Pilot wave theory, fundamental forces

[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.

 

[Maaneli]

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

I don't see how your explanation in post #100 is relevant. In post #100, you say that your use of the language of 4-D hypersurfaces in your theory was misleading, not that your admission that your theory has a foliation-like structure (in the sense that Tumulka defines it) was misleading. They seem to me to be different issues.

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.

I think the meaning of "foliation-like" is evident: it just means that you have a structure which is akin to how spacetime is foliated by a time parameter t which orders the spacelike level surfaces of Euclidean space (and thus any particle trajectories on that spacetime). And your foliation-like structure is (again) just this: 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.

 

[Maaneli]

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.)

I'm not sure I understand the reasoning there. Do you want your theory to make mathematical (if not physical) sense for any signature of the metric in flat space only, or also curved space? If flat space only, then OK, that sounds reasonable. But if also curved space, then I don't understand why you would want to retain symmetry Lorentz invariance when Lorentz symmetry is not even a symmetry of curved spacetime.

 

[Maaneli]

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.

But I am arguing that when you allow for superluminal signaling using nonequiibrium measurements in your theory, your theory seems to violate the relativity of simultaneity.

 

[Maaneli]

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?

Two events are simultaneous if they occur at the same time. The relativity of simultaneity asserts that two events which are simultaneous one reference frame, are not necessarily simultaneous in any other frame. In other words, simultaneity is not absolute, but depends on an observer's reference frame. Now, in the nonlocal (to be more precise) signaling scenario I considered, the entangled particles A and B are simultaneously forced into definite spin orientations in ALL reference frames. In other words, the absolute simultaneity implied by nonlocal signaling from quantum nonequilibrium, is frame independent. So I conclude that this nonlocal signaling violates the relativity of simultaneity. Am I missing something here?

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.

True, but the relativity of simultaneity is nevertheless a consequence of the metrical structure of Minkowski spacetime, whereas it seems to me that your theory (by virtue of the addition of a synchronization parameter which makes the dynamics of a system of N spacetime particle coordinates nonlocal) can violate this consequence of the metrical structure of Minkowski spacetime, when you allow for nonlocal signaling via subquantum measurements. I can make this claim more precise, if you like.

 

[Maaneli]

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?

I think you've been misunderstanding me. I NEVER claimed that your theory failed to preserve symmetry Lorentz invariance. I simply pointed out that in your theory, in order to preserve symmetry Lorentz invariance for deBB particle dynamics in spacetime, you have to incorporate something IN ADDITION to the SR metrical structure, namely, a foliation-like structure involving the external synchronization parameter s.

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.

Yes, I am also perplexed at how s can be both external like absolute time and internal like proper time, and that's probably because I haven't thought enough about your argument yet. But thanks for the references.

 

[Demystifier]

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.

OK, now I have seen the figures in the Maudlin book, so I can make comments.

I still don't see how you can draw a SINGLE spacelike slice. Indeed, Maudlin himself says:
"If these are the only constraints that our coordinate frame must meet, the we have a very wide range of choices. One such choice is depicted in figure 2.5."
Therefore, I don't see how figure 2.5 shows you are able to draw SINGLE spacelike slice.

Perhaps your idea is that the spacelike slice is FLAT and ORTHOGONAL to the point of intersection with a particle trajectory? Yes, you can do that if there is ONLY ONE trajectory? But what if there are two trajectories (two particles)? Will the flat slice orthogonal to one trajectory be also orthogonal to the other trajectory? In general, it will not. Therefore, you cannot make a meaningfull SINGLE slice in that way.

Or perhaps your idea is that the spacelike slice is flat and lies on the (imagined) flat spacelike line that connects two particles at points of same s? Yes, you can do that as well, but only if you have only two particles. If you have more than two particles, the idea fails again. (That is why my figure shows 3 particles.)

 

[Demystifier]

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.

Have you forgoten again that I mainly consider Klein-Gordon equation? Let us do simpler things first.

Then maybe you can show me how your relativistic deBB theory can be written in the coordinate-independent GC formulation.

As I said two times already (and now I am repeating it the third time), I will not do it before you show me how Bohm's nonrelativistic theory for particles without spin can be written in SOME coordinate-independent formulation (coordinate with respect to 3-space).

 

[Demystifier]

I don't see how your explanation in post #100 is relevant.

Well, my post #100 is more about psychology than about physics. But if you still miss the point of it, just forget it. It is not essential at all.

And your foliation-like structure is (again) just this: 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.

Sorry, but I simply do not accept that such a structure should be called "foliation-like". I see nothing foliation-like in it. This structure is a relation between two points, and two points do not make a surface. At least not unless you introduce some ADDITIONAL structure (not only the parameter s and the many-time wave function), which I don't.

 

[Demystifier]

I'm not sure I understand the reasoning there. Do you want your theory to make mathematical (if not physical) sense for any signature of the metric in flat space only, or also curved space? If flat space only, then OK, that sounds reasonable. But if also curved space, then I don't understand why you would want to retain symmetry Lorentz invariance when Lorentz symmetry is not even a symmetry of curved spacetime.

Well, a curved spacetime with signature (+---) also contains a Lorentz symmetry. More precisely, a local Lorentz symmetry. Does it help?

 

[Demystifier]

Two events are simultaneous if they occur at the same time. The relativity of simultaneity asserts that two events which are simultaneous one reference frame, are not necessarily simultaneous in any other frame. In other words, simultaneity is not absolute, but depends on an observer's reference frame. Now, in the nonlocal (to be more precise) signaling scenario I considered, the entangled particles A and B are simultaneously forced into definite spin orientations in ALL reference frames. In other words, the absolute simultaneity implied by nonlocal signaling from quantum nonequilibrium, is frame independent. So I conclude that this nonlocal signaling violates the relativity of simultaneity. Am I missing something here?

OK, that's clear enough. And you are right, nonlocal signaling violates the relativity of simultaneity. Yet, in the next post I explain why it is NOT in contradiction with metrical structure of Minkowski spacetime.

 

[Demystifier]

... the relativity of simultaneity is nevertheless a consequence of the metrical structure of Minkowski spacetime

No, this is not true. What is true is that the metrical structure of Minkowski spacetime implies relativity of simultaneity IF THERE IS NO ANY OTHER STRUCTURE. But in the case we are considering there is another structure. And this additional structure is not the parameter s (as you might naively think), but the non-local wave function. (Or the scalar potential in the classical setting discussed in http://xxx.lanl.gov/abs/1006.1986.)
And yet, you can see that this nonlocal wave function (or the scalar potential) is compatible with the metrical structure of Minkowski spacetime and does not introduce a foliation-like structure.

Perhaps it is also possible to derive relativity of simultaneity from the assumptions of
1) metrical structure of Minkowski spacetime
and
2) locality.
(I am not sure about that assertion, which is why I say "Perhaps".) But it surely cannot be derived from 1) alone.

 

[Demystifier]

One additional comment. The simplest way to see that the parameter s by itself does not introduce any additional PHYSICAL structure is to consider the case of TWO CLASSICAL PARTICLES THAT DO NOT INTERACT WITH EACH OTHER. Even in this case one can describe both trajectories by parameterizing them with a common parameter s, and even in this case one can say that points of equal s have something to do with absolute simultaneity. Yet, it should be obvious that such "absolute simultaneity" does not have any physical meaning. Instead, a genuine new PHYSICAL structure is provided by the nonlocal wave function (scalar potential), and this structure does not have an explicit dependence on s.

 

[Demystifier]

Maaneli, I think I know what your problem is. It seems that you think that one cannot calculate the trajectories in spacetime without using the parameter s. But this is simply wrong. The trajectories in spacetime can be calculated even without the parameter s. See Eq. (30) and the discussion around it in
http://xxx.lanl.gov/abs/quant-ph/0512065

The trajectories in spacetime are integral curves of the (conserved) vector current, and it is a well-known fact in differential geometry that integral curves of vector fields are well-defined objects in a coordinate-free formulation of differential geometry.

See also some possibly illuminating high-school basics here:
http://en.wikipedia.org/wiki/Parametric_equation

For some basics on integral curves in both coordinate and coordinate-free languages see
http://en.wikipedia.org/wiki/Integral_curve

For more advanced (coordinate-free) differential geometry of curves see
http://en.wikipedia.org/wiki/Regular_parametric_representation

All this may be helpful if you plan to do your "homework" in #127. But for pedagogical purposes, I will not do it for you. I think I gave you many hints here, which should be enough.

 

[Maaneli]

Have you forgoten again that I mainly consider Klein-Gordon equation? Let us do simpler things first.

It's really not that difficult to go from the Dirac and Pauli equation to the KG and Schroedinger equation. But if you REALLY need me to, I'll just show the Schroedinger case.

As I said two times already (and now I am repeating it the third time), I will not do it before you show me how Bohm's nonrelativistic theory for particles without spin can be written in SOME coordinate-independent formulation (coordinate with respect to 3-space).

Wow, way to quote me out of context! I'll also repeat myself for the third time: Are you familiar with Geometric Calculus? If not, then just say so and I'll show you in detail how it is used to formulate the nonrelativistic Schroedinger equation. But if so, then let me know to what extent so that I don't have to explain all the operators and notation when writing down the formulation.

By the way, just a heads up - I'll be out of town for a week or so starting tomorrow, and I may not have enough internet access to reply to your other posts, and your future reply to this post, until I'm back home.

 

[Demystifier]

But if you REALLY need me to, I'll just show the Schroedinger case.

Good, thanks! I am looking forward to see it.

Are you familiar with Geometric Calculus?

Yes I am. OK, it is not that I use it every day, so it may take some time to remind myself of some details. But I don't expect any serious difficulties from my side.

If not, then just say so and I'll show you in detail how it is used to formulate the nonrelativistic Schroedinger equation. But if so, then let me know to what extent so that I don't have to explain all the operators and notation when writing down the formulation.

Fair enough! I think it would be sufficient to outline the main steps in it, and perhaps to omit some details. But I have only one wish. I would prefer a formulation in which Schrodinger equation is NOT DERIVED FROM A RELATIVISTIC EQUATION, but considered as a "fundamental" equation by its own. Maybe it looks paradoxical, but in this form it will be much easier for me to generalize it to the relativistic case. (You will see why.)

By the way, just a heads up - I'll be out of town for a week or so starting tomorrow, and I may not have enough internet access to reply to your other posts, and your future reply to this post, until I'm back home.

OK, thanks for the note!

 

[Maaneli]

Maaneli, I think I know what your problem is. It seems that you think that one cannot calculate the trajectories in spacetime without using the parameter s. But this is simply wrong. The trajectories in spacetime can be calculated even without the parameter s. See Eq. (30) and the discussion around it in
http://xxx.lanl.gov/abs/quant-ph/0512065

The trajectories in spacetime are integral curves of the (conserved) vector current, and it is a well-known fact in differential geometry that integral curves of vector fields are well-defined objects in a coordinate-free formulation of differential geometry.

See also some possibly illuminating high-school basics here:
http://en.wikipedia.org/wiki/Parametric_equation

For some basics on integral curves in both coordinate and coordinate-free languages see
http://en.wikipedia.org/wiki/Integral_curve

For more advanced (coordinate-free) differential geometry of curves see
http://en.wikipedia.org/wiki/Regular_parametric_representation

All this may be helpful if you plan to do your "homework" in #127. But for pedagogical purposes, I will not do it for you. I think I gave you many hints here, which should be enough.

OK, back.

Thanks for the links, but no, I never thought that it was impossible to compute trajectories without the s parameter.

 

[Maaneli]

OK, now I have seen the figures in the Maudlin book, so I can make comments.

I still don't see how you can draw a SINGLE spacelike slice. Indeed, Maudlin himself says:
"If these are the only constraints that our coordinate frame must meet, the we have a very wide range of choices. One such choice is depicted in figure 2.5."
Therefore, I don't see how figure 2.5 shows you are able to draw SINGLE spacelike slice.

Perhaps your idea is that the spacelike slice is FLAT and ORTHOGONAL to the point of intersection with a particle trajectory? Yes, you can do that if there is ONLY ONE trajectory? But what if there are two trajectories (two particles)? Will the flat slice orthogonal to one trajectory be also orthogonal to the other trajectory? In general, it will not. Therefore, you cannot make a meaningfull SINGLE slice in that way.

Or perhaps your idea is that the spacelike slice is flat and lies on the (imagined) flat spacelike line that connects two particles at points of same s? Yes, you can do that as well, but only if you have only two particles. If you have more than two particles, the idea fails again. (That is why my figure shows 3 particles.)

Yes, for more than one particle, the hypersurfaces can't be flat. So modify the drawing of these hypersurfaces by making sure that they are only locally flat and orthogonal to the point of intersection with each particle trajectory. An example of how this looks can be found on slide #4 of Tumulka's talk:

http://www.math.rutgers.edu/~tumulka/talks/penn09.pdf

 

[Maaneli]

No, this is not true. What is true is that the metrical structure of Minkowski spacetime implies relativity of simultaneity IF THERE IS NO ANY OTHER STRUCTURE. But in the case we are considering there is another structure. And this additional structure is not the parameter s (as you might naively think), but the non-local wave function. (Or the scalar potential in the classical setting discussed in http://xxx.lanl.gov/abs/1006.1986.)
And yet, you can see that this nonlocal wave function (or the scalar potential) is compatible with the metrical structure of Minkowski spacetime and does not introduce a foliation-like structure.

Perhaps it is also possible to derive relativity of simultaneity from the assumptions of
1) metrical structure of Minkowski spacetime
and
2) locality.
(I am not sure about that assertion, which is why I say "Perhaps".) But it surely cannot be derived from 1) alone.

OK, I think I agree with your correction to my statement, in light of your theory. Though, without the example of your theory, it would be hard to see the flaw in my assertion, as there is no other known dynamical structure that is consistent with the metrical structure of Minkowski spacetime, and yet violates the relativity of simultaneity.

 

[Maaneli]

Hrvoje,

See the attachment for the GC formulation of the nonrelativistic Schroedinger equation.

 

[Demystifier]

Yes, for more than one particle, the hypersurfaces can't be flat. So modify the drawing of these hypersurfaces by making sure that they are only locally flat and orthogonal to the point of intersection with each particle trajectory. An example of how this looks can be found on slide #4 of Tumulka's talk:

http://www.math.rutgers.edu/~tumulka/talks/penn09.pdf

Sure, you can do that. But as I already stressed, such a foliation is not unique. In this sense, the particle trajectories do not define a foliation. At best, they define AN INFINITE CLASS of foliations. But a single particle also defines an infinite class of foliations, and I don't think that it conflicts with relativity in any meaningful sense.