Continuous Geometries with a Transition Probability

John Von Neumann

1981년 1월 · American Mathematical Society: Memoirs of the American Mathematical Society 252권 · American Mathematical Soc.

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Axioms where are motivated by quantum mechanics are formulated for a probability-logic system. It is shown that these axioms characterize precisely those lattices which are lattices of all projections in a irreducible von Neumann algebra of type II1 or I[subscript]N, N [greater-than or equal to] 4 in Hilbert spaces of arbitrary dimension and real or complex scalars.

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John von Neumann was a Hungarian-born American mathematician with a minor interest in economics. He earned a degree in mathematics from the University of Budapest and one in chemical engineering from the Hochschule in Zurich. He moved to the United States in 1930 and three years later joined the Institute for Advanced Study at Princeton, where he was the youngest member of the small but select group that included Albert Einstein (see Vol. 5) and Enrico Fermi (see Vol. 5). Von Neumann founded the branch of economics known as game theory. He published a seminal paper on the subject in 1928 and later collaborated with Oskar Morgenstern in developing and expanding it. Game theory analyzes individual choice in situations of risk, where the participants do not control or know the probability distributions of all variables on which the outcome of their acts depends. In 1944, von Neumann and Morgenstern published their classic Theory of Games and Economic Behavior, which achieved immediate prominence because it dealt with bargaining strategies under conditions of conflict. Game theory was applied to a number of situations, such as wage negotiations between unions and management. It was also popular with the Pentagon, which used it to model superpower confrontations that reflected outcomes based on the selection of various military options. Game theory was only one of von Neumann's many interests. In 1943 he worked on the Manhattan Project and was instrumental in developing the atomic bomb. He also helped develop ENIAC, the first electronic computer. Because of his work with the atomic bomb, he received a presidential appointment to the Atomic Energy Commission in 1955. In 1957, at the age of 53, his illustrious career ended when he died tragically of bone cancer.