I PBR & Relativity: Wave Function Uniqueness?

[atyy]

In relativity, the wave function in different frames is not unitarily related, because the wave function collapses at different times. For example, in a Bell experiment, in a frame in which Alice and Bob measure simultaneously, there is no collapse. But in a frame in which Alice measures before Bob, Alice collapses the wave function. Here it seems that there are multiple wave functions for the same situation, but PBR says that the true state (including hidden variables) implies a unique wave function. Does PBR apply, because the true state is different in each frame (a state requires a choice of simultaneity), or does PBR not apply?

 

[Morbert]

but PBR says that the true state (including hidden variables) implies a unique wave function

I think this is probably too strong a statement. Instead I think the PBR theorem only implies the wavefunction cannot simply represent knowledge of an underlying physical state. We do not have to commit to a real wavefunction if we do not attempt to model an underlying physical state. We can for example interpret the wavefunction as representing knowledge of experimental outcomes.

 

[Demystifier]

The PBR theorem considers a single measurement by a single observer. It does not consider two or more measurements at different times, so the collapse does not play any role in the theorem. Hence the PBR theorem as such cannot say anything about multiple wave functions associated with different observers. But it would be interesting to study generalizations.

 

[atyy]

I think this is probably too strong a statement. Instead I think the PBR theorem only implies the wavefunction cannot simply represent knowledge of an underlying physical state. We do not have to commit to a real wavefunction if we do not attempt to model an underlying physical state. We can for example interpret the wavefunction as representing knowledge of experimental outcomes.

Yes, I mean assuming hidden variables.

 

[Demystifier]

Yes, I mean assuming hidden variables.

If there are hidden variables a'la Bohmian particle positions, then there is no wave function collapse of the full wave function of the universe. What collapses are the conditional wave functions (*). Even with an absolute Newtonian time, different observers naturally define different conditional wave functions. That's because they condition on different available knowledge they have access to. So yes, there are multiple conditional wave functions, but they are fully compatible with the fact that there is only one wave function of the universe.

(*) For the notion of conditional wave function see e.g. pages 23-24 in my lecture
http://thphys.irb.hr/wiki/main/images/e/e6/QFound4.pdf

 

[atyy]

If there are hidden variables a'la Bohmian particle positions, then there is no wave function collapse of the full wave function of the universe. What collapses are the conditional wave functions (*). Even with an absolute Newtonian time, different observers naturally define different conditional wave functions. That's because they condition on different available knowledge they have access to. So yes, there are multiple conditional wave functions, but they are fully compatible with the fact that there is only one wave function of the universe.

(*) For the notion of conditional wave function see e.g. pages 23-24 in my lecture
http://thphys.irb.hr/wiki/main/images/e/e6/QFound4.pdf

Yes, but the full Bohmian wave function is not the quantum state. The quantum state is the collapsed wave function, and PBR applies to the quantum state.

 

[Demystifier]

Yes, but the full Bohmian wave function is not the quantum state. The quantum state is the collapsed wave function, and PBR applies to the quantum state.

Fine. Then I propose the following conjecture (as a generalization of the PBR theorem to multiple observers):
When there are multiple ontic observers $o_1,o_2,...$, then wave function $\psi_{o_k}$ (associated with any observer $o_k$) is ontic and uniquely defined by $o_k$ and other ontic stuff.

 

[atyy]

Fine. Then I propose the following conjecture (as a generalization of the PBR theorem to multiple observers):
When there are multiple ontic observers $o_1,o_2,...$, then wave function $\psi_{o_k}$ (associated with any observer $o_k$) is ontic and uniquely defined by $o_k$ and other ontic stuff.

Hmmm, so the wave function is still epistemic in some sense? I guess the Bohmian conditional wave function is epistemic?

 

[Demystifier]

Hmmm, so the wave function is still epistemic in some sense? I guess the Bohmian conditional wave function is epistemic?

Yes, the conditional wave function can be considered "epistemic" in the sense that it depends on knowledge of the observer. But if knowldge itself is represented by something ontic (e.g. the ontic state of the brain), then the conditional wave function is naturally viewed as ontic.

 

[Sunil]

Does PBR apply, because the true state is different in each frame (a state requires a choice of simultaneity), or does PBR not apply?

PBR presupposes realism. It is essentially a theorem which tells that if there is some reality, then the wave function should be part of it.

But if you presuppose realism, you have to reject Einstein causality anyway (except you go completely insane accept great conspiracy of superdeterminism) and to go back to a preferred frame. So, PBR is irrelevant for interpretations which want fundamental relativistic symmetry instead of a preferred frame.

Anyway, PBR is a misguided impossibility theorem proving the impossibility of things already known to exist. The counterexample is Caticha's entropic dynamics:

Caticha, A. (2011). Entropic Dynamics, Time and Quantum Theory, J. Phys. A 44 , 225303, arxiv:1005.2357

The error in the theorem is a misguided definition of psi-ontology: There should be no overlap for different wave functions. But in an objective epistemic interpretation, where the state is defined by objective information about the preparation procedure and its result, this objective incomplete information is part of reality (not of the system, but so what) too.

 

[atyy]

Yes, the conditional wave function can be considered "epistemic" in the sense that it depends on knowledge of the observer. But if knowldge itself is represented by something ontic (e.g. the ontic state of the brain), then the conditional wave function is naturally viewed as ontic.

So if consciousness is due to the ontic state of the brain, consciousness causes collapse?

 

[Demystifier]

The error in the theorem is a misguided definition of psi-ontology

A definition may indeed be misguided, but I wouldn't call it an "error in the theorem". In the worst case it can make the theorem irrelevant or uninteresting, but not wrong.

 

[atyy]

PBR presupposes realism. It is essentially a theorem which tells that if there is some reality, then the wave function should be part of it.

But if you presuppose realism, you have to reject Einstein causality anyway (except you go completely insane accept great conspiracy of superdeterminism) and to go back to a preferred frame.

Yes, for the discussion here, let's assume reality and reject Einstein causality (to discuss PBR).

 

[Demystifier]

So if consciousness is due to the ontic state of the brain, consciousness causes collapse?

Yes. But it should be stressed once again that it's collapse of the conditional wave function, while the full wave function remains intact.

 

[Sunil]

Instead I think the PBR theorem only implies the wavefunction cannot simply represent knowledge of an underlying physical state. We do not have to commit to a real wavefunction if we do not attempt to model an underlying physical state. We can for example interpret the wavefunction as representing knowledge of experimental outcomes.

No. PBR seems to imply this but fails. The problem is that knowledge of the preparation procedure is also part of reality. Last but not least, the measurement devices and the record about the preparation procedure are part of reality. Even more, the mind having that incomplete knowledge is also part of reality too. So, if reality is fixed completely, the incomplete information about the system in that mind is fixed too, thus, the corresponding pure state of the quantum system is fixed too. So, it is psi-ontological by definition of psi-ontology.

 

[Sunil]

A definition may indeed be misguided, but I wouldn't call it an "error in the theorem". In the worst case it can make the theorem irrelevant or uninteresting, but not wrong.

Same situation as with von Neumann's impossibility theorem. Also not wrong but misleading, given the counterexample.

Except that Bohmian mechanics came later, while the counterexample for PBR, Caticha's entropic dynamics, came even before the theorem. A side effect of publish or perish science - people simply cannot know all relevant papers even in the domain they are working.

 

[atyy]

Yes. But it should be stressed once again that it's collapse of the conditional wave function, while the full wave function remains intact.

Wonderful, so Bohmian mechanics proves the Copenhagen "knowledge" interpretation, and even von Neumann's interpretation (my personal favourite).

Ok, just to make sure I understand you correctly - in Bohmian mechanics, the conditional wave function is different for different observers because different Copenhagen observers who are part of Bohmian reality are still free to decide when the Copenhagen measurement occurs?

 

[Sunil]

Ok, just to make sure I understand you correctly - in Bohmian mechanics, the conditional wave function is different for different observers because different Copenhagen observers who are part of Bohmian reality are still free to decide when the Copenhagen measurement occurs?

They have different splits between the classical and the quantum part. The classical part is the one where they can see the trajectory, the quantum part is the part where they don't have information about trajectory, but only about the wave function. So Wigner's friend may be yet in the quantum part of Wigner, thus, giving him a cat-like wave function, simply because his real trajectory is not known to Wigner, while the friend himself knows his own trajectory well.

 

[Demystifier]

Wonderful, so Bohmian mechanics proves the Copenhagen "knowledge" interpretation, and even von Neumann's interpretation (my personal favourite).

Ok, just to make sure I understand you correctly - in Bohmian mechanics, the conditional wave function is different for different observers because different Copenhagen observers who are part of Bohmian reality are still free to decide when the Copenhagen measurement occurs?

Yes, except that I would put "free to decide" in quotation marks, because the observers obey the deterministic Bohmian laws too. A freedom to decide is just al illusion emerging from coarse graining.

 

[martinbn]

In relativity, the wave function in different frames is not unitarily related, because the wave function collapses at different times. For example, in a Bell experiment, in a frame in which Alice and Bob measure simultaneously, there is no collapse. But in a frame in which Alice measures before Bob, Alice collapses the wave function. Here it seems that there are multiple wave functions for the same situation, but PBR says that the true state (including hidden variables) implies a unique wave function. Does PBR apply, because the true state is different in each frame (a state requires a choice of simultaneity), or does PBR not apply?

Isn't this just an argument showing that there is no collapse even when one of them mesures first.

 

[atyy]

Yes, except that I would put "free to decide" in quotation marks, because the observers obey the deterministic Bohmian laws too. A freedom to decide is just al illusion emerging from coarse graining.

I think I roughly see what you mean especially from @Sunil's Wigner friend in Bohmian mechanics example. But in the relativistic Bell experiment, it seems that there are not different classical-quantum cuts, just a change of frame? If we assume emergent relativity in Bohmian mechanics, what is happening there?

 

[Demystifier]

But in the relativistic Bell experiment, it seems that there are not different classical-quantum cuts, just a change of frame? If we assume emergent relativity in Bohmian mechanics, what is happening there?

That's certainly interpretation dependent. If collapse is observer-independent as in GRW theory, then you are right. But if collapse depends on what information an agent does or does not possesses (as in QBism, for instance), then it cannot be just a change of frame.

In the Bohmian interpretation, the collapse of the conditional wave function depends on the choice of conditioned variables, that is, particle positions treated as known. Since it's natural to identify the conditioned variables with the variables of an agent, such as agent's apparatus variables or agent's brain variables, it follows that in Bohmian interpretation it's not just a change of frame.

But Bohmian mechanics allows you to choose the conditioned variables in many different ways. One interesting choice is to condition on all agents at once, which, I think, is something that does not have an analog in QBism or von Neumann collapse by consciousness. In this way you obtain collapse without being forced to choose either one agent or the other, so it's a kind of collapse that does "not depend" on choice of the agent.

 

[Demystifier]

They have different splits between the classical and the quantum part. The classical part is the one where they can see the trajectory, the quantum part is the part where they don't have information about trajectory, but only about the wave function. So Wigner's friend may be yet in the quantum part of Wigner, thus, giving him a cat-like wave function, simply because his real trajectory is not known to Wigner, while the friend himself knows his own trajectory well.

Well said! I would only add that the friend knows his own trajectory only in a coarse grained sense. He does not know the precise Bohmian trajectories of individual particles, for otherwise he could prove that the Bohmian interpretation is right.

 

[atyy]

In the Bohmian interpretation, the collapse of the conditional wave function depends on the choice of conditioned variables, that is, particle positions treated as known. Since it's natural to identify the conditioned variables with the variables of an agent, such as agent's apparatus variables or agent's brain variables, it follows that in Bohmian interpretation it's not just a change of frame.

What I'm thinking of is that in Copenhagen, the collapse is typically frame dependent. If there is emergent relativitvistic quantum theory in Bohmian mechanics, then wouldn't it be possible to derive the frame-dependent collapse of Copenhagen from Bohmian mechanics?

 

[Demystifier]

What I'm thinking of is that in Copenhagen, the collapse is typically frame dependent. If there is emergent relativitvistic quantum theory in Bohmian mechanics, then wouldn't it be possible to derive the frame-dependent collapse of Copenhagen from Bohmian mechanics?

It depends. How exactly do you formulate frame dependence in Copenhagen? Clearly, you must use relativistic quantum theory. How do you formulate Schrodinger evolution (before collapse) in relativistic theory? Do you use many-time formalism? Or do you use many states, one for each possible choice of the time coordinate?

 

[atyy]

It depends. How exactly do you formulate frame dependence in Copenhagen? Clearly, you must use relativistic quantum theory. How do you formulate Schrodinger evolution (before collapse) in relativistic theory? Do you use many-time formalism? Or do you use many states, one for each possible choice of the time coordinate?

I'm not familiar with the many time formalism, so I guess it is many states. But I haven't heard that term before, so an example of what I'm thinking of is the picture in Fig 1 of https://arxiv.org/abs/0706.1232.

 

[Demystifier]

I'm not familiar with the many time formalism, so I guess it is many states. But I haven't heard that term before, so an example of what I'm thinking of is the picture in Fig 1 of https://arxiv.org/abs/0706.1232.

In the item 1. below the picture it is said that Lorentz covariance is true only at the level of probabilities, not at the level of states. Since Bohmian and Copenhagen interpretations predict the same probabilities (of measurement outcomes), I think it answers your question.

 

[atyy]

In the item 1. below the picture it is said that Lorentz covariance is true only at the level of probabilities, not at the level of states. Since Bohmian and Copenhagen interpretations predict the same probabilities (of measurement outcomes), I think it answers your question.

ie. in Bohmian mechanics, if there is emergent relativity in the true ether frame, then one can use any Copenhagen frame (with different collapse in each frame) to obtain the same answer as the true ether frame?

 

[Demystifier]

ie. in Bohmian mechanics, if there is emergent relativity in the true ether frame, then one can use any Copenhagen frame (with different collapse in each frame) to obtain the same answer as the true ether frame?

If by "answer" you mean probability of the measurement outcome, then yes.

 

[Demystifier]

Isn't this just an argument showing that there is no collapse even when one of them mesures first.

Which interpretation of QM without collapse do you have in mind? If it's statistical ensemble, is it the Ballentine's version or the vanhees's version? Or perhaps you mean many worlds? Or what?

My point is that you cannot just say that there is no collapse without putting it into a consistent framework of thinking.

 

[atyy]

Fine. Then I propose the following conjecture (as a generalization of the PBR theorem to multiple observers):
When there are multiple ontic observers $o_1,o_2,...$, then wave function $\psi_{o_k}$ (associated with any observer $o_k$) is ontic and uniquely defined by $o_k$ and other ontic stuff.

In some sense wouldn't a generalized theorem violate the spirit of the original PBR, since the observer dependence seems to still argue for an "epistemic" view of the wave function?

If by "answer" you mean probability of the measurement outcome, then yes.

Considering the different wave functions for different frames in relativity, would you add choice of reference frame to your conjectured generalization of PBR?

 

[Demystifier]

In some sense wouldn't a generalized theorem violate the spirit of the original PBR, since the observer dependence seems to still argue for an "epistemic" view of the wave function?

If the ultimate goal of PBR theorem is to prove that Bohmian mechanics is the only interpretation that makes sense, then no.

Considering the different wave functions for different frames in relativity, would you add choice of reference frame to your conjectured generalization of PBR?

I wouldn't, but perhaps someone would.

 

[Sunil]

What I'm thinking of is that in Copenhagen, the collapse is typically frame dependent. If there is emergent relativitvistic quantum theory in Bohmian mechanics, then wouldn't it be possible to derive the frame-dependent collapse of Copenhagen from Bohmian mechanics?

No. In Bohmian mechanics one needs a preferred frame in the relativistic context. As in every realist or causal interpretation. This is not in conflict with minimal relativity which talks only about observables, only with fundamental relativity which forbids hidden preferred frames.

All you can do is to use BM for different frames, which will show different and incompatible trajectories, and show that observable probabilities do not depend on this, so that one cannot tell by observation which of the many Bohmian versions is the correct one. And then try again the old positivist trick that once they are not observable they do not exist at all.

 

[atyy]

No. In Bohmian mechanics one needs a preferred frame in the relativistic context. As in every realist or causal interpretation. This is not in conflict with minimal relativity which talks only about observables, only with fundamental relativity which forbids hidden preferred frames.

All you can do is to use BM for different frames, which will show different and incompatible trajectories, and show that observable probabilities do not depend on this, so that one cannot tell by observation which of the many Bohmian versions is the correct one. And then try again the old positivist trick that once they are not observable they do not exist at all.

If I understood correctly, @Demystifier gave the opposite answer in post #27. In emergent relativity, there is a preferred frame due to the underlying Bohmian mechanics. In that preferred frame (invisible to the Copenhagen observer), one can derive the quantum formalism for a Copenhagen observer who happens to use the preferred frame. Because of emergent relativity, the quantum formalism will predict the same probabilities for measurement outcomes regardless of which frame the Copenhagen observer uses.

 

[Demystifier]

If I understood correctly, @Demystifier gave the opposite answer in post #27. In emergent relativity, there is a preferred frame due to the underlying Bohmian mechanics. In that preferred frame (invisible to the Copenhagen observer), one can derive the quantum formalism for a Copenhagen observer who happens to use the preferred frame. Because of emergent relativity, the quantum formalism will predict the same probabilities for measurement outcomes regardless of which frame the Copenhagen observer uses.

I don't see that as opposite. What @Sunil says is fully compatible with my claims. And I also agree with you. The only thing I don't understand is where is the conflict between your and Sunil's claims. It looks to me as if we are all saying the same thing, but from slightly different perspectives.

 

[Demystifier]

If the ultimate goal of PBR theorem is to prove that Bohmian mechanics is the only interpretation that makes sense, then no.

PBR = Proving Bohm Rationally

 

[atyy]

PBR = Proving Bohm Rationally

I wish there were a counterpart showing that the wave function must be epistemic. Then since the wave function must be real and must be epistemic, there must be two wave functions as in Bohmian mechanics.

 

[Demystifier]

I wish there were a counterpart showing that the wave function must be epistemic. Then since the wave function must be real and must be epistemic, there must be two wave functions as in Bohmian mechanics.

That would be nice, but I'm afraid that Everetians would say that this proves their interpretation too.

 

[atyy]

That would be nice, but I'm afraid that Everetians would say that this proves their interpretation too.

It's not yet a coherent interpretation (though I respect Wallace's work immensely), so that should be ok.

 

[Sunil]

If I understood correctly, @Demystifier gave the opposite answer in post #27. In emergent relativity, there is a preferred frame due to the underlying Bohmian mechanics. In that preferred frame (invisible to the Copenhagen observer), one can derive the quantum formalism for a Copenhagen observer who happens to use the preferred frame. Because of emergent relativity, the quantum formalism will predict the same probabilities for measurement outcomes regardless of which frame the Copenhagen observer uses.

And where do you see a difference?

This is a general property of emergent relativistic symmetry. Once it emerged, observers cannot see the differences between the frames. But once it is only emergent, and not fundamental, there is a preferred frame for the realists.

Maybe my "BM for different frames" was misleading. This should be understood like "the Lorentz ether theory for different frames". One may be a proponent of BM/Lorentz ether or whatever has a preferred frame but one may not know what is the preferred frame in the actual situation. So, one would have to make guesses, what if the preferred frame is CMBR with harmonic time, what if CMBR with comoving proper time, or whatever else could be a reasonable candidate for a preferred frame. These would be, essentially, different physical theories with different hypotheses about what is the preferred frame. For BM, they would predict different trajectories.

I wish there were a counterpart showing that the wave function must be epistemic. Then since the wave function must be real and must be epistemic, there must be two wave functions as in Bohmian mechanics.

No. PBR is simply a total failure, as explained and shown by Caticha's entropic dynamics. Which does not have two wave functions, its wave function is purely epistemic.

 

[PeterDonis]

Caticha's entropic dynamics

Is this what you are referring to?

https://arxiv.org/abs/1908.04693

 

[Morbert]

No. PBR seems to imply this but fails. The problem is that knowledge of the preparation procedure is also part of reality. Last but not least, the measurement devices and the record about the preparation procedure are part of reality. Even more, the mind having that incomplete knowledge is also part of reality too. So, if reality is fixed completely, the incomplete information about the system in that mind is fixed too, thus, the corresponding pure state of the quantum system is fixed too. So, it is psi-ontological by definition of psi-ontology.

This argument would move us away from PBR and towards a more generic debate. See for example section II "Exactly How Quantum States Fail to Exist" in Fuch's paper https://arxiv.org/pdf/1612.07308.pdf

 

[atyy]

And where do you see a difference?

This is a general property of emergent relativistic symmetry. Once it emerged, observers cannot see the differences between the frames. But once it is only emergent, and not fundamental, there is a preferred frame for the realists.

Maybe my "BM for different frames" was misleading. This should be understood like "the Lorentz ether theory for different frames". One may be a proponent of BM/Lorentz ether or whatever has a preferred frame but one may not know what is the preferred frame in the actual situation. So, one would have to make guesses, what if the preferred frame is CMBR with harmonic time, what if CMBR with comoving proper time, or whatever else could be a reasonable candidate for a preferred frame. These would be, essentially, different physical theories with different hypotheses about what is the preferred frame. For BM, they would predict different trajectories.

Thanks, I understood you the second time round.

 

[Sunil]

Is this what you are referring to?

https://arxiv.org/abs/1908.04693

Yes.

 

[atyy]

No. PBR is simply a total failure, as explained and shown by Caticha's entropic dynamics. Which does not have two wave functions, its wave function is purely epistemic.

In Caticha's Entropic Dynamics, it seems the observer retains a special status? Is it similar in spirit to the derivations of quantum mechanics given by Hardy or by Chiribella and colleagues?

https://arxiv.org/abs/quant-ph/0101012
Quantum Theory From Five Reasonable Axioms
Lucien Hardy

https://arxiv.org/abs/1011.6451
Informational derivation of Quantum Theory
G. Chiribella, G. M. D'Ariano, P. Perinotti

It's rather amazing to me that Bohmian mechanics is supposed to be some limit of Entropic Dynamics.

 

[Demystifier]

PBR is simply a total failure, as explained and shown by Caticha's entropic dynamics.

I think it's an overstatement. PBR rules out a large class of theories that naively someone might consider reasonable. But on the other hand, it's good to know that there are also other definitions of "epistemic" theories that are not ruled out by PBR.

 

[Sunil]

I think it's an overstatement. PBR rules out a large class of theories that naively someone might consider reasonable. But on the other hand, it's good to know that there are also other definitions of "epistemic" theories that are not ruled out by PBR.

Sorry, no, this is what is done by Bell's theorem or by Kochen-Specker. They rule out certain interesting classes of theories, namely theories with Einstein causality and theories without contextuality. These would be features many people would like to have, so even if we have counterexamples with BM and other realist theories, the theorems remain useful.

But Caticha's entropic dynamics is not some somehow extravagant "other definition" of a psi-epistemic theory, it is psi-epistemic in the most natural, straightforward and beautiful way, there is nothing to object for a proponent of psi being epistemic. Nobody would try to construct something "more epistemic" or "epistemic in a more natural meaning" or so. The problem with PBR is simply a misguided definition of psi-ontology, and this makes the theorems (there are already many variants) worthless.

(Thinking about the difference between objective and subjective Bayesian - no. An objective Bayesian interpretation is by definition psi-ontological - the incomplete information is objective, means, part of reality. And if we have an objective Bayesian interpretation, we can also reinterpret this as subjective, so PBR cannot be used to forbid subjective Bayesian too.)

 

[Demystifier]

But Caticha's entropic dynamics is not some somehow extravagant "other definition" of a psi-epistemic theory, it is psi-epistemic in the most natural, straightforward and beautiful way, there is nothing to object for a proponent of psi being epistemic.

OK, some basic questions on ED. Is Schrodinger equation postulated or derived from something more fundamental? Is there a conceptual difference between pure and mixed states?

 

[Sunil]

In Caticha's Entropic Dynamics, it seems the observer retains a special status? Is it similar in spirit to the derivations of quantum mechanics given by Hardy or by Chiribella and colleagues?

No, Hardy as well as Chiribella et al. use a non-realistic approach, that means, the axioms are only about preparations/observables without any reference to something really existing inside the quantum system.

Instead, Caticha assumes that the real state of the quantum system is defined by some configuration, and presupposes a configuration space.

The observer plays a less important role than one may think, because behind this is the objective Bayesian interpretation. The question of the objective Bayesian interpretation is what is the rational choice given a certain amount of information. Subjective Bayesians care only about the internal consistency of their choices of probability. So, if there is no information about a dice, the objective Bayesian interpretation prescribes 1/6 for all, while the subjective prescribes nothing.

It's rather amazing to me that Bohmian mechanics is supposed to be some limit of Entropic Dynamics.

A nice result, but not that surprising, because all the realist interpretations use the same probability flow on the configuration space. So the average velocity in the stochastic variants is the same Bohmian velocity.

That's why I tend to think that what really matters is the relation to fundamental relativity. You cannot give the probability flow as defined by the Schrödinger equation for $|\psi|^2$ a physical interpretation without throwing away Einstein causality and returning to a preferred frame. This is the most important splitting line between different interpretations. If the Bohmian velocity is only an average or a deterministic velocity already does not change much - Einstein causality forbids this.

 

[Sunil]

OK, some basic questions on ED. Is Schrodinger equation postulated or derived from something more fundamental? Is there a conceptual difference between pure and mixed states?

The Schrödinger equation is derived. Essentially in a similar way as in Nelsonian stochastics, except that what is used is the scheme of entropic inference developed by the objective Bayesians.

This gives an equation for probability density $\rho(q,t)$ and the phase $\Phi(q,t)$. Then, one can see that $\psi(q,t) = \sqrt{\rho}\exp(\frac{i}{\hbar}\phi)$ fulfills the Schrödinger equation.

I have not seen a place where mixed states are considered, but I think that they are different follows from standard QM mathematics.