Pilot wave theory, fundamental forces

[Frame Dragger]

Maaneli

You clearly have a strong grasp of the pilot wave theory, can you explain your understanding to a waitress? I'm concerned about time in the quantum world, not gravity. Gravity in quantum terms, if you forget the standard model that will be proven accurate but hugely misinterpreted eventually, can be easily described in quantum-relavatistic terms that are equivalent to recent experiments where blobs of oil find their way around a maze. Mass creates a potential difference in the background fabric. The mechanism is beyond current theories, but it's most certainly not any Higgs particle that imparts mass to itself. Until we accept that the background fabric is more than minowski spacetime, and that our post-enlightenment view is a barrier in terms of understanding that our physics is looking at a holographic plate from the perspective of both the surface image and the projected image. We need to connect the holographic principle with non locality in QM. We need to really understand the inside out view we have of reality where relativity says that there is no such thing as time for photons and electrons.

We think of relativity as the enemy of QM. In reality, Einstein gave us a theory that was ahead of it's time.

All fine notions, if true, but science does not embrace the assumption first and then seek to disprove. The process may be slow, and further retarded by the near-impossiblity of testing some notions (The Holographic Principle is fascinating, but there is NOTHING to show it's true yet). There is a difference between the physics and the metaphysics, and it's just that fine line (clear and bright though it is) that has had Insturmentalism and TCI as the primary working notions in QM. They work. It's implicit in many such interpretations that the reality they describe is not a perfect or absolute description, but the work is done in the math, and for that steps must be taken in sequence. If you want breakthroughs in unifying posits, conjectures and notions... the breakthrough is going to emerge from experimental evidence or a mathematical/computing breakthrough.

Barring that, this is a field of incrementalism because it works.

EDIT: Other than the unecessary comment about female waitstaff being less than crisp, why should a coherent theory of QM or SR/GR or what supercedes them be explicable to a layperson at all? I'm fairly sure that THP is beyond the average career waiter/waitress unless you use shadow-on-landscape analogies and leave it at that.

 

[Maaneli]

it does not

http://arxiv.org/PS_cache/quant-ph/pdf/0607/0607124v1.pdf

Since the existence of a time foliation would be against the spirit of relativity, several attempts have been undertaken at obtaining a relativistic Bohm-like theory without a time foliation. I briefly describe four such proposals in this subsection, items (i)–(iv) below. However, (i)–(iii) are not satisfactory theories, and (i) and (iv) both involve some foliation-like structure, something just as much against the spirit of relativity as a time foliation.

(i) Synchronized trajectories [11, 21, 56]. Define a path s 7→ X(s) in (space-time)N as the integral curve of a vector field jψ on (space-time)N, with jψ a suitably defined current vector field obtained from a wave function ψ on (space-time)N. The path
X(s) =(X1(s), . . . ,XN(s)) defines N paths in space-time, parametrized by a joint parameter s, which are supposed to be the particle world lines. This approach is based on a naive replacement of space with space-time. Apparently, it does not possesses any equivariant measure, and thus does not predict any probabilities.
Moreover, it does introduce a foliation-like structure: The joint parametrization defines a synchronization between different world lines, as it defines which point on one world line is simultaneous to a given (spacelike separated) point on a second world line. Indeed, the synchronization is encoded in the world lines since, if N non-synchronous points X1(s1), . . . ,XN(sN) on the N world lines are chosen, then the integral curve s → Y (s) of jψ starting from Y (0) =(X1(s1), . . . ,XN(sN)) will generically lead to different world lines than X.

11.-Berndl, K., Durr, D., Goldstein, S., Zangh`ı, N.: Nonlocality, Lorentz invariance, and Bohmian quantum theory. Phys. Rev.A 53: 2062–2073(1996).
21.-Dewdney, C., Horton, G.: A Non-Local, Lorentz-Invariant, Hidden-Variable Interpretation of Relativistic Quantum Mechanics Based on Particle Trajectories. J. Phys. A: Math. Gen. 34: 9871–9878 (2001).
56.-Nikolic, H.: Relativistic Quantum Mechanics and the Bohmian Interpretation. Foundations of Physics Letters 18: 549–561 (2005).

Good timing, I was just about to cite this paper and this exact section, as this synchronized trajectories approach to Bohm-Dirac theory is exactly what Demystifier proposes in equations (17)-(19) in his paper.

So, Demystifier, you asked if I see the need for a preferred foliation in equations (17)-(19), and my answer would be yes, because of the reasoning given above by Tumulka.

Another problematic issue for the synchronized trajectories approach seems to be the statement that it "apparently does not possesses an equivariant measure". I'm sure Demystifier will disagree with that, so perhaps it's best to go straight to the argument given in the 1996 Berndl et. al paper, for why this is so:

-Berndl, K., Durr, D., Goldstein, S., Zangh`ı, N.: Nonlocality, Lorentz invariance, and Bohmian quantum theory. Phys. Rev.A 53: 2062–2073(1996).
http://arxiv.org/abs/quant-ph/9510027

They argue (see section 4) that the reparameterization invariant Dirac current velocity (see equation 32), is not Lorentz invariant because it is not of the form J_k/rho for more than one particle. Thus, equivariance does not hold in any obvious way for a multi-particle, multi-time Bohm-Dirac theory.

On the other hand, Demystifier seems to suggest that equivariance does hold by writing the multi-time, multi-particle Dirac wavefunction in polar form, assuming that the polar decomposition of the N-particle multi-time Dirac equation goes through, and then looking at the relativistic continuity equation for the multi-time, multi-particle Dirac four-current (which seems to imply equivariance). It is unclear to me which argument is correct, and hopefully Demystifier will address Berndl et. al's point.

 

[Maaneli]

Maaneli

You clearly have a strong grasp of the pilot wave theory, can you explain your understanding to a waitress? I'm concerned about time in the quantum world, not gravity. Gravity in quantum terms, if you forget the standard model that will be proven accurate but hugely misinterpreted eventually, can be easily described in quantum-relavatistic terms that are equivalent to recent experiments where blobs of oil find their way around a maze. Mass creates a potential difference in the background fabric. The mechanism is beyond current theories, but it's most certainly not any Higgs particle that imparts mass to itself. Until we accept that the background fabric is more than minowski spacetime, and that our post-enlightenment view is a barrier in terms of understanding that our physics is looking at a holographic plate from the perspective of both the surface image and the projected image, we will hide under Bohr's clever arguments. We need to connect the holographic principle with non locality in QM. We need to really understand the inside out view we have of reality where relativity says that there is no such thing as time for photons and electrons.

We think of relativity as the enemy of QM. In reality, Einstein gave us a theory that was ahead of it's time.

Not sure why I was singled out, but I basically agree with Frame Dragger's response to your waitress question.

 

[Demystifier]

Yoda jedi and Maaneli,

The question of equivariant probability density is indeed the crucial question. My answer to this question is better explained in my second paper
http://xxx.lanl.gov/abs/0904.2287 [to appear in Int. J. Mod. Phys. A]
Appendix B.
The point is the following. There is no equivariance in the sense of Eq. (127). However, there IS equivariance in the sense of Eq. (125).
Berndl et al consider only the equivariance of the form of (127) [actually generalized to the case of many particles] and do not consider the equivariance of the form of (125) [which can also be generalized to the case of many particles]. Therefore, their conclusion that there is no equivariance has only a partial validity. The crucial difference between (127) and (125) is that the latter treats time and space on an equal footing (which is very relativistic in spirit), while the former does not treat time and space on an equal footing.

To conclude, my claim is that time should be treated on an equal footing with space, and that this, among other things, solves the problem of equivariance.

 

[Demystifier]

i
Moreover, it does introduce a foliation-like structure:

Yes, but it does not introduce a PREFERRED foliation-like structure. Instead, such a structure is determined dynamically, through the choice of initial conditions.

It is analogous to the fact that a planet also defines a particular Lorentz frames (the one with respect to which it is at rest), but it does not mean that classical laws of physics describing the motion of the planet are not relativistic covariant.

 

[Demystifier]

On the other hand, Demystifier seems to suggest that equivariance does hold by writing the multi-time, multi-particle Dirac wavefunction in polar form, assuming that the polar decomposition of the N-particle multi-time Dirac equation goes through, and then looking at the relativistic continuity equation for the multi-time, multi-particle Dirac four-current (which seems to imply equivariance).

That's not what I suggest at all. I do not use the Dirac current. Not even for spin-1/2 particles. (See Sec. 3.4 and Appendix A of the second paper.)

 

[Maaneli]

That's not what I suggest at all. I do not use the Dirac current. Not even for spin-1/2 particles. (See Sec. 3.4 and Appendix A of the second paper.)

I think that may be a semantic misunderstanding, because all I meant is that you use a four-current whose divergence is equal to zero, namely, the relativistic continuity equation obtained from the Dirac equation under polar decomposition (equation 17 of the first paper).

 

[Maaneli]

Yes, but it does not introduce a PREFERRED foliation-like structure. Instead, such a structure is determined dynamically, through the choice of initial conditions.

It is analogous to the fact that a planet also defines a particular Lorentz frames (the one with respect to which it is at rest), but it does not mean that classical laws of physics describing the motion of the planet are not relativistic covariant.

But aren't you introducing an absolute simultaneity surface (a hypersurface across which all the particle positions are simultaneously defined, even at spacelike separations), by virtue of the fact that you have to synchronize the initial positions of the particles at a common time s, and that this synchronization has to hold for all future s, even when they are spacelike separated? And isn't that simultaneity surface unique?

Also, the issue (in my view at least) is not whether the equations of motion are relativistically covariant, but whether the spacetime structure introduced is consistent with "fundamental Lorentz invariance" (which I take to mean the constraints on dynamics imposed by the causal structure of Minkowski spacetime). And a simultaneity surface that introduces spacelike causal influences does not seem to me to be consistent with fundamental Lorentz invariance in that sense.

Also, when considering the possibility of nonequilibrium particle distributions in the multi-time Bohm-Dirac theory (assuming also for the moment that such a theory is in fact equivariant), I don't see anything in the synchronized trajectories approach that stops it from allowing superluminal signaling, as Valentini has demonstrated is possible with nonequilibrium particle distributions; and superluminal signaling is the most explicit violation of fundamental Lorentz invariance that I can possibly think of.

 

[Demystifier]

I think that may be a semantic misunderstanding, because all I meant is that you use a four-current whose divergence is equal to zero, namely, the relativistic continuity equation obtained from the Dirac equation under polar decomposition (equation 17 of the first paper).

The semantic misunderstanding must be more than that, because in that paper I do not use Dirac equation, but Klein-Gordon equation.

Besides, not also that, even though I use a four-current whose divergence is equal to zero, that's not exactly the content of Eq. (17). Instead, Eq. (17) is a combination of TWO facts, one that the divergence of the four-current is zero, and the other that psi does not depend on s. In other words, the left-hand side of (3) contains two zeros, i.e., (3) is a consequence of the trivial fact that 0+0=0.

Anyway, that's not essential. The essential stuff is presented in post #64.

 

[Maaneli]

The semantic misunderstanding must be more than that, because in that paper I do not use Dirac equation, but Klein-Gordon equation.

Crap, you're right! I must have confused the equations in your paper with those of Berndl and co.. My mistake.

 

[Demystifier]

But aren't you introducing an absolute simultaneity surface
(a hypersurface across which all the particle positions are simultaneously defined, even at spacelike separations),

No, because n points (for each value of s) do not define u surface.

by virtue of the fact that you have to synchronize the initial positions of the particles at a common time s,

The parameter s is not time.

and that this synchronization has to hold for all future s,

The synchronization changes with s.

And isn't that simultaneity surface unique?

No, for two reasons. First, because n points do not define a surface uniquely. Second, because even these n points depend on the initial conditions at s=0.

Also, the issue (in my view at least) is not whether the equations of motion are relativistically covariant, but whether the spacetime structure introduced is consistent with "fundamental Lorentz invariance" (which I take to mean the constraints on dynamics imposed by the causal structure of Minkowski spacetime).

"Fundamental Lorentz invariance" is not the same as constraints on dynamics imposed by the causal structure of Minkowski spacetime. At least, your terminology is not standard.

Also, when considering the possibility of nonequilibrium particle distributions in the multi-time Bohm-Dirac theory (assuming also for the moment that such a theory is in fact equivariant), I don't see anything in the synchronized trajectories approach that stops it from allowing superluminal signaling, as Valentini has demonstrated is possible with nonequilibrium particle distributions;

With that I agree.

and superluminal signaling is the most explicit violation of fundamental Lorentz invariance that I can possibly think of.

As I already explained, here you are using a non-standard terminology. Lorentz invariance is the principle that the laws of physics do not depend on the choice of the Lorentz frame of coordinates. That's all. Superluminal signaling is consistent with Lorentz invariance.

See again the first Objection and Response in the attachment of post #109 in
https://www.physicsforums.com/showthread.php?t=354083

 

[yoda jedi]

Yes, but it does not introduce a PREFERRED foliation-like structure.

agreed.

(never stated "preferred", anyway)

I've read the first paper before, and I liked it very much. But I still don't understand how you've managed to get around the need for a preferred frame "or" spacetime foliation, in your effort to construct a fundamentally Lorentz invariant deBB dynamics.

just juxtaposed by you.

 

[Maaneli]

No, because n points (for each value of s) do not define u surface.

But the parameter s must be a universal value for the world lines of the n points to be synchronized. And if that is the case, then for any instant of s, there exists a spacelike simultaneity hypersurface across which the synchronized particles co-determine each others velocities.

The parameter s is not time.

But as a "joint parameter", it plays precisely the role of a universal time parameter for the evolution of the particle spacetime coordinates. Yes, I realize that the wavefunction on configuration spacetime doesn't depend on s, but that doesn't mean that s cannot also be interpreted as a time parameter (even if it is a fictitious one).

As I already explained, here you are using a non-standard terminology. Lorentz invariance is the principle that the laws of physics do not depend on the choice of the Lorentz frame of coordinates.

Lorentz invariance is already implied by constraining particle dynamics with the causal structure of Minkowski spacetime. As for my criterion for "fundamental Lorentz invariance" being non-standard, maybe so, but I know other researchers in the field (Valentini, Towler, and Tumulka) who I think would agree with it. Moreover, there exist relativistic pilot-wave theories that only make use of the causal structure of Minkowski spacetime in defining the particle dynamics, and are regarded as concrete examples of pilot-wave theories that achieve fundamental Lorentz invariance. Examples include Euan Squires' local light-cone synchronization model, and the nonlocal model by Tumulka and Goldstein which makes use of opposite arrows of time on the light-cones of N-particles.

Superluminal signaling is consistent with Lorentz invariance.

See again the first Objection and Response in the attachment of post #109 in
https://www.physicsforums.com/showthread.php?t=354083

Superluminal signaling by tachyons (which you discuss in that O an R paper as an example of why superluminal signaling is compatible with special relativity) is indeed consistent with Lorentz invariance, but only because tachyons are specifically predicted by the equations of special relativity. By contrast, the possibility of nonequilibrium particle distributions, and the possible superluminal signaling that results from such distributions, is not specifically predicted by the equations of special relativity, nor does it have any relation to tachyons. Moreover, unlike tachyons, superluminal signaling by quantum nonequilibrium violates the causal structure of Minkowski spacetime.

Also, you mentioned something in that O and R paper about superluminal signaling in a QFT being undetectable at the macroscopic classical level, because quantum correlations are destroyed by decoherence. But any pilot-wave theory with a preferred frame or a foliation-like structure (including the synchronized trajectories approach) permits the possibility of the superluminal signaling from nonequilibrium matter distributions being detectable at the macroscopic classical level.

 

[Maaneli]

Yoda jedi and Maaneli,

The question of equivariant probability density is indeed the crucial question. My answer to this question is better explained in my second paper
http://xxx.lanl.gov/abs/0904.2287 [to appear in Int. J. Mod. Phys. A]
Appendix B.
The point is the following. There is no equivariance in the sense of Eq. (127). However, there IS equivariance in the sense of Eq. (125).
Berndl et al consider only the equivariance of the form of (127) [actually generalized to the case of many particles] and do not consider the equivariance of the form of (125) [which can also be generalized to the case of many particles]. Therefore, their conclusion that there is no equivariance has only a partial validity. The crucial difference between (127) and (125) is that the latter treats time and space on an equal footing (which is very relativistic in spirit), while the former does not treat time and space on an equal footing.

To conclude, my claim is that time should be treated on an equal footing with space, and that this, among other things, solves the problem of equivariance.

Seems reasonable, except that Berndl et al also consider the velocity equation (33)/(34), which is equivalent in form to (123) in your paper, and has an associated probability density that does treat time and space on equal footing. They also conclude that (33)/(34) is statistically transparent, and that they will elaborate on this in a future paper (but unfortunately they never did).

 

[Demystifier]

But the parameter s must be a universal value for the world lines of the n points to be synchronized. And if that is the case, then for any instant of s, there exists a spacelike simultaneity hypersurface across which the synchronized particles co-determine each others velocities.

That is true. However, my point is that there is an INFINITE number of such hypersurfaces. Neither of them is preferred. Moreover, you don't need such a hypersurface at all to calculate the trajectories or anything else.

But as a "joint parameter", it plays precisely the role of a universal time parameter for the evolution of the particle spacetime coordinates. Yes, I realize that the wavefunction on configuration spacetime doesn't depend on s, but that doesn't mean that s cannot also be interpreted as a time parameter (even if it is a fictitious one).

You are right. The parameter s can be interpreted as a sort of time. However, this is more like Newton absolute time, note like Einstein relativistic time. That's what I meant when I said that "s is not time". Note also that the separation between different points on a single Bohmian trajectory may be spacelike in some cases, which is another reason why it may be misleading to call it "time".

Lorentz invariance is already implied by constraining particle dynamics with the causal structure of Minkowski spacetime.

I would say that Lorentz invariance is a necessary but not sufficient assumption to constrain particle dynamics with the causal structure of Minkowski spacetime. Would you agree with that?

By contrast, the possibility of nonequilibrium particle distributions, and the possible superluminal signaling that results from such distributions, is not specifically predicted by the equations of special relativity, nor does it have any relation to tachyons. Moreover, unlike tachyons, superluminal signaling by quantum nonequilibrium violates the causal structure of Minkowski spacetime.

Well, all my discussion is (tacitly) restricted to the case of quantum equilibrium.

Also, you mentioned something in that O and R paper about superluminal signaling in a QFT being undetectable at the macroscopic classical level, because quantum correlations are destroyed by decoherence. But any pilot-wave theory with a preferred frame or a foliation-like structure (including the synchronized trajectories approach) permits the possibility of the superluminal signaling from nonequilibrium matter distributions being detectable at the macroscopic classical level.

True, but as I said, all my discussion is (tacitly) restricted to the case of quantum equilibrium.

 

[Demystifier]

Seems reasonable,

I'm glad that you think so, because it is the most important part of my idea.

except that Berndl et al also consider the velocity equation (33)/(34), which is equivalent in form to (123) in your paper, and has an associated probability density that does treat time and space on equal footing.

I wouldn't say it is really equivalent. The crucial difference is that their v^0 is positive, while my v^0 does not need to be.

They also conclude that (33)/(34) is statistically transparent, and that they will elaborate on this in a future paper (but unfortunately they never did).

Their statistical transparency is a consequence of the crucial difference above. The problem with it is that their (33)/(34) do not work for bosons.

 

[Maaneli]

I would say that Lorentz invariance is a necessary but not sufficient assumption to constrain particle dynamics with the causal structure of Minkowski spacetime. Would you agree with that?

Well, I would say it depends on what precisely you mean by Lorentz invariance. If you mean *fundamental* Lorentz invariance, then I would say it depends on what you mean by fundamental Lorentz invariance. And it seems that you and I have different notions of what fundamental Lorentz invariance could mean exactly. Perhaps this is not so surprising, as Berndl et al. point out that it is notoriously difficult to make the notion of fundamental Lorentz invariance precise. Nevertheless, I will reiterate that fundamental Lorentz invariance (in my view) is the Lorentz invariance implied by the causal structure of Minkowski spacetime in classical Einstein-Minkowski special relativity, and not simply invariance of the equations of motion under coordinate transformations. (After all, when it is asked whether deBB theory is fundamentally compatible with "special relativity", the latter is generally implied to mean the classical Einstein-Minkowski formulation of special relativity, where Lorentz invariance of the equations of motion is a consequence of the structure of Minkowski spacetime, rather than some independent constraint on the equations of motion). And any quantum theory which is fundamentally Lorentz invariant must, by this definition, be consistent with and keep the causal structure of Minkowski spacetime unmodified and unappended (e.g. Sutherland's causally symmetric Bohm model). Hence, any quantum theory which modifies or appends the causal structure of Minkowski spacetime (whether by a preferred foliation or a foliation-like synchronization parameter) is not fundamentally Lorentz invariant.

By the way, after our first set of exchanges, I found a paper by Tim Maudlin in "Bohmian Mechanics and Quantum Theory: An Appraisal" where he essentially shares and defends my view on the meaning of fundamental Lorentz invariance. It is entitled, "Space-Time in the Quantum World":

http://books.google.com/books?id=EF...&resnum=4&ved=0CBoQ6AEwAw#v=onepage&q&f=false

Maudlin's views in that paper are encapsulated in the following statements:

In saying that we ought to frame the [relativistic constraint in terms of space-time structure, I also mean to rule out formulations based on coordinate transformations. This point is often overlooked because inconsistency with Special Relativistic space-time structure (i.e. Minkowski space-time) turns out to be equivalent to invariance under the Lorentz transformations.

and

So Lorentz invariance turns out to be a roundabout route to a more fundamental property: the essential fact about Lorentz invariant theories is that their dynamics depend only on the Special Relativistic metrical structure.

Well, all my discussion is (tacitly) restricted to the case of quantum equilibrium.

In that case, it will be very interesting to see how superluminal signaling and all that jazz will work if you allow for nonequilibrium distributions. You mentioned that in your theory, there are an infinite number of spacelike simultaneity hypersurfaces across which the synchronized particles co-determine each others velocities. So will the superluminal signaling occur along all of those hypersurfaces? And would you still regard superluminal signaling in your theory as consistent with your definition of fundamental Lorentz invariance? If not, would you then assert that your theory is only fundamentally Lorentz invariant (by your definition) in the special case of quantum equilibrium?

 

[Demystifier]

Maaneli, you are correct that I use a different definition of "fundamental" Lorentz invariance than you (and Maudlin) do. Let me refer to this (your and Maudlin's) definition as CAUSAL Lorentz invariance.

You are also correct that my theory is not causal Lorentz invariant.

However, my point is that I do not see any particular motivation for retaining causal Lorentz invariance. For me, the only reason why I want Lorentz invariance is SYMMETRY, so for me the covariance with respect to coordinate transformations and the absence of a preferred frame is enough. Let me refer to it as SYMMETRY Lorentz invariance.

Let us also not use the unfair and vague expression "fundamental" Lorentz invariance any more.

Now let me answer you questions, having the definitions above in mind.

"So will the superluminal signaling occur along all of those hypersurfaces?"
- Yes.

"And would you still regard superluminal signaling in your theory as consistent with your definition of fundamental Lorentz invariance?"
- I would regard it consistent with symmetry Lorentz invariance.

 

[Demystifier]

So Lorentz invariance turns out to be a roundabout route to a more fundamental property: the essential fact about Lorentz invariant theories is that their dynamics depend only on the Special Relativistic metrical structure.

This is a nonsense. No dynamical theory in physics depends ONLY on the Special Relativistic metrical structure. They all depend on something additional as well. Perhaps he meant something else here, but I cannot figure out what.

Consider, for example, electromagnetic field. It certainly depends on something which is not the principle of Special Relativistic metrical structure. Let us also consider a particular solution of Maxwell equations. For a particular solution, there may exist a particular Lorentz frame in which only electric field is nonzero, while magnetic field is zero. Would you say that this means that Maxwell theory is not "fundamentally" Lorentz invariant? I hope you would not.

 

[Maaneli]

Consider, for example, electromagnetic field. It certainly depends on something which is not the principle of Special Relativistic metrical structure. Let us also consider a particular solution of Maxwell equations. For a particular solution, there may exist a particular Lorentz frame in which only electric field is nonzero, while magnetic field is zero.

Yeah, the specification of initial and boundary conditions on the field. But even those conditions are constrained by the SR metrical structure.

 

[Demystifier]

Yeah, the specification of initial and boundary conditions on the field. But even those conditions are constrained by the SR metrical structure.

The ONLY constraint (by the SR metrical structure) on initial conditions (in classical electrodynamics) is that the conditions must be specified on a hypersurface which is SPACELIKE.

By contrast, in the relativistic covariant (RC) version of Bohmian mechanics (BM), there is no such constraint. But in a sense, the absence of such a constraint makes the theory even "more relativistic", in the sense that the difference between space and time is even "more relative", or more precisely that time is treated on an equal footing with space.

But perhaps the best way to explain in what sense RC BM is Lorentz invariant is through analogy with the textbook nonrelativistic BM. Namely, nonrelativistic BM is invariant with respect to rotations in 3-dimensional space (group SO(3)). Whatever you call this form of invariance ("fundamental", "non-fundamental", "symmetry", "covariance", ... whatever) in nonrelativistic BM, RC BM is invariant with respect to Lorentz transformations in 4-dimensional spacetime (group SO(1,3)) IN EXACTLY THE SAME SENSE. In other words, you have the following symmetries:
- nonrelativistic BM: SO(3)
- RC BM: SO(1,3)
and this is essentially THE ONLY difference between nonrelativistic BM and RC BM. You obtain RC BM from nonrelativistic BM by a replacement SO(3) -> SO(1,3).

 

[Demystifier]

Also a comment on the Valentini's idea that nonequilibrium BM may be used for superluminal signalling. Recently I have found a way to use nonlocal correlations for superluminal signalling EVEN IN EQUILIBRIUM:
http://xxx.lanl.gov/abs/1006.0338
And it works equally well in the Bohmian and the many-world interpretation.

I would like to see your opinion.

 

[Demystifier]

To further clarify the issue of relativistic BM, let me make some additional (possibly summarizing) remarks.

You start from relativistic causality as the starting requirement. From this, it follows that
1. Lorentz covariance can be derived from it.
2. Time is NOT treated on an equal footing with space.

On the other hand, I start from SO(1,3) symmetry group for spacetime as the starting requirement. From this, it follows that
1. Lorentz covariance can be derived from it.
2. Time SHOULD be treated on an equal footing with space if no further axioms are introduced.
3. Relativistic causality does not need to be obeyed.

So in essence, you insist on relativistic causality, which is inconsistent with treating time on an equal footing with space. By contrast, I insist on treating time on an equal footing with space, which, in general, is inconsistent with relativistic causality. One cannot have both, so one must decide. In BM, only the second option seems possible.

Calling one of the two approaches "fundamental" does not help.

 

[Maaneli]

The ONLY constraint (by the SR metrical structure) on initial conditions (in classical electrodynamics) is that the conditions must be specified on a hypersurface which is SPACELIKE.

I don't see why this is inconsistent with Maudlin's statement. But perhaps it helps to elaborate on what Maudlin said before and after that statement of his:

Minkowski space-time is homogeneous and, with respect to all time-like (or spacelike) directions, isotropic. Like Euclidean space, Minkowski space-time admits of global rectilinear orthogonal coordinate systems. Since the space-time is homogeneous, the space-time structure itself looks the same when expressed in a coordinate dependent form relative to rectilinear orthogonal coordinate systems whose origins are shifted with respect to one another. And since it is isotropic (in the time-like directions), the space-time structure also is the same when expressed in a coordinate dependent form relative to rectilinear orthogonal coordinate systems whose time axes are rotated with respect to one another. (And similarly for rotations of the spatial axes.) This means that the metric, expressed in terms of the coordinates, takes the same functional form for all Lorentz frames, and hence is invariant under the Lorentz transformations. This invariance is a consequence of the global symmetries of the metrical structure of Minkowski space-time. Any theory which is invariant under those same transformations displays the same symmetries, and so does not postulate any new space-time structure. So Lorentz invariance turns out to be a roundabout route to a more fundamental property: the essential fact about Lorentz invariant theories is that their dynamics depend only on the Special Relativistic metrical structure. And once put this way, all reference to coordinate systems and coordinate transformations may be dropped. Given, for example, a coordinate-free formulation of a theory, we may ask whether it postulates only the relativistic space-time structure or whether it posits more.

the difference between space and time is even "more relative", or more precisely that time is treated on an equal footing with space.

Can you explain again what exactly you mean by 'treating time on an equal footing with space'? Do you just mean treating time as another spatial coordinate (such as in the 4-vector), and associating a linear operator with it?

But perhaps the best way to explain in what sense RC BM is Lorentz invariant is through analogy with the textbook nonrelativistic BM. Namely, nonrelativistic BM is invariant with respect to rotations in 3-dimensional space (group SO(3)). Whatever you call this form of invariance ("fundamental", "non-fundamental", "symmetry", "covariance", ... whatever) in nonrelativistic BM, RC BM is invariant with respect to Lorentz transformations in 4-dimensional spacetime (group SO(1,3)) IN EXACTLY THE SAME SENSE. In other words, you have the following symmetries:
- nonrelativistic BM: SO(3)
- RC BM: SO(1,3)
and this is essentially THE ONLY difference between nonrelativistic BM and RC BM. You obtain RC BM from nonrelativistic BM by a replacement SO(3) -> SO(1,3).

If by "textbook [is there even a textbook version?] nonrelativistic BM" you mean the first-order pilot-wave dynamics, then yes, its dynamics is invariant under rotations in Euclidean 3-space. But perhaps it is also worth recognizing that this invariance is a consequence of the theory's natural kinematics being actually Aristotelian, rather than Galilean. And this difference, one could argue, makes it unnatural to force Lorentz invariance onto the theory.

 

[Maaneli]

Also a comment on the Valentini's idea that nonequilibrium BM may be used for superluminal signalling. Recently I have found a way to use nonlocal correlations for superluminal signalling EVEN IN EQUILIBRIUM:
http://xxx.lanl.gov/abs/1006.0338
And it works equally well in the Bohmian and the many-world interpretation.

I would like to see your opinion.

Sounds eyebrow raising. But unfortunately, I can't read it because the PDF link is not working for me.

 

[Maaneli]

However, my point is that I do not see any particular motivation for retaining causal Lorentz invariance. For me, the only reason why I want Lorentz invariance is SYMMETRY, so for me the covariance with respect to coordinate transformations and the absence of a preferred frame is enough.

Well, the advantages of retaining causal Lorentz invariance, and the potential problems with symmetry Lorentz covariance are pointed out by Maudlin:

The advantage of a coordinate-free formulation of compatibility with Relativity is twofold. First, it is immediately extendible to General Relativity. One can ask whether the dynamics of a theory postulate more than the metrical structure of a General Relativistic space-time just as one can ask for Minkowski space-time. Formulations in terms of invariance of coordinate based theories under transformations between rectilinear orthogonal reference systems cannot be extended to the General Relativistic context, where no such reference frames exist. Lacking the global symmetries of the Special Relativistic metric, no invariance can guarantee compatibility with relativistic space-time.

But the deeper advantage of the elimination of criteria based on coordinate transformations is simply that coordinate systems do not, in any deep sense, exist. My room may be fully of air molecules and electromagnetic fields, but is not criss-crossed by coordinate curves. Insofar as coordinate systems can be given any physical significance, it is either directly by appeal to the space-time structure (as one can define rectilinearity and orthogonality, and hence rectilinear orthogonal systems by appeal to the metrical structure) or else by appeal to the (possible or actual) results of assigning numbers to space-time points by using, for example, clocks and rods.

Let us also not use the unfair and vague expression "fundamental" Lorentz invariance any more.

OK.

"So will the superluminal signaling occur along all of those hypersurfaces?"
- Yes.

This is odd. Are you saying that superluminal signaling occurs *simultaneously* along all those hypersurfaces? Or just that the hypersurface along which the signaling occurs is frame-dependent, and that signaling can be observed along anyone of the hypersurfaces, given the appropriate reference frame? If the former, then it sounds like superluminal signaling from nonequilibrium could provide an operational definition of absolute simultaneity for ALL reference frames (which of course contradicts the relativity of simultaneity). And if the latter, then I think your theory could run into certain ontological paradoxes like those found by Maudlin in Fleming's hyperplane-dependent relativistic QM, where photons can have a certain polarization with respect to one reference frame, but no such polarization with respect to another reference frame.

 

[Demystifier]

So Lorentz invariance turns out to be a roundabout route to a more fundamental property: the essential fact about Lorentz invariant theories is that their dynamics depend only on the Special Relativistic metrical structure. And once put this way, all reference to coordinate systems and coordinate transformations may be dropped. Given, for example, a coordinate-free formulation of a theory, we may ask whether it postulates only the relativistic space-time structure or whether it posits more. [/B][/I]

Good! Because the covariant BM I am talking about depends only on the special relativistic metrical structure (except, of course, for the initial conditions) and can be written in a coordinate-free formulation. (If you want me to explicitly write it this way, I will do it for you after you write for me the nonrelativistic BM in 3-space-coordinate-free formulation.)

Can you explain again what exactly you mean by 'treating time on an equal footing with space'?

I have explained it elsewhere. See
http://xxx.lanl.gov/abs/1002.3226
page 5, item 2)

But perhaps it is also worth recognizing that this invariance is a consequence of the theory's natural kinematics being actually Aristotelian, rather than Galilean. And this difference, one could argue, makes it unnatural to force Lorentz invariance onto the theory.

Being natural or not, I claim that it is possible.

 

[Demystifier]

Sounds eyebrow raising. But unfortunately, I can't read it because the PDF link is not working for me.

Write arXiv:1006.0338 in Google!

 

[Demystifier]

Are you saying that superluminal signaling occurs *simultaneously* along all those hypersurfaces? Or just that the hypersurface along which the signaling occurs is frame-dependent, and that signaling can be observed along anyone of the hypersurfaces, given the appropriate reference frame?

I think the best answer is - neither. I am saying that hypersurfaces are completely irrelevant objects here that play no role in the formulation of the theory. I have already explained it to you few months ago in this thread. Superluminal signalling does not occur along hypersurfaces. It occurs between pointlike particles. It is much easier to understand all this if you look at the EQUATIONS that define the theory and try to figure out by yourself what these equations really mean. Then it will become clear to you why it is more confusing than useful to think in terms of hypersurfaces.

 

[Demystifier]

Or let me use an analogy with nonrelativistic BM. A point is space is denoted as r=(x,y,z). Consider two particles with space positions {\bf r}_1 and {\bf r}_2 at a given time t. There is a Cartesian frame (given by a rotation of the original Cartesian frame) in which {\bf r}_1 and {\bf r}_2 have the same value of z. In this frame, we say that the interaction between these two particles is z-taneous. Does it lead to any paradoxes? Does it mean there is a preferred z-coordinate? Does it mean there is a preferred foliation of space into 2-surfaces? Whatever your answer is, the same answer applies to analogous questions in relativistic-covariant BM. And if you still don't get it, then look at the equations of relativistic-covariant BM again.