Is the Quantum Hamilton-Jacobi Equation just the Classical Hamilton-Jacobi Equation Properly accounting for Magnetic Interactions?

Wikipedia has a nice description of the quantum potential of Bohmian Mechanics, and the more I read it, the more I think it can fit in with the idea that quantum behavior is just the classical electrodynamical consequences of elementary particles being charges circulating at the speed of light.   In the quantum potential article, it is already discussed that the quantum potential can be equated to the internal energy of the zitterbewegung.  What better way for this to influence the motion than via magnetic interactions?

However, de Broglie-Bohm theory is, like quantum mechanics generally, a non-local theory.  Classical electrodynamics is of course Einstein-local, but it's also widely recognized that time-symmetric electrodynamics, while remaining Einstein local, has a substantial similarity to non-locality. Wheeler and Feynman famously used time-symmetric interactions with the rest of the universe to explain radiation reaction. So, the classical time-symmetric Hamilton-Jacobi equation accounting for magnetic zitterbewegung interactions may lead to a theory that has the apparent non-locality of quantum mechanics.

So far I've been focusing on just one small part of the total magnetic interaction; the part that's radially-directed and has a non-vanishing average and falls off inverse-squarely at long distance.  There are many other terms that would also need to be taken into account, particularly at distances where the separation is not large compared to the Compton radii of the interacting particles.  On the other hand, those that fall off less than inverse squarely will have a longer-range character, and when treated time-symmetrically, can plausibly interact significantly and seemingly non-locally with, say, slits in an obstructing plane.


A couple of days ago I got my paper returned from Physics Letters A.  Here is what the editor said:

Ms. Ref. No.:  PLA-D-14-02082
Title: Radially Acting, Relative Spin Phase and Polarization Modulated, Inverse Square Law Magnetic Force Between Spinning Charged Particles
Physics Letters A

Dear Mr. Lush,

Thank you for submitting your work to our journal.
I have read your manuscript with interest.
I regret that it is not suitable for publication in Physics Letters A since it does not satisfy our criteria of urgency and timeliness.
Please consider submitting your work to a journal which has a more pedagogical bias.

Yours sincerely,

{name removed)
Editor


I think I will take the kind editor's advice and submit it to a different journal, but first I'm going to add some speculation about the quantum force and the quantum Hamilton-Jacobi equation.

Is the Quantum Force of Bohmian Mechanics the Magnetic Force Between Spinning Dirac Particles?

In Bohmian mechanics there is a quantum potential, and from its gradient
can be obtained a "quantum force."  A few days ago I calculated the quantum force for the ground state of hydrogen and got that it's equal and opposite the Coulomb force between two charges.   This is consistent with what I remembered about Bohmian mechanics, that in the s states the electron is stationary.  This seemed far-fetched to me.  However, it is not entirely unlike my recent finding regarding the magnetic force between two spinning charged particles where their spin is a consequence of circulatory motion of the charges at the speed of light.

In any case, if we have two particles that are going around in little circles at the speed of light (what I called zitter particles in my new paper, but Rivas refers to in the electron case as a Dirac electron, if I understand him correctly) then it's easy to have that the magnetic force between them exactly cancels the Coulomb force expected if the were static charges, if the circular motion centers stay fixed.  The phase relationship between the two particles' positions in their circulatory motions however must be evaluated accounting for the propagation delay.  This means that the phase relationship including retardation changes with separation, modulating the inverse square law magnetic force radially-acting part. The modulation wavelength is the Compton wavelength.

It's not obvious to me how this might correspond precisely to the quantum force of Bohmian mechanics but it's at least reminiscent of it.  Of course, a classical force corresponding to the quantum force cannot cancel the Coulomb force precisely everywhere.          

My new paper on relative spin phase modulated magnetic force

When I was thinking it was necessary to take a difference between retarded and advanced fields to have a time-symmetric electrodynamics that's Lorentz covariant, I investigated whether a magnetic interaction between point charges moving in little circles at or near the speed of light could play the role of the Coulomb force in binding atoms.  Really it can't, but it's remarkable how close it comes.

I needed a force that was radial and of the same strength and range dependence as the Coulomb force.  Rivas shows in his book, Kinematical Theory of Spinning Particles, how the electric acceleration field of a point charge moving in a circle of an electron Compton wavelength radius has an average that is inverse square like the Coulomb force and of the same strength.  (That is pretty surprising given that the acceleration field falls off explicitly only directly inversely with distance, hence it's characterization as the radiation field.)  But, since the magnetic field in Gaussian units is just the electric field crossed by a unit vector, the magnetic acceleration field for a charge doing the relativistic circular motion has the same strength as the electric field.

If we have a magnetic field that's just as strong as a Coulomb field, and the charge moving in it is moving at the speed of light, then v/c for the charge is one and the resulting magnetic force via the Lorentz force law is just as strong as the Coulomb force.  So, if we suppose the test charge is going in the same type of little circle as the field-generating charge, if the two motions are exactly aligned and in phase, then it turns out there's a purely radial component of the force that's constant in time and just as strong as the Coulomb force. If the circular motions of the two particles are out of phase, then the strength varies sinusoidally with the phase difference, and so it could either double or cancel the electric force.  But the phase difference has to include the time delay of propagation from one circulating particle to the other. (It's important to realize that the charges aren't orbiting around each other, but each doing their little  circular motions separately, with the circle centers many (Compton wavelength) diameters apart.)  So there can be a very substantial influence on the motion of the center of the test charge's circular motion due to the magnetic force, if it is already also moving because of the electric field, due to the relative orientation of the charges plane of motion (which translates to the spin polarization) and the difference in phase of the internal motions of their spins.  The phase of the zitterbewegung motion of the Dirac electron has already been shown by David Hestenes to correspond to the phase of the wave function of a free electron.  So for bound particles, it doesn't seem unreasonable to suppose that the wavefunction involves the relative zitterbewegung phase of the interacting particles.

In the process of putting the story together about this magnetic force as an alternative to the Coulomb force in the time-symmetric picture, I realized (see the previous post) that I missed a sign change in going from retardation to advancement of the magnetic field, so that it does not change sign, and so the magnetic force does not need to replace the Coulomb force but only augment it, in order to plausibly explain quantum behavior.

I'm looking forward to understanding how this picture plays out, but that will take a while, so for now I am putting out what I have.  It will appear on arxiv tomorrow, but I have already posted it to Researchgate here.
          

Correction

A few days ago I realized I'd overlooked a different way out of the problem that time-advanced magnetic fields want to reverse sign compared to retarded ones, and so causing magnetic effects to cancel out in the time-symmetric picture. The cancellation doesn't happen because the sign on the unit vector from the source to field point, that is crossed onto the electric field to obtain the magnetic field, also changes sign in going from retardation to advancement.  That is, in the retarded case we have B = n x E, but in the advanced case this becomes  B = -n x E.   The unit vector n here changes sign because it originates as the gradient of the retarded or advanced time, so the sign change that changes retardation to advancement applies directly to it.

This means the problem I've been working to overcome for the last eight or ten months does not even exist.  In particular, the cancellation of the strong magnetic force that I was getting in the time symmetric picture does not occur. So, it can account for preon binding (for charged preons, at least) just fine. What I have been saying recently, that Lorentz covariance requires magnetic forces be preserved in the time-symmetric picture, if there is one, is perfectly true, but it is also completely consistent with electrodynamics and time-symmetric electrodynamics under the usual assumption that the retarded and advanced solutions are summed rather than differenced.  Writing it up for publication forced me to realize this, because when I went through, as a simple demonstration of the problem, calculating the magnetic moment from a static current loop from the retarded potentials (see, e.g., Landau and Lifshitz  Eq. 66.2) I discovered that it absolutely does not change sign when going over to the advanced case.

Now, everything is better except the idea that the electron g factor of (about) 2 can be explained as the difference between retarded only and time symmetric electrodynamics.  I still think this is a very attractive idea, but the story isn't as compelling because it isn't true as I said that in the conventional time-symmetric picture the g factor would be zero, i.e., that the electron magnetic moment would vanish.  In the usual picture (attributable to Dirac 1938) the time symmetric field is the mean of the retarded and advanced fields, which leads to a g-factor of one.

Maybe I should give up my obsession with the electron g factor, now that I am routinely thinking of the electron as a composite particle.  If it has a positive charge part with opposite but smaller spin, doesn't that give a g factor larger than unity?  Maybe g factor two is easy to get in the preon model.  I am still used to thinking of an electron as a structureless object so this kind of thinking is not natural.  Maybe g-factor two is simply a confirmation that it's a composite particle.

Later I think I will look more at the g factor of a composite electron, but for right now I'm trying to complete something entirely new to put on arxiv and hopefully soon after submit to a journal.  Hopefully I will upload it to arxiv within a day or two.   Before that happens, I am also planning on revising my kinematics arxiv paper to remove the new section I just added a few weeks ago.  Probably I will do that later today.  I can't let it stay up there long knowing it is dead wrong.

    

Why the electron g-factor is 2

Today I realized (I think) that if electrodynamics is time-symmetric, and if the magnetic force does not flip its sign for the advanced magnetic forces compared to retarded magnetic forces, then this will naturally double the strength of the magnetic forces at scales where the retarded and advanced interactions are experienced close to in phase.  So the electron magnetic field, if produced by moving charge, will be twice what it would be for retarded only forces.  So perhaps the electron g-factor being (about) 2 can be taken as confirmation that electrodynamics is time-symmetric.

After mulling it over for a while today, I decided to do a quick update to my arxiv paper to include this observation. I added a new section V that consists of 3 paragraphs with no equations. It will post tomorrow (as v8) if I don't change it further and reset the clock.  (I also corrected Eq. (8), which did not affect any subsequent results.  It may become relevant in the next update (v9) however.  I have a lot of material towards a new version beyond v7/v8, but it was inconclusive until the new ideas of the last few weeks, which tentatively seem to be panning out nicely.  I have only been working on it again for the last few days, though.  Prior to that I was unusually tied up for several weeks with my engineering job on a hot project.)

Another thing I want to mention, that I was being coy about in my last post, is how it might be possible to have electrical velocity fields invert sign between retardation and advancement, and still have apparent electrostatic forces between (apparently) stationary charges.  The way it might possible is if what we take as electrostatic electric fields and forces are actually time averages of  electric acceleration fields.  Martin Rivas (citation will be in the new posted version, and is in some of the earlier versions already) has already shown how if the electron is modeled as a circulating point charge moving at the speed of light (and it will still be true for asymptotically close to the speed of light), then the time-averaged acceleration field is Coulomb-like by several Compton radii away (when the radius of the circular motion is the Compton wavelength).  Also, for the ultra-relativistic charge, the velocity field collapses to a point and so doesn't contribute to the average Coulomb-like field.

I spent today trying to modify Rivas' calculation to see if I can get a similar result in spite of switching the sign of the advanced forces.  It seems intuitively that it couldn't but I am encouraged, as far as I got today. It doesn't make it vanish identically, as my intuition predicts it should have.  This is a very preliminary observation, so maybe it will fall apart, but it shouldn't take long to get an answer one way or the other.  I have another reason to be optimistic, though, because I also tried yesterday adding sign-reversed (compared to the usual) advanced fields into my attempted derivation of anti-Euler forces from the velocity magnetic field terms, and now it does seem to be emerging.  I have spent six long months trying to get this with no prior success, so it seems very encouraging. This is also only a preliminary observation that could evaporate.  I still have a lot of work to do before I can have something to submit to a journal, but I feel like I'm making serious progress again finally, after months of getting nowhere fast.

      

About the sign of advanced magnetic forces

In time-symmetric electrodynamics, it is always assumed that the sign of the electric force is the same for the advanced force as for the retarded force.  This must be so (one would think) because in the inertial reference frame where an infinitely heavy charge is at rest, a test charge held initially at rest and then released would experience no net Coulomb force if it were otherwise.

It follows from this seemingly necessary choice that the sign of the magnetic force in a different inertial reference frame, where the heavy charge is in motion, will invert for advanced compared to retarded magnetic forces.  So, in time symmetric electrodynamics, magnetic forces tend to cancel out.  This appears at least at first glance to keep the magnetic force that seems to correspond to my predicted anti-centrifugal force from being able to overcome Coulomb repulsion.

On account of the considerations above, I have recently been going carefully over how to get the time-advanced and time-retarded fields and forces by Lorentz transforming from the reference frame where the field-source charge is stationary.  (The retarded case is already analyzed quite a bit in the appendices of my arxiv paper.)  There haven't been any surprises there, but a few days ago I began to realize that the derivation of the magnetic force as the anti-Coriolis force of the Thomas precession doesn't care whether the field or force is retarded or advanced.  It can't change sign from retardation to advancement because neither has entered into the derivation in any way.  Thus, if the anti-Coriolis force is a real force, then either time-advanced electromagnetic forces cannot exist, or the sign of the Coulomb force must flip between retardation and advancement.  Consistency of the force law with relativistic kinematics demands this, if I am correct.  

Superstrong electromagnetic interactions

Since I've given up trying to put out a separate paper quickly on the superstrong magnetic force between highly accelerating ultrarelativistic charges, as I said in the previous post,  and have gone back to trying to finish my more general paper on the relationship between electrodynamics and relativistic kinematics, I should report out a little on what I found and didn't find.

The strong attractiveness of the magnetic force in the retarded magnetic acceleration field is already shown in the version posted on arxiv.  What I was trying to determine was whether there's an obvious way there can be net attraction in the time symmetric case, as considered by Schild, where the magnetic force due to the advanced field tends to cancel the force due to the retarded field.  My idea was that in the ultrarelativistic case the delay and advance angles approach 90 degrees, so maybe it might be possible to change the phase relationship so that the net force is strongly attractive on average.

It turned out that although I was able show tentatively that the retarded and advanced forces don't have to exactly cancel and can easily exceed in magnitude Coulomb repulsion, I wasn't able to generate a net attractive force.  I may try further later but for now I have gone back to trying to finish the more general argument.

What I did was to assume two charges were circularly orbiting each other in an approximately circular orbit with an orbit diameter smaller than one one-hundredth the size of a proton, and at a velocity very close to the speed of light.  I wrote a matlab program to calculate the full retarded and advanced, and non-radiative and radiative, electric and magnetic fields at the position of one particle due to the other, and accounting for delay and advancement, where the motion was assumed to be circular and periodic, but allowing the accelerations to depart from the strict centripetal acceleration of a pure circular orbit.  That is, I let the non-centripetal acceleration affect the fields but not the orbit.  Then I looked at the induced acceleration of one particle due to the other, and attempted to construct a configuration where the motions of each particle induced by the other would be consistent.  I totally ignored radiation damping, as did Schild, although it's enormous in this configuration.

It turned out to be pretty simple to build a configuration where the motions seem approximately consistent in the time-symmetric electrodynamic sense.  A lot more work would be needed to determine if this is a real or meaningful result, and I don't mean to assert that it is.  If I had more confidence I could build something convincingly meaningful in a reasonable amount of time, I'd continue to work on it, but for now I think my time is better spent elsewhere.

To illustrate what I'm trying to describe, I captured a plot from my matlab program, see below.   Clicking on the figure should expand it.  The top two strip plots are what is used to calculate the full em field at the position of the second (test) particle, and then the bottom two plots are the acceleration induced on the test particle.  The scales are not very meaningful because the magnitudes depend on how close the velocity is to the speed of light, and the orbital radius, and the invariant particle masses, and in a complicated way.


The top two strip plots show the motion of the charge the generates the field that the test particle moves in.  That's the field source particle.  The field that the test particle generates isn't allowed to affect the source particle here.  The top plot shows that I arbitrarily imposed a strong radial oscillating (at the orbital period) acceleration on top of the constant radial centripetal acceleration in it's circular circular orbit.   The second plot is showing that there's no comparatively significant motion in the tangential or axial directions.  That's just the noise level when some large positive and negative numbers got added together, at matlab default precision.

Then the time retarded and advanced fields at the test particle position, moving oppositely in a nominally circular orbit, are calculated and the corresponding acceleration of the test particle due to their sum is plotted on the two lower strips. What's interesting to me is that the oscillating radial acceleration of the source particle has induced a similar radial acceleration in the test particle, out of phase such that if it were allowed to act back on the source particle has a hope of leading to a consistent periodic motion, perhaps.  There is also a tangential acceleration induced, but it's much smaller in magnitude.  The smaller magnitude is in part at least due to the difference in relativistic mass along track versus cross-track.

This would be a rabbit hole to pursue seriously, that one might never emerge from.  But it would be fun.

A week or so ago I googled "preon binding force" and turned up an article by Jogesh Pati, the originator of the term "preon," according to wikipedia:

http://adsabs.harvard.edu/abs/1981PhLB...98...40P

Magnetism as the origin of preon binding

Pati, Jogesh C.

Physics Letters B, Volume 98, Issue 1-2, p. 40-44.

It is argued that ordinary ``electric''-type forces - abelian or nonabelian - arising within the grand unification hypothesis are inadequate to bind preons to make quarks and lepton unless we proliferate preons. It is therefore suggested that the preons carry electric and magnetic charges and that their binding force is magnetic. Quarks and leptons are magnetically neutral. Possible consistency of this suggestion with the known phenomena and possible origin of magnetic charges are discussed.

(The article can be downloaded without fee here.)


So, apparently, I am not the first to think preons might be bound magnetically.  However, in order to achieve magnetic binding, the above article postulates that preons possess magnetic charges, which are not required by the mechanism I propose.

I decided to write a short paper on how electrical charges even of like polarity can be magnetically bound according classical electrodynamics, without going extensively into the relativistic kinematics arguments, to submit to a journal as soon as possible.  I thought I could just excerpt that part of my paper as it's currently posted on arxiv, but now I'm wanting to elaborate a little bit further, taking better account of retardation and perhaps looking at how it acts in time symmetric electrodynamics (i.e., allowing for time-advanced as well as time-retarded interaction).  Properly accounting for retardation makes things much more complicated and possibly intractable, but it is impossible to argue that it's negligible in this case.  It is thus not going as quickly as I'd initially hoped.