How does Bohmian Mechanics actually replicate QM?

[Son Goku]

TL;DR

How can a deterministic theory give 1-random frequencies?

I was recently trying to understand how Bohmian Mechanics could model quantum theory. In an old lecture of Sidney Coleman's called "Quantum Theory with the Gloves off" available here:
https://www.damtp.cam.ac.uk/user/ho/Coleman.pdf
He shows with a "physicist's proof" that QM predicts truly random limiting frequencies.

I wondered how Bohmian Mechanics could replicate this. A truly random string is often called 1-random or Kolmogorov–Levin–Chaitin random and it seemed impossible for a truly deterministic theory to replicate this. I know Bohmian Mechanics has the equilibrium assumption, but the above suggests that:
(a) A system will only approximately enter equilibrium with some "non-Born" fluctuations in the probabilities.
(b) The only way for a system to be exactly in equilibrium is if the Bohmian particle position was coupled to a truly random oracle. Which is really just displacing the fundamental randomness.

A recent theorem by Klaas Landsmann seems to confirm this. It's in "Undecidability, Uncomputability, and Unpredictability", eds. A. Aguirre, Z. Merali, D. Sloan, pp. 17-46. Available here:
https://fqxi.org/community/forum/topic/3425

With this it seems non-relativistic QM has no deterministic models and so Bohmian Mechanics is not truly an interpretation of QM, but a competing theory.

This is ignoring QFT where separate theorems block the existence of any deterministic model.

 

[Demystifier]

Randomness in Bohmian mechanics (BM) is really pseudo-randomness, as in a deterministic pseudo-random generator. Metaphorically speaking, BM is a deterministic pseudo-random generator in which agents know the algorithm but don't know the seed. I this sense BM is indeed a competing theory in principle, but for practical purposes it is "just" an interpretation.

 

[Son Goku]

Thanks, I was looking for a Bohmian response.

I assume then the standard Bohmian view is that most systems rapidly thermalize and approach equilibrium closely so that non-Born fluctuations are small. As opposed to them being actually in equilibrium, since you'd be back to fundamental randomness then as per the above.

 

[Demystifier]

Thanks, I was looking for a Bohmian response.

I assume then the standard Bohmian view is that most systems rapidly thermalize and approach equilibrium closely so that non-Born fluctuations are small. As opposed to them being actually in equilibrium, since you'd be back to fundamental randomness then as per the above.

There are actually two schools of thought on that, the Valentini et al school that it rapidly approaches the equilibrium, and the Durr et al school that it is always in equilibrium. For a review see https://www.mdpi.com/1099-4300/20/6/422

 

[Son Goku]

Perfect, so all the above theorem adds is that in the Durr et al case Bohmian Mechanics is fundamentally random since equilibrium would need to be sourced by a truly random oracle.

 

[Son Goku]

For anybody interested Landsmann has a nice account of the theorem here:
https://arxiv.org/abs/2202.12279

See the start of section 4 for a very clear explanation. Bohmian Mechanics in equilibriym basically factors the randomness via the position variable.