COMMENTS TO THE AUTHOR:
Reviewer #2: My problem with this paper remains the same. It is a highly contrived model of a system and process that likely has no meaningful relationship with reality. It predicts no new, interesting and unexpected results whereby it might be tested. I regard the likelihood that others will find it of sufficient interest to invest serious effort in extending it as little to none.
The author states his motivation as being to record his efforts in the (unlikely) event that others may want to pursue the topic as he has. That's fine. But it is not a justification for publication in a journal such as FOOP (where real trees are killed to preserve the work of authors). Work such as this properly belongs on the arXiv server, where it is available to others, but does not clutter up the pages of a journal aimed at general interest articles about the foundations of physics. I suggest the author pursue "publication" at arXiv.
In response, I want to state some of my thoughts about whether or not what I found and wrote about is likely to have a "meaningful relationship with reality". I agree with Reviewer 2 that it's not at all obvious it has such. Further, I think that in order for such a meaningful relationship to exist, a certain unlikely condition will have to hold that if true will have far-reaching implications.
In the original version of my paper there was no mention of a possible connection to Bohm's quantum force. The paper merely derived (based strongly on prior results by Rivas) the magnetic force between relativistically-circulating charges and noted that part of it could either cancel or double the Coulomb force expected if the charges were static, depending on the phase difference in the circulatory motions of the charges, and where the phase difference includes the light propagation delay from one charge to the other. At that point I knew that the magnetic force could cancel the expected Coulomb force, but I hadn't brought to mind the similarity this presents to the Bohmian quantum force, so the paper I initially submitted to Physics Letters A didn't make this connection. I simply thought it of interest that such a strong radial and separation-modulated magnetic interaction could exist classically, where generally in atomic theory magnetic spin-spin and spin-orbit couplings are only a small correction to the Coulomb interaction, When PLA rejected the paper I was going to leave it at that, at least for a time, given that I had already put it on arxiv, and that I had completed the due diligence expected in submitting it for review. But the PLA editor did make an encouraging remark, and before long my memory was jogged by a book review by Peter Holland posted to arxiv about the Bohmian quantum force, and so I decided it was worth a try with Foundations of Physics (FOOP).
The initial paper I submitted to FOOP was only slightly modified from the PLA submittal, to include mention of a possible connection to Bohm's quantum force. The title was, "Relative Spin Phase Modulated Radial Inverse Square Law Magnetic Force," a shortened version of the PLA submitted title. FOOP then asked for a revision to make it more suitable for a journal on the foundations of physics. I was happy about this outcome, but I would have been at least as happy to have it published as it was. It put me on the spot to explain to skeptical real physicists why it could be important. The resulting version didn't have any additional physics beyond the initial FOOP submittal, but it tried to explain how it might be important, and included speculation that it might be more interesting if considered from the point of view of time-symmetric electrodynamics.
So, the first review by FOOP was actually the third version, and the second revison compared to the submittal to PLA (which actually corresponds to arxiv version 2). The FOOP pre-review revision is arxiv version 5, and has the title of the final submittal, which refers to Bohm's quantum force. However at this point the idea that the magnetic force I found could account easily and directly for Bohm's force had already fallen apart, as it was apparent that the sign of the force would invert due to change of separation about two orders of magnitude smaller than a Bohr radius. I was expecting initially it would invert at close to a Bohr radius separation. I didn't note the problem explicitly at the time, although it would have been better if I had.
At this point Reviewer #1, who I have to think is Rivas (suggesting a reviewer was required, and I suggested Rivas) said it was already a known result, and that it should not be published unless I extended it by calculating the Poynting vector and show that the system didn't radiate. I didn't expect that would be the result of such a calculation, so I didn't undertake it, and thought hard instead about what possibly could be the meaning of the new force, if any. I had an idea that perhaps the rapid sign variation could be avoided by supposing the net force was a sum of time retarded and advanced parts, and that in certain states of relative motion the advanced and retarded points could have a phase relationship so that the net magnetic force could exactly cancel the net electric force, as the Bohmian quantum force does in s states. I got an initial result that one of two modulated factors could be held constant for particular relative velocities that could be related to the Bohr velocities. A few days later I was able to relate it also to the de Broglie wavelength. I felt like putting this result on arxiv as soon as possible, so I did, and then I submitted almost the same version to FOOP, as the due date for the requested revision had arrived. In addition to relating the de Broglie wavelength to a modulation of the magnetic force, the new version discussed the problem of the too-rapid sign changes with change in interparticle separation. However while the new revision was being reviewed by FOOP I continued to work on a further improved version. I didn't feel like I had a working model for the Bohmian quantum force. I had only related the de Broglie wavelength to a modulation of the envelope of a rapidly-varying time-symmetric net force, where I had hoped to obtain a constant force. Also, I was getting a modulation with a wavelength of twice the de Broglie length.
Over the next few weeks I managed to improve the situation somewhat. I related the de Broglie wavelength exactly to a modulation caused by time dilation of the circulatory motion of a charge with a non-zero average motion. Also, while the other modulation applied to radial motion only, the modulation due to time dilation was valid for general motion, and when considered in conjunction with the modulation due to radial motion resulted in a modulation with the de Broglie wavelength for general motion. It was about a month into the review process, but I wanted to let FOOP know I had an improved version, so I posted it to arxiv and wrote a letter informing FOOP it was there in preliminary form. Shortly after sending the letter I received the second set of reviews from FOOP, and the provisional acceptance with the decision of "Minor Revisions". Actually the only revision explicitly requested was that I change the title, from Reviewer 2. Reviewer 1 said the new comments "increased the scientific value" of the paper and that it was now acceptable for publishing, I didn't receive any new comments from Reviewer 3. The letter from FOOP gave me two months to do another revision, which I took to be a tacit acknowledgement of my letter requesting that I be allowed to do another significant revision.
When I evaluated the effect of relative average motion of the circulating charges, I considered the problem from the point of view of the (test) charge circulating around a fixed point while the charge creating the field that acts on the test circulating is the one with net motion. I wanted to get the same result in the reference frame where the test charge has net motion and the field-creating charge circulates around a fixed point. The derivation of the field in the paper (as done by Rivas) assumes the motion is around a fixed point, so switching to the other frame violated that assumption, which didn't worry me much but could have been objected to. Mainly, I wanted to get more confidence in the results I was getting by getting them in a different reference frame. I had initially meant to do it in the frame where the source charge has no average motion, but it was not as obvious to me how to do it as it was in the other frame.
When I recalculated the force with relative motion in the source rest frame, I was unable to reconcile the two results exactly. They should not be identical, but I expected them to be consistent with the Lorentz transformation law on the four-force. I was not able to reconcile them exactly by the new deadline, but the derivation in the source rest frame is simpler and preserves the modulation relationship to the de Broglie wavelength, so I submitted it. Both versions are now posted to arxiv. Version 8 is newer than the FOOP revision 2 that was provisionally accepted, but will be superseded by arxiv version 9 which is identical to the final submittal to FOOP except it's in two-column format (in part to avoiding being mistaken for the version that I gave the copyright to FOOP). This current version link will point there after tomorrow when it becomes public, and I don't plan on making any more updates.
In conclusion I can say that I've been unable to make convincing sense out this magnetic force between relativistic circulating charges, but I continue to think it has some relationship to reality. That it is so easily related to the de Broglie wavelength should be enough to make it worthy of further consideration, it seems to me if not to Reviewer #2. However it does not seem to be equatable to the Bohmian quantum force as I had originally supposed. But as I remark in the final version of my paper, there is one obvious electromagnetic force that could cancel the Coulomb force as the Bohmian quantum force does in quantum s states. It is the time-advanced electric force. In the case where the time advanced Coulomb force exactly cancels the retarded Coulomb force, then a force such as the one of my paper is needed to replace the Coulomb attraction.