Archive for December, 2017

## note 395(6)

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

395(6).pdf

## note 395(6)

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

395(6).pdf

## note 395(5)

Yes, it is due to isotropic averaging, the final version of the calculations being given in Note 395(7), using a Taylor series expansion. To first order in <delta r dot delta r> the result is Eq. (10) of Note 395(7). This gives the Lamb shift using mode theory.

Date: Sat, Dec 30, 2017 at 6:34 PM
Subject: note 395(5)
To: Myron Evans <myronevans123>

Where does the correction of 1/3 in eq. (5) come from? Is it from the isotropic averaving in note (4)?

Horst

## Maclaurin series expansion of the Vacuum Correction

This is given by the general formula (5). Its use in quantum physics is exemplified by the accurate calculation of the Lamb shift. The vacuum corrections to the Newtonian and Coulombic potentials are worked out with well known expressions for the Dirac delta function, Eqs. (19) and (20). Mean square fluctuations in the vacuum can be worked out classically using the statistical mechanics of vacuum particles and Monte Carlo and molecular dynamics methods. This method can be used for any scalar function of relevance in physics. The Maclaurin expansion was inferred by Colin Maclaurin, who became a full professor of mathematics at Marechal College Aberdeen at the age of 19.

a395thpapernotes5.pdf

## New Method of Calculating Vacuum Corections

This is a general and powerful method which results in Eq. (8), valid for any scalar function. It applies to scalar and vector potentials and fields such as E, B, and g. Eq. (27) defines the vacuum driving force for the vacuum electric field. This can be amplified by Euler Bernoulli resonance. The Euler Bernoulli equation defines the circuit design needed to greatly amplify a vacuum fluctuation of type (27). This is very important for practical applications. There have been several experimental demonstrations of such resonance effects in a circuit, notably the demonstration by the Alex Hill group to the US Navy in 2005. Since then thee has been tremendous theoretical and experimental progress.

a395thpapernotes4.pdf

## 395(3): Effect of Vacuum Fluctuations on Spin Spin Structure

This note calculates the effect of vacuum fluctuations on spin spin fine structure using the results of UFT394. After checking the hand calculations with computer algebra some very interesting graphics can be drawn of the interaction energy on the classical level. The total interaction energy being the sum of the dipole and contact terms. Various well known quantization schemes of spin spin spectra can also be utilized to investigate the effect of the vacuum. The scheme used in this note is the well known one of aligning the spins in the Z axis. What is actually observed in fine structure must already incorporate the effect of the ubiquitous vacuum. So it is possible in theory to find the vacuum fluctuations experimentally from spectral analysis. This is true for all physics. What is actually observed experimentally must always contain a contribution from the vacuum, as in the anomalous g factor of the electron and the Lamb shift.

a395thpapernotes3.pdf

## 395(2): Electron Electron Spin Spin Interaction

This note gives the well known calculations for electron electron spin spin interaction theory in ESR in preparation for the calculations of the effect of the vacuum planned for Note 395(3). The usual theory develops the contact term with the Dirac delta function, but this is a dubious procedure, because ordinary differentiation shows that the contact term is zero. Therefore in the next note it will be developed with the results of UFT394 using vacuum fluctuations, an entirely new idea in physics, and the Dirac delta function not used.

a395thpapernotes2.pdf