Is the wavefunction ontological or epistemological?

Part 2: Bohm, von Neumann, and Bell

Quantum mechanics predicts only probabilities and the result of any particular experiment cannot be computed. This led Einstein to believe that quantum mechanics is incomplete. Maybe there is a “sub-quantum” world which explains the peculiarities of the quantum world and maybe quantum randomness is only a result of us “macroscopic bulls” destroying the “sub-quantum china” in the “sub-quantum shop” during measurement. This explanation is usually called a “hidden variable theory”.

Inspired by the early pilot wave theory of de Broglie, Bohm constructed a classical deterministic explanation of quantum mechanics and John Bell studied it in detail. At that time, there was popular an impossibility theorem for hidden variables by von Neumann and Bohm’s model seemed to represent a direct counterexample to von Neumann’s theorem.

As a first task, Bell proceeded to investigate the apparent contradiction and found an unjustified assumption in von Neumann’s theorem, assumption which is obeyed by quantum mechanics but may not necessarily be obeyed by hidden variable models.

The von Neumann’s assumption is as follows:

<A+B> = <A>+<B>

meaning the average of the sum is the sum of the averages for any two observables A and B.

To justify why von Neumann’s assumption is dubious, Bell noted that it means that we are comparing results from different measurements (which may correspond to incompatible experimental setups) and we are reasoning “counterfactually” which is not amenable to experimental verification. Hence the demand is ad-hoc and there is no need to be forced on a hidden variable model of quantum mechanics. Coming back to Bohm’s theory, Bell noticed that for a 2-particle system there is a very complicated dependency between particles, and he wondered if it is still possible to construct hidden variable models when imposing Einstein’s locality condition.

So Bell assumed this from the EPR paper: “On the other hand, since at the time of measurement the two systems no longer interact, no real change can take place in the second system in consequence of anything that may be done to the first system”.

The way Bell derived his famous result was using not the EPR setting, but another experimental setup proposed by Bohm (and the experiment is usually called EPR-B) of two separating electrons in a state of zero total spin. For each electron, there are deterministic hidden variable models which recover the predictions of quantum mechanics, but what can be said about correlations of measurement results? Bell showed that quantum mechanics correlations are higher than the correlation resulting from any hidden variable models, and hence, there are no locally causal hidden variable explanations possible for quantum mechanics. Sometimes this is stated as an impossibility of local realism.

In particular, the EPR argument is not a disproof of the completeness of quantum mechanics because of the faulty locality assumption which prevents achieving the complete quantum predictions.

In the next post I’ll dive more into Bell’s theorem and Bell’s inequalities.