Hidden variables

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Under hidden variables or hidden variables ( hidden parameters ) is meant in some deterministic interpretations of quantum mechanics occurring sizes, which is awarded to physical reality and with the aid of the "pure" coincidence in the non-deterministic standard interpretation of quantum mechanics is to be attributed to deterministic mechanisms. Such interpretations are usually accompanied by a philosophical realism , so that such interpretations are also referred to as realistic interpretations of quantum mechanics .


The parameters are called hidden because they do not appear in the standard interpretation of quantum mechanics and consequently no measurement method can be derived from this standard interpretation. So if they exist, they would be hidden in the standard interpretation. This does not mean that hidden variables cannot, in principle, be measured . In principle, it cannot be ruled out that a measurement method can be derived from a deterministic theory with hidden parameters. On the other hand, there are deterministic theories (such as the De Broglie-Bohm theory ) which can be shown to make exactly the same empirical predictions as non-relativistic standard quantum mechanics, so that their hidden parameters cannot in principle be measured.

A distinction is made between theories with local and non-local hidden variables:

  • Theories with local hidden variables always fulfill Bell's inequality , provided that it can be ensured that the variables that determine the behavior of the particles to be measured are statistically independent of those that determine the measurement setting on the other detector .
  • However, quantum mechanics violates Bell's inequality , in accordance with the results of the Aspect experiment on the Einstein-Podolsky-Rosen paradox , named after Alain Aspect . Therefore, there can be no description of reality with local hidden variables unless one also assumes that the measuring devices in Bell experiments can never be set independently of the systems observed with them.
The best-known theory with non-local variables is the previously mentioned De Broglie-Bohm theory by Louis de Broglie and David Bohm . It is a deterministic theory in which the quantum mechanical wave function is regarded as a "guide wave " for unobservable particle trajectories. The De Broglie-Bohm theory is, however, also a non-relativistic theory; a satisfactory extension for the relativistic case is still pending.


The name hidden variables comes from John von Neumann , who in his book The Mathematical Foundations of Quantum Mechanics from 1932 believed that he could prove that such theories can be mathematically excluded. Criticism of this was expressed by the philosopher Grete Hermann in 1935 (almost completely ignored at the time) and, with considerably more attention, by John Stewart Bell in 1966.

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Individual evidence

  1. Without this independence, it cannot be assumed - as is necessary for the derivation of Bell's inequality - that the probability distribution of the measurement results on one detector does not depend on the measurement settings on the other ( remote context independence ), cf. Shimony, Abner: Bell's Theorem . In: Edward N. Zalta (Ed.): The Stanford Encyclopedia of Philosophy . September 21, 2017 (English, stanford.edu ). In such LHV theories, the dynamic laws would be locally realistic and deterministic, but the initial conditions would be such that the Bell inequality is still violated. Very few physicists take this approach.
  2. Johann von Neumann, The mathematical foundations of quantum mechanics, Springer 1932. He discusses the problem on p. 109 and gives his proof of impossibility in Chapter VI on the measurement process.
  3. Bell, On the problem of hidden variables in quantum mechanics , Reviews of Modern Physics, Volume 38, 1966, pp. 447-452. Bell called von Neumann's proof later even as stupid (foolish). See also Jeffrey Bub, Von Neumann's 'No Hidden Variables' Proof: A Re-Appraisal , 2010.