Von Neumann’s Hilbert Space Formalism

Anthony Duncan A, Michel Janssen B

Published Aug 17, 2023• 1 citation

Abstract

Abstract This chapter covers the completion of the formalism of quantum mechanics and its statistical interpretation in the first two installments of a remarkable trilogy of work by von Neumann in 1927. In the first, von Neumann introduces the modern Hilbert space formalism and demonstrates the equivalence of matrix and wave mechanics by showing that they are just two different instantiations of Hilbert space, one being the space of square-summable sequences, the other being the space of square integrable functions. In the second, drawing on work by von Mises on the foundations of probability theory, he provides a careful formulation of the statistical interpretation of this formalism. In doing so, he introduces such staples of modern quantum mechanics as density operators characterizing quantum statistical ensembles and the distinction between pure and mixed states.

DOI: 10.1093/oso/9780198883906.003.0010

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