Where does non-locality originate in dBB theory?

[Demystifier]

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

Perhaps I'm wrong, but the theory of quantum measurements in that paper also seems fishy to me.

 

[zenith8]

Perhaps there is nothing wrong with [a preferred foliation], 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.

Does it really look ugly? But if you insist that all reference frames are equivalent, then:

(1) you get causal paradoxes over who measured things first (like someone was moaning about earlier in relation to the EPR experiment). These don't appear if you have a preferred frame.

(2) either (in Minkowski spacetime) there is no 'temporal becoming' since everything exists simultaneously as a 4d worldtube, or (in Einstein 3+1 spacetime) things pop in and out of reality as you switch reference frames (uh??) and objects undergo (reciprocal!) physical length contraction just because they are in relative motion for no readily apparent reason.

Those sound pretty ugly to me - at least philosphically.

In the Lorentzian interpretation with a preferred frame you have a causal explanation for length contraction/time dilation, you have temporal becoming, you don't get the causal paradoxes, and you have complete agreement with experiment..

I accept that historically people have thought preferred frames unnecessary (because that our condition of being in quantum equilibrium means we can't detect it..) but that viewpoint was developed for a local physics. With our new non-local universe, it might be worth looking again at preferred frames (since the `ether' or absolute space or whatever you want to call it is presumably the medium in which the nonlocal interactions are absolutely simultaneous..)

 

[Demystifier]

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.

Again I agree, and I am certainly not strictly against the idea of a preferred frame. After all, I have published a lot of papers with a preferred frame by myself. Yet, the idea that the preferred frame can be eliminated seems even more attractive to me. Besides, I find it very challenging as a research direction. In any case, we can certainly make consensus that various different approaches should be studied.

 

[inflector]

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

What does "preferred foliation" mean? I've seen the term used here quite a bit but never explained. I tried googling it but didn't find any definitions just plenty of usage.

I have a very very rough idea what a foliation means but don't understand the term "preferred foliation."

 

[Demystifier]

Does it really look ugly? But if you insist that all reference frames are equivalent, then:

(1) you get causal paradoxes over who measured things first (like someone was moaning about earlier in relation to the EPR experiment). These don't appear if you have a preferred frame.

(2) either (in Minkowski spacetime) there is no 'temporal becoming' since everything exists simultaneously as a 4d worldtube, or (in Einstein 3+1 spacetime) things pop in and out of reality as you switch reference frames (uh??) and objects undergo (reciprocal!) physical length contraction just because they are in relative motion for no readily apparent reason.

Those sound pretty ugly to me - at least philosphically.

In the Lorentzian interpretation with a preferred frame you have a causal explanation for length contraction/time dilation, you have temporal becoming, you don't get the causal paradoxes, and you have complete agreement with experiment..

I accept that historically people have thought preferred frames unnecessary (because that our condition of being in quantum equilibrium means we can't detect it..) but that viewpoint was developed for a local physics. With our new non-local universe, it might be worth looking again at preferred frames (since the `ether' or absolute space or whatever you want to call it is presumably the medium in which the nonlocal interactions are absolutely simultaneous..)

I would say that the unpleasent features you discuss are counterintuitive, not ugly. But of course, this is all subjective, and I am not trying to change your opinion. I am just trying to explain to you the way I think.

 

[Maaneli]

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

You're welcome. I don't see why equation 60 implies a preferred frame. The psi^bar_f is not the complex conjugate of the psi_i.

 

[Maaneli]

Again I agree, and I am certainly not strictly against the idea of a preferred frame. After all, I have published a lot of papers with a preferred frame by myself. Yet, the idea that the preferred frame can be eliminated seems even more attractive to me. Besides, I find it very challenging as a research direction. In any case, we can certainly make consensus that various different approaches should be studied.

OK, fair enough.

 

[Maaneli]

Perhaps I'm wrong, but the theory of quantum measurements in that paper also seems fishy to me.

Which parts?

 

[Maaneli]

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

Well if there is no other way to do it, then the assumption of a stochastic process would be well-justified, I think.

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 )

Interesting, I wasn't aware of this. Can you give a ref. about the Stueckelberg equation predicting a continuous mass spectrum?

 

[Demystifier]

Interesting, I wasn't aware of this. Can you give a ref. about the Stueckelberg equation predicting a continuous mass spectrum?

It's trivial, you can easily show it by yourself. Just recall that mass^2 are eigenvalues of the operator \partial^{\mu}\partial_{mu} and consider solutions with the dependence on s of the form exp(i const s).

I don't want to comment the Sutherland's paper any more, because I have not yet studied it carefully. I hope I will find time to do it properly in the near future.

 

[Demystifier]

Well if there is no other way to do it, then the assumption of a stochastic process would be well-justified, I think.

As usual, It's hard to disagree with your well balanced statements.

 

[Demystifier]

OK, I have finished a more careful reading of the Shaterland's paper, so let me make some comments.

First, I have some inessential "technical" objections:
1) Contrary to the statement in the paper, Eq. (37) is NOT correct in standard QM. (A correct version would involve density matrices which generalize the notion of wave functions.)
2) Negative probabilities, as such, do not make sense.

Now to the point. Even if some details are incorrect (which I think they are), it seems that the main idea MIGHT WORK. But how exactly is that possible? Well, the idea is just an attempt to exploit a well-known loophole of the Bell theorem: the SUPERDETERMINISM loophole. Namely, if everything, including our "free" decisions, is actually determined by physical laws, then, AT LEAST IN PRINCIPLE, it is possible to get Bell correlations without nonlocality. The standard Bohm interpretation is also superdeterministic, but it still does not contain sufficiently many hidden-variables to avoid nonlocality. To overcome this, Shaterland adds ADDITIONAL hidden variables, the wave functions psi_f. There is no doubt that you can avoid nonlocality by adding a sufficient number of superdeterministic hidden variables. The difficult part is to do it in a relatively simple way, and that's what Shaterland attempts to do. His attempt can be seen as a combination of transactional and Bohmian interpretation.

This approach can be compared with that of 't Hooft, who is trying to construct local superdeterministic hidden variables that, at first sight, do not even resemble QM.

 

[Demystifier]

You're welcome. I don't see why equation 60 implies a preferred frame. The psi^bar_f is not the complex conjugate of the psi_i.

You are right, the equation is covariant, but his notation (the use of label 0) is very confusing.

Nevertheless, he does not give a covariant version of (42).

In addition, (42) does not look "causally symmetric", but I guess it can be justified by the thermodynamic arrow of time.

 

[zonde]

Does it really look ugly? But if you insist that all reference frames are equivalent, then:

(1) you get causal paradoxes over who measured things first (like someone was moaning about earlier in relation to the EPR experiment). These don't appear if you have a preferred frame.

(2) either (in Minkowski spacetime) there is no 'temporal becoming' since everything exists simultaneously as a 4d worldtube, or (in Einstein 3+1 spacetime) things pop in and out of reality as you switch reference frames (uh??) and objects undergo (reciprocal!) physical length contraction just because they are in relative motion for no readily apparent reason.

Those sound pretty ugly to me - at least philosphically.

In the Lorentzian interpretation with a preferred frame you have a causal explanation for length contraction/time dilation, you have temporal becoming, you don't get the causal paradoxes, and you have complete agreement with experiment..

Neo-Lorentzian interpretation makes exactly the same predictions as special relativity at least as long as you do not invoke some FTL stuff. So if you have paradoxes in SR (actually you don't) then you have exactly the same paradoxes in Neo-Lorentzian interpretation. The only difference is that Neo-Lorentzian interpretation is more intuitive then SR.

 

[Demystifier]

The only difference is that Neo-Lorentzian interpretation is more intuitive then SR.

This is not the only difference. The other difference is that it is also mathematically less elegant than SR. And this is indeed why the SR view is more popular, because in modern theoretical physics mathematical elegance is more appreciated than intuitivity.

 

[zonde]

This is not the only difference. The other difference is that it is also mathematically less elegant than SR. And this is indeed why the SR view is more popular, because in modern theoretical physics mathematical elegance is more appreciated than intuitivity.

Can you explain?
Do you mean that with Neo-Lorentzian interpretation you should always stick to one preferred reference frame?
Because as I see once you have established transformation in certain group of different reference frames that is intrinsically symmetric it afterwards does not matter (mathematically) how you have done that.
It's like that the way you prove theorem does not change the theorem itself.

 

[Maaneli]

It's trivial, you can easily show it by yourself. Just recall that mass^2 are eigenvalues of the operator \partial^{\mu}\partial_{mu} and consider solutions with the dependence on s of the form exp(i const s).

Right, I see. My reading indicates though that proponents of Stueckelberg such as Kyprianidis, Horwitz and Piron, etc., don't think that this creates problems for the empirical predictions of such theories.

 

[Maaneli]

First, I have some inessential "technical" objections:
1) Contrary to the statement in the paper, Eq. (37) is NOT correct in standard QM. (A correct version would involve density matrices which generalize the notion of wave functions.)

Eq. (37) is not true even for ideal measurements?

Negative probabilities, as such, do not make sense.

Formally, I don't see anything wrong with negative probabilities in the context of Sutherland's theory. Also, negative probabilities exist in classical statistical physics as well. See for example the backwards Kolmogorov equation.

to the point. Even if some details are incorrect (which I think they are), it seems that the main idea MIGHT WORK. But how exactly is that possible? Well, the idea is just an attempt to exploit a well-known loophole of the Bell theorem: the SUPERDETERMINISM loophole. Namely, if everything, including our "free" decisions, is actually determined by physical laws, then, AT LEAST IN PRINCIPLE, it is possible to get Bell correlations without nonlocality. The standard Bohm interpretation is also superdeterministic, but it still does not contain sufficiently many hidden-variables to avoid nonlocality. To overcome this, Shaterland adds ADDITIONAL hidden variables, the wave functions psi_f. There is no doubt that you can avoid nonlocality by adding a sufficient number of superdeterministic hidden variables. The difficult part is to do it in a relatively simple way, and that's what Shaterland attempts to do. His attempt can be seen as a combination of transactional and Bohmian interpretation.

This approach can be compared with that of 't Hooft, who is trying to construct local superdeterministic hidden variables that, at first sight, do not even resemble QM.

I think your characterization of Sutherland's theory is exactly right. So, if you think Sutherland's proposal works for the examples that he considers, then here you have an example of a hidden variables theory whose dynamics is relativistically covariant, and does not require a preferred frame or a synchronization parameter.

 

[Demystifier]

Eq. (37) is not true even for ideal measurements?

Not even then, but it is an inessential technicality.

Formally, I don't see anything wrong with negative probabilities in the context of Sutherland's theory. Also, negative probabilities exist in classical statistical physics as well. See for example the backwards Kolmogorov equation.

Maybe, but this is inessential technicality too.

I think your characterization of Sutherland's theory is exactly right. So, if you think Sutherland's proposal works for the examples that he considers, then here you have an example of a hidden variables theory whose dynamics is relativistically covariant, and does not require a preferred frame or a synchronization parameter.

I think that his proposal MIGHT work, provided that some details are better developed.
But I don't plan to do it. Instead, soon I will upload on the arXiv something similar but, I believe, even better: A local relativistic-covariant theory of particle trajectories that does not contain more hidden variables than my relativistic-covariant version of Bohm theory.

 

[Demystifier]

Right, I see. My reading indicates though that proponents of Stueckelberg such as Kyprianidis, Horwitz and Piron, etc., don't think that this creates problems for the empirical predictions of such theories.

That's true, but I never understood why do they think so.

 

[Maaneli]

I think that his proposal MIGHT work, provided that some details are better developed.
But I don't plan to do it. Instead, soon I will upload on the arXiv something similar but, I believe, even better: A local relativistic-covariant theory of particle trajectories that does not contain more hidden variables than my relativistic-covariant version of Bohm theory.

Cool, does your local relativistic-covariant theory also account for Bell nonlocality?

 

[Maaneli]

That's true, but I never understood why do they think so.

From one of Kyprianidis's papers, he writes:

"Horwitz and Piron [8] assign a definite mass only to non-interacting free particles while systems in presence of interaction are “off-mass shell” states. The latter approach has a profound physical meaning and it can be immediately associated with de Broglie’s “variable mass”, which is the rest mass additioned by the Quantum Potential. From this point of view, it is quite natural to seek for a description of quantum phenomena in terms of an indefinite mass theory, since quantum particles are never free but always submitted to the quantum potential. Therefore the Klein—Gordon theory must correspond to a specific restriction on the set of possible physical solutions, and consequently to a specific selection of processes out of the multiplicity of the processes described by the generalized Schrödinger equation."

Kyprianidis then goes on to suggest another way to deal with this indefinite mass problem.

 

[Demystifier]

Cool, does your local relativistic-covariant theory also account for Bell nonlocality?

Here it is:
http://xxx.lanl.gov/abs/1010.2082

It reproduces all predictions of QM. However, it is not completely local; it requires initial nonlocal correlations between the particle positions. Yet, it is much more local than standard Bohmian mechanics in the sense that nonlocal forces can be eliminated by appropriate choice of parameterization of the particle trajectories. ALL nonlocality is encoded in the initial conditions.

 

[unusualname]

aren't simplistic particle trajectories ruled out by bell type inequalities?

eg http://arxiv.org/abs/0903.3878

and

This subject [Bohmian Mechanics] was assessed by the NSF of the USA as follows [Cushing, J. T., review of Bohm, D., and Hiley, B., The Undivided Universe, Foundations of Physics, 25, 507, 1995.] "...The causal interpretation [of Bohm] is inconsistent with experiments which test Bell's inequalities. Consequently...funding...a research programme in this area would be unwise"..

and don't you have to chuck in a spin component to make it (more) consistent? (and hence use a C^2 representation for the wavefunction)

quote pasted from:
http://www.mth.kcl.ac.uk/~streater/lostcauses.html#XI

 

[zenith8]

aren't simplistic particle trajectories ruled out by bell type inequalities?

eg http://arxiv.org/abs/0903.3878

and

and don't you have to chuck in a spin component to make it (more) consistent? (and hence use a C^2 representation for the wavefunction)

quote pasted from:
http://www.mth.kcl.ac.uk/~streater/lostcauses.html#XI

Both Streater and the NSF are just wrong. Streater isn't usually wrong when he can be bothered to make an effort but he's just completely lazy here. It's clear that he's read about two paragraphs worth of stuff about deBB then just unloads his prejudices about what he thinks the theory is about in order to amuse the undergraduate viewers of his website..

Ilja Schmelzer has a full rebuttal of the whole of Streater's 'lost causes' page http://ilja-schmelzer.de/realism/dBBarguments.php" .

 

[unusualname]

ok, thanks for the link, I guess you can always argue that dBB predicts same measurements as standard non-relativistic QM.

 

[LukeD]

Here it is:
http://xxx.lanl.gov/abs/1010.2082

It reproduces all predictions of QM. However, it is not completely local; it requires initial nonlocal correlations between the particle positions. Yet, it is much more local than standard Bohmian mechanics in the sense that nonlocal forces can be eliminated by appropriate choice of parameterization of the particle trajectories. ALL nonlocality is encoded in the initial conditions.

Hey hey hey! This paper exactly describes what I was talking about on page one of this thread (my third post)! This is just the Quantum Trajectory Method done relativistically. So the integral curves then do interact locally? (except for the initial conditions)

hallelujah
You should know the integral curves are continuous versions of the Consistent Histories studied by Robbert Griffiths (I had him for a class as an undergrad). It might help you to read up on them. I don't recommend his book though it's not very well written. There are only a handful of books written on the subject though.

Basically, what Griffiths concluded is that consistent histories are never unique. There are always other histories that describe the same situation, and the descriptions can never be taken to be simultaneously true. It's a perfect description of quantum complementarity

 

[LukeD]

Oh, maybe you haven't yet noticed that you can get all of the dynamics just from the initial set of integral curves and the initial |\Psi|^2 (at least non-relativistically you can do this. it might break down somewhere in the relativistic case). I noticed that your paper gets all of its dynamics from the wavefunction without any back reaction from the integral curves. I guess you probably cannot tell me if those interactions are purely local. However, you claim that the velocities can be made to come from a local differential equation. I cannot see this myself, but if it is true then I suppose that means that the interaction between |\Psi^2| and the integral curves must be local (since the wavefunction can always be decomposed into those two components)

You've probably noticed though that the integral curves are defined everywhere in configuration space. For a 2-particle system, there is an integral curve going through (x1, x2) and an integral curve going through (x1, x2+\epsilon) so it doesn't make any sense to try to separate the positions of the particles - both of their positions are simultaneously needed to talk about the state in configuration space. So the fact that the velocity of one particle depends strongly on the position of the other particles for an entangled state isn't a problem. Of course it does! That's essentially what entanglement means and there's nothing non-local with it in this picture of integral curves through configuration space.

 

[unusualname]

in the paper I linked to http://arxiv.org/abs/0903.3878 , the author claims that a Bell-CHSH inequality shows that trajectories fail in the non-relativistic case. Don't you have to introduce a spin component even in this case, ie modify the pilot wave to a C^2 valued function including the spin?

 

[ueit]

Here it is:
http://xxx.lanl.gov/abs/1010.2082

It reproduces all predictions of QM. However, it is not completely local; it requires initial nonlocal correlations between the particle positions. Yet, it is much more local than standard Bohmian mechanics in the sense that nonlocal forces can be eliminated by appropriate choice of parameterization of the particle trajectories. ALL nonlocality is encoded in the initial conditions.

IMHO you are wrong when describing the initial correlations between particle positions as non-local. I think that the source of this misunderstanding comes from a wrong view of what "initial conditions" mean.

The "initial conditions" in an experiment are nothing but the position/momenta of the particles at the beginning of the experiment. They are by no means "initial" for the system itself which has a history going back to the Big-Bang. The situation is analogous with an astronomical observation of the solar system. The position/momenta of the planets are correlated, but is this evidence for non-locality? I think not.

I think that the "initial nonlocal correlations" would necessarily appear in any deterministic theory of motion where particles interact at a distance and they are a direct consequence of the evolution of the system from the Big-Bang till now.

 

[Demystifier]

IMHO you are wrong ...

IMHO, you have not actually read the paper (only the abstract).

 

[ueit]

IMHO, you have not actually read the paper (only the abstract).

I have read it. In the conclusion you state:

To provide consistency with statistical predictions of QM, one must assume that the a priori probabilities of initial particle positions Xμ a (0) are given by (15). Thus, all nonlocality can be ascribed to initial nonlocal correlations between the particle spacetime positions.

Why do you think that the particles having a certain distribution is a sign of non-locality? It can be a result of the Big-Bang itself, or it can be a consequence of the past interactions, prior to the beginning of the experiment.

 

[Demystifier]

Why do you think that the particles having a certain distribution is a sign of non-locality? It can be a result of the Big-Bang itself, or it can be a consequence of the past interactions, prior to the beginning of the experiment.

First, you should note that "initial" in this paper refers to s=0, not to t=0. Thus, "initial" does not necessarily mean "at the big bang", or "at a spacelike hypersurface". In fact, a part of the "initial conditions" may even be in the future. See the picture on page 8 of the attachment in
https://www.physicsforums.com/blog.php?b=2240

Second, what initial conditions have to do with nonlocality? Well, if initial conditions involve nonlocal correlations (which they do, according to the paper), then there is something nonlocal about that. It is not much more than a purely linguistic tautology: if something is nonlocal (whatever that means), then it is nonlocal.

But what causes this nonlocality? Well, the paper does not attempt to answer this question. Just like most physical theories do not attempt to answer why the initial conditions are such as they are.

 

[Demystifier]

The position/momenta of the planets are correlated, but is this evidence for non-locality? I think not.

Good analogy. If you think that way, you may think of the theory (in the paper) as being completely local.

I think that the "initial nonlocal correlations" would necessarily appear in any deterministic theory of motion where particles interact at a distance and they are a direct consequence of the evolution of the system from the Big-Bang till now.

Perhaps it is true that you will get SOME initial nonlocal correlations. But can you get initial nonlocal correlations EXACTLY EQUAL TO THOSE PREDICTED BY QM? I don't think so.

 

[LukeD]

Demystifier: Could you respond to my post at the top of this page (page 12)? Maybe I'm mistaken, but it really seems to me like your paper is describing exactly what I was asking about on page 1.

 

[Demystifier]

Hey hey hey! This paper exactly describes what I was talking about on page one of this thread (my third post)! This is just the Quantum Trajectory Method done relativistically. So the integral curves then do interact locally? (except for the initial conditions)

hallelujah
You should know the integral curves are continuous versions of the Consistent Histories studied by Robbert Griffiths (I had him for a class as an undergrad). It might help you to read up on them. I don't recommend his book though it's not very well written. There are only a handful of books written on the subject though.

Basically, what Griffiths concluded is that consistent histories are never unique. There are always other histories that describe the same situation, and the descriptions can never be taken to be simultaneously true. It's a perfect description of quantum complementarity

I don't know what exactly do you mean by "quantum trajectory method", but my approach does not have much in common with consistent histories of Griffiths.

Yet, the answer to your question is - yes. More precisely, different particles do not have a direct mutual interaction at all. Instead, each particle is guided by another local current. Yet, each of these currents is calculated from the common non-separable wave function.

EDIT: If by quantum trajectory method you mean the method by Lopreore and Wyatt, then my approach does not have much to do with it either.

 

[LukeD]

Yet, the answer to your question is - yes. More precisely, different particles do not have a direct mutual interaction at all. Instead, each particle is guided by another local current. Yet, each of these currents is calculated from the common non-separable wave function.

Ok, I see what you're saying. But I think your idea has much more in common with both Consistent Histories and the Quantum Trajectory Method than you think. The complete set of trajectories IS a Consistent History. It matches the mathematical definition. There is really no way around that.

As far as the Quantum Trajectory Method - it works in non-relativistic QM, but I don't know about relativistic QM. You haven't taken advantage of the method, but I'm dying to know if you can do it relativistically. In the non-relativistic theory, you can separate the N-particle wavefunction \Psi(q1,...qN,t) into 2 parts, \rho = |\Psi(q1,...,qN,t)|^2 and N velocity fields, one for each particle, v_i(q1,...,qN,t). You can then solve for 2 local, coupled differential equations that describe the evolutions of \rho and the velocity fields as functions of time.
You already have the velocity fields, but you're still using the full wave function to describe their evolution. The velocity fields exist as a "component" of the wavefunction. You can pull them out.

 

[Demystifier]

As far as the Quantum Trajectory Method - it works in non-relativistic QM, but I don't know about relativistic QM.

I far as I understand it, the quantum trajectory method works only for wave equations which are first-order equations in time derivatives, and only for wave functions without spin. The relativistic Klein-Gordon equation does not satisfy the first condition, while the relativistic Dirac equation does not satisfy the second condition.

 

[Demystifier]

The complete set of trajectories IS a Consistent History. It matches the mathematical definition. There is really no way around that.

OK, but in the consistent history approach no set of consistent histories takes a preferred role. So if my trajectories are one such set, there is still a plenty of other such sets, so my set does not play any particularly important role in the CH approach.

 

[LukeD]

OK, but in the consistent history approach no set of consistent histories takes a preferred role. So if my trajectories are one such set, there is still a plenty of other such sets, so my set does not play any particularly important role in the CH approach.

Correct. In particular, when you have spin, the set of trajectories in position space is non-unique. This means that there are multiple sets of trajectories that will answer all your position questions. You said that the Quantum Trajectory Method does not work when you have spin. I don't understand why it doesn't. Could you point me towards a paper?

If you want to answer questions in a different basis (say momentum), you can do it with the dBB paths, but it's not very straight forward (as you know, momentum is not just mv). It's much easier to just construct trajectories in momentum space. Of course you cannot combine your conclusions between the position and momentum space representations because the operators don't commute. Inserting a position measurement into your momentum space trajectories would cause an interaction that changes the dynamics.

So I think you already know this, but the point I'm trying to make is that your trajectories don't have a special role aside from the fact that they answer all your position questions. (The position trajectories also satisfy a simple differential equation. It's not so nice for trajectories in other bases)

 

[Demystifier]

You said that the Quantum Trajectory Method does not work when you have spin. I don't understand why it doesn't. Could you point me towards a paper?

I don't know any paper claiming this, that is only my own conclusion based on my superficial understanding of QT method. It seems to me that you cannot reconstruct the 2-COMPONENT wave function from trajectories and density (involving a summation over 2 components). Do you know a paper claiming the opposite?

 

[LukeD]

Ah ok, I see the problem. Well, I will think about this and see if I can come up with a solution. I suspect that the answer might be that you need to enlarge your configuration space to include both position and spin (though the treatment of spin is certainly not trivial. a naive approach seems unlikely to work. the solution needs to be firmly grounded in Consistent Histories)

I've barely found a single paper on the Quantum Trajectory method. I've never seen a treatment of spin.

 

[Ilja]

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.

Much before there is Ockham's razor, which suggests, in my opinion, not to introduce entities like whole four-dimensional spacetime into existence without necessity, if a three-dimensional space does the same job equally well, and without any fatalistic implications.

Then, there are not too many possibilities. If you are really looking for a preferred foliation, the harmonic condition gives a very nice candidate, which gives all the properties necessary for preferred frames, namely an essential simplification for the equations of GR.

 

[Demystifier]

Ilja, your arguments certainly make sense. Yet, such a view is not without difficulties. For example, consider a spacetime with a horizon, e.g., a black hole. Is there a coordinate singularity at the horizon in harmonic coordinates? I am afraid there is, which constitutes a problem.

 

[Ilja]

... consider a spacetime with a horizon, e.g., a black hole. Is there a coordinate singularity at the horizon in harmonic coordinates? I am afraid there is, which constitutes a problem.

Let's distinguish here two cases: First, my own theory of gravity (http://ilja-schmelzer.de/gravity), which is slightly different from GR. In fact, the only reason to modify the GR equations was to obtain the harmonic condition as an Euler-Lagrange equation. But the Lagrange formalism has an inherent "action equals reaction" symmetry, so the influence of the metric on the harmonic coordinates leads to a backward influence of the harmonic coordinates on the metric. As a consequence, if the coordinates become infinite, this influence would also become infinite - and, as a consequence, it does not happen. So, in this theory the collapse stops shortly before horizon formation and the theories allows for stable gravastars slightly greater than their horizon size. So, no problem here.

But the problem is also not really a problem in GR + harmonic condition. Here, indeed, during the collapse the harmonic time becomes infinite. But now we have to interpret this. In the Lorentz interpretation, the preferred coordinate defines the true time, all what really exists is what exists now in terms of true time. Instead, GR proper time is a particular, philosophically unimportant showing of particular devices named clocks.

Now, clock time dilation can become infinite, so that the integral defining proper time τ along a path from now to infinite true time may be finite. But this is of no philosophical importance, because τ has no such importance. Simply the numbers shown by clocks never (in true time t) become greater than a given number τ_∞. Big deal.