- #1

Son Goku

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- TL;DR Summary
- 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.

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.