The decision by the journal was, "Minor Revisions". So, my paper is provisionally accepted by Foundations of Physics. I have until December 7 to submit the revised version, all of which I may need as I'm planning to add some material in addition to even the current arxiv version. The current arxiv version 8 is already significantly beyond what I sent them in early July as the second revision, which was similar to version 6.

The addition I'm working on beyond what's in version 8 is to derive the de Broglie wavelength in the rest frame of the field source particle. Currently it is only done is the rest frame of the test particle. This will provide more confidence in the correctness of the original (version 8) result. I don't see how anybody can object to that. Even if the journal doesn't want to print it, I want to have it and I will post it on arxiv.

I sent an email to the journal immediately after uploading version 8 to arxiv, to let them know I'd resolved the problem of obtaining only half the de Broglie wavelength, and wanted to submit another revision that would be similar to the arxiv version 8. Also, versions 6 and 7 didn't take proper account of the interparticle separation, R, dependence on time in Equation (33), but this is corrected in version 8. I didn't receive an explicit direction one way or the other but the email is in the saved correspondence, so I take the decision of "minor revisions" as tacit permission to provide the improvements.

## Sunday, October 11, 2015

## Sunday, September 27, 2015

### A Better Analysis of the De Broglie Wavelength Relationship to the Zitterbewegung

This is the latest version of my paper on the magnetic force between Dirac particles due to the zitterbewegung, which as of this writing is version 8 and was posted late last week. This version is more successful in getting the de Broglie wavelength exactly as a modulation of the magnetic force for both non-radial and radial relative motion between Dirac particles.

The new version solves the problem of getting a modulation that had a wavelength of half the de Broglie wavelength for radial relative motion. It also corrects a mistake of the previous two versions, and in what was sent to Foundations of Physics as a requested revision (which was similar to arxiv version 6). I overlooked the time dependence of the interparticle separation in deriving the preferred velocity and its relationship to the Bohr velocity. The new version recovers the idea that there is a preferred radial velocity where the advanced magnetic force will cancel the retarded magnetic force, but it is not completely satisfying in that it requires spiral motion.

After I posted the new version to arxiv, I sent the journal an email to let them know in particular about the error in what I sent them. I'd only realized I'd made the mistake about five days previously. I haven't heard anything back yet about how they want to handle it, or if they're still interested, but it has only been a couple of days. Meanwhile, the status is still "under review".

The new version solves the problem of getting a modulation that had a wavelength of half the de Broglie wavelength for radial relative motion. It also corrects a mistake of the previous two versions, and in what was sent to Foundations of Physics as a requested revision (which was similar to arxiv version 6). I overlooked the time dependence of the interparticle separation in deriving the preferred velocity and its relationship to the Bohr velocity. The new version recovers the idea that there is a preferred radial velocity where the advanced magnetic force will cancel the retarded magnetic force, but it is not completely satisfying in that it requires spiral motion.

After I posted the new version to arxiv, I sent the journal an email to let them know in particular about the error in what I sent them. I'd only realized I'd made the mistake about five days previously. I haven't heard anything back yet about how they want to handle it, or if they're still interested, but it has only been a couple of days. Meanwhile, the status is still "under review".

## Saturday, September 12, 2015

### Another Connection to the de Broglie Wavelength

I found another connection to the de Broglie wavelength, when I took account of time dilation due to relative motion between Dirac particles. Assuming (after various authors) that a Dirac particle (such as the electron) consists of a point charge moving in a circulatory motion at the speed of light so that it has angular momentum of h-bar over two, then supposing one Dirac particle is stationary while another is moving, there will be a difference in frequency of the circulatory motion of one particle relative to the other. It's easy to calculate both the frequency of the circulatory motion and the amount of time dilation, and so also to get a difference in frequency due to the time dilation. The frequency difference depends on the speed of the relative motion. It turns out that the de Broglie wavelength can be equated to the time traveled by the moving particle during one period of the difference frequency due to time dilation.

I put a new version of my paper on arxiv to capture just this observation, about three weeks ago. Then I had to get about preparing a revised version of the same paper for re-submittal to Foundations of Physics. I had a lot of consternation in getting the revision done, because I'm still sorting things out about this on several fronts. The extension of the paper to include the connection to the de Broglie wavelength is unsatisfying in several ways, that I will perhaps describe in more detail later. The worst way is that I'm so far only able to make some sense of things when the spins are aligned; if they are opposite then the model seems to break down. But also, up to my re-submittal deadline, I was unable to directly explain how the observation about the de Broglie wavelength being related to time dilation could be related and reconciled to my other observation about the de Broglie wavelength as described in my previous post. So, I didn't include it in the resubmittal. (The journal would probably have given me an extension to the deadline, but I was eager to get a review of what I had so far, and had no idea when I'd be able to figure things out better. I'd already spent considerable effort trying to understand it, with scant success.) Now I want to report that I've made better sense of at least this bit, and was able to combine the new result with my previous derivation of the de Broglie wavelength, which I think is well motivated but gave a result too small by a factor of a half, and get the right result.

I'll be working on a new revision to post on arxiv, hopefully within a week or two, that will make better sense than the one I link to above, which is v7. So, v8 will be a better version, and I'm also planning to rewrite the narrative somewhat beyond what I did for the re-submittal to FOOP. Also at that time I will contact the journal to tell them I made a little more progress, to see if they're interested.

## Wednesday, July 15, 2015

### Did I explain the de Broglie wavelength?

I've been working for the last week or so to extend my paper (although not in the direction suggested by Reviewer 1) and I think I've made some real progress. I was thinking that I might be able to obtain the Bohr velocity as the speed where the phase difference between the retarded and advanced zitter motions is pi. This would tend to cancel the zbw magnetic force. It turned out it worked; I got (pi over 2 times) the Bohr velocity if the separation is the Bohr radius. It's a completely distinct relationship between range and velocity than the circular orbit Coulomb attraction one, so it's not a trivial thing.

A couple of days later I got to thinking that probably the de Broglie wavelength relationship was implicit in what I had done, and when I looked at my equations, I could see it right away. So I cleaned it up a little and stuck it up on arxiv. It's got an extraneous factor of a half in it, however, that I hope to resolve soon. Also all I did was add a new section (V) and one sentence to the abstract. I'll be making a more extensive revision soon enough for resubmittal to FOOP and will replace it again in any case, and certainly I will relace it promptly if I can resolve the excess factor of a half.

As of this writing the version is 6. Here is the link to the newest version: http://arxiv.org/abs/1409.8271

A couple of days later I got to thinking that probably the de Broglie wavelength relationship was implicit in what I had done, and when I looked at my equations, I could see it right away. So I cleaned it up a little and stuck it up on arxiv. It's got an extraneous factor of a half in it, however, that I hope to resolve soon. Also all I did was add a new section (V) and one sentence to the abstract. I'll be making a more extensive revision soon enough for resubmittal to FOOP and will replace it again in any case, and certainly I will relace it promptly if I can resolve the excess factor of a half.

As of this writing the version is 6. Here is the link to the newest version: http://arxiv.org/abs/1409.8271

## Sunday, July 5, 2015

### Brief Response to Reviewer 3

Yesterday I posted the three reviews of my paper sent me last week by Foundations of Physics. I'm responding to Reviewer 3 first because I have no substantial disagreement with this review, and it raises an interesting question, which I will enjoy expanding on in my resubmission. Here is the Reviewer 3 comment:

The paper presents the computation of a magnetic force between two Dirac particles where the spin is interpreted as coming from Zitterbewegung. The idea follows from the earlier work by Rivas.

Here are my remarks.

1) Many of the interesting questions are left out and only mentioned as "beyond the scope of the paper" - this makes the results rather restricted. I'd suggest that the author explains why this is not touched.

2) I have some doubts how this results affects interpretation of quantum mechanics.

The result is nice and (as a matter of fact) rather easy to obtain.

It might be interesting to see whether such computations would make any predictions that would enable to falsify it (say in atom spectroscopy).

3) I fail to see the relevance of the part of Section 6 (evaluation of the quantum force).

Would the author, please, relate it to the other results ?

Mixing the nonrelativistic quantum mechanics with relativistic computation of classical force is slightly dangerous.

4) I would suggest the author to correct the language - there are many prepositions missing, some of the sentences are also a bit awkward. Example " p3 lines 6-7 "has also" should read: "has also been...". P11 line 38 "An evaluation" etc etc.

In response to remark 1, I will look into not simply stating something is beyond the scope of the paper. I think in at least most cases there is a good reason why whatever is being referred to isn't relevant to the paper or otherwise shouldn't be included, and that reason can (perhaps should have been) stated. I thank Reviewer 3 for the helpful remark.

I think remark 2 is most interesting. Rather than trying to define an experiment that might falsify the hypothesis that Bohm's quantum force is a consequence of the magnetic force, though, it will be more to the immediate point to better determine what the correspondence is. I already point out the quantum force cannot be directly equated to (even just one part of) the magnetic force between two Dirac particles. What's needed here is to derive the quantum force from the classical Hamilton-Jacobi equation by incorporation of the magnetic interaction. Bohm's original paper shows how the Schroedinger equation can be put in a form that can be regarded as the classical Hamilton-Jacobi equation with an additional term he named the quantum mechanical potential. The quantum force is then derived in the usual fashion, as a gradient of the quantum potential. So, it should be possible, if the hypothesis is correct, to derive the quantum potential from the magnetic interaction between two Dirac particles.

A derivation of the quantum potential from classical principles would be a very powerful result that would greatly improve my paper, but I doubt I can do it within the two months proposed by FOOP for the creation of a revised submission. Not that I'm unwilling to try, but one reason the journal might want to publish is so that other people who might be more qualified than I am to carry this out can attempt it if they're interested.

In remark 3,

The paper presents the computation of a magnetic force between two Dirac particles where the spin is interpreted as coming from Zitterbewegung. The idea follows from the earlier work by Rivas.

Here are my remarks.

1) Many of the interesting questions are left out and only mentioned as "beyond the scope of the paper" - this makes the results rather restricted. I'd suggest that the author explains why this is not touched.

2) I have some doubts how this results affects interpretation of quantum mechanics.

The result is nice and (as a matter of fact) rather easy to obtain.

It might be interesting to see whether such computations would make any predictions that would enable to falsify it (say in atom spectroscopy).

3) I fail to see the relevance of the part of Section 6 (evaluation of the quantum force).

Would the author, please, relate it to the other results ?

Mixing the nonrelativistic quantum mechanics with relativistic computation of classical force is slightly dangerous.

4) I would suggest the author to correct the language - there are many prepositions missing, some of the sentences are also a bit awkward. Example " p3 lines 6-7 "has also" should read: "has also been...". P11 line 38 "An evaluation" etc etc.

In response to remark 1, I will look into not simply stating something is beyond the scope of the paper. I think in at least most cases there is a good reason why whatever is being referred to isn't relevant to the paper or otherwise shouldn't be included, and that reason can (perhaps should have been) stated. I thank Reviewer 3 for the helpful remark.

I think remark 2 is most interesting. Rather than trying to define an experiment that might falsify the hypothesis that Bohm's quantum force is a consequence of the magnetic force, though, it will be more to the immediate point to better determine what the correspondence is. I already point out the quantum force cannot be directly equated to (even just one part of) the magnetic force between two Dirac particles. What's needed here is to derive the quantum force from the classical Hamilton-Jacobi equation by incorporation of the magnetic interaction. Bohm's original paper shows how the Schroedinger equation can be put in a form that can be regarded as the classical Hamilton-Jacobi equation with an additional term he named the quantum mechanical potential. The quantum force is then derived in the usual fashion, as a gradient of the quantum potential. So, it should be possible, if the hypothesis is correct, to derive the quantum potential from the magnetic interaction between two Dirac particles.

A derivation of the quantum potential from classical principles would be a very powerful result that would greatly improve my paper, but I doubt I can do it within the two months proposed by FOOP for the creation of a revised submission. Not that I'm unwilling to try, but one reason the journal might want to publish is so that other people who might be more qualified than I am to carry this out can attempt it if they're interested.

In remark 3,

## Saturday, July 4, 2015

### New Reviews

I have received three reviews from Foundations of Physics on my paper about the similarity of the magnetic force between relativistically-circulating charges to the quantum force of Bohmian mechanics. I didn't get a final decision; they requested "major" revisions. I'm willing to make any reasonable revisions, but I don't think certain of the revisions requested by two of three reviewers are well justified or reasonable. So I don't consider the revisions I'll be making all that major.

Maybe I will discuss my disagreements and objections to the reviews briefly here soon but not in this post. For now I will just post the reviews. In any case I will be creating both a revised version of the paper and a specific response to the reviews, as requested by FOOP, and can post that response when it's available. Also, I might mention that I recently posted on arxiv a version of the paper that is identical to the submittal to FOOP that the reviews apply to, except for being in two-column APS-like format rather than the FOOP format. So, the page references in the reviews may not line up exactly. Here's the (re-)submittal that was reviewed: http://arxiv.org/abs/1409.8271v5

Now the reviews:

Maybe I will discuss my disagreements and objections to the reviews briefly here soon but not in this post. For now I will just post the reviews. In any case I will be creating both a revised version of the paper and a specific response to the reviews, as requested by FOOP, and can post that response when it's available. Also, I might mention that I recently posted on arxiv a version of the paper that is identical to the submittal to FOOP that the reviews apply to, except for being in two-column APS-like format rather than the FOOP format. So, the page references in the reviews may not line up exactly. Here's the (re-)submittal that was reviewed: http://arxiv.org/abs/1409.8271v5

Now the reviews:

## Friday, March 20, 2015

### Status Update and What Martin Rivas Did

I haven't posted in a while because I've been preoccupied rewriting my paper at the request of Foundations of Physics. They still haven't sent it to referees but they did send it to a member of the editorial board, apparently, and gave it a revise-before-review status, in mid-January, with a due date of 20 March for a new version. I guess I won't reproduce here what the board member said, which was brief, but the general idea was that it was not written appropriately for a journal on the foundations of physics and would I please make it so. This I was entirely willing and happy to do, as it was originally written for a letters journal (Physics Letters A, as I have already posted about), and not tailored much for FOOP. I probably would have done more on this account, except when I got the rejection letter from PLA I was only two days away from departing on a month-long overseas assignment for my engineering job which I knew would be demanding, and didn't have time to revise it much, but at the same time wanted to have somebody else looking at it based on what I thought were encouraging words from the PLA editor in spite of not wanting to publish it. It was nice on that assignment to not have to worry about it or work on it and yet my project could still make progress by having it be considered by illustrious others. Another factor is that it's a difficult chore for me to build a narrative around an analysis, for an audience that I know is generally much more sophisticated than I am. I'm happier just pushing the equations around.

In any case I have given it my best attempt, I feel, and used the available time as best I could, rewriting it to a large extent, and then going over it a lot of times and long past the point where I was getting tired of re-reading words that I am still not all that happy with. Hopefully in the end it will sink or swim based on the physics contained within.

About the physics of my paper, what I wanted to mention is that since I noticed how the magnetic force between to classical spinning particles could apparently cancel the Coulomb force, and even before I realized that that is similar to the Bohmian quantum force, I knew that this result followed very readily from work done by Rivas at least ten years ago. So far as I have been able to determine he has not published explicitly that magnetic forces can cancel Coulomb forces, but some of his figures that show numerical modeling results, both in his book and published papers, are certainly very close to realizing this to be the case. It's surprising to me that these results seemingly have not attracted more attention. So, one of the things I attempted to do in the re-write is to be more clear about how much of what I wrote in the first version of the paper is actually prior work of Rivas in showing that the time-averaged acceleration field of a luminally-circulating charge can reconstitute, as it were, the usual electric and magnetic fields of a static charge and magnetic dipole. I found this result astonishing from the first time I really noticed it in his book, which was after I decided that I had to calculate these fields for myself. But, I was only planning on calculating the velocity fields and for the case of motion that was only asymptotically close to luminal, not exactly luminal. When I found that Rivas had already addressed this problem, I saw right away that he had both done a far better job than I probably could ever have done, and also that the result was quite surprising and not at all what I was anticipating.

As I try to say more clearly in the new version, given that Rivas has already shown that the average electric acceleration fields of a luminally circulating charge are identical to the Coulomb field of a static charge, and given that with luminal motions for both the field source charge and the charge being acted on, it should not be surprising that the magnetic interaction can have an inverse square dependence on interparticle separation and be of similar magnitude to the Coulomb interaction. It is less obvious, to me at least, that it should end up being radial. That it should also depend on the phase differences in the luminal motions and that this difference has to account for retardation is at least obvious once pointed out. So, perhaps what I foind is not news after all and if so, I will be happy to see Martin Rivas get 100% of the credit he deserves. On the other hand, if I truly am the first to notice this, I will be very proud, but still want to say that I probably would never have figured this out on my own, and that Rivas deserves most of the credit for it. I hope I have made that clear enough in the version I sent to FOOP last Monday.

I posted a new version on arxiv that is not quite the final version that I sent to FOOP, but fairly close. The new arxiv version has all of the minor math corrections I found (none of which affected the original thesis) and the general overall rewrite. The FOOP version has still a newer title, however, and at least another week's worth of word-smithing which I think (or hope at least) further improved its read-ability. Here's the current arxiv version: Similarity of the Magnetic Force between Dirac Particles to the Quantum Force of Bohmian Mechanics

In any case I have given it my best attempt, I feel, and used the available time as best I could, rewriting it to a large extent, and then going over it a lot of times and long past the point where I was getting tired of re-reading words that I am still not all that happy with. Hopefully in the end it will sink or swim based on the physics contained within.

About the physics of my paper, what I wanted to mention is that since I noticed how the magnetic force between to classical spinning particles could apparently cancel the Coulomb force, and even before I realized that that is similar to the Bohmian quantum force, I knew that this result followed very readily from work done by Rivas at least ten years ago. So far as I have been able to determine he has not published explicitly that magnetic forces can cancel Coulomb forces, but some of his figures that show numerical modeling results, both in his book and published papers, are certainly very close to realizing this to be the case. It's surprising to me that these results seemingly have not attracted more attention. So, one of the things I attempted to do in the re-write is to be more clear about how much of what I wrote in the first version of the paper is actually prior work of Rivas in showing that the time-averaged acceleration field of a luminally-circulating charge can reconstitute, as it were, the usual electric and magnetic fields of a static charge and magnetic dipole. I found this result astonishing from the first time I really noticed it in his book, which was after I decided that I had to calculate these fields for myself. But, I was only planning on calculating the velocity fields and for the case of motion that was only asymptotically close to luminal, not exactly luminal. When I found that Rivas had already addressed this problem, I saw right away that he had both done a far better job than I probably could ever have done, and also that the result was quite surprising and not at all what I was anticipating.

As I try to say more clearly in the new version, given that Rivas has already shown that the average electric acceleration fields of a luminally circulating charge are identical to the Coulomb field of a static charge, and given that with luminal motions for both the field source charge and the charge being acted on, it should not be surprising that the magnetic interaction can have an inverse square dependence on interparticle separation and be of similar magnitude to the Coulomb interaction. It is less obvious, to me at least, that it should end up being radial. That it should also depend on the phase differences in the luminal motions and that this difference has to account for retardation is at least obvious once pointed out. So, perhaps what I foind is not news after all and if so, I will be happy to see Martin Rivas get 100% of the credit he deserves. On the other hand, if I truly am the first to notice this, I will be very proud, but still want to say that I probably would never have figured this out on my own, and that Rivas deserves most of the credit for it. I hope I have made that clear enough in the version I sent to FOOP last Monday.

I posted a new version on arxiv that is not quite the final version that I sent to FOOP, but fairly close. The new arxiv version has all of the minor math corrections I found (none of which affected the original thesis) and the general overall rewrite. The FOOP version has still a newer title, however, and at least another week's worth of word-smithing which I think (or hope at least) further improved its read-ability. Here's the current arxiv version: Similarity of the Magnetic Force between Dirac Particles to the Quantum Force of Bohmian Mechanics

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