Where does non-locality originate in dBB theory?

[inflector]

The top three images show the time-dependence of the square of the wave field (taken to be the physical object mathematically represented by the Schroedinger wave function).

The bottom three images show the time-dependence of the density \rho of particles (deliberately taken to be NOT equal to the square of the wave function, at least initially). The trajectories of these particles are being influenced by the changing shape of the wave field.

Note how the particle density evolves in the course of time.

There is a prize for the first person who posts telling me how that explains the Born rule. No clues now, it's not easy.

It sure looks like the density matching the square of the wave function simply emerges from the dynamics of the effects of the wave field. The densities go from one that doesn't match the wave function squared at all to one that closely tracks by the third column. Which means that the Born Rule is emergent in dBB. Right?

 

[LukeD]

Anyone else?

What about my try?

My only disagreement with this viewpoint that it is a proof of the Born rule is that if we have the wavefunction, then we already have the distribution. Instead, I like to view this result about the Born distribution being an equilibrium as beig in terms of a distribution and a conservative velocity field (the 2 things we need to get the wavefunction and describe the dynamics of dBB). Then, if we have a small rouge sample (a small sample of the total distribution that does not obey the |\psi|^2 statistics) and if it is so small that it's existence doesn't effect the dynamics of the larger distribution, the "rouge sample"'s statistics eventually approach that of the larger distribution.

In other words, perturbations on the |\psi|^2 distribution get smoothed out until they have no effect if the perturbations are small enough that we can use the unperturbed \frac{\nabla S}{m} velocity

 

[Demystifier]

I haven't been here for a couple of days. Many questions have been asked during this time, and some of them have been answered. Unfortunately, I don't have time to answer all these questions by myself. So please, if you still want me to answer some PARTICULAR questions, draw my attention to them again.

Thanks!

 

[ThomasT]

I haven't been here for a couple of days. Many questions have been asked during this time, and some of them have been answered. Unfortunately, I don't have time to answer all these questions by myself. So please, if you still want me to answer some PARTICULAR questions, draw my attention to them again.

Thanks!

Thank you. Just point us to the best sources that you know for learning dBB, especially for non-experts if possible, and then you probably won't have to, er, endure, any of our questions for at least a few days. Possibly weeks, who knows.

 

[ThomasT]

It sure looks like the density matching the square of the wave function simply emerges from the dynamics of the effects of the wave field. The densities go from one that doesn't match the wave function squared at all to one that closely tracks by the third column. Which means that the Born Rule is emergent in dBB. Right?

This seems generally/essentially right to me. But I'm generally/essentially ignorant about this stuff, so hopefully Zenith8, or Demystifier, or DA, or nismaratwork, or Maaneli, or LukeD ... or somebody, will give the definitive answer.

My two cents, on looking at this again, is that it's assumed, via Bohmian mechanics, that the evolution of the wave field is determining the particle density. Particle density, ie., the probabiltiy of detection, is directly proportional to wave amplitude. The Born rule in qm says that the probability of a particular detection at a particular position at a particular time is the square of the amplitude of the wavefunction at that position at that time. At t - 4pi the particle density evolution matches the wave field evolution. Hence, the Bohmian mechanical evolution has reproduced, in a more 'natural' way equivalent to the evaluation of qm wavefunctions via the Born rule in standard qm, the probability of detection at a particular position and time.

I think this is essentially equivalent to what LukeD said. But I'm not sure, so hopefully LukeD will clarify.

Edit: I thought I might add that, apparently, the Born rule is a very deep physical insight. And the fact that it emerges more or less 'naturally' in dBB is, well, sort of exciting.

 

[zenith8]

It sure looks like the density matching the square of the wave function simply emerges from the dynamics of the effects of the wave field. The densities go from one that doesn't match the wave function squared at all to one that closely tracks by the third column. Which means that the Born Rule is emergent in dBB. Right?

Correct! If you happen to live in the same country as me, then I'll send you a coconut.

If the electron density distribution is not equal to the square of the wave field, and the system is evolving according to the laws of QM, then it will become so distributed over the course of time. Once so distributed, it will stay like that for ever. Psi^2 is the only distribution with this property.

This is entirely analagous to the usual approach to thermal equilibrium, which is why one talks about 'quantum equilibrium' and 'quantum-nonequilibrium'

The reason why the Born rule seems to be true whenever we make a measurement now is that the universe has a long and violent astrophysical history and it has had plenty of time to come into equilibrium. The current state of the universe is analagous to the the usual thermodynamical `heat death' - except it's just happened quicker.

Note that this has potential observable consequences. Look far back in time enough (e.g. by looking at the cosmic microwave background) and maybe you'll find something that isn't in equilibrium yet..

Just to make this result aesthetically pleasing, look at http://www.tcm.phy.cam.ac.uk/~mdt26/raw_movie.gif" from the same site I gave earlier. This shows the whole process of approaching equilibrium for a system in a 2D box. The cool thing is effect of nodes. Do you see all those little vortices moving around, stirring everything up? Those are the nodes - the singularities in the velocity field where the wave field goes to zero.. The more of them you have, the more chaotic the system, and the quicker the approach to equilibrium. Cool, or what?

from: http://rspa.royalsocietypublishing.o...3/253.full.pdf

Is the Born rule truly helpful here?

Good link. Indeed, as far as I know, Valentini and Westman were the first people to do calculations like the above..

My only disagreement with this viewpoint that it is a proof of the Born rule is that if we have the wavefunction, then we already have the distribution.

No you don't. That's a postulate of orthodox QM, not of de Broglie-Bohm..

Thank you. Just point us to the best sources that you know for learning dBB, especially for non-experts if possible, and then you probably won't have to, er, endure, any of our questions for at least a few days. Possibly weeks, who knows.

The http://www.tcm.phy.cam.ac.uk/~mdt26/pilot_waves.html" has a full graduate lecture course, popular lectures, links to pretty much every deBB paper ever published including lots of review articles. I would start there.

Start with the `Supplementary Popular Lecture' called 'The return of pilot waves, or why Bohr, Heisenberg, Pauli, Born, Schrödinger, Oppenheimer, Feynman, Wheeler, von Neumann and Einstein were all wrong about quantum mechanics' - very funny.

By the way, a good intro into the relativistic non-locality stuff that this thread is supposed to be about is in Lecture 5 of the course.

OK - next question. Demystifier's turn, I think. I have to go away for the day.

 

[Demystifier]

Thank you. Just point us to the best sources that you know for learning dBB, especially for non-experts if possible, and then you probably won't have to, er, endure, any of our questions for at least a few days. Possibly weeks, who knows.

In my opinion, the best source is the book by P. Holland. But if it is too much for a start, then you can start, e.g., with these two reviews:
http://www.tcm.phy.cam.ac.uk/~mdt26/local_papers/bohm_hiley_kaloyerou_1986.pdf
http://www.tcm.phy.cam.ac.uk/~mdt26/local_papers/passon_2006.pdf

 

[nismaratwork]

Thank you to both Zenith8, and Demystifier... I have a lot of reading to do.

 

[unusualname]

For those interested in the historical origins and early objections to de Broglie's pilot wave idea you shouldn't miss:

Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference (pdf)

this is a draft of the 533 page book published recently by Cambridge University press

 

[inflector]

Correct! If you happen to live in the same country as me, then I'll send you a coconut.

Just credit me one virtual coconut and I'll be fine. I spent 6 years in the U.S. Virgin Islands so I have had my share of the real ones.

What I find remarkable is how this is the first time I have heard of this result. Seriously? The Born Rule just falls out of the dynamics of dBB? That is an astounding result. The fact that I haven't heard of this before despite my interest in dBB for at least 18 months is just testimony to the effects of the status quo on people's thinking.

 

[unusualname]

I fail to see what is surprising about deterministic particle trajectories guided by a Schrodinger Equation having (and evolving to) an invariant probability density given by |Psi|^2. Since we already know that |Psi|^2 is the conserved probability density for the SE!

 

[inflector]

I fail to see what is surprising about deterministic particle trajectories guided by a Schrodinger Equation having (and evolving to) an invariant probability density given by |Psi|^2. Since we already know that |Psi|^2 is the conserved probability density for the SE!

What's surprising is NOT the result

\rho = |\psi|^2

which is, as you stated, what we already knew.

What is surprising is how this behavior—which we already know is true through experimental observation—emerges from the equations for the dynamic guidance of the particles.

Consider the second paper Demystifier linked to above:

http://www.tcm.phy.cam.ac.uk/~mdt26/local_papers/passon_2006.pdf

In this paper, it states three postulates for dBB theory. The third is:

3. The position-distribution, \rho, of an ensemble of systems which are described by the wave function, \psi, is given by |\psi|^2. This postulate is called the quantum equilibrium hypothesis.

So in dBB the Born rule is postulated.

Note from: http://www.math.ru.nl/~landsman/Born.pdf on the last page, the first two sentences of the last paragraph of the paper:

In most interpretations of quantum mechanics, some version of the Born rule is simply postulated. This is the case, for example, in the → Consistent histories interpretation, the → Modal interpretation and the → Orthodox interpretation.

I always found the mere postulation of the Born Rule to be unnatural and evidence of a weakness in the theories that required it.

So now with these results from Valentini et al. and Towler more recently, we have a deep and important postulate emerging from only the first two postulates of dBB thereby removing the need for the third postulate entirely. So now rather than relying on three postulates, dBB only requires two, the wave function and the guidance equation alone since the Born Rule emerges naturally from the first two postulates.

You don't find this surprising?

 

[zenith8]

I fail to see what is surprising about deterministic particle trajectories guided by a Schrodinger Equation having (and evolving to) an invariant probability density given by |Psi|^2. Since we already know that |Psi|^2 is the conserved probability density for the SE!

It might be obvious to you, Mr. name, but it sure isn't obvious to anyone else. Take the recently published "Compendium of Quantum Physics" (published by Springer in 2008) - available at Google books http://books.google.co.uk/books?id=...resnum=1&ved=0CC4Q6AEwAA#v=onepage&q&f=false". At the end of the article on the Born rule, the author states (p. 69):

"The conclusion seems to be that no generally accepted derivation of the Born rule has been given to date, but this does not imply that such a derivation is impossible in principle."

What he does not mean is "I've considered the deBB explanation of this but it's not generally accepted." He just (of course) doesn't mention it - though he mentions most of the other 'interpretations'. That a guy can do that in a well-researched Encyclopedia article is just scandalous but entirely typical.

So, if it's so obvious, can you derive it mathematically? How would you go about doing that?

So now with these results from Valentini et al. and Towler more recently, we have a deep and important postulate emerging from only the first two postulates of dBB thereby removing the need for the third postulate entirely. So now rather than relying on three postulates, dBB only requires two, the wave function and the guidance equation alone since the Born Rule emerges naturally from the first two postulates.

Inflector - I entirely agree with you. Funnily enough, even the guidance equation doesn't really have to be postulated (though there are some subtleties). If you simply state that probability means 'probability of a particle being at x' as opposed to 'probability of a particle being found at x in a suitable measurement' then the guidance equation is essentially just a transcription of the usual expression for the probability current (which is non-zero only in a non-stationary state with a moving charge distribution, obviously)...

Early presentations of the deBB theory did indeed include \rho = |\Psi|^2 as a postulate, but it is now widely understood that this is not necessary. That doesn't stop people still writing this in review articles though, as you've seen.

 

[inflector]

Take the recently published "Compendium of Quantum Physics" (published by Springer in 2008) - available at Google books http://books.google.co.uk/books?id=...resnum=1&ved=0CC4Q6AEwAA#v=onepage&q&f=false". At the end of the article on the Born rule, the author states (p. 69):

"The conclusion seems to be that no generally accepted derivation of the Born rule has been given to date, but this does not imply that such a derivation is impossible in principle."

The Born Rule chapter in that compendium appears to be the same as the one I quoted from and linked to earlier at http://www.math.ru.nl/~landsman/Born.pdf. The author must have posted an excerpt of his chapter separately.

Early presentations of the deBB theory did indeed include \rho = |\Psi|^2 as a postulate, but it is now widely understood that this is not necessary. That doesn't stop people still writing this in review articles though, as you've seen.

Interesting, I didn't know that. Thanks for the help and updates.

 

[zenith8]

The Born Rule chapter in that compendium appears to be the same as the one I quoted from and linked to earlier at http://www.math.ru.nl/~landsman/Born.pdf. The author must have posted an excerpt of his chapter separately.

Ah, sorry.. I didn't make that connection.

 

[unusualname]

It might be obvious to you, Mr. name, but it sure isn't obvious to anyone else. Take the recently published "Compendium of Quantum Physics" (published by Springer in 2008) - available at Google books http://books.google.co.uk/books?id=...resnum=1&ved=0CC4Q6AEwAA#v=onepage&q&f=false". At the end of the article on the Born rule, the author states (p. 69):

"The conclusion seems to be that no generally accepted derivation of the Born rule has been given to date, but this does not imply that such a derivation is impossible in principle."

What he does not mean is "I've considered the deBB explanation of this but it's not generally accepted." He just (of course) doesn't mention it - though he mentions most of the other 'interpretations'. That a guy can do that in a well-researched Encyclopedia article is just scandalous but entirely typical.

So, if it's so obvious, can you derive it mathematically? How would you go about doing that?

No I probably couldn't derive it, but I'm pretty sure that this type of result is common in ergodic dynamical systems, admittedly the general mathematical theorems from that subject aren't always easily applicable to real-world examples (or in fact even to many simple toy systems), but it's often difficult to get analytic rigour in theoretical physics (these days)

By "not surprising" I just mean it's common with chaotic deterministic dynamics to have the phase space trajectories evolve to an invariant probability density, but I may have misunderstood the technicalities which are being addressed here with regard to specifying the problem. I do agree that mathematically it is a difficult result to derive, but it sounds a bit like the proof that the Lorentz Attractor exists (Tucker ~1999) which was difficult but no one was really surprised.

 

[DevilsAvocado]

So now rather than relying on three postulates, dBB only requires two, the wave function and the guidance equation alone since the Born Rule emerges naturally from the first two postulates.

You don't find this surprising?

Yes I do. I find it very surprising, but maybe in a 'slightly' different manner...

Postulate 3
. . .
However, while it is ensured that the quantum equilibrium hypothesis is satisfied for a configuration which is |ψ|² distributed once, it is by no means clear why any configuration should be accordingly distributed initially.
. . .
However, there exists a more convincing approach to justify the quantum equilibrium hypothesis. Work by Dürr/Goldstein/Zanghi (1992) shows, that the quantum equilibrium hypothesis follows by the law of large numbers from the assumption that the initial configuration of the universe is "typical" for the |\Psi|^2 distribution (with \Psi being the wavefunction of the universe).
. . .
According to this view the quantum equilibrium hypothesis is no postulate of the de Broglie-Bohm theory but can be derived from it.

??

The initial configuration of the universe is "explained" by the law of large numbers?

The law of large numbers: The expected value of a single die roll is 3.5, with the accuracy increasing as more dice are rolled.

I have to tell you that chance that the initial configuration of the universe can be "explained" by the law of large numbers, is the same as a dice will actually show 3.5, if you just roll it long enough...

This does not look like derivation, but a mathematical variant of the dreadful http://en.wikipedia.org/wiki/Antropic_principle" .

 

[DevilsAvocado]

For those interested in the historical origins and early objections to de Broglie's pilot wave idea you shouldn't miss:

Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference (pdf)

this is a draft of the 533 page book published recently by Cambridge University press

THANKS unusualname! Extremely interesting, goes straight into the HDD archive!

 

[DevilsAvocado]

I’ll be back later, h**l of a lot to read...

 

[zenith8]

Yes I do. I find it very surprising, but maybe in a 'slightly' different manner...??

The initial configuration of the universe is "explained" by the law of large numbers?

The law of large numbers: The expected value of a single die roll is 3.5, with the accuracy increasing as more dice are rolled.

I have to tell you that chance that the initial configuration of the universe can be "explained" by the law of large numbers, is the same as a dice will actually show 3.5, if you just roll it long enough...

This does not look like derivation, but a mathematical variant of the dreadful http://en.wikipedia.org/wiki/Antropic_principle" .

There are two 'schools of thought' on this in the deBB community.

(1) Sheldon Goldstein's crew believe that the initial conditions of the universe were such that, because our universe is 'typical', the particles were distributed as \Psi^2 right from the Big Bang. They aggressively promote the view that anyone who believes otherwise is an idiot.

(2) Almost everyone else (including, for what it's worth, me) believes that one should show that \rho=\Psi^2 arises dynamically, irrespective of the initial conditions. This is what Valentini, Westman, Towler et al. appear to have done.

To me it's completely obvious that the second view is the correct one - assumptions about initial conditions can always be wrong.

You seem to be mixing up these two diametrically opposed point of view.

 

[inflector]

(1) Sheldon Goldstein's crew believe that the initial conditions of the universe were such that, because our universe is 'typical', the particles were distributed as \Psi^2 right from the Big Bang. They aggressively promote the view that anyone who believes otherwise is an idiot.

Not that anyone cares about my opinion, but this seems self-evidently to be a crock of male bovine excrement.

Unless you can establish that the dynamics tend towards an equilibrium of \rho=|\Psi|^2 then the theory rests on shaky ground. That's the whole reason, I never liked the \rho=|\Psi|^2 distribution having to be specified as a postulate.

Who knows what the initial conditions were?

Besides, anytime I see an aggressive promotion that "anyone who believes otherwise is an idiot," on any subject, I have found that this generally indicates a lack of confidence and is often a substitute for intelligence and insight.

 

[zenith8]

Not that anyone cares about my opinion, but this seems self-evidently to be a crock of male bovine excrement.

Unless you can establish that the dynamics tend towards an equilibrium of \rho=|\Psi|^2 then the theory rests on shaky ground. That's the whole reason, I never liked the \rho=|\Psi|^2 distribution having to be specified as a postulate.

Who knows what the initial conditions were?

Besides, anytime I see an aggressive promotion that "anyone who believes otherwise is an idiot," on any subject, I have found that this generally indicates a lack of confidence and is often a substitute for intelligence and insight.

Well, quite..

 

[zonde]

If the electron density distribution is not equal to the square of the wave field, and the system is evolving according to the laws of QM, then it will become so distributed over the course of time. Once so distributed, it will stay like that for ever. Psi^2 is the only distribution with this property.

This is fine when you consider electrons. This is not fine if you consider photons.
Electrons are not moving at c and they can get feedback from environment and as a consequence undergo rapid decoherence. Photons on the other hand move at c.

There is even something from experiments that as I see directly contradicts with such picture for photons. Would you care to go into analysis of HOM effect?

 

[Demystifier]

There are two 'schools of thought' on this in the deBB community.

(1) Sheldon Goldstein's crew believe that the initial conditions of the universe were such that, because our universe is 'typical', the particles were distributed as \Psi^2 right from the Big Bang. They aggressively promote the view that anyone who believes otherwise is an idiot.

(2) Almost everyone else (including, for what it's worth, me) believes that one should show that \rho=\Psi^2 arises dynamically, irrespective of the initial conditions. This is what Valentini, Westman, Towler et al. appear to have done.

To me it's completely obvious that the second view is the correct one - assumptions about initial conditions can always be wrong.

You seem to be mixing up these two diametrically opposed point of view.

I agree that the Valentini et al camp uses a much nicer way of promoting their views than the Goldstein et al camp. Yet, I think that each of the approaches has certain advantages over the other. For example, in the relativistic-covariant Bohmian mechanics I am promoting, the Goldstein et al approach seems to work much better.

 

[DevilsAvocado]

Who knows what the initial conditions were?

Cosmologists.

See http://en.wikipedia.org/wiki/Wilkinson_Microwave_Anisotropy_Probe" for more info.

 

[zenith8]

Cosmologists.
See http://en.wikipedia.org/wiki/Wilkinson_Microwave_Anisotropy_Probe" for more info.

Quite right. Which is why Valentini et al. are looking for deviations in the CMB which are a signature of quantum nonequilibrium predicted by de Broglie-Bohm theory (this will probably have to wait for the higher accuracy results of Planck in 2012 or whenever).

See his talk at "http://www.vallico.net/tti/deBB_10/conference.html" " entitled "In search of a breakdown in quantum theory" for the details.

 

[DevilsAvocado]

Quite right. Which is why Valentini et al. are looking for deviations in the CMB which are a signature of quantum nonequilibrium predicted by de Broglie-Bohm theory (this will probably have to wait for the higher accuracy results of Planck in 2012 or whenever).

See his talk at "http://www.vallico.net/tti/deBB_10/conference.html" " entitled "In search of a breakdown in quantum theory" for the details.

Cool. I must read faster... information over-load right now... hopefully I’ll be back in a couple of days with some comments worth reading...

 

[inflector]

Cosmologists.

See http://en.wikipedia.org/wiki/Wilkinson_Microwave_Anisotropy_Probe" for more info.

Well since we are currently lacking a coherent theory of quantum gravity, even cosmologists are guessing and back-extrapolating from when the universe was about 487,000 years old.

 

[inflector]

Iin the relativistic-covariant Bohmian mechanics I am promoting, the Goldstein et al approach seems to work much better.

Interesting. Can you explain what about the Goldstein approach makes it work better with your relativistic- covariant Bohmian mechanics?

 

[DevilsAvocado]

Well since we are currently lacking a coherent theory of quantum gravity, even cosmologists are guessing and back-extrapolating from when the universe was about 487,000 years old.

Well, the cosmologists are not the only ones 'guessing', right? At least they have 'some' data!

 

[nismaratwork]

Quite right. Which is why Valentini et al. are looking for deviations in the CMB which are a signature of quantum nonequilibrium predicted by de Broglie-Bohm theory (this will probably have to wait for the higher accuracy results of Planck in 2012 or whenever).

See his talk at "http://www.vallico.net/tti/deBB_10/conference.html" " entitled "In search of a breakdown in quantum theory" for the details.

Would the absence of those artifacts at the proper resolution falsify dBB?

 

[Maaneli]

Would the absence of those artifacts at the proper resolution falsify dBB?

There is no "proper" resolution. The absence of nonequilibrium signatures at a particular length and time scale would only put constraints on the possibility of nonequilibrium. DeBB theory, as it currently stands, does not require that nonequilibrium states should have existed in the early universe, only that they are possible states of the early universe.

However, it's worth mentioning that Valentini himself would find the deBB theory scientifically implausible if, after a 1,000 years of searching the universe, no evidence of quantum nonequilibrium were ever found.

 

[Maaneli]

Interesting. Can you explain what about the Goldstein approach makes it work better with your relativistic- covariant Bohmian mechanics?

Yes, Hrvoje, I'm also curious to know how the Typicality approach helps your covariant theory more than dynamical relaxation does.

 

[Demystifier]

Yes, Hrvoje, I'm also curious to know how the Typicality approach helps your covariant theory more than dynamical relaxation does.

It's actually very simple. In nonrelativistic BM, relaxation does not work for a stationary wave function which does not depend on t. Likewise, in relativistic BM with spacetime probabilistic interpretation, relaxation does not work for a wave function which does not depend on s. And of course, the relativistic wave function (in my approach) never depends on s.

 

[Maaneli]

It's actually very simple. In nonrelativistic BM, relaxation does not work for a stationary wave function which does not depend on t. Likewise, in relativistic BM with spacetime probabilistic interpretation, relaxation does not work for a wave function which does not depend on s. And of course, the relativistic wave function (in my approach) never depends on s.

I see your point, but then there seems to me a contradiction in your relativistic theory, since, in the NR limit of your theory, you get back the usual nonrelativistic deBB wavefunction where psi dynamically evolves with respect to the single time t. So it's as if in the relativistic case of your theory, you won't get dynamical relaxation, while in the nonrelativistic case, you would expect to get dynamical relaxation (with respect to t). That would suggest to me that one just has to think harder about how dynamical relaxation can occur in the relativistic case of your theory.

 

[zenith8]

That would suggest to me that one just has to think harder about how dynamical relaxation can occur in the relativistic case of your theory.

What about Holland's result that, if you consider the non-relativistic spin 1/2 theory as the limiting case of the relativistic Dirac theory, then this fixes the guidance equation uniquely (recalling there is a 'gauge freedom' in the standard one) and that this unique equation has a 'spin term' in addition to the gradient of the phase? With such a guidance equation, the electrons are no longer at rest in the stationary wave function case.

 

[Demystifier]

That would suggest to me that one just has to think harder about how dynamical relaxation can occur in the relativistic case of your theory.

I agree. And I would be very happy if I were not the only guy who is actually doing it (thinks hard about that).

 

[Demystifier]

What about Holland's result that, if you consider the non-relativistic spin 1/2 theory as the limiting case of the relativistic Dirac theory, then this fixes the guidance equation uniquely (recalling there is a 'gauge freedom' in the standard one) and that this unique equation has a 'spin term' in addition to the gradient of the phase? With such a guidance equation, the electrons are no longer at rest in the stationary wave function case.

The Holland's approach does not work for
1) particles with other spins (0, 1, ...)
2) systems of more than one entangled particles with spin 1/2

 

[zenith8]

The Holland's approach does not work for
1) particles with other spins (0, 1, ...)
2) systems of more than one entangled particles with spin 1/2

OK. Can you explain (2)?

 

[Demystifier]

OK. Can you explain (2)?

The Holland's paper
http://xxx.lanl.gov/abs/quant-ph/0305175
does not discuss the many-particle case at all.

A many-particle case (for spin 1/2) is studied in
http://xxx.lanl.gov/abs/quant-ph/9801070
where it was found necessary to introduce a preferred foliation of spacetime, which is certainly not unique.

The only (currently known) way to avoid preferred foliation is the evolution with respect to a scalar parameter s. But then, as I explained, the natural probabilistic interpretation does not seem compatible with the idea of dynamical relaxation towards the equilibrium. A typicality approach works much better.

 

[zenith8]

The Holland's paper
http://xxx.lanl.gov/abs/quant-ph/0305175
does not discuss the many-particle case at all.

A many-particle case (for spin 1/2) is studied in
http://xxx.lanl.gov/abs/quant-ph/9801070
where it was found necessary to introduce a preferred foliation of spacetime, which is certainly not unique.

The only (currently known) way to avoid preferred foliation is the evolution with respect to a scalar parameter s. But then, as I explained, the natural probabilistic interpretation does not seem compatible with the idea of dynamical relaxation towards the equilibrium. A typicality approach works much better.

For the many-particle case see section 10.5 of Bohm and Hiley's book or, for example, Timko and Vrscay's 'spin-dependent Bohmian electronic trajectories for helium' available at Found. Phys. 39, 1055 (2009) or on the usual web page.

Anyway, what's wrong with preferred foliations? Perfectly compatible with all known experimental results - it's just the neo-Lorentzian interpretation of relativity, no?

 

[Demystifier]

Anyway, what's wrong with preferred foliations? Perfectly compatible with all known experimental results - it's just the neo-Lorentzian interpretation of relativity, no?

Perhaps there is nothing wrong with it, but looks ugly. Too many possibilities are allowed, so how to know which foliation is the right one? In the absence of direct experimental evidence for a theory, simplicity and mathematical elegance should be the main guiding principles.

Besides, a preferred foliation is certainly not in the spirit of the Holland's approach that was first mentioned by you.

 

[Maaneli]

What about Holland's result that, if you consider the non-relativistic spin 1/2 theory as the limiting case of the relativistic Dirac theory, then this fixes the guidance equation uniquely (recalling there is a 'gauge freedom' in the standard one) and that this unique equation has a 'spin term' in addition to the gradient of the phase? With such a guidance equation, the electrons are no longer at rest in the stationary wave function case.

Btw, Holland wasn't the first to recognize this in the literature. Hestenes and Gurtler did way back in the 70's:

Consistency in the formulation of the Dirac, Pauli, and Schroedinger theories
Journal of Mathematical Physics, 16 573–584 (1975).
http://geocalc.clas.asu.edu/pdf/Consistency.pdf

Also, I independently derived this result as an undergrad, which suggests that it's been independently rediscovered by others countless times.

 

[Maaneli]

Perhaps there is nothing wrong with it, but looks ugly. Too many possibilities are allowed, so how to know which foliation is the right one? In the absence of direct experimental evidence for a theory, simplicity and mathematical elegance should be the main guiding principles.

But if you take the possibility of quantum nonequilibrium seriously, then there's nothing fundamentally problematic about that underdetermination of foliations - we just happen to be stuck in a special state (the quantum equilibrium state) that prevents us from observing the correct foliation.

 

[Maaneli]

The only (currently known) way to avoid preferred foliation is the evolution with respect to a scalar parameter s.

Though it isn't popular among deBB theorists, another logical possibility is to introduce retrocausation a la Sutherland's model:

Causally Symmetric Bohm Model
Rod Sutherland
http://arxiv.org/abs/quant-ph/0601095

 

[Demystifier]

But if you take the possibility of quantum nonequilibrium seriously, then there's nothing fundamentally problematic about that underdetermination of foliations - we just happen to be stuck in a special state (the quantum equilibrium state) that prevents us from observing the correct foliation.

I completely agree. Yet, it does not change the fact that the theory itself is ugly. It's hard to take seriously a theory that looks ugly, unless there is a direct experimental evidence supporting the theory.

For example, there are many alternatives to the classical Einstein theory of gravity, compatible with existing experimental data. Yet, the Einstein theory is the most popular. Why? Because neither of the alternatives is so elegant.

Another example is the Standard Model of elementary particles. It is in perfect agreement with all experiments. Yet, many physicists search for alternatives (GUTs, supersymmetries, strings, ...). Why? Because the Standard Model is terribly ugly.

 

[Maaneli]

I agree. And I would be very happy if I were not the only guy who is actually doing it (thinks hard about that).

Yeah, I'm thinking about it. One possibility I have in mind is to allow that a nonequilibrium version of your relativistic psi does initially depend on s, and then a stochastic Markov process dynamically relaxes the wavefunction to an equilibrium state with respect to s (much like in the Parisi-Wu approach to stochastic quantization). Then, it is only in this stochastic equilibrium state that your relativistic psi appears to be independent of s, and thus not allow for relativistic nonequilibrium states thereafter.

On the other hand, if you want to insist on deterministic dynamics, you might insist that your relativistic psi should always depend on s, in which case, your relativistic deBB theory becomes a deBB version of the Stueckelberg proper time formulation of relativistic QM.

 

[Demystifier]

Though it isn't popular among deBB theorists, another logical possibility is to introduce retrocausation, a la Sutherland's model:

Causally Symmetric Bohm Model
Rod Sutherland
http://arxiv.org/abs/quant-ph/0601095

Thanks, I didn't know about this. But it also seem to require a preferred frame [Eq. (60)].

 

[Demystifier]

Yeah, I'm thinking about it. One possibility I have in mind is to allow that a nonequilibrium version of your relativistic psi does initially depend on s, and then a stochastic Markov process dynamically relaxes the wavefunction to an equilibrium state with respect to s (much like in the Parisi-Wu approach to stochastic quantization). Then, it is only in this stochastic equilibrium state that your relativistic psi appears to be independent of s, and thus not allow for relativistic nonequilibrium states thereafter.

That seems interesting, but I don't like the idea that I must add a stochastic process by hand.

On the other hand, if you want to insist on deterministic dynamics, you might insist that your relativistic psi should always depend on s, in which case, your relativistic deBB theory becomes a deBB version of the Stueckelberg proper time formulation of relativistic QM.

Irrespective of dBB, the Stueckelberg equation does not seem to be in agreement with observations. In particular, we do not observe a continuous mass spectrum. (See however
http://xxx.lanl.gov/abs/0801.4471 )

 

[Maaneli]

I completely agree. Yet, it does not change the fact that the theory itself is ugly. It's hard to take seriously a theory that looks ugly, unless there is a direct experimental evidence supporting the theory.

For example, there are many alternatives to the classical Einstein theory of gravity, compatible with existing experimental data. Yet, the Einstein theory is the most popular. Why? Because neither of the alternatives is so elegant.

Another example is the Standard Model of elementary particles. It is in perfect agreement with all experiments. Yet, many physicists search for alternatives (GUTs, supersymmetries, strings, ...). Why? Because the Standard Model is terribly ugly.

I would agree that it is reasonable to take more seriously alternative models, if those alternative models can make all the same predictions as the standard theory, but with fewer and more physically plausible assumptions. However, I still think it's dubious to say that the standard deBB theory is hard to take seriously because it has this feature which seems "ugly" (or even fugly) to you.

In the 19th century, positivistic physicists like Mach criticized Boltzmann's statistical mechanics on similar grounds, saying for example that for molecules in thermal equilibrium, one could double the number of particles composing a gas, but halve their volume and masses (or something like that), and make all the same predictions. Of course, we now know that Mach's criticism is wrong because we understand (and can empirically observe) that equilibrium dynamics masks important microscopic details of particle dynamics, and that equilibrium dynamics is only a special case of a more general nonequilibrium dynamics. So even though Boltzmann's statistical mechanics has this feature which would probably seem ugly to you if you were living in that time, we can see that nature can still conform to such ugly features.