Status Report and a Hypothesis about Why the Electron g-factor is 2

After months of feeling stuck and confused, I have finally been making real progress towards getting my paper on the origin of the magnetic force revised so that I can re-submit to a journal.  I saw a different way of going about things, compared to how I was doing it previously, and it really seems to have paid off with new insights.

One new insight is that the force I've been investigating for the last 4 years, the anti-Coriolis force due to Thomas precession of the rest frame of a field-source charged particle as observed from an inertial frame, is not the known magnetic force.  The most obvious reason it cannot be the magnetic force is that it vanishes when there is no acceleration of the source particle, whereas the magnetic force needs only a translating, non-accelerating charge to set up a magnetic field that can act on a second translating charge.  A second problem is that the force (as described in my paper) has a mass ratio factor, of the ratio of the mass of the particle being acted on to that of the field-source particle, that is not seen in Maxwell-Lorentz electrodynamics.  However, due to the fact that most electrodynamical observations are based on electron currents interacting with other electron currents, so that the mass ratio factor would be unity,  I was ignoring this problem for the time being.  Happily, though, I now believe I have a new understanding of how the magnetic force can arise kinematically as a consequence of Thomas precession, and as an anti-Coriolis force, that does not have these difficulties. 

The way around these aforementioned difficulties is to recognize that even if the field-source particle is non-accelerating, because of assumed essentially infinite mass or other constraint (such as being confined to an electrically-neutral wire),  its rest frame will undergo Thomas precession from the point of view of a test particle that's being accelerated toward the field-source particle by Coulomb attraction.  A physicist-observer in the rest frame of the test particle will notice that although the source particle rest frame is rotating, there is apparently no Coriolis force acting on the test particle in the source particle rest frame, and so deduce that there must be an anti-Coriolis force acting on the test particle there.  In this case, however, the numerator mass factor needed in the anti-Coriolis force is cancelled by the denominator mass factor of the same particle acceleration that induces the Thomas precession of the source-particle rest frame.  This obtains a magnetic force that agrees with observation in not involving a mass ratio between the field-source and test particles, nor an acceleration of the source particle with respect to an inertial reference frame. 

I hope to be able to provide this description quantitatively fairly soon.  I will update the paper on arxiv as soon as possible.  I may even do a retraction that acknowleges the known problems of the current version and outlines the new approach and findings, in a week or so, prior to posting the full revision.  In the meantime, though, I want to mention the new idea I am having about how all of this might relate to the known (so-called) "non-classical" gyromagnetic ratio of the electron.  It is of course a well-established fact that the electron produces intrinsic magnetic moment very close to twice as effectively as expected from classical electrodynamics, given its intrinsic angular momentum and charge-to-mass ratio, and assuming it has no complex structure.  The idea is, this other force that I've been studying that is the anti-Coriolis force due to Thomas precession of a field-source rest frame when the field-source particle is accelerating under Coulomb attraction or repulsion to the test particle, and a relativistic-kinematics necessity just as is the magnetic force, will effectively double the magnetic field strength for an electron that is generating magnetic moment, at least for one model of the electron spin,.  (In the zitterbewegubg model, the electron spin is a consequence of a relativistic circularatory motion of the point charge electron, with radius of the motion equal to the electron Compton wavelength.)  Thus the g-factor of two is to be expected, at least according to this model, which I have found compelling for other reasons.

Why There Can Be No Rotational Inertial Forces in Thomas Precessing Reference Frames

There is a simple and compelling argument for why the Thomas precession (TP) does not give rise to rotational inertial forces such as the centrifugal, Coriolis, or Euler forces, for hypothetical observers or systems that are undergoing TP.  There can be no forces of rotational inertia in such reference frames because the rate if any of TP is entirely observer dependent.  An inertial frame observer sees  TP of a gyroscope only if the gyroscope is simultaneously translating and accelerating transversely (i.e., crosswise) to the relative translation.  The amount of TP, that is, its angular velocity or rate of precession, is then readily calculated as proportional to the vector cross product of the gyroscope's relative translational velocity and acceleration. Thus, because the TP is calculated by the external, inertial frame, observer and depends entirely on the relative translation and acceleration, it is not uniquely defined from the point of view of an observer co-moving with the gyroscope.  Other observers in inertial reference frames moving at different velocities relative to gyroscope observe a different rate of TP of the gyroscope.  Indeed, from the point of view of an observer moving with the gyroscope, there is no TP of the gyroscope at all, since the relative translation and acceleration both vanish.  Therefore the co-moving observer cannot experience rotational inertial effects such as the centrifugal or Coriolis pseudoforces.

It is a very short step from making this observation to understanding the origin of the magnetic force, and probably the strong and weak forces as well. 

A sign error in my arxiv paper

I just found a sign error in my latest arxiv version (v3) of my paper:http://arxiv.org/abs/1108.4343v3. The second sign from the right on Eq. (20) is wrong.  The gamma^2 factor in the denominator of the first term on the right of equation of Eq 21 has to move to the numerator. Also then the equation is an approximation but still good to order (v/c)^2, which is all it ever was good to, at best.

My philosophy is: finding errors is  always a good thing, especially when you can find them yourself.

Sorry for letting it slip through.

I was actually looking for an error, because things were not making sense as it was.  With this correction, and with what follows it but is unposted so far, I hope to be able to show soon that the magnetic force between two particles of equal mass can be derived exactly and generally from an assumption of no pseudoforces in Thomas-precessing particle rest frames. 

As version 3 of the paper correctly shows, the complete magnetic force is present in the anti-Coriolis force of the Thomas precession, but there are extra terms as well.  These originate because the velocity in a particle rest frame is relative to the particle, while the magnetic force is proportional to the (other) particle velocity in the inertial frame.  This leads to an additional vector triple product term (the last term in the right of Eq. (19)) which then can be split into a sum of two vector terms with dot-product factors (Eq. (20)).  One of them  contributes a gamma-squared factor (that now has to move to the numerator but can still be subsumed into the full relativistic electrodynamic description, I think)  and the other vanishes in the circular orbit case (only).  So, if the magnetic force is to result exactly with no leftover terms, something has to cancel that leftover term, or it must get subsumed somehow into the proper relativistic electrodynamic descrption of the electromagnetic two-body problem in an inertial frame.  This one I think will get cancelled by the anti-Euler force, in the case of non-circular motion.   I got a suitable term from the anti-Euler force, but it was the wrong sign to cancel, hence I was hunting for a sign error. 

I may be days away from having everything worked out and ready to get into shape for re-submission.  This will take at least a month but perhaps I can post something sooner in rough form if I get everything to work out, and then do a final clean-up prior to submission. 

No evidence of ethics in Romney's behavior

Mitt Romney says, he paid the legally-required amount of taxes, and we wouldn't want a president who paid more than the legal amount, would we?

Personally I'd be quite comfortable with a president who figured his taxes simply and without inventing heroic dodges to avoid paying his fair share, like whatever he did to accumulate tens of millions in his IRA, when the yearly contribution restrictions make it nearly impossible. 

Romney apparently believes only a fool would fail to exploit every loophole.  It does not even occur to him there are more noble ways of thinking.

This modus operandi is well evidenced in his business dealings as described here: http://www.bloomberg.com/news/2012-07-15/romney-s-bain-yielded-private-gains-socialized-losses.html . 

There is no trace of introspection or social concern in this man.  We certainly don't want him for our President.

New version of my paper explaining the origin of the magnetic force

Last week I posted a version of my paper linking the magnetic and strong forces with the Thomas precession on arxiv here.

The revision was needed to correct several errors and misunderstandings, as I previously described.

The new version doesn't attempt to extend the analysis, it just corrects the recognized errors and reflects my better current understanding.  It contains an Errata section identifying the corrections.

I'm still working on extending the analysis beyond the bound, circular orbit case.   Any case other than this must invoke an anti-Euler force as well as becoming generally more relativistic in nature.  The circular orbit case is greatly simplified by the fact that the force is perpendicular to the velocity, and so the relativistic gamma (or Lorentz) factor doesn't come much into play.  Also, the Euler force doesn't come into play since the angular velocity of the Thomas precession is constant.  So, relaxing the restriction to circular motion makes the analysis much more complicated, but it should also become much more persuasive if it all comes together in a consistent way, as I believe it must.

Also, I'm still uncertain whether an anti-Euler force implies the existence of hitherto unknown forces, or merely leads to previously-known forces such as the magnetic and weak forces.  As I  mentioned a couple of posts ago, it gives rise to force terms at order (v/c)^2 that are of course at the same order as the the magnetic force.  But, the full relativistic treatment of the classical two-body electromagnetic problem for (spinless) charged particles already has a lot of terms at this order that are not directly recognizable as magnetic force related terms.  So, the anti-Euler force terms could simply correspond to terms already present in inertial-frame electrodynamics.  Showing this to be the case will make the derivation quite persuasive, seems to me.  On the other hand, if there are new forces at the same order as the magnetic force, consistency with observations to date demands they would have to average to zero or be otherwise difficult to notice.  Perhaps if they were difficult to notice but gave a hint of where one might look for confirmation, that would be of interest.  However at the moment I'm optimistic that the anti-Euler force at order (v/c)^2 will simply be relatable to electrodynamics as it is already understood.  The anti-Euler effects at higher order in v/c however I think will have to correspond to either the weak or new forces.
   

I Haz Cites

One of them was expected and I posted about it previously (but see also here), but the other was completely unexpected.

Vladimir Hnizdo's paper, "Spin-orbit coupling and the conservation of angular momentum"  (http://arxiv.org/abs/1103.4092) has now been published in European Journal of Physics (subscription or payment required).  This paper cites two of my arxiv papers (0905.0927       and 0709.0319) and also acknowledges me as raising the issue the paper addresses.

The other citation is in a commissioned review article, "Resource Letter EM-1: Electromagnetic Momentum," (http://ajp.aapt.org/resource/1/ajpias/v80/i1/p7_s1) by David J. Griffiths, in American Journal of Physics.  (There is apparently no arxiv e-print but the references may be found here.  The citation of my paper is also noted here.) It appeared in January but I just found out about it last week.  My paper is cited as one of three examples of ongoing research related to "hidden" momentum.  My arxiv paper is in response to one of the other two articles, that was published in American Journal of Physics, that I commented on, and as I posted about previously, my comment was published in the journal, followed by a response by the original paper authors that seemed to negate my comment. Then, when I wrote a further response, it was rejected.

I don't mean to fault American Journal of Physics particularly for not publishing my paper that is now being cited by their own comissioned review article, as I understand they have many many submissions and as a pedagogical journal have to be averse to controversy.  However I am very pleased that this paper is being cited and by such a distinguished author.  It is especially nice because I had essentially given up on that paper and had no further plans to submit it anywhere, and yet it took a lot of time to do the work and write and I thought it obtained an interesting result.  Now it seems reasonable to think I was not the only one to find the results interesting (and in fairness, two of three of the referees were positive about it). 

I do hope and even expect that eventually I will be getting my work published in mainstream peer-reviewed journals, but until it is, getting cited in mainstream journals is a pretty good substitute.  If it was a choice to get published but not cited, versus not published but cited by such distinguished authors, I would have to choose the latter.

Status of my paper on the origin of the magnetic and strong forces

I posted my paper, "Does Thomas precession cause rotational pseudoforces in particle rest frames?" (http://arxiv.org/abs/1108.4343) in August 2011 and updated it in early September 2011.  The update added the assessment of whether the kinematically-necessary anti-centrifugal force of the Thomas precession could plausibly correspond to the strong force that accounts for quarks combining to form nucleons.  As I described in earlier posts, it has been submitted to several journals, but not peer-reviewed.  Physica Scripta sent it out for review, but then withdrew it when the first referee objected because I called it a plausibilty argument.  Since then I've been working on a more convincing, particularly more relativistically precise, argument that I hope the journals will consider more suitable for publication.  It is taking quite a while to carry this out, but in the process I have found a number of flaws in the paper as currently posted that I want to acknowledge my awareness of.

Fixing the flaws in the paper will help make the next version more convincing, especially by removing the restriction of the current version to bound motion.   The approach I'm taking is not just to fix the flaws, however.  It is rather to present a completely rigorous electrodynamic analysis for general motion to order (v/c)^2 (and to (v/c)^4 in the case of circular bound motion), which is much more complicated than the minimally-relativistic treatment I posted on arxiv and previously submitted, and so is taking quite a while.

I'm still in the process of carrying out the analysis, and don't yet have a complete and convincing argument, but I have a few things I think are worth reporting prior to availibilty of a new version. 

I also want to mention that I found some related prior work after posting the current arxiv versions, and there is a paper that appeared on arxiv shortly after mine, by Royer (http://arxiv.org/abs/1109.3624), that also describes the magnetic force as the anti-Coriolis force of the Thomas precession.  One prior paper is by Bergstrom, "On the Origin of the Magnetic Field".  Another is "New perspectives on the classical theory of motion, interaction and geometry of space-time," by A. R. Hadjesfandiari. (http://vixra.org/pdf/1011.0058v1.pdf)

My interpretation of all of these papers is that they have a different perspective than mine.  They are concerned with the dynamics or kinematics of the particle experiencing a magnetic or general electrodynamic force and as mediated by a given electromagnetic field, whereas my approach is more oriented around a direct two-particle interaction, and the kinematical consequences of requiring physics to provide consistent descriptions from the point of view of various observers, in particular an observer moving with the particle that is the source of the field.  

The Bergstrom paper, from the early 1970s, attempts to explain the magnetic force as a Coriolis force.  The magnetic force however is  properly an anti-Coriolis force (in that it accounts for the absense of a Coriolois force in the rest frame of the particle that is the source of the magnetic field).  Bergstrom can't get to this result however because he is using the incorrect formula for the angular velocity of the Thomas precession, as in Moller and many subsequent textbooks, which leads to a sign error.  This also hampered me for about three years, until I realized the sign had to be wrong, and then immediately after remembered that Malykin had said exactly that in his review paper (cited in my paper). 

Royer neatly avoids the problem of the what is the proper Thomas precession angular velocity, by doing the analysis directly with successive Lorentz transforms.  That leads to the correct interpretation that the magnetic force is an anti-Coriolis force.  However, Royer mentions that there is no anti-centrifugal force expected, as I agree for his approach.  That is, there is no anti-centrifugal force in the electromagnetic field.  The need for the anti-centrifugal force doesn't become obvious until one tries to describe an electrodynamic interaction in the rest reference frame of the source particle.



I found the Hadjesfandiari paper shortly after finishing the second (and currently-posted) version of my paper.  When I realized that the Lorentz force must be incomplete, omitting as it does both anti-centrifugal and anti-Euler forces, I googled "Lorentz force incomplete" and turned it up.  (It is from late 2010 and so is prior to mine being posted, although there is evidence on the web of me having the basic idea in the summer of 2008.  That was when I noticed that the expected Coriolis force on a moving charge in the rest frame of a charged particle (of equal mass) has the same form and magnitude as the magnetic force.) 


Now about my paper, the first thing I want to mention is that I now believe that the restriction I made to bound motion was based on an error of understanding and is unnecessary.  It seemed at the time though to solve a problem I was having with getting the magnetic force to equate exactly to the anti-Coriolis force of the Thomas precession.  The problem is that in the laboratory frame where (say) the center of mass of two interacting charged particles is stationary, the magnetic force on either particle depends only on the velocity of that particle relative to the local magnetic field.  The source-particle velocity enters through the magnetic field, but not in the interaction of the other particle with the magnetic field.  But, in the source-particle rest frame the expected (but absent) coriolis force is based on the relative velocity of the non-source particle to the source particle, which is equal to the velocity difference between the particles in the lab frame.  This led to some stray terms that made the analysis seem incorrect, but I could see they would vanish for bound motion, so I decided to make that assumption in order to get a working paper. Now however I am fairly certain that those extra terms are just part of the electrodynamics in the lab frame.  Showing this will be something that makes my whole thesis much more convincing, but I am still working out the details.  Since there is of course no restriction to bound motion in the laws of electrodynamics, it isn't very good to have to make it for my argument that the magnetic force is an anti-Coriolis force of the Thomas precession.


Another flaw with the current version of my paper is that it fails to be sufficiently cognizant of the general necessity of including the anti-Euler force.  I mention that there must be an anti-Euler force, and that there isn't one currently in electrodynamics (although it may correspond to the weak force), but I failed to take to heart that it needed to be included in the analysis unless the motion is restricted to zero radial motion between the particles.  The next version will be at least cognizant of this fact, and possibly may include the anti-Euler force explictly to order (v/c)^2.  I have tentatively obtained a description of it to order (v/c)^2, which I would like to publish as soon as possible.  It's quite simple to describe to this order, although it has a very complex description when higher order terms are included.  These high-order terms will become significant at nuclear scales, which opens up the possibility of the anti-Euler force accounting for the weak force, which also has a complicated description and is significant only at nuclear scale.  However the anti-Euler force if I have things right also has a effect at order (v/c)^2 that I'm very eager to incorporate in my positronium atom model, as I have mentioned in at least one other post here.  At order (v/c)^2 the anti-Euler force can be relevant at the atomic scale.

Still another problem with the current version is that it isn't sufficiently careful about keeping the order of the analysis consistent. This leads to a confusing and unconvincing handling of the centrifugal force. The centrifugal force is an order (v/c)^4 effect, while the magnetic force is a (v/c)^2 effect, so it would have been better to throw the former away sooner than to carry it along as long as I did. On the other hand, working the analysis at order (v/c)^4 is complicated but possible in the case of circular motion (where the delay equation can be solved exactly), and it becomes clear that the Lorentz force as we know it today does not account for the needed anti-centrifugal force. This is an analysis I hope to include in a future version, but it may not be the next. Getting everything to work out exactly at order (v/c)^4 has eluded me so far and I don't want to hold up the next version just for that.


Finally, I might mention that thanks to a friend I'm now aware of a pedagogical write-up on the Thomas precession (in use at UC Berkeley) that explicitly states there is no centrifugal force experienced by an observer in a Thomas-precessing frame, due to the Thomas precession (http://bohr.physics.berkeley.edu/classes/221/0708/notes/thomprec.pdf.)  This answers the titular question of my paper in the negative.  The question was really only meant to be rhetorical, in any case. I think it should be rather obvious that if the Thomas precession is to be a non-trivial effect, it must not give rise to rotational pseudoforces in the particle rest frame.  The point of asking the question was that if there are indeed no rotational pseudoforces in Thomas-precessing particle rest frames, kinematics requires the presence of compensatory forces in the inertial laboratory frame.  Seeing the statement explicitly in teaching materials should be justification for somebody to ask what are the kinematical implications of the absence of centrifugal forces.  I believe they are profound and warrant further attention.

Why the links no longer work

Many of the links to the literature on this blog's homepage no longer function, unfortunately, because of an apparent clamp-down by the original journals, or at least some of them, which forced the people operating the sites where they resided to take them down or behind a firewall. 

I think it is especially unfortunate that the journal publishers are so touchy in the case of the sort of physics literature relevant to my work.  It is often decades old and probably not of very wide interest.

A hint of quantization as a classical-physics consequence of intrinsic spin

Equation (1) is a requirement that must be satisfied in order for the total angular momentum of the quasiclassical positronium atom to be a constant of the motion, assuming that the electron and positron are initially oriented so that their components perpendicular to the orbital angular momentum are anti-parallel.  In Eq. (1), L is the vector total orbital angular momentum (the sum of the orbital angular momenta of the electron and positron), n is a unit vector in the direction from the positron to the electron, and the intrinsic spin vectors are s.  The subscripts on the spin quantities indicate whether the spin is that of the electron or positron, and further whether it is the vector component parallel or perpendicular to L.

Equation (1): A condition for constancy of total angular momentum

Equation (1) is fairly easily obtained based on Thomas's equation of motion of the spin (which can alternatively be obtained from the Bargmann-Telegdi-Michel covariant equation of spin motion) specialized to the positronium atom, as is done in my positronium paper separately for the electron and positron, and then taking their difference to obtain an equation of relative motion for the spins.  Then putting in the initial condition that is the main result in my positronium paper, that antiparallel L-perpendicular spin components are a condition for total angular momentum constancy, Eq. (1) results.  If Eq. (1) could be satisfied at all points on the orbit (that is, as the vector n traces out the relative particle positions around the orbit) then the needed relative orientation would be maintained and constant angular momentum would be maintained.  (Total angular momentum constancy is in turn a necessary condition for nonradiativity.)   

To understand how Eq. (1) comes close to obtaining a quantum condition from classical electrodynamics with intrinsic spin,

How Existence of Intrinsic Spin Might Explain the Non-Classical Character of Atomic Radiation

Bohr's Correspondence principle tells us that in the limit of large quantum numbers, quantum physics will agree with classical physics. For example, the simple Rutherford atom model of hydrogen, where a point charge electron orbits a point charge proton in a classical Keplerian orbit, will increasingly agree with observation, in terms of rate of radiative decay and frequency of transition radiation, as the electron orbital energy and angular momentum is raised. (The study of so-called Rydberg atoms, that is, atoms where the outer electron is excited to a high energy level compared to the rest, confirms this.) The frequency of the electromagnetic radiation becomes in the limit of large quantum number simply the frequency of revolution of the electron in its classical orbit. The rate of energy loss is also calculable from the radiation intensity predicted for an accelerating point charge according to classical electrodynamics.

At lower energy levels, and apart from the issue of quantized energy levels, the classical model of radiative decay of the system of bound point charges deviates significantly from observation. There are at least two obvious differences from what the classical model predicts. The transition time between two defined energy levels is less than the classical model predicts, and the frequency of the radiation is inconsistent with the classical expectation. The classical expectation is that the radiation of decay between two energy levels will start at the orbital frequency of the higher energy level and end at the (higher) orbital frequency of the lower. Observation however shows that transition radiation is essentially monochromatic, with the frequency predictable based on simply the energy difference between the levels. These seemingly inexplicable differences with the expectations of classical physics led to some despair at the time of their discovery. The initial triumph of quantum theory, even the early quantum theory of Bohr and Sommerfeld, and later the modern version developed particularly by Heisenberg, Schroedinger, and Dirac, was its ability to accurately describe atomic spectra including these features.

Understanding the Origin of Quantum Behavior by Study of the Positronium Atom

For several years now I have been of a belief that if there is an explanation of quantum behavior in the classical electrodynamics consequences of the existence of intrinsic spin, then the Positronium atom is the ideal system for its elucidation. This belief grew out of my study of spin-orbit coupling in the quasiclassical hydrogen atom,

Status Update

I am disappointed that Physica Scripta has withdrawn my paper from consideration after receiving a response from only one referee. That referee observed that I state in my concluding remarks that the analysis should be taken as only a plausibilty argument, and therefore it should not be published on that grounds alone.

I want to say that although I do consider that paper to be only a preliminary analysis, and that there are relativistic effects omitted that could negate it (which is why I say it is only a plausibilty argument) it is nonetheless a quantitative argument that both the magnetic and strong forces are direct consequences of the Thomas precession. As such I believe it is of sufficient interest to warrant publication in its own right, or at least review.

That said, I'm not entirely upset because it was just my first attempt, and the next will be much more relativistically precise.