# Is Pilot Wave Theory contextual? How?

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?

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.

Does Bohmian mechanics really use the above postulate?

No it does not. Prediction of non-position variables work very differently in Bohmian Mechanics than in regular QM. It starts with the observation - upon reviewing many different real experimental techniques - that all of these experiments can be predicted in position space. There are two categories of justification for this: One is simple but impossible to calculate, the other is more intricate but tractable.

1. Simple, but impossible to calculate: In the end, the detector will show a signal on some device which is read by a human (often called a "pointer" in the quantum foundations lingo) and the detector positions can be predicted as a position measurement. Through this any measurement can be predicted in principle... but no measurement can be predicted in practice. That's why I find (2) more satisfying
2. In practice you never need to wait until the end of the experimental process to model in position space. Take for example the momentum measurements done at particle accelerators including at the LHC: Being immersed in a constant B field, charged particles move in a circle perpendicular to the field. Multiple positions are measured by silicon detectors and fit to a circle, whose radius is used to infer a momentum. This is tractable and can be used to make predictions directly. Some experiments are easier than this, some harder, but all I have encountered do not need to delay the prediction to when the signal is macroscopic.

It has been proven that measurements in the style of (2) reproduce the same results as the fourier transform (in the case of momentum) or for spin. I'm sure it's been done for other variables too but I haven't seen the proofs personally. The proof is pretty cool actually. Check out Dürr and Teufel's book, chapter 9.4, for the momentum equivalence for a common class of momentum measurements, for example. The book can be found on known websites which host textbook pdfs free of charge.

Does this mean that the Kochen Specker theorem rules out Bohmian mechanics?

Well, since it doesn't use the postulate you recommended, in some sense this question is already answered. But I'll add a bit by just pointing out that Bohmian Mechanics does not attribute classical properties (like momentum, energy, spin, etc) to particles at any time. You could try to define something like that by taking momentum to be mass $$\times$$ bohmian velocity, but there is at best an indirect correspondence of this to real measurement outcomes. Bohmian Mechanics drops the majority of the measurement formalism of QM: no projectors, no observable operators or eigenvalue problems. The cost is just that the predictions for non-position observables require more calculation... although since the equivalence is proven for many circumstances in practice you can almost always safely take e.g. $$|\langle p | \psi \rangle |^2$$ as a shortcut to doing the full calculation in position space.