One significant passage is its mathematical argument against the idea of
hidden variables. Von Neumann's claim rested on the assumption that any linear combination of
Hermitian operators represents an observable and the expectation value of such combined operator follows the combination of the expectation values of the operators themselves. • For an
observable R, a function f of that observable is represented by f(R). • For the sum of observables R and S is represented by the operation R+S, independently of the mutual
commutation relations. • The correspondence between observables and Hermitian operators is one to one. • If the observable R is a
non-negative operator, then its expected value \langle R\rangle\geq0. • Additivity postulate: For arbitrary observables R and S, and real numbers a and b, we have \langle aR+bS\rangle=a\langle R\rangle+b\langle S\rangle for all possible ensembles. Von Neumann then shows that one can write : \langle R\rangle=\sum_{m,n}\rho_{nm} R_{mn}=\mathrm{Tr}(\rho R) for some \rho , where R_{mn} and \rho_{nm} are the matrix elements in some basis. The proof concludes by noting that \rho must be Hermitian and non-negative definite (\langle \rho\rangle\geq0 ) by construction.
Rejection This proof was rejected as early as 1935 by
Grete Hermann who found a flaw in the proof. The additive postulate above holds for quantum states, but it does not need to apply for measurements of dispersion-free states, specifically when considering non-commuting observables. Thus there still the possibility that a hidden variable theory could reproduce quantum mechanics statistically. 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. == Reception ==