Mathematical Foundations of Quantum Mechanics
It is an important early work in the development of the mathematical formulation of quantum mechanics.[1] The book mainly summarizes results that von Neumann had published in earlier papers.[2] Von Neumman formalized quantum mechanics using the concept of Hilbert spaces and linear operators.[10] For von Neumann, this meant that the statistical operator representation of states could be deduced from the postulates.Consequently, there are no "dispersion-free" states:[a] it is impossible to prepare a system in such a way that all measurements have predictable results.[10] Von Neumann's argues that if dispersion-free states were found, assumptions 1 to 3 should be modified.[11] Von Neumann's concludes:[12] if there existed other, as yet undiscovered, physical quantities, in addition to those represented by the operators in quantum mechanics, because the relations assumed by quantum mechanics would have to fail already for the by now known quantities, those that we discussed above.It is therefore not, as is often assumed, a question of a re-interpretation of quantum mechanics, the present system of quantum mechanics would have to be objectively false, in order that another description of the elementary processes than the statistical one be possible.This proof was rejected as early as 1935 by Grete Hermann who found a flaw in the proof.[13] Thus there still the possibility that a hidden variable theory could reproduce quantum mechanics statistically.[11] In 1952, David Bohm constructed the Bohmian interpretation of quantum mechanics in terms of statistical argument, suggesting a limit to the validity of von Neumann's proof.[9][10] Bell showed that the consequences of that assumption are at odds with results of incompatible measurements, which are not explicitly taken into von Neumann's considerations.[2] A review by Jacob Tamarkin compared von Neumann's book to what the works on Niels Henrik Abel or Augustin-Louis Cauchy did for mathematical analysis in the 19th century, but for quantum mechanics.