Pilot wave theory, fundamental forces

[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.

 

[Maaneli]

Good! Because the covariant BM I am talking about depends only on the special relativistic metrical structure (except, of course, for the initial conditions)

Well I'm not so sure that it depends only on the special relativistic metrical structure. You have to use a foliation-like structure, namely, a synchronization parameter, to preserve the Lorentz covariance of the particle dynamics. And this synchronization parameter is something additional to the SR metrical structure, rather than something naturally implied by the SR metrical structure.

... 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.)

Unless you have some specific point to make (in which case, please just be explicit about it), I don't understand why you're asking me to write down said nonrelaivistic deBB formulation. The discussion here is about relativistic deBB theories. And I am not the one claiming to have a formulation of deBB theory which is compatible with SR. I am simply pointing out a condition that I think any such alleged theory should satisfy. Namely, I share the view of Maudlin that

A theory is compatible with Relativity if it can be formulated without ascribing to space-time any more or different intrinsic structure than the (special or general) relativistic metric.

On the other hand, you have a different view, in which you reject the idea of using only the intrinsic structure of the (special or general) relativistic metric, in order to claim that a (deBB) theory is compatible with Relativity. And you claim to have a covariant deBB theory which you say you can write in a coordinate-free formulation, and which thus shares the advantages of a coordinate-free formulation of SR (as characterized by Maudlin). Fine. Then show us how you do it, and show us that it is consistent with general covariance. That, I think, would significantly help the plausibility of your theory.

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

Thanks, that's what I thought it meant. And in that case, I would argue that, contrary to your summary point #2 in your post #83, a theory which is 'causally Lorentz invariant' does indeed treat space and time on equal footing. An example of such a theory is this:

Two Arrows of Time in Nonlocal Particle Dynamics
Authors: Roderich Tumulka
http://lanl.arxiv.org/abs/quant-ph/0210207

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

Yes, but lots of things are possible in physics. What's important, IMHO, is how plausibly you can motivate the reasons for retaining symmetry Lorentz invariance.

 

[Maaneli]

Write arXiv:1006.0338 in Google!

No worries, I acquired a copy shortly after I wrote that. I'll get back to you on it when I have time to read it.

 

[Maaneli]

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.

Now hang on - In an earlier post of yours, you replied to my question "So will the superluminal signaling occur along all of those hypersurfaces?" with an unqualified *yes*. Did you just not read the question I asked before you answered it?

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.

OK, if I look at the equations, particularly the trajectory equation for X_a(s), it would seem that I could draw those trajectories on a spacetime-like diagram, where the universal parameter s is on the y-axis and the particle position 4-vector X is on the x-axis. For an N particle system, X_a, it would seem that I could then draw a unique simultaneity hypersurface across the particle velocities are instantaneously co-determined, just as one can do so in the standard Bohm-Dirac theory where the universal time t is analogous to s, and the position 3-vector x_a is analogous to X_a. But I guess you would say that such a spacetime-like diagram is only fictitious, and that the real particle dynamics is seen in spacetime where there is no such hypersurface. OK, so now the fact that superluminal signaling occurs in spacetime between pointlike particles would seem to violate the relativity of simultaneity (in the sense that if I make a standard quantum measurement of the spin orientation of a particle A which is entangled with a space-like separated particle B, the latter of whose spin orientation is being continuously monitored with a nonequilibrium pointer, I will see an instantaneous change in the spin orientation of B, and this will look the same regardless of what Lorentz frames I choose for A and B). So in that sense, (and correct me if you think I am wrong) it would seem that your theory is not compatible with all the postulates of special relativity, when you allow for nonequilibrium distributions.

 

[Demystifier]

Well I'm not so sure that it depends only on the special relativistic metrical structure. You have to use a foliation-like structure, namely, a synchronization parameter, to preserve the Lorentz covariance of the particle dynamics. And this synchronization parameter is something additional to the SR metrical structure, rather than something naturally implied by the SR metrical structure.

No, there is no foliation-like structure. The synchronization parameter is NOT something additional to the SR metrical structure, just as time in nonrelativistic BM is NOT something additional to the 3-space rotational-symmetry structure. Read my post #90. There is no much point is answering your other questions before you understand this. I am convinced that, when you understand this, you will withdraw most of your other questions.

Unless you have some specific point to make (in which case, please just be explicit about it), I don't understand why you're asking me to write down said nonrelaivistic deBB formulation. The discussion here is about relativistic deBB theories. And I am not the one claiming to have a formulation of deBB theory which is compatible with SR.

You just don't get it. My point is that relativistic-covariant BM in 4-dimensional spacetime is ANALOGOUS to nonrelativistic BM in 3-dimensional space. I am just trying to make you understand this ANALOGY, because when you do, you will suddenly say: "Oh, THAT is what you meant. Now I get it. In fact, it is trivial." But it is essential that you see this analogy by yourself, while I can only guide you in the right direction. And at the moment, it seems to me that you don't have a clue what I am talking about, because you are not able to see the analogy. And that is probably because you are unable to think of time as just another "space" coordinate.

To help you think in the correct way, let me suggest you a mental trick. For a moment, FORGET that the spacetime metric has the form (+---). Instead , think of metric as just any metric, which can be (++++), (++--), or whatever. In fact, simply don't think about metric at all. Just pretend that you have a 4-dimensional space with some unspecified metric. Or if it is easier for you, just pretend that the metric is (++++). And forget that one of the coordinates is called "time". (Who cares about names, anyway?) And NOW try to understand again what equations of relativistic-covariant BM are actually saying. This trick works for many physicists, so it could work for you as well.

 

[Demystifier]

One additional way to guide your thinking. Don't think about special relativity as Einstein did in 1905. Think about special relativity as Minkowski did few years later. When Minkowski discovered the spacetime view of the Einstein special theory of relativity, Einstein was not able to see much sense in it. It took a lot of time before Einstein understood the advantage of the Minkowski spacetime view. But when he finally did, it open the door for discovering general relativity. Without Minkowski view, Einstein would never discover general relativity.

Likewise, it is impossible to understand relativistic-covariant BM using only Einstein 1905 view of special relativity. Time is the 4-th dimension, and it is Minkowski, not Einstein, who first understood it. Without FULLY appreciating the point that time is (almost) nothing but the 4-th dimension, it is impossible to fully understand relativistic-covariant BM. If you say "Yes, I know that time is the 4-th dimension, but still time is not really the same as space." - then you probably don't get it yet. Try the mental trick in my previous post above.

 

[fundamentally]

If you want to understand the source of relativity, then you must understand the quaternion waltz c = ab/a. With real's and complex numbers c equals b, but with quaternions and octonions the imaginary part of b is affected. In quaternionic Hilbert space the combination of a unitary transform (that moves the state around in Hilbert space) and an observation of position always involves a quaternion waltz! So it must be fundamental to physics. When you analyse the effects then you will discover the source of relativity!

 

[Demystifier]

So in that sense, (and correct me if you think I am wrong) it would seem that your theory is not compatible with all the postulates of special relativity, when you allow for nonequilibrium distributions.

Maybe it is not compatible with all postulates of the original 1905 Einstein special theory of relativity. But I don't care much about the 1905 formulation, as long as I have a formulation which I find much better, such as Minkowski formulation I mentioned in the post above. The theory of relativity neither started nor ended with Einstein.

For a difference between different views of relativity, see also the Mike Towler lectures on deBB.

 

[fundamentally]

You can specify an infinite dimensional separatable Hilbert space over the real's, the complex numbers, the quaternions and according to Horwitz with some trouble also over the octonions. Quantum logical propositions can be represented in the closed subspaces of such a Hilbert space. It is an enlightening experience to try to prove "All items in universe influence each other's positions" by implementing this in the Hilbert space. You must first implement the items, then the position of the items. Next the universe of items and finally the influences. In this way you will encounter many aspects of quantum physics. If you do it properly, then you will find the source of gravity.

See http://www.scitech.nl/English/Science/Exampleproposition.pdf

 

[Demystifier]

Yes, but lots of things are possible in physics. What's important, IMHO, is how plausibly you can motivate the reasons for retaining symmetry Lorentz invariance.

Plausibility is a subjective thing. Some find the collapse postulate plausible, some don't. Some find nonrelativistic BM plausible, some don't. Likewise, some find symmetry Lorentz invariance plausible (see e.g. http://xxx.lanl.gov/abs/1006.5254), some don't.

It is up to me to explain why it is plausible TO ME, but it is up to the others to decide if it is also plausible TO THEM. I cannot ask from others to accept that it is plausible. But at least I can ask from others to understand my ideas properly.

 

[Demystifier]

Now hang on - In an earlier post of yours, you replied to my question "So will the superluminal signaling occur along all of those hypersurfaces?" with an unqualified *yes*. Did you just not read the question I asked before you answered it?

When I said "Yes", I made a mistake by accepting your suggestion to talk in the language of hypersurfaces. This is an unnatural language for the theory, which makes it easy to say something inconsistent when you try to use this language. The theory does not contain hypersurfaces. Thus, even though it is not impossible to talk about this theory in terms of hypersurfaces (which I tried), it makes more confusion than clarification. I will try to avoid it in the further discussions.

Roughly, it reminds me to explanations of special relativity (in classical mechanics) in terms of a preferred Lorentz frame. As Lorentz has shown, it is possible as well (the Lorentz "eather"). Yet, it introduces more confusion than clarification. It is a very unnatural way to talk about special relativity and it is better to avoid it.

 

[Demystifier]

For an N particle system, X_a, it would seem that I could then draw a unique simultaneity hypersurface across the particle velocities are instantaneously co-determined

No, you could not. Or if you think that you could, can you show such a picture here (as an attachment)?

OK, so now the fact that superluminal signaling occurs in spacetime between pointlike particles would seem to violate the relativity of simultaneity.

I have responded to this type of arguments in my
http://xxx.lanl.gov/abs/1002.3226
(second half of page 3 and the beginning of page 4).

Let me rephrase what I have written there. If that counts as violation of relativity of simultaneity (which I claim it shouldn't), then one can argue that subluminal (i.e., SLOWER than light) signaling also violates the relativity of simultaneity. Here is why: Let the communication be achieved by a messsage particle slower than light. Then there is a particular Lorentz frame in which the particle is at rest. Then I can say that this particular Lorentz frame defines a preferred notion of simultaneity. And then the relativity of simultaneity is violated.

Can you find a mistake in this argument on subluminal signals? I bet you can. But then, can you find a similar mistake in the argument on superluminal signals? If not, see the reference above.

 

[Demystifier]

And you claim to have a covariant deBB theory which you say you can write in a coordinate-free formulation, and which thus shares the advantages of a coordinate-free formulation of SR (as characterized by Maudlin). Fine. Then show us how you do it, and show us that it is consistent with general covariance. That, I think, would significantly help the plausibility of your theory.

As I said, I will do it. But to be sure that you understand my notation (which is rather abstract in the coordinate-free language), I want you first to write the NONRELATIVISTIC BM in a language that does not depend on SPACE-coordinates. I am sure you think that at least this nonrelativistic task can be accomplished. So please do it, just for the sake of fixing the notation. You do that easy job first, and then I will do the hard one. (Although, as you will see, this hard job is not hard at all. It is trivial. But I cannot be sure that you will understand it before you do your easy part of the job.)

Another way of saying this is that s is for relativistic 4-dimensional BM what is t for nonrelativistic 3-dimensional BM. To better understand what I mean by that, see also my most recent paper
http://xxx.lanl.gov/abs/1006.1986

Another useful observation is that s is a generalization of the concept of proper time (and I hope that you will agree that proper time does not ruin relativity in any relevant sense). This is also explained in more detail in the paper above (the Appendix).

 

[Demystifier]

See also the picture on page 8 of the attached talk (that I will present in the Towler Institute this summer). Can you draw the preferred foliation for these trajectories?

 

[Maaneli]

No, there is no foliation-like structure. The synchronization parameter is NOT something additional to the SR metrical structure, just as time in nonrelativistic BM is NOT something additional to the 3-space rotational-symmetry structure. Read my post #90. There is no much point is answering your other questions before you understand this. I am convinced that, when you understand this, you will withdraw most of your other questions.

I think it's obvious that the synchronization parameter is NOT something found in the SR metrical structure, just as an absolute time coordinate is NOT something found in the Euclidean metric. And I did read your post #90 (again), but it is not relevant to this point. Also, this is the second time that you're being inconsistent in your own characterization of your own theory, because when Yoda Jedi pointed out to you a section in one of Tumulka's papers which mentions that a relativistic theory such as yours involves a foliation-like structure, you did not get defensive. You simply agreed. If you don't remember, then let me remind you:

------------------

Yoda Jedi: (Quoting Tumulka) "Moreover, it does introduce a foliation-like structure"

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

------------------

Perhaps you just didn't/don't know what is meant by a 'foliation-like' structure, in which case, let me spell it out for you:

(i) Synchronized trajectories [11, 21, 56]. Define a path s → X(s) in (space-time)N as the integral curve of a vector field j^ψ on (space-time)N , with j^ψ 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.
http://arxiv.org/abs/quant-ph/0607124

Is its clear now? I hope so.

You just don't get it. My point is that relativistic-covariant BM in 4-dimensional spacetime is ANALOGOUS to nonrelativistic BM in 3-dimensional space. I am just trying to make you understand this ANALOGY, because when you do, you will suddenly say: "Oh, THAT is what you meant. Now I get it. In fact, it is trivial." But it is essential that you see this analogy by yourself, while I can only guide you in the right direction. And at the moment, it seems to me that you don't have a clue what I am talking about, because you are not able to see the analogy. And that is probably because you are unable to think of time as just another "space" coordinate.

No, you misunderstood my comments (or maybe I wasn't clear enough). I get that you want the synchronization parameter s to be analogous to the absolute time t, and the 4-vector X_N to be analogous to the 3-vector x_N, for N particles. Here, maybe you'll also recall this exchange:

------------

Maaneli: ... by virtue of the fact that you have to synchronize the initial (spacetime) positions of the particles at a common time s,

Demystifier: The parameter s is not time.

Maaneli: 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).

Demystifier: 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 [which I already implied by saying it is a UNIVERSAL time parameter].

-------------

What I didn't get is why you needed me to write down the coordinate-free formulation of nonrelativistic deBB, BEFORE you write down the coordinate-free formulation of your relativistic deBB theory. But now I see that you just wanted to point out that it would be analogous. Well, I was not objecting that it would be analogous, and so I just didn't see the need for me to do it before you write your relativistic theory in said form.

 

[Maaneli]

One additional way to guide your thinking. Don't think about special relativity as Einstein did in 1905. Think about special relativity as Minkowski did few years later. When Minkowski discovered the spacetime view of the Einstein special theory of relativity, Einstein was not able to see much sense in it. It took a lot of time before Einstein understood the advantage of the Minkowski spacetime view. But when he finally did, it open the door for discovering general relativity. Without Minkowski view, Einstein would never discover general relativity.

Likewise, it is impossible to understand relativistic-covariant BM using only Einstein 1905 view of special relativity. Time is the 4-th dimension, and it is Minkowski, not Einstein, who first understood it. Without FULLY appreciating the point that time is (almost) nothing but the 4-th dimension, it is impossible to fully understand relativistic-covariant BM. If you say "Yes, I know that time is the 4-th dimension, but still time is not really the same as space." - then you probably don't get it yet. Try the mental trick in my previous post above.

Yeah, I know the history and get your point.