Pilot wave theory, fundamental forces

[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]

 

[Dmitry67]

Demistifier,

There are 2 different problems as I understand:
1. Even in proton the number of quarks is not well defined. It is 3 if we use 'hard' measurements.
2. Some particles has non-integer particle content even for the 'hard' measurements
Check the list:
http://en.wikipedia.org/wiki/List_of_mesons
Kl, Ks, Eta prime with all that sqrt(2) and sqrt(6) in denominator

 

[Demystifier]

Demistifier,

There are 2 different problems as I understand:
1. Even in proton the number of quarks is not well defined. It is 3 if we use 'hard' measurements.
2. Some particles has non-integer particle content even for the 'hard' measurements
Check the list:
http://en.wikipedia.org/wiki/List_of_mesons
Kl, Ks, Eta prime with all that sqrt(2) and sqrt(6) in denominator

1. As far as I know, the number 3 is well defined.
2. According to the tables you gave, these particles are superpositions of 2-particle states, and a superposition of 2-particle states is a 2-particle state itself. And number 2, as far as I know, is a well-defined number as well.

I thought that you are talking about phenomena such as Bjorken scaling in which the number of particles appears to change as you change energy, but now I see that you talk about something much more trivial. As long as we talk about QM (rather than QFT), all these states have a well defined number of particles (3 for proton and 2 for mesons you mentioned), so the Bohmian interpretation also says that the number of particles is well defined (3 for proton and 2 for mesons). You must have misunderstood something about elementary QM, but I don't know what. To be sure, the numbers sqrt(2) and sqrt(6) are normalization factors, NOT the numbers of particles. Even MWI says that the number of particles is well defined for these states (3 for proton and 2 for mesons).

Before asking a question on BM, one should first know the corresponding basics of standard QM. In other words, one should know what one is talking about. You are often pretending that you understand some aspects of "ordinary" QM even when you don't. Perhaps such bluffing works for those who don't know you well, but it doesn't work for me.

 

[Demystifier]

Maaneli, for a concise (and possibly more illuminating) summary of relativistic Bohmian mechanics, see also Sec. 2.1 of my last paper
http://xxx.lanl.gov/abs/1007.4946

 

[Demystifier]

Let me also present a brief comment on the Valentini's
http://xxx.lanl.gov/abs/0812.4941 [Phys. Lett. A 228 (1997) 215]
argument against Lorentz invariant BM.

Valentini correctly observes that the nonrelativistic BM has an Aristotelian (rather than Galilean) symmetry, which is related to the fact that the wave function (viewed as the fundamental "force") determines velocity (rather than acceleration). From this, he argues that attempts to generalize the theory to a Lorentz-invariant theory are misleading. However, that argument is wrong.

Let me explain why. He interprets Lorentz invariance as a generalization of Galilean invariance. On the other hand, in modern (Minkowski-like) view of relativity, Lorentz invariance is actually a generalization of ROTATIONAL invariance, not of Galilean invariance. And rotational invariance certainly IS a symmetry of nonrelativistic BM. Therefore, it is natural to search for a Lorentz-invariant generalization of BM.

 

[Maaneli]

Hrvoje,

Thanks for all your comments. I'm extremely busy at the moment and may not be able to reply for some time, possibly not until we meet at the Italy conference. Just letting you know.Maaneli

 

[Demystifier]

Thanks, Maaneli.

 

[Ilja]

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

There is a counter-argumentation by Bell, "how to teach special relativity", where he argues
that it is, instead, the Minkowski interpretation which causes much more confusion.

It is published in "speakable and unspeakable".

 

[bohm2]

Let me also present a brief comment on the Valentini's
http://xxx.lanl.gov/abs/0812.4941 [Phys. Lett. A 228 (1997) 215]
argument against Lorentz invariant BM.

Valentini correctly observes that the nonrelativistic BM has an Aristotelian (rather than Galilean) symmetry, which is related to the fact that the wave function (viewed as the fundamental "force") determines velocity (rather than acceleration). From this, he argues that attempts to generalize the theory to a Lorentz-invariant theory are misleading. However, that argument is wrong.

Let me explain why. He interprets Lorentz invariance as a generalization of Galilean invariance. On the other hand, in modern (Minkowski-like) view of relativity, Lorentz invariance is actually a generalization of ROTATIONAL invariance, not of Galilean invariance. And rotational invariance certainly IS a symmetry of nonrelativistic BM. Therefore, it is natural to search for a Lorentz-invariant generalization of BM.

I'm guessing that even if Valentini's argument isn't convincing, Albert's "narrative argument" still holds?

What it is for a theory to be metaphysically compatible with special relativity (which is to say: what it is for a theory to be compatible with special relativity in the highest degree) is for it to depict the world as unfolding in a four-dimensional Minkowskian space-time. And what it means to speak of the world as unfolding within a four-dimensional Minkowskian space-time is (i) that everything there is to say about the world can straightforwardly be read off of a catalogue of the local physical properties at every one of the continuous infinity of positions in a space-time like that, and (ii) that whatever lawlike relations there may be between the values of those local properties can be written down entirely in the language of a space-time [like] that—that whatever lawlike relations there may be between the values of those local properties are invariant under Lorentz-transformations...

What we do have (on the other hand) is a very straightforward trick by means of which a wide variety of theories are radically non-local and (moreover) are flatly incompatible with the proposition that the stage on which physical history unfolds is Minkowski-space can nonetheless be made fully and trivially Lorentz-invariant; a trick (that is), by way of which a wide variety of such theories can be made what you might call formally compatible with special relativity...

As things stand now we have let go not only of Minkowski-space as a realistic description of the stage on which the world is enacted, but (in so far as I can see) of any conception of that stage whatever. As things stand now (that is) we have let go of the idea of the world’s having anything along the lines of a narratable story at all! And all this just so as to guarantee that the fundamental laws remain exactly invariant under a certain hollowed-out set of mathematical transformations, a set which is now of no particularly deep conceptual interest, a set which is now utterly disconnected from any idea of an arena in which the world occurs.

I wonder if Albert would be satisfied even with a narrative, "realist" Lorentz invariant Bohmian formulation (assuming that is even possible)?

‘Special Relativity as an Open Question’
http://books.google.ca/books?id=ENp...ABQ&sqi=2&ved=0CEoQ6AEwAA#v=onepage&q&f=false

Physics and narrative
http://philosophyfaculty.ucsd.edu/faculty/wuthrich/PhilPhys/AlbertDavid2008Man_PhysicsNarrative.pdf

 

[Demystifier]

There is a counter-argumentation by Bell, "how to teach special relativity", where he argues
that it is, instead, the Minkowski interpretation which causes much more confusion.

It is published in "speakable and unspeakable".

I agree that Minkowski interpretation is confusing when you hear about it for the first time. But it is equally confusing (when you hear about it for the first time) that all 3 directions of space are on equal footing, and that there is no absolute up and down in space, and that people living in Australia can walk on Earth without falling down away from earth.

 

[Demystifier]

I'm guessing that even if Valentini's argument isn't convincing, Albert's "narrative argument" still holds?

I wonder if Albert would be satisfied even with a narrative, "realist" Lorentz invariant Bohmian formulation (assuming that is even possible)?

‘Special Relativity as an Open Question’
http://books.google.ca/books?id=ENp...ABQ&sqi=2&ved=0CEoQ6AEwAA#v=onepage&q&f=false

Physics and narrative
http://philosophyfaculty.ucsd.edu/faculty/wuthrich/PhilPhys/AlbertDavid2008Man_PhysicsNarrative.pdf

Let me just comment that Lorentz invariant version of BM without preferred frame is not narrative.