Bohmian interpretation and schroedingers cat

vanesch

Staff Emeritus
Science Advisor
Gold Member
5,007
16
1. In the 'Undivided Universe', Bohm & Hiley seem to be saying that the particle is not only guided by the quantum potential (which isn't exactly the wavefunction, but very related), but even more importantly, by the classical potential. The quantum potential is usually very small compared to the classical potential. As far as I understood.
Yes, that is to say that in many cases, we can just do classical physics. The quantum potential is only important in those cases where we have a deviation between classical and quantum predictions.

2. In measurement, it would seem to me, the measurement device's visible output is (also) determined by the measured particle, not only the measured wavefunction. Measurement devices in BM are not special, they are physical objects like any other.
This is the circular thing in BM: by definition, one calls the "measurement output" as the *particle state* of the measurement apparatus. Its quantum state is as in MWI: that is: the wavefunction associated to the measurement act undergoes, just as in MWI, a superposition of outcomes. Only, the *particle state* of the measurement apparatus, which is a function of the wavefunction of the "device under test", its own wavefunction and its (unknowable) initial particle states, will go into a particular particle configuration. It is this particle configuration which is called "the outcome of measurement". If we would have called its wavefunction the "outcome of measurement" we would be in exactly the same situation as in MWI. It is the fact of focussing on the particle state of a measurement apparatus that we are able, in BM, to give a definite outcome to a measurement.

My question here: won't then the consequences also have an influence on the wavefunction, that is, even if the particle does not have an influence on its own wavefunction, won't it, via the measurement, which can potentially drive any decision? I haven't yet found a text in the above book that would be specific on this question (not looking at the mathematics).
No, the wavefunction is strictly independent of the particle world. The wavefunction in BM follows ALWAYS the schroedinger equation (just as in MWI), and never undergoes any "projection" (which would in that case give to BM exactly the same conceptual problems as in CI: why a "different" dynamics for a "measurement" than for a "physical interaction" ?).

But in BM, the wavefunction branches that are not "in resonance" with the particle state will not influence (in most cases) the particle dynamics anymore. So, concerning the particle dynamics, we can, after a measurement, just as well consider a projection: it won't influence the particle dynamics (the quantum potential) much.
This is a bit similar as considering the EM interaction of one charged particle upon another: the EM radiation that has been emitted "away" from the other particle won't influence it anymore, so in a calculation of the motion of the particles, we can just as well ignore this EM radiation. But does that mean that it doesn't exist ? I ask the same question to the Bohmians: does the fact that certain branches in the wavefunction have no influence anymore on the particle dynamics imply that they don't exist ?

3. In general, it seems that the quantum potential is a "guiding principle", but not "just" a guiding principle. As such, it plays a major role in the book. But I personally see this as similar to the stars and planets in relation to gravity. The "real" thing are the stars and planets, but that doesn't mean that gravity isn't also real and important. (Even though B&H don't use this example.)
It is stronger than this, because (at least in a Newtonian frame) the gravity interaction can be DEDUCED from the particle dynamics. There is no dynamical state to gravity itself. But not so for the wavefunction in BM: its dynamical state is NOT "recorded" in the particle states and hence, it lives its own life. But more so: in Newtonian dynamics, gravity is influenced by the particles. Not so in BM: the wavefunction is not influenced by the particles.

4. It does appear, in BM, with its emphasis on ontology, that "Hilbert space" specifically is regarded an arbitrary mathematical tool to *describe* that which eventually leads to the quantum potential, rather than thought of as a real physical space. However the quantum potential is thought of as related to a physical "force".
I know, but I think that this is a rethorical weakness of BM: IF one admits the ontological existance of the wavefunction itself, then the particle dynamics doesn't solve all of the conceptual difficulties of MWI, because that wavefunction with all its branches is "still out there". So in BM, one tries to minimise the ontology of the wavefunction and to maximize the particle ontology.
 
479
10
It is stronger than this, because (at least in a Newtonian frame) the gravity interaction can be DEDUCED from the particle dynamics. There is no dynamical state to gravity itself. But not so for the wavefunction in BM: its dynamical state is NOT "recorded" in the particle states and hence, it lives its own life. But more so: in Newtonian dynamics, gravity is influenced by the particles. Not so in BM: the wavefunction is not influenced by the particles.
I know, but I think that this is a rethorical weakness of BM: IF one admits the ontological existance of the wavefunction itself, then the particle dynamics doesn't solve all of the conceptual difficulties of MWI, because that wavefunction with all its branches is "still out there". So in BM, one tries to minimise the ontology of the wavefunction and to maximize the particle ontology.
For the entire universe the wavefunction should be stationary because of energy conservation. In this case there is no dynamics associated with it. It is a constant, eternal field. Therefore BM could be completely described by the particle configuration.

It is also possible in this case that the universal wavefunction can be deduced from the particle configuration so one could do away with it.
 

vanesch

Staff Emeritus
Science Advisor
Gold Member
5,007
16
For the entire universe the wavefunction should be stationary because of energy conservation.
I don't see why the wavefunction of the universe should be an eigenfunction of the hamiltonian, and not a superposition of them.
 
261
37
Vanesch: Re your post #21: yes, that's one reason I'm not a Bohmian anymore...

RK
 
479
10
I don't see why the wavefunction of the universe should be an eigenfunction of the hamiltonian, and not a superposition of them.
http://en.wikipedia.org/wiki/Ground_state" [Broken]:

As an eigenstate of the Hamiltonian, a stationary state is not subject to change or decay (to a lower energy state). In practice, stationary states are never truly "stationary" for all time. Rather, they refer to the eigenstate of a Hamiltonian where small perturbative effects have been ignored. The language allows one to discuss the eigenstates of the unperturbed Hamiltonian, whereas the perturbation will eventually cause the stationary state to decay. The only true stationary state is the ground state.
For an infinitely old universe (presupposed by QM) I would expect to find it in its ground state. So, no wavefunction dynamics. The wavefunction for the ground state can also be determined from the particle configuration.
 
Last edited by a moderator:
I think something here still isn't clear about BM. Perhaps, Vanesh, your description is then not fully describing the role of the particles.

Are the classical potentials (and thereby the classical forces between particles) determined by the particles, and independent of the wavefunction?

If not, what difference then would the particles make at all?

If yes, does it mean that the particle of the measured object has an influence on the measurement device which is independent of the wavefunction?

And it follows, our human experience, and the actions we then take, are based also on the particle positions, rather than the wavefunction alone.
 
[Continued from the previous message]

I don't think it is correct to say that in BM there is a disregard for the wavefunction.

For example, on page 58 of "The Undivided Universe", B & H write:

In our interpretation of the quantum theory, we see that the interaction of parts is determined by something that cannot be described solely in terms of these parts and their preassigned interrelationships. Rather it depends on the many-body wave function (which, in the usual interpretation, is said to determine the quantum state of the system). This many-body wave function evolves according to Schrödinger's equation. Something with this kind of dynamic significance that refers directly to the whole system is thus playing a key role in the theory. We emphasize that this is the most fundamentally new aspect of the quantum theory.
(The italics are original.)

However what is measured and humanly experienced are the positions of the particles. It is simply not true that these would be the "shadows" of the wave function, since the wave function leaves trillions and trillions of possibilities which multiply each second, and the particles would decide the single specific outcome. It would be like saying the chess rules completely determine each chess game, which isn't true even though the chess rules give only about 40 possibilities each move, very, very limiting compared to the wave function.
 

vanesch

Staff Emeritus
Science Advisor
Gold Member
5,007
16
However what is measured and humanly experienced are the positions of the particles.
INDEED. And this comes awfully close to MWI, where one says:
"what is measured and humanely experienced is ONE of the branches".

I admit directly that there is a difference: the Bohmian view allows for a "single, common, objective" experience, while MWI needs to lock them up in different "subjective" experiences.

It is simply not true that these would be the "shadows" of the wave function, since the wave function leaves trillions and trillions of possibilities which multiply each second, and the particles would decide the single specific outcome. It would be like saying the chess rules completely determine each chess game, which isn't true even though the chess rules give only about 40 possibilities each move, very, very limiting compared to the wave function.
I know. The difference between BM and MWI is that in BM, there is one unique "token" (branch indicator!) for all subjective experience, which consists of "the particles" (which are branch indicators in fact). While in MWI, each subjective experience "traces its own" path throughout the branchings (its "consistent history").

Even probability-wise there are similarities:
while BM has some difficulties indicating (although there is work in this direction, I know) why the "probability distribution of the particles" must initially be in agreement with the norm of the wavefunction, or the Born rule in position representation (a very strange "initial condition"), MWI has some difficulties telling us why the probabilities of "an individual experience" must be distributed according to the Born rule, and just posits it (although, here too, there is work in progress).
 

Fra

3,073
142
MWI has some difficulties telling us why the probabilities of "an individual experience" must be distributed according to the Born rule, and just posits it
But I think this is an issue for foundational QM itself right? Regardless of interpretation and is IMO one of the challanges in trying to understand QM at the next level. Unlike the interpretation only-issues, I find this interesting.

They way I see it, this contains two issues.

1) How is "probability" defined in this context. (Ie. how can we apply the probability formalisms from mathematics to physics).

2) Given that the first problem is solve, prove that the born rule holds.

But I think the two issues are entangled and need to solved together.

Vanesch, I I'm not into MWI, but parts of what you say is in line with my thinking too, and you seem to more or less adhere to a kind of "relational view" of QM (or?), and that makes me wonder:

Doesn't you find the the whole standard construction of the frequentist interpretation of probability, to sort of at least conceptually or philosophically, clash with the relational thinking? I mean, it means there must be someone (or something) that is physically responsible for the event counting? And this something must be able to ultimately be able to encode alot of information right?

Ie. the question is wether the whole concept of probability as used in standard QM, is relationally designed? I think it's not.

Do you agree or disagree?

/Fredrik
 
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).

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.
 

vanesch

Staff Emeritus
Science Advisor
Gold Member
5,007
16
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.
 
Last edited:
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:

-------
"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."
-------

Also, you seem to get confused about the deterministic nature of BM by the mention of Brownian motion. The relevant quote:

-------
"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]."
-------

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

Science Advisor
Insights Author
2018 Award
9,839
2,844
But Brownian motion is also fundamentally deterministic, isn't it?
 
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

Staff Emeritus
Science Advisor
Gold Member
5,007
16
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

3,073
142
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
 
Last edited:

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving

Hot Threads

Top