Pilot wave theory, fundamental forces

[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.

 

[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.

 

[Demystifier]

Hrvoje,

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

Thanks, but that is not enough. Let me repeat (in a more precise form) what I asked you to do:
1. Write the MANY-particle nonrelativistic Schroedinger equation in a coordinate free formulation. (For simplicity, you can take V=U=A=0.)
2. Write the corresponding equations for BOHMIAN TRAJECTORIES in a coordinate free formulation.

And THEN I will generalize it to the relativistic case.

Or alternatively, skip all that and jump to my post #143 below. It should be obvious from it that relativistic BM can be written in a coordinate-free form, so that neither of us needs to write anything more about it.

 

[Demystifier]

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.

So, does it mean that you agree that WITH example of my theory there IS a known dynamical structure that is consistent with the metrical structure of Minkowski spacetime, and yet violates the relativity of simultaneity?

By the way, one can introduce such a structure even in classical local relativistic mechanics. Consider two twins who initially have the same velocity and same position, and their clocks show the same time. After that, they split apart, and each has a different trajectory, independent of each other. Yet, one can consider pairs of points on two trajectories which have THE SAME VALUE OF PROPER TIME (showed by a local clock on each trajectory). Such a structure (defined at least mathematically, if not experimentally) also can be said to violate relativity of simultaneity, in a way very similar to that of my theory. Of course, there is a difference, but the similarity may be illuminating too.

 

[Demystifier]

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

But then you must be missing something really obvious. Since I cannot guess what, let me remind you about a few (obvious) facts:

Mathematics:
1. A divergence of a scalar function is a vector field.
2. A vector field is a coordinate-free entity.
3. Integral curves of a vector field are also coordinate-free entities.
4. Projections of a curve on lower-dimensional surfaces are also coordinate-free entities.

Physics:
1. In relativistic QM (of spin-0 particles), the phase of the wave function is a scalar function (living in the 4n-dimensional configuration space).
2. Relativistic Bohmian trajectories in the 4n-dimensional configuration space are integral curves of the vector field given by the divergence of the phase of the wave function.
3. n relativistic Bohmian trajectories in the 4-dimensional spacetime are projections of a trajectory in 2. on n 4-dimensional surfaces.

Do you have problems to understand any of the facts above?
If not, then isn't it obvious that relativistic BM can be written in a coordinate-free form?
If so, do I still need to write it explicitly?

 

[Maaneli]

Thanks, but that is not enough. Let me repeat (in a more precise form) what I asked you to do:
1. Write the MANY-particle nonrelativistic Schroedinger equation in a coordinate free formulation. (For simplicity, you can take V=U=A=0.)
2. Write the corresponding equations for BOHMIAN TRAJECTORIES in a coordinate free formulation.

And THEN I will generalize it to the relativistic case.

Or alternatively, skip all that and jump to my post #143 below. It should be obvious from it that relativistic BM can be written in a coordinate-free form, so that neither of us needs to write anything more about it.

We can forgo it. It is trivial to write the deBB formulation in the GC formulation, and I agree that your theory can also be written in an coordinate-free form using GC.

 

[Maaneli]

So, does it mean that you agree that WITH example of my theory there IS a known dynamical structure that is consistent with the metrical structure of Minkowski spacetime, and yet violates the relativity of simultaneity?

Yes, it would seem that I would have to agree with that.

 

[Demystifier]

So, it seems that we reached the agreement now, right?

See also private messages.

 

[Maaneli]

So, it seems that we reached the agreement now, right?

On those issues, yes.

I still have to go through the material you sent me though to understand how the s parameter can play the role of both a Newtonian time, as well as a proper time.

Also, I am curious about how one might physically interpret s. Is there some physical clock that can operationally define durations of s? And if so, how does that clock differ from a clock that operationally defines durations of the proper time t?

Also, from what I recall, Berndl et al's attempts to treat time and space on equal footing were applied only to a relativistic pilot-wave theory involving the Dirac equation. By contrast, your work seems to only have been applied thus far to the Klein-Gordon equation. Have you tried yet to extend your approach to the Dirac equation, and if so, are there any new obstacles that result from trying to do so? Do you run into the same problems that Berndl et al faced?

 

[Demystifier]

I still have to go through the material you sent me though to understand how the s parameter can play the role of both a Newtonian time, as well as a proper time.

Also, I am curious about how one might physically interpret s. Is there some physical clock that can operationally define durations of s? And if so, how does that clock differ from a clock that operationally defines durations of the proper time t?

I said something about all that in
http://xxx.lanl.gov/abs/1006.1986
mainly in the classical context. In short, if you can measure particle trajectories directly (which in classical physics you can), then you can have a clock that measures s, and another clock that measures proper time tau. However, since there are no nontrivial classical scalar potentials in nature (even though the principle of relativity allows them), s and tau turn out to be essentially the same in most cases of practical interest. Yet, see Eq. (80) showing that s of many particles is a kind of average tau.

Also, from what I recall, Berndl et al's attempts to treat time and space on equal footing were applied only to a relativistic pilot-wave theory involving the Dirac equation. By contrast, your work seems to only have been applied thus far to the Klein-Gordon equation. Have you tried yet to extend your approach to the Dirac equation, and if so, are there any new obstacles that result from trying to do so? Do you run into the same problems that Berndl et al faced?

First, my approach is based on their very general equation (31), which can be applied to both fermions and bosons.

Second, I have explicitly studied fermions (Dirac equation) as well.
See
http://xxx.lanl.gov/abs/0904.2287 [Int. J. Mod. Phys. A25:1477-1505, 2010]
Sec. 3.4 and Appendix A.
See also the attachment in
https://www.physicsforums.com/showpost.php?p=2781627&postcount=103
pages 28-32.

Since I use spacetime probability (not space probability), I do no face the problems of Berndl et al. See also the attachment above, pages 12-13.

 

[Dmitry67]

I have few questions about BM

1. Are particles (in BM sense, hidden particles riding the wave) inside, say, u-quark, are different from particles inside, say, electron?

2. In BM, what are Kl and Ks mesons? Or Eta mesons? How many BM particles are inside them? (because in QM this number is not integer)

 

[Demystifier]

1. Are particles (in BM sense, hidden particles riding the wave) inside, say, u-quark, are different from particles inside, say, electron?

The particles by themselves are the same, but they behave differently because they are guided by different wave functions.

2. In BM, what are Kl and Ks mesons? Or Eta mesons? How many BM particles are inside them? (because in QM this number is not integer)

In the Bohmian interpretation of QFT, the number of particles is actually infinite (but integer). However, the influence of most of them is usually negligible.

When you say that the number of particles is not integer, you actually mean that the AVERAGE number of particles is not integer. You DON'T mean that the state is an eigenstate of the number operator with a non-integer eigen-value. For example, in a superposition |2> + |3> the average number of particles is 2.5, while the number of Bohmian particles is 2+3=5. Yet, when the number of particles is DIRECTLY (strongly) measured, then experiment gives either 2 or 3 (not 2.5), and either 2 or 3 Bohmian particles have a non-negligible influence on the measuring apparatus.

Indeed, this is a general feature of Bohmian mechanics that, without measurements, gives results that do not agree with experimental results, and yet gives the same measurable predictions as standard theory when the effects of measurement are taken into account.

For those who are interested in details (which Dmitry isn't), they can find them in
http://xxx.lanl.gov/abs/0904.2287 [Int. J. Mod. Phys. A25:1477-1505, 2010]