# How does Bohmian Mechanics actually replicate QM?

• A
• Son Goku
In summary: However, as the system approaches equilibrium, the fluctuations from the position variable are too small to be detected.
Son Goku
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.

Demystifier and gentzen
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.

atyy and 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.

atyy
Son Goku said:
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

atyy, gentzen and 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.

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.

## 1. How does Bohmian Mechanics explain the probabilistic nature of quantum mechanics?

Bohmian Mechanics, also known as the pilot-wave theory, posits that particles have definite positions and trajectories at all times, unlike in standard quantum mechanics where they exist in a state of superposition. However, the particles are guided by a hidden wave, known as the "pilot wave," which determines their behavior and results in the probabilistic outcomes observed in quantum experiments.

## 2. Does Bohmian Mechanics violate the principles of relativity?

No, Bohmian Mechanics is fully consistent with the principles of relativity. The theory is based on a non-local hidden variable, which means that the particles' behavior is not determined by their local properties but by the pilot wave that exists everywhere in space. This allows for the theory to be Lorentz-invariant and maintain the principles of relativity.

## 3. How does Bohmian Mechanics handle entanglement?

In Bohmian Mechanics, entangled particles are still connected by the pilot wave, which ensures that they remain correlated and exhibit non-local behavior. However, the theory does not require the concept of "spooky action at a distance" as in standard quantum mechanics, as the particles' behavior is still determined by their local properties and the pilot wave.

## 4. Can Bohmian Mechanics be tested experimentally?

Yes, there have been several experiments conducted to test the predictions of Bohmian Mechanics, and they have shown good agreement with the standard quantum mechanical predictions. However, there are some proposed experiments that could potentially distinguish between the two theories, such as the detection of the pilot wave itself.

## 5. What are the implications of Bohmian Mechanics for our understanding of reality?

Bohmian Mechanics challenges the traditional interpretation of quantum mechanics, which suggests that particles do not have definite properties until they are observed. Instead, the theory suggests that particles have well-defined trajectories and positions at all times, which may have implications for our understanding of causality and determinism in the universe.

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