The book mainly summarizes results that von Neumann had published in earlier papers.[4][5][6][7][8] Its main significance may be its argument against the idea of hidden variables, on thermodynamic grounds. However, this idea was refuted as early as 1935 by Grete Hermann, but the critique remained unknown until J.S. Bell's rediscovery in 1966.[9]
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.[9]Bell shows that the consequences of that assumption are at odds with results of incompatible measurements, which are not explicitly taken into von Neumann's considerations.
^von Neumann, J. (1927). "Mathematische Begründung der Quantenmechanik [Mathematical Foundation of Quantum Mechanics]". Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse: 1–57.
^von Neumann, J. (1927). "Wahrscheinlichkeitstheoretischer Aufbau der Quantenmechanik [Probabilistic Theory of Quantum Mechanics]". Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse: 245–272.
^von Neumann, J. (1927). "Thermodynamik quantenmechanischer Gesamtheiten [Thermodynamics of Quantum Mechanical Quantities]". Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse. 102: 273–291.
^von Neumann, J. (1929). "Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren [General Eigenvalue Theory of Hermitian Functional Operators]". Mathematische Annalen: 49–131.
^von Neumann, J. (1931). "Die Eindeutigkeit der Schrödingerschen Operatoren [The uniqueness of Schrödinger operators]". Mathematische Annalen. 104: 570–578. doi:10.1007/bf01457956. S2CID120528257.