Very interesting and important development, demonstrating the power of computer algebra. It shows for example that if the Dirac delta function is not used, del squared f in Eq. (13) vanishes if f is the Coulomb potential, and there is no Lamb shift. Also, there is no second order correction to the magnetic dipole potential. Eq. (13) and its generalization Eq. (14) are the equations used to calculate the Lamb shift, so there can be great confidence in their application to physics in general.
Precise Meaning of the Tensorial Taylor Expansion
To: Myron Evans <myronevans123>
I worked out the full eq.(14) by Maxima code. It can be programmed quasi-recursive. It comes out that all odd exponents of (delta r * del) give zero results (up to eq. o34 of the protocol). As examples I used the vector potential of a magnetic dipole and the Coulomb potential. In both cases the second-order terms vanish due to the symmetry
<dX^2> = <dY^2> = <dZ^2> = 1/3 * <dr^2>.
In constrast to the earlier method with 1/x expansion, now the orders of expansion are complete, for example fourth order gives all fourth order terms, and no terms of lower or higher order. I will send over some plot examples later.
Horst
Am 02.01.2018 um 13:53 schrieb Myron Evans:
In this note it is demonstrated by detailed calculation that the tensorial Taylor expansion (1) is the same as the very condensed notation vector Taylor expansion (7). The clearest expression however is Eq. (14), in Cartesian components. Although Eq. (14) looks complicated it is easily worked out with computer algebra, thus eliminating human error. The meaning of isotropic averaging is most clearly explained with Cartesian components as in Eqs. (8) to (10). This method can be used to find the effect of the vacuum to any order of the Taylor series on any scalar function of of physics. The Lamb shift is explained with this general and powerful method, used to second order with f being the Coulomb potential between the proton and electron in the H atom. In reading around this subject using google, and doing a literature search, I found all kinds of obscurities and amateurish errors, sloppy notation, sloppy articles, and so on. Horst Eckardt, Douglas Lindstrom and I always aim for maximum precisipn and clarity. This is achieved only after years of hard work and multiple cross checks. This is why ECE theory is so spectacularly successful, and in my opinion, why wikipedia and much of standard physics is such a dismal, dogmatic failure. Every single wikipedia article I have studied had to be entirely rewritten because of errors and obscurities. In other areas wikipedia may be fairly useful, but in theoretical physics it is not of much help, and a decade ago, was used to launch a personal attack on myself, using abuse and fraud. This attack failed completely. Google is useful, because it is effectively a huge library, but there is no quality control, and some of the stuff dredged up by Google is appallingly bad. Other stuff is good. One just has to use great care and experience, and above all, work it all out for yourself as in this note.