Many thanks again. I did a literature search about the Taylor expansion used in the Lamb shift calculation. This is the vector Taylor expansion in three dimensions and tensorial algebra and directional derivatives are needed in general. The result to second order is the same as the Lamb shift calculation, but I need to clarify the meaning of the three dimensional Taylor expansion wth a few examples. It contains a lot more information than the usual one dimensional Taylor expansion of the textbooks. I have been working on this today and will continue tomorrow. The three dimensional Taylor expansion is correctly given in Eq. (3) of Note 395(7). The result (10) is correct, and can be applied to any scalar function f. You might like to experiment with a few functions (ppotentials adn fields of various kinds) and at a late stage the higher order terms can be added. These higher order terms have to be worked out carefully. Having read around the subject today it has become clear that the notation used in the three dimensional Taylor series is tensorial. So delta r dot del in Eq. (3) has a special tensorial meaning, so does (delta r dot del) squared and so on. I will clarify this tomorrow in the final note for UFT395.
To: Myron Evans <myronevans123>
I changed the factors and the operators del^4 and del^6 as discussed. The single terms generated by del^2, del^4, del^6 are quite complex as expected. Now we have the surprising result that there are no fluctuations in 2nd, 4th and 6th order, probably the vector potential is free of fluctuations. Did we ever inspect the Coulomb potential concerning fluctuations? Perhaps we should do this.
Horst
Am 31.12.2017 um 11:16 schrieb Myron Evans:
Very interesting result. Eq. (7) means del squared (del squared A sub X) i.e. that the second derivatives inside the first bracket on the right hand act on the second bracket, generating fourth order derivatives of various kinds. Del is a vector operator, so del squared = del dot del, and del fourth = (del dot del)(del dot del) = (del squared) squared. Note that Eq. (17) of Note 395(70 means that there should be a factor (1/9) multiplying the second term on teh RHS of Eq. (5) of Note 395(6), and a facor 1 / 81 multiplying the second term, so it is a rapidly converging series.
Date: Sat, Dec 30, 2017 at 9:38 PM
Subject: note 395(6)
To: Myron Evans <myronevans123>I calculated the fluctuation terms of the magnetic vector potential. There is no fluctuation in 2nd order. The fourth and sixth order do contribute.
How do I have to interpret eq. (7)? Is this a product of derivatives which leads to mixed derivatives of type
d^2/dX^2 d^2/dY^2
etc.? Then I have to modify my calculations.Horst