Hidden Variables in Quantum Mechanics

The Hidden Variables section in Ballentine’s Statistical Interpretation of Quantum Mechanics examines the possibility of supplementing quantum mechanics with additional parameters (hidden variables) that determine the outcome of individual measurements, rather than relying on probabilistic quantum states.

Von Neumann’s Theorem

Von Neumann’s theorem aimed to show that no hidden-variable theory could reproduce all the statistical predictions of quantum mechanics. His proof relied on the assumption that expectation values should be additive:

⟨A + B⟩ = ⟨A⟩ + ⟨B⟩.

This condition holds for quantum ensembles but assumes it should also apply to hidden-variable models.

Key Issues in Von Neumann’s Proof

  • Noncommuting Observables: Quantum mechanics includes observables that do not commute, such as position q and momentum p. Von Neumann’s theorem does not properly account for these cases.
  • Strong Assumption of Linearity: Expectation value linearity does not necessarily hold in hidden-variable models.
  • Implication: If Von Neumann’s assumptions were correct, quantum mechanics would be the only possible theory, ruling out hidden variables.

Bell’s Rebuttal

John Bell revisited Von Neumann’s proof and identified its flaws. He pointed out that the assumption:

⟨A + B⟩ = ⟨A⟩ + ⟨B⟩

is not valid for hidden-variable theories since it assumes quantum mechanical averages apply to individual measurements.

Bell’s Key Arguments

  • Hidden-Variable Theories Do Exist: Bohmian mechanics (Bohm, 1952) reproduces all quantum statistical predictions.
  • Misinterpretation of Linearity: Expectation values of noncommuting variables need not sum linearly.
  • Constructing a Working Hidden-Variable Model: Bell provided counterexamples demonstrating that hidden variables could exist.

Bell’s Theorem: A Stronger No-Go Result

While Bell criticized Von Neumann’s proof, he later formulated Bell’s theorem, which provided a stronger argument against local hidden-variable theories. His theorem is based on Bell inequalities,