John von Neumann


John von Neumann : biography

28 December 1903 – 8 February 1957

In the 1936 paper on analytic measure theory, von Neumann used the Haar theorem in the solution of Hilbert’s fifth problem in the case of compact groups.

Ergodic theory

Von Neumann made foundational contributions to ergodic theory, in a series of articles published in 1932, which have attained legendary status in mathematics.Two famous papers are below: . . . Of the 1932 papers on ergodic theory, Paul Halmos writes that even "if von Neumann had never done anything else, they would have been sufficient to guarantee him mathematical immortality". By then von Neumann had already written his famous articles on operator theory, and the application of this work was instrumental in the von Neumann mean ergodic theorem.Michael Reed, Barry Simon, "Functional Analysis: Volume 1", Academic Press; Revised edition (1980)

Operator theory

Von Neumann introduced the study of rings of operators, through the von Neumann algebras.D.Petz and M.R. Redi, , in The Neumann compendium, World Scientific, 1995, pp. 163–181 ISBN 9810222017. A von Neumann algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator.

The von Neumann bicommutant theorem shows that the analytic definition is equivalent to a purely algebraic definition as an algebra of symmetries.

The direct integral was introduced in 1949 by John von Neumann. One of von Neumann’s analyses was to reduce the classification of von Neumann algebras on separable Hilbert spaces to the classification of factors.

Lattice theory

Garrett Birkhoff writes: "John von Neumann’s brilliant mind blazed over lattice theory like a meteor". Von Neumann worked on lattice theory between 1937 and 1939. Von Neumann provided an abstract exploration of dimension in completed complemented modular topological lattices: "Dimension is determined, up to a positive linear transformation, by the following two properties. It is conserved by perspective mappings ("perspectivities") and ordered by inclusion. The deepest part of the proof concerns the equivalence of perspectivity with "projectivity by decomposition"—of which a corollary is the transitivity of perspectivity."

Additionally, "[I]n the general case, von Neumann proved the following basic representation theorem. Any complemented modular lattice L having a "basis" of n≥4 pairwise perspective elements, is isomorphic with the lattice ℛ(R) of all principal right-ideals of a suitable regular ring R. This conclusion is the culmination of 140 pages of brilliant and incisive algebra involving entirely novel axioms. Anyone wishing to get an unforgettable impression of the razor edge of von Neumann’s mind, need merely try to pursue this chain of exact reasoning for himself—realizing that often five pages of it were written down before breakfast, seated at a living room writing-table in a bathrobe."

Mathematical formulation of quantum mechanics

Von Neumann was the first to rigorously establish a mathematical framework for quantum mechanics, known as the Dirac–von Neumann axioms, with his 1932 work Mathematische Grundlagen der Quantenmechanik.

After having completed the axiomatization of set theory, von Neumann began to confront the axiomatization of quantum mechanics. He immediately realized, in 1926, that a quantum system could be considered as a point in a so-called Hilbert space, analogous to the 6N dimension (N is the number of particles, 3 general coordinate and 3 canonical momentum for each) phase space of classical mechanics but with infinitely many dimensions (corresponding to the infinitely many possible states of the system) instead: the traditional physical quantities (e.g., position and momentum) could therefore be represented as particular linear operators operating in these spaces. The physics of quantum mechanics was thereby reduced to the mathematics of the linear Hermitian operators on Hilbert spaces.