Dear friends,

Let me summarize briefly below, my journey over the past 2.5 years thorough which I arrived at the near-final paper http://vixra.org/pdf/1710.0159v4.pdf.

Very briefly, in December 2015, as memorialized in this forum, I took on the problem of trying to find a spacetime metric from which the electrodynamic Lorentz Force motion could be derived as entirely geodesic motion, using least action variation, in the exact same way that gravitational motion is obtained. As shown in section 2, I was able to do so mathematically, using the metric (3.3). But there were three problems: First, while the derivation worked mathematically, the metric I used appeared to be dependent upon the properties of individual charges travelling through the electromagnetic field, which is physically impermissible in field theory. The only way to overcome this was via an “inequivalence principle” laid out in section 5, which led inexorably to the finding that electromagnetic interactions between like charges dilate time, just as do gravitational interactions and special relativistic motion. Second, the Lorentz motion contained an additional A^2 term, see (2.12), which needed to be removed in some appropriate fashion. Third, interrelated, the metric had the rather unusual property of being quadratic in d\tau, see (3.4) and (3.5). This needed to be understood in a physically sensible way, and the having very same A^2 term in (3.5) made such an understanding challenging.

Now, I have long known, as do many others, that Dirac’s equation is an “operator square root” of the spacetime metric equation. So, about a year ago it occurred to me that if the A^2 term could be removed, then the quadratic line element (3.5) was in fact merely pointing toward a richer form of Dirac’s equation, which I have named the “hyper-canonical” Dirac equation. I had had a number of false starts over 18 months trying to remove the A^2 term. But last summer I finally realized that the correct way to do so was to employ Heisenberg’s equation of motion and Ehrenfest’s Theorem, see section 8, and to then use the two covariant gage conditions (9.4), (9.5) which are essentially the Lorenz gauge applied to both the “expected value of the spacetime divergence” and the “spacetime divergence of the expected value,” of the gauge fields. Not only did this remove the A^2 term noted above in a generally covariant manner, but the way in which it removed two degrees of freedom from the gauge field led precisely to the known properties of massless photons with two helicity states, see (15.1) through (15.3).

With this development, the challenging quadratic metric (3.5) became (11.3) which could be put into a form (11.5) that is very reminiscent of the “square root” from which Dirac’s equation emanates. Starting in section 13, using an electromagnetic tetrad analogous to the tetrads used for Dirac’s equation in curved spacetime, and after also adding a “spin connection” following the usual protocol to maintain proper spacetime covariance, I was able to obtain a “hyper-canonical” Dirac equation (18.8) which in momentum space is (18.10), see also (19.18). This is where I was in the Fall of 2017.

From there, I began the painstaking process of extracting the Hamiltonian from this new Dirac equation. This was a challenge mainly because this new Hamiltonian, as you can see in its final form complete form (21.1), contains a very large number of terms which I needed to check and recheck multiple times until I was satisfied that all had been correctly calculated. With the Hamiltonian finally obtained, the past few months I was able in section 23 to show how under the specific conditions where all external fields other than a constant magnetic field are “turned off,” this new Dirac equation inherently contains a magnetic moment anomaly.

This IMHO is hugely important, because if you understand the usual Dirac’s equation, you will understand that it only predicts a g-factor g=2 for the charged leptons, and no magnetic moment anomaly. But in the natural world of course, there is an anomaly which to first order loops is equal to the Schwinger factor \alpha/2\pi, where \alpha ~ 1/137.036 is the running low-probe electromagnetic coupling. The only known way to explain this anomaly is to introduce renormalization theory whereby some infinities are subtracted from other infinities. Beyond this ugliness that Dirac and Feynman and others complained about, renormalization theory is a separate appendage to – not an intrinsic part of – Dirac theory. This suggests that something is missing from the standard Dirac equation. So, at (23.5) through (23.9), I finally show how the anomaly is now made intrinsic to Dirac theory via the “hyper-canonical” Dirac equation and its Hamiltonian, and how infinite-number renormalization is thereby rendered entirely unnecessary.

Of course, it is always desirable for any new theory to offer some testable predictions. Over the past several weeks, I added sections 24 through 26 which lay out six distinct types of experiment through which this may all be tested. And with all of that, I look forward to your public or private feedback. If private, my email address is yablon@alum.mit.edu.

Best regards to all,

Jay