The Kochen Specker theorem says that hidden variable theories must be contextual. I'm not seeing anything in the definition of Bohmian mechanics that makes the hidden variable variable assignments dependent on the measurements. Bohmian mechanics seems to revolve around the Newtonian idea that hidden variables are assigned irrespective of measurements, and measurements simply reveal their values, thereby solving the measurement problem..
Bohmian mechanics treats $|\psi (x, y, z) |^2$ as a classical probability distribution, being acted upon by a classical potential $V$ and a quantum potential $Q$. There is some position hidden variable that this probability distribution describes.
This is sufficient to explain all quantum mechanical experiments involving a position measurement. But so far, this does not explain energy or momentum measurement probabilities. One solution can be to add this as a postulate:
If the pilot wave, corresponding to an ensemble, is $|\psi\rangle$, then as the system reaches the "Born equilibrium", each particle settles into a hidden variable state $a$ of observable $A$ with probability $|\langle a|\psi\rangle|^2$, where $|a\rangle$ is an eigenstate of $A$
Does Bohmian mechanics really use the above postulate?
If it has this postulate, then it can explain all quantum measurement statistics. But then again, this postulates seems inconsistent because it violates the Kochen-Specker theorem. This postulate is simultaneously assigning hidden variables to non-commuting observables in a non-contextual (measurement independent) way
Does this mean that the Kochen Specker theorem rules out Bohmian mechanics?
Best Answer
The contextuality in Bohmian mechanics indeed occurs when measuring observables that are not position or momentum. Bohmian mechanics does not use the postulate in the question, there is no general assumption that there are hidden variables corresponding to all observables, in particular there is no "hidden angular momentum" or "hidden spin" variable. The position of the particle is a hidden variable (and hence also its momentum in some sense), but there aren't any more hidden variables.
Spin in Bohmian mechanics is usually explained by saying it is just a contextual position measurement: Spin measurements like the Stern-Gerlach experiment are based on observing different outcomes for the position measurement and then inferring that this is due to different outcomes for a spin measurement. Bohmian mechanics essentially denies this: A Stern-Gerlach apparatus is not a measurement of an intrinsic quality of a particle called "spin", it's just a position measurement: The Stern-Gerlach apparatus essentially splits the wavefunction of a spin-1/2 particle into two parts, and then the Bohmian trajectories just follow either the "spin-up" branch or the "spin-down" branch. Which branch you get just depends on the initial position of the particle, not on some sort of additional value for $\sigma_z$ it might carry. See "The Pilot-Wave Perspective on Spin" by Norsen for a longer explanation of this idea, and chapter 7 of "Bohmian Mechanics" by Tumulka for a more general discussion of observables in Bohmian mechanics.
This is contextual because this means that the Bohmian explanation for spin measurements depends on the exact specifics of the measurement apparatus since Bohmian mechanics in the end needs to reduce essentially all measurements of observables that aren't position to position measurements, and how exactly position relates to the measurement is not independent of the measurement apparatus.