Pilot wave theory, fundamental forces

[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.

 

[Maaneli]

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.

Minkowski's formulation also predicts the relativity of simultaneity.

Yeah, I know the lecture you're referring to.

 

[Maaneli]

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.

But if you want to try and convince OTHER people that your theory should be accepted for its claims (which I am assuming you want to do at the TTI conference), then it is to your advantage to consider and address aspects of your theory that OTHER people might find implausible. You might even show that by retaining symmetry Lorentz invariance, one can solve certain difficult physics problems in a novel and simple way, than when using the standard approach. That would certainly help to convince OTHER people of the plausibility of your theory.

 

[Maaneli]

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

I'm not sure that a picture is necessary. All I am thinking of is a standard spacetime-like diagram on which one would draw the world lines of particle trajectories (just like for the standard nonrelativistic deBB theory), but where s plays the role of t, and the 4-vector X plays the role of the 3-vector x. It seems evident to me that since s is a universal 'time' parameter for the particle trajectories, then at any instant of s, I should be able to draw a single spacelike slice across all the particle world lines. In other words, the spacetime-like diagram can be thought of as composed of a series of spacelike hyperplanes stacked up in the +s direction.

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.

If communication is superluminal, then there is a Lorentz frame in which it is instantaneous. If the communication is instantaneous in one Lorentz frame, then it is not instantaneous in any other Lorentz frame. Therefore, there is a preferred Lorentz frame with respect to which the communication is instantaneous.


I don't understand the reasoning in that assertion, nor how it applies to the example I gave involving signaling with quantum nonequilibrium. In my example, instantaneous signaling (and hence violation of relativity of simultaneity) does not occur in just one preferred Lorentz frame, but rather in *all* Lorentz frames.

 

[Maaneli]

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.)

If I am not familiar with the notation you use, then I'll ask you to questions to understand it, or I'll look up the notation and learn it for myself. In any case, I know that Geometric Algebra provides a coordinate-free formulation of the Schroedinger-Pauli equation and the Dirac equation. Is that the notation you would use? If so, then go for it, as I am already familiar with the notation.

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).

I read that Appendix, but I'll have to think about it a bit more to understand it physically. It's odd though that s can be a generalization of proper time, and yet play the role of an absolute time parameter.

 

[Maaneli]

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?

Thanks, but as you can infer from my description in post #108, the spacetime-like diagram I have in mind is not the same as the picture on page 8 of your talk.

 

[Demystifier]

Thanks, but as you can infer from my description in post #108, the spacetime-like diagram I have in mind is not the same as the picture on page 8 of your talk.

It would be much easier to comment it if you could DRAW your diagram. (As I have drawn mine.)

 

[Demystifier]

I read that Appendix, but I'll have to think about it a bit more to understand it physically. It's odd though that s can be a generalization of proper time, and yet play the role of an absolute time parameter.

The analogy with Eq. (8) should be helpful.

 

[Demystifier]

If I am not familiar with the notation you use, then I'll ask you to questions to understand it, or I'll look up the notation and learn it for myself. In any case, I know that Geometric Algebra provides a coordinate-free formulation of the Schroedinger-Pauli equation and the Dirac equation. Is that the notation you would use? If so, then go for it, as I am already familiar with the notation.

It is irrelevant what notation I would use. It was YOU who insisted on coordinate free formulation, so I assume that you know what YOU mean by coordinate-free formulation. Since my only motivation for using coordinate-free formulation is to convince YOU, then I will use YOUR formalism, whatever that will be.

Learning is an active process, and sometimes the best way to learn something is to do it by yourself. This is such a case, which is why, for pedagogical purposes, I insist that you first make the coordinate-free formulation of nonrelativistic BM, where "coordinate" refers to space coordinates of the 3-dimensional space. Or if you are not sure that this can be done, then I ask you: Does it mean that you are not sure that nonrelativistic BM is "fundamentally" rotational invariant?

 

[Demystifier]

And I did read your post #90 (again), but it is not relevant to this point.

As long as you think so, there will be no much progress in our discussion. The SIMPLEST way to understand my ideas is through the analogy with 3 dimensional space (+ external Newton time). Almost any objection on my 4-dimensional theory (+ external s) has an analogue in this well-understood 3-dimensional theory. Since the 3-dimensional theory is conceptually much simpler (but technically almost identical), it is much easier to solve any problem in 4D theory by translating it to the analogous problem in 3D theory. It is a mental trick that makes "hard" problems trivial. See also the last pargraph in #94.

Also, this is the second time that you're being inconsistent in your own characterization of your own theory

Yes , I admit that. See my post #100 for the explanation.

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.[/B]

The second sentence is correct. The first is neither correct nor wrong because the authors do not explain what they mean by "foliation-like". It they had written instead "It does introduce a unique foliation structure", then it would be wrong, but the authors were aware of this, which is why they have not wrote it.

 

[Demystifier]

But if you want to try and convince OTHER people that your theory should be accepted for its claims (which I am assuming you want to do at the TTI conference), then it is to your advantage to consider and address aspects of your theory that OTHER people might find implausible. You might even show that by retaining symmetry Lorentz invariance, one can solve certain difficult physics problems in a novel and simple way, than when using the standard approach. That would certainly help to convince OTHER people of the plausibility of your theory.

With that, I completely agree. I am trying my best here.

Let me try with another plausible argument:
I want a theory that makes mathematical (if not physical) sense for ANY signature of the metric. That is, not only Minkowski signature (+---), but also Euclidean signature (++++), "two-time" signature (++--), or whatever. For example, the Einstein equation (in GENERAL relativity) is such a theory. But then the theory cannot rest on the concepts such as light-cones and relativistic causality, because these concepts do not make sense for arbitrary signature. This is a motivation to insist only on symmetry Lorentz invariance, and not on causality Lorentz invariance.

Do you find it plausible? (I do.)

 

[Demystifier]

Minkowski's formulation also predicts the relativity of simultaneity.

In a sense it does, but it is not one of its axioms. On the other hand, it is manifest that my theory is compatible with all axioms of Minkowski's formulation.

 

[Demystifier]

I don't understand the reasoning in that assertion, nor how it applies to the example I gave involving signaling with quantum nonequilibrium. In my example, instantaneous signaling (and hence violation of relativity of simultaneity) does not occur in just one preferred Lorentz frame, but rather in *all* Lorentz frames.

Then I probably misunderstood you (which is probably my fault). But if it occurs in ALL Lorentz frames, then I don't understand how is it incompatible with relativity of simultaneity? And if it is not, then what exactly is the problem?

Note also that an axiom that says something about "relativity of simultaneity" does not treat time on an equal footing with space. That is because the concept of "simultaneity" refers to time and not to space.

 

[Demystifier]

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.

THAT is the way of thinking I am trying to force you to use! The ANALOGY!
Now let us continue in the same spirit:

Yet, it is obvious that absolute time coordinate is NOT something that ruins the rotational symmetry of the Euclidean metric. (Time is EXTERNAL with respect to Euclidean space.) For the same reason, the synchronization parameter cannot be something that ruins the Lorentz symmetry of the SR metrical structure. (The synchronization parameter is EXTERNAL with respect to Minkowski space.)

Do you get it now?

Or perhaps you are confused how can s be both external (like absolute time in Newtonian mechanics), and internal (like proper time)? In that case, read http://xxx.lanl.gov/abs/1006.1986 , especially paragraphs around Eqs. (8), (11), (12)-(13), Appendix, last paragraph of Sec. 2, and Sec. 4.4. This paper is written rather pedagogically and is intended to teach people a lot about relativity.

 

[Maaneli]

It would be much easier to comment it if you could DRAW your diagram. (As I have drawn mine.)

OK, I'm not sure why it's difficult to see what I have in mind, but nevertheless, the diagram I have in mind is essentially the same as figures 2.4 (page 36), 2.5 (page 37), and 2.6 (page 39) in Maudlin's book (they all show in the free access parts of this link),

Quantum Nonlocality and Relativity
http://books.google.com/books?id=dB...&resnum=4&ved=0CDAQ6AEwAw#v=onepage&q&f=false

but where each instant of t is replaced with each instant of your s parameter, and the x-axis (which in Maudlin's diagram represents Euclidean 3-space) represents spacetime instead of Euclidean space. The particle trajectories in figure 2.5 are then trajectories in Minkowski spacetime, and are parameterized by your absolute time s. Does that help?

 

[Maaneli]

It is irrelevant what notation I would use. It was YOU who insisted on coordinate free formulation, so I assume that you know what YOU mean by coordinate-free formulation. Since my only motivation for using coordinate-free formulation is to convince YOU, then I will use YOUR formalism, whatever that will be.

And I already suggested which coordinate-independent formulation to use. Geometric Calculus. So go for it. And, if you don't mind, try and generalize this coordinate-independent formulation of your theory to the a deBB analogue of the semiclassical Einstein equation.

Learning is an active process, and sometimes the best way to learn something is to do it by yourself. This is such a case, which is why, for pedagogical purposes, I insist that you first make the coordinate-free formulation of nonrelativistic BM, where "coordinate" refers to space coordinates of the 3-dimensional space.

Please be honest: Are you familiar with Geometric Calculus (GC) and the GC formulation of QM? Or do you know of any other coordinate-free formulation of QM? If no (or if so) to both, then just say so and I'll be happy to write down for you the Dirac equation and Schroedinger-Pauli equation in the coordinate-independent GC formulation. Then maybe you can show me how your relativistic deBB theory can be written in the coordinate-independent GC formulation.