Pilot wave theory, fundamental forces

[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]