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In Bohmian Mechanics, there is an explicit notion of non-locality where one measurement outcome affects the other in quantum entanglement.

In the theory, is this influence traveling through some medium, even if superluminal? Or is it genuinely instantaneous?

Or is the theory silent on either option?

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    $\begingroup$ As Andrew’s answer says, “the equation of motion for (say) $\vec{q}^1$ at time $t$ depends on the positions of all the other particles $\vec{q}^2, \cdots \vec{q}^N$ at time $t$”. This is “true instantaneous action at a distance” as far as I’m concerned. $\endgroup$
    – Ghoster
    Commented May 7 at 6:44
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    $\begingroup$ @Cory As explained on "de Broglie–Bohm" Wikipedia, a relativistic version also exists (by just straightforwardly applying it to relativistic quantum field theory, see: arxiv.org/abs/quant-ph/0202104 and arxiv.org/abs/quant-ph/0407089 ). But even if you restrict yourself to the nonrelativistic case there is still nothing else going on then in nonrelativistic schroedinger wavefunctions! So please explain, what has BM got to do with it? $\endgroup$
    – Jos Bergervoet
    Commented May 7 at 6:44
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    $\begingroup$ Just curious: Are you a physics student or a philosophy student? (Anyone interested in physics is welcome here, regardless of what they are studying.) I see that you prefer the Stanford Encyclopedia of Philosophy to physics articles in Wikipedia. $\endgroup$
    – Ghoster
    Commented May 7 at 6:48
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    $\begingroup$ @Ghoster I am neither, I’m a software engineer and I’m interested in physics, philosophy and math. Also, those SEP articles are arguably more reliable than Wikipedia, are they not? Of course both can be useful. Just to also add, that particular article is written by a physicist. $\endgroup$
    – user401242
    Commented May 7 at 7:34
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    $\begingroup$ I'm a theoretical physicist and also worked in engineering, therefore I do not know what you mean with "seemingly instantaneous action", also not if you replace that with "influence". I only see a differential equation for $\psi$, and that describes everything. The words you use have no meaning if you can't relate them to the equations that comprise the theory. $\endgroup$
    – Jos Bergervoet
    Commented May 7 at 7:40

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First, the non-locality in Bohmian mechanics is not due to entanglement or to measurement. It is simply baked into the Bohmian approach from the beginning. It is true that is a hidden variable theory that gets around Bell's inequalities by being a non-local hidden variable theory. But, the non-locality is not "due to" entanglement.

Bohmian mechanics for $N$ particles in $3$ spatial dimensions is based on $3N+1$ equations in $3N+1$ variables.

  • $3N$ of the variables are the trajectories of the $N$ particles, $\vec{q}^k(t)$, where $k=1, 2, \cdots, N$, and the vector symbol $\vec{}$ indicates that $\vec{q}^k$ is a vector in ordinary 3-space.

  • The remaining variable is the pilot wave $\Psi(\vec{q}_1, \vec{q}_2, \cdots, \vec{q}_N, t)$.

First, there is the equation for the pilot wave $\Psi$, which is just the ordinary Schrodinger equation $$ i \hbar \frac{\partial \Psi(q, t)}{\partial t} = H \Psi(\vec{q}_1, \vec{q}_2, \cdots, \vec{q}_N, t) $$ Then there are equations of motion for each of the $N$ particles $$ m_k \frac{d \vec{q}^k}{dt} = \hbar \nabla_k {\rm Im}\ {\rm ln} \Psi(\vec{q}_1, \vec{q}_2, \cdots, \vec{q}_N, t) $$ Finally, the Born rule is recovered by arguing that the distribution of particles reaches an equilibrium distribution $|\Psi|^2$.

The nonlocality in Bohmian mechanics arises from the second equation; the fact that the equation of motion for (say) $\vec{q}^1$ at time $t$ depends on the positions of all the other particles $\vec{q}^2, \cdots \vec{q}^N$ at time $t$. Whether you want to say $\Psi$ is a "medium through which the non-locality travels" is up to you... in light of the lessons of special relativity and the aether, I am inclined not to introduce a loaded term like "medium" and to simply trust the equations to describe the non-intuitive structure they are describing.

In response to some discussion in the comments, I'll note my discussion is essentially just a summary of the wikipedia article.

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  • $\begingroup$ If the nonlocality arises from the equation for $\psi$ than it is the same nonlocality that we already had in the Schroedinger equation without Bohmian Mechanics. Or do you mean there is anything new here? (That's asking in fact what you mean when writing "arises from the second equation", I guess...) $\endgroup$
    – Jos Bergervoet
    Commented May 7 at 6:50
  • $\begingroup$ Bohmian Mechanics is an interpretation of quantum mechanics. The fundamentals of the mathematical formalism are obviously the same @JosBergervoet But it puts a picture as to what the equations actually mean or describe in reality. Hence why it’s called an “interpretation”. $\endgroup$
    – user401242
    Commented May 7 at 7:35
  • $\begingroup$ The word "interpretation" means nothing. Bohmian mechanics makes an addition to the theory. It adds the classical point particle to a wave function. But that particle does nothing on its own, it just follows the wave's guidance, so there are no new influences (or actions) added to those already in the wave function. (And the answer above actually fails in explaining this clearly). $\endgroup$
    – Jos Bergervoet
    Commented May 7 at 7:45
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    $\begingroup$ @alanf You are referencing an answer with a score of zero, and asserting that a rather recent paper is correct while two encyclopedias are wrong, without explaining why you think so. I’m sorry, but I don’t find that convincing. $\endgroup$
    – Ghoster
    Commented May 7 at 8:34
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    $\begingroup$ @alanf According to section 7, Valentini has been claiming since 1991 that (his nonequilibrium version of) Bohmian mechanics disagrees with standard QM. If 33 years later he still hasn’t convinced the physicists who write encyclopedias that this is how Bohmian mechanics should be understood, I consider him a contrarian. I wasn’t aware until now that people who do Bohmian mechanics don’t even agree on how to do it, or on whether or not it agrees with QM. His nonequilibrium approach should be called Valentinian mechanics to reduce confusion. $\endgroup$
    – Ghoster
    Commented May 7 at 9:15