Is the wavefunction ontological or epistemological?
Part 2: Bohm, von Neumann, and
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,
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,
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.
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”.
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
is not a disproof of the completeness of quantum mechanics because of the
faulty locality assumption which prevents achieving the complete quantum
In the next post I’ll dive more into
theorem and Bell’s inequalities.