an aggregate of 0-, 1- and 2-cells and first assumes one curve to be a chain (joining the 0-cells), the other to be a co-chain (joining the 2-cells). The idea of homology (as defined by means of ‘integral functions’) and the law of antisymmetry then serve to pass from these specialized to arbitrary curves. This definition is here contrasted with another that arose from ‘topologizing’ the procedure by which the author in hisIdee der Riemannschen Fläche (1913) constructed the Abelian integrals of the first kind: it makes no use of triangulation, but instead of the coverage of the surface by neighborhoods and their topological mapping onto circular disks. Finally the paper indicates the steps by which one proves the coincidence of both definitions.