nLab hidden variable theory




physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes

theory (physics), model (physics)

experiment, measurement, computable physics



In its standard formulation, quantum physics is a probabilistic theory of nature, since it provides only probabilities (in fact probability amplitudes) for values of observable in experiment. This is in contrast to classical mechanics, which (on classical pure states) predicts events with certainty.

However, on very small scales classical mechanics predicts the wrong events, while quantum mechanics and quantum field theory predicts the right probabilities.

On the other hand, it is well known that upon “coarse graining”, which means after averaging over certain details, also classical mechanics induces a probabilistic theory of nature, namely statistical mechanics/thermodynamics.

Therefore it is natural to speculate that maybe also quantum physics is the statistical mechanics of a more refined theory of as yet unseen “degrees of freedom of nature” on small scales, which does predict events with certainty on very small scales, and which reduces to quantum mechanics on larger scales as one averages over these unseen new degrees of freedom.

These “unseen degrees of freedom” are usually called hidden variables, and a theory which is non-probabilistic but designed to reproduce quantum mechanics as its statistical coarse grained theory are called hidden variable theories. (These are hence one potential interpretation of quantum mechanics.)

There have been various attempts to construct such hidden variable theories. However, there are also theorems about the characteristic properties of quantum mechanics which assert (under some (natural) assumptions, of course) that there cannot be a hidden variable theory.

These theorems are

One well-developed attempt to construct a hidden variable theory is Bohmian mechanics; this makes hidden variables out of the entire wavefunction and violates the assumption of locality.


Hidden variables (and their non-existence) are already mentioned in:

Textbook account:

See also:

The original article on Bell's theorem:

  • John Bell, On the Einstein Podolsky Rosen paradox, Physics 1 (1964) 195-200 [pdf]

The original article on the Kochen-Specker theorem is

Discussion of hidden variable theories in terms of quantum logic is in section 5 of

  • Gianpiero Cattaneo, Maria Luisa Dalla Chiara, Roberto Giuntini and Francesco Paoli, Quantum Logic and Nonclassical Logics, p. 127 in Kurt Engesser, Dov M. Gabbay, Daniel Lehmann (eds.) Handbook of Quantum Logic and Quantum Structures: Quantum Logic, 2009 North Holland

Last revised on October 11, 2023 at 08:04:44. See the history of this page for a list of all contributions to it.