Bohmian interpretation and schroedingers cat

[vanesch]

My understanding is that BM explains the probabilities as resulting from the dynamics of chaotic motion (referring to Brownian motion as an example in some respects).

Nope. BM is strictly deterministic: it is Newtonian dynamics with added forces! The only probabilistic aspect in BM (which make it coincide with the statistical predictions of QM) is that one needs to consider the *initial conditions* (the positions of the particles) as distributed statistically according to the norm squared of the initial wavefunction. If you don't do that, BM is blatantly in contradiction with QM. And if you do so, you can show that this statistical sample evolves and has the same statistical properties as predicted by QM. In other words, if (by hand) you put in the initial statistical distribution of the particles in agreement with the norm squared of the wavefunction in position representation, then this property is conserved under Bohmian dynamics.

The "wiggles" you see in typical Bohmian particle traces are not random Brownian motion, but are very precise Newtonian-like dynamics.

However, and maybe this is what you are referring to, there has been some work (don't have a reference handy but there are articles on the arxiv about this) that shows that even if you start from different initial distributions, that after a long time the probability distributions of the particles seem to evolve in the direction of those given by the reduced density matrices for sub-systems. That's very interesting because that's what is also at the basis of classical statistical mechanics: that one can make the equiprobability hypothesis because any initial particle distribution will quickly evolve for its low-order correlation functions into an ensemble that is compatible with the equiprobability hypothesis.

BM therefore, in theory, does not have to state the probabilities as an assumption, and also allows for the possibility that under very specific (unknown) circumstances these might be different than in standard QM.

Normally not! If you do not make the assumption of an initial statistical uncertainty in agreement with QM, you get blatantly different results out, and BM would be easily rejected. At least, naively in a simple application. If you take into account a long evolution, and you only look at subsystems, then there is, as I said, some work that indicates that we get out the same statistics as QM.

 

[colorSpace]

Nope. BM is strictly deterministic: it is Newtonian dynamics with added forces! The only probabilistic aspect in BM (which make it coincide with the statistical predictions of QM) is that one needs to consider the *initial conditions* (the positions of the particles) as distributed statistically according to the norm squared of the initial wavefunction. If you don't do that, BM is blatantly in contradiction with QM. And if you do so, you can show that this statistical sample evolves and has the same statistical properties as predicted by QM. In other words, if (by hand) you put in the initial statistical distribution of the particles in agreement with the norm squared of the wavefunction in position representation, then this property is conserved under Bohmian dynamics.

The "wiggles" you see in typical Bohmian particle traces are not random Brownian motion, but are very precise Newtonian-like dynamics.

Wrong. While BM is deterministic (without being strict about it), this is not in contradiction to what I said. "The Undivided Universe" tells a different story than you.

Here a quote from B & H:

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"What we have to explain then is why P should tend to approach |ψ|2 in typical situations that are currently treated in physics (i.e. situations in which the quantum laws are valid). In this chapter [Chapter 9] we shall give such an explanation showing that one can understand how an arbitrary probabilistic density, P, may approach |ψ|2 even on the basis of our deterministic theory because the latter leads to chaotic motion under a wide range of conditions. We shall then show how the overall statistical approach may be generalised to include, not only what are usually called pure states, but also what are usually called mixed states (which are at the basis of quantum statistical mechanics). Finally we shall extend this study and show how the approach of P to |ψ|2 could further be justified on the basis of an underlying stochastic process in the movement of particles."
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Also, you seem to get confused about the deterministic nature of BM by the mention of Brownian motion. The relevant quote:

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"We have thus far been explaining quantum probabilities in terms of chaotic motions that are implied by the quantum laws themselves, with pure ensembles representing chaotic motions of the particles and mixed ensembles bringing in also chaotic variations in the quantum field. Whenever we have statistical distributions of this kind, however, it is always possible that these chaotic motions do not originate in the level under investigation, but rather that they arise from some deeper level. For example, in Brownian motion, small bodies which may contain many molecules undergo chaotic velocity fluctuations as a result of impacts originating at a finer molecular level. If we abstract these chaotic motions and consider them apart from their possible causes we have what is called a stochastic process which is treated in terms of a well-defined mathematical theory [5]."
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However, and maybe this is what you are referring to, there has been some work (don't have a reference handy but there are articles on the arxiv about this) that shows that even if you start from different initial distributions, that after a long time the probability distributions of the particles seem to evolve in the direction of those given by the reduced density matrices for sub-systems. That's very interesting because that's what is also at the basis of classical statistical mechanics: that one can make the equiprobability hypothesis because any initial particle distribution will quickly evolve for its low-order correlation functions into an ensemble that is compatible with the equiprobability hypothesis.

Why would you mention this as an "however" ?

Normally not! If you do not make the assumption of an initial statistical uncertainty in agreement with QM, you get blatantly different results out, and BM would be easily rejected. At least, naively in a simple application. If you take into account a long evolution, and you only look at subsystems, then there is, as I said, some work that indicates that we get out the same statistics as QM.

So you agree that BM, in theory, does not have to state the probabilities as an assumption, and also allows for the possibility that under very specific (unknown) circumstances these might be different than in standard QM ?

 

[Demystifier]

But Brownian motion is also fundamentally deterministic, isn't it?

 

[colorSpace]

BTW, Vanesh, since you like Hilbert space, you might find this quote interesting, in chapter 15.8 of TUU:

"We are now ready to extend the model of a particle in our interpretation so that it can be included within the framework of Hilbert space."

:)

 

[vanesch]

So you agree that BM, in theory, does not have to state the probabilities as an assumption, and also allows for the possibility that under very specific (unknown) circumstances these might be different than in standard QM ?

Well, it is not clear to me in how much that this is still ongoing research, or a final established result, that "most of the time" starting with just ANY distribution, you arrive onto the QM distributions.

However, consider the following. We already KNOW that IF you start with initial distributions given by the quantum distributions, that you THEN always follow the quantum distributions (that's about the basic theorem that justified BM in the beginning: IF you take as initial distributions the initial wavefunction norm squared, then this remains conserved through dynamics).

Now, IF it would turn out that with a DIFFERENT initial distribution you obtain a DIFFERENT "midway" statistical distribution than that given by QM, that would mean that your "midway" statistical distribution is SENSITIVE to the initial distribution (because we already know that IF it has the QM distribution from the start, it cannot deviate from it midway). So I don't see what would be the importance of a result showing that your distribution is sensitive to initial conditions...
The most interesting result (and I know that some work has been done on that, but I don't know how conclusive it is) would be that we are essentially INDEPENDENT of the initial distribution. In that case, it cannot be anything else but the quantum distribution (given that we already know ONE initial condition where this is going to be the case, namely the initial quantum distribution).

 

[Fra]

If you take into account a long evolution, and you only look at subsystems, then there is, as I said, some work that indicates that we get out the same statistics as QM.

I always try to seek similarities, rather than differences, and I can help thinking this sounds very familiar, and I wonder if this BM view may connect to how I like to think of it - although the clothing of words, and thinking is different.

In a relational view as I see it, the prior information to which everything relates (in that view), is formed from it's interaction history ~ evolution. This prior structure is what is partially retained from the interaction history. And in a sense I think of this as an equilibration process, which "selects" the most favourable prior.

In principle, and in the general I case I think the prior is also dynamical. And in this view I see ordinary QM, as the idealisation where the implicit prior is in equilibrium with the environment. It means that the local environment are all in a limited "agreement", which also explains why the expectations in such case happen to be exact. Because the local group of "observers" has developed a collective - semiobjective - reference.

If it is not, then I doubt ordinary QM formalism would make sense - like, if the evolved prior is not yet in equilibrium with the environment, then it's expectations will be wrong, which results in a further deformation of the observers microstructure. But due to the thinkg works in reality, we are highly unlikely to observe such "extremely far from equilibrium" states in normal situations. Except possible in very extreme and twisted situations. Planck domain physics and similar stuff maybe?

Ie. if consider that statistics is always conditional on the structure of the underlying event space, then maybe ordinary QM statistics corresponds to the case where the underlying event space or probabiltiy space is equilibrated. But in the cases where it's not, the normal formalism fails. this seems to be at the root of some of QM assumptions, we assume that there exists a well defined and objective hilbert space and event space.

If the bohmians object to this, in the general case - for reasons I may not comlpetely understand - I think I might share their "conclusion" by other ways of reasoning?

Then maybe a solution to the sound "bohmian objections" may have a satisfactory possible resolution in the relational view?

/Fredrik