Wednesday, May 22, 2024

What happens to particles after they have been interacting according to Bohm?

 Once more, I am trying to better understand the Bohmian or pilot wave approach to quantum mechanics. And I came across this technical question, which I have not been able to successfully answer from the literature:

Consider a particle, described by a wave function \(\psi(x)\) and a Bohmian position \(q\) that both happily evolve in time according to the Schrödinger equation and the Bohmian equation of motion along the flow field. Now, at some point in time, the (actual) position of that particle gets recorded, either using a photographic plate oder by flying through a bubble chamber or similar. 

Unless I am not mistaken, following the "having a position is the defining property of a particle"-mantra, what is getting recorded is \(q\). After all, the fact, that there is exactly one place on a photographic place that gets dark was the the original motivation of introducing the particle position denoted by \(q\). So far, so good (I hope).

My question, however, is: What happens next? What value of \(q\) am I supposed to take for the further time evolution? I see three possibilities:

  1. I use the \(q\) that was recorded.
  2. Thanks to the recording, the wave function collapses to an appropriate eigenstate (possibly my measurement was not exact, I just inferred that the particle is inside some interval, then the wave function only gets projected to that interval) and thanks to the interaction all I can know is that \(q\) is then randomly distributed according to \(|P\psi|^2\) (where \(P\) is the projector) ("new equilibrium").
  3. Anything can happen, depending on the detailed inner workings and degrees of freedom of the recording device, after all the Bohmian flow equation is non-local and involves all degrees of freedom in the universe.
  4. Something else
All three sound somewhat reasonable, but upon further inspection, all of them have drawbacks: If option 1 were the case, that would have just prepared the position \(q\) for the further evolution. Allowing this to happen, opens the door to faster than light signalling as I explained before in this paper. Option 2 gives up the deterministic nature of the theory and allows for random jumps of the "true" position of the particle. This is even worse for option 3: Of course, you can always say this and think you are safe. If there are other particles beyond the one recorded and their wave functions are entangled, option 3 completely gives up on making any prediction about the future also of those other particles. Note that more orthodox interpretations of quantum mechanics (like Copenhagen, whatever you understand under this name) does make very precise predictions about those other particles after an entangled one has been measured. So that would be a shortcoming of the Bohmian approach.

I am honestly interested in the answer to this question. So please comment if you know or have an opinion!