nLab John von Neumann

John von Neumann (German: Johann von Neumann; Hungarian: Neumann János Lajos) was a Hungarian mathematician.

He defended his PhD thesis in 1925 advised by Lipót Fejér, with the title Az általános halmazelmélet axiomatikus felépítése (Axiomatic construction of general set theory), which introduced the NBG set theory, as well as classes and von Neumann ordinals.

• Wikipedia entry

Selected works

On set theory:

• J. v. Neumann, Eine Axiomatisierung der Mengenlehre, Journal für die reine und angewandte Mathematik 154 (1925), 219–240. doi.

PhD thesis (journal version):

• J. v. Neumann, Die Axiomatisierung der Mengenlehre, Mathematische Zeitschrift 27 (1928), 669–752. doi.

On Hermitian operators and introducing the formal definition of Hilbert spaces:

• John von Neumann, §I in: Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren, Math. Ann. 102 (1930) 49–131 [doi:10.1007/BF01782338]

On the mathematical foundations of quantum mechanics based on Hilbert spaces of quantum states:

• David Hilbert, John von Neumann, Lothar W. Nordheim, Über die Grundlagen der Quantenmechanik, Math. Ann. 98 (1928) 1–30 [doi:10.1007/BF01451579]

• John von Neumann:

  Mathematische Grundlagen der Quantenmechanik (German) (1932, 1971) [doi:10.1007/978-3-642-96048-2]

  Mathematical Foundations of Quantum Mechanics Princeton University Press (1955) [doi:10.1515/9781400889921, Wikipedia entry]

Co-introducing Jordan algebras as algebras of quantum observables:

• Pascual Jordan, John von Neumann, Eugene Wigner, On an algebraic generalization of the quantum mechanical formalism, Ann. Math. 35 (1934) 29-64 [jstor:1968117, doi:10.1007/978-3-662-02781-3_21]

On quantum logic:

• Garrett Birkhoff, John von Neumann: The logic of quantum mechanics, Annals of Mathematics 37 823-843 (1936) [doi:10.2307/1968621, pdf]

Introducing the theory of what came to be known as von Neumann algebra factors:

The classification of factors into types I, II, III and the construction of examples not of type I:

• Francis J. Murray, John von Neumann, On Rings of Operators, Annals of Mathematics, Second Series, 37 1 (1936) 116-229 [doi:10.2307/1968693, jstor:1968693]

Discussion of traces on these factors:

• Francis J. Murray, John von Neumann, On rings of operators. II, Trans. Amer. Math. Soc. 41 (1937) 208-248 [doi:10.2307/1989620, jstor:1989620]

On isomorphism of factors and proof of a single isomorphism class of approximately finite type $II_1$ factors:

• Francis J. Murray, John von Neumann, On rings of operators. IV, Annals of Mathematics, Second Series, 44 4 (1943) 716-808 [doi:10.2307/1969107, jstor:1969107]

On decomposing von Neumann algebras as a direct integral of factors:

• John von Neumann, On rings of operators, reduction theory, Annals of Mathematics Second Series, Vol. 50, No. 2 (1949) [jstor:1969463]

Recollection of the history which made von Neumann turn to discussion of these “factors”, motivated from considerations in the foundations of quantum mechanics and quantum logic:

• Miklos Rédei, Why John von Neumann did not Like the Hilbert Space formalism of quantum mechanics (and what he liked instead), Studies in History and Philosophy of Modern Physics 27 4 (1996) 493-510 [doi:10.1016/S1355-2198(96)00017-2]

category: people