## Program for Averaging

It could but well worth the effort because the man square fluctuations can be modelled in several ways. I trust that your family is well, and best wishes from all at AIAS. The graphics will also be very interesting

o: EMyrone@aol.com
Sent: 29/11/2017 10:28:05 GMT Standard Time
Subj: Re: [Spamverdacht] Checking 393(6)

In a program for averaging, the rules for cross-correlation etc. have to be included. This could become a bit complicated. I will see if it is possible with reasonable effort.
From Thursday to Monday I will not be at home for family reasons, I hope I can do something in-between.

Horst

Am 29.11.2017 um 11:17 schrieb EMyrone:

Many thanks indeed for all this work. I will write out the final result for the total expression for E. It would be very useful to write code to average the final result by computer. I will average it by hand first. The calculations are straightforward but tedious. The results are full of interest

To: EMyrone
Sent: 28/11/2017 16:28:28 GMT Standard Time
Subj: Re: Checking 393(6)

I had to check the averaging operations by hand. All seems o.k. up to eq. (25). From the 8 contributions of (16), I obtain

E_1 ~ 1/r^5 * (5/3 r (p*r) <delta r*delta r> + 1/9 p <(delta r*delta r)^2> + 1/3 <delta r*delta r> r.

The last term (from contribution 6) seems to be missing in your eq. (26), and for the first term I obtained the factor 5/3 instead of 7/3.
In eq. (15) the factor should be 11/6 because a 4 is already below the fraction bar. The total result (27) will change.
Computer algebra obtains for E_1 the simplified expression (without averaging):

where dr^2 means bold delta r * bold delta r, and dr, r and p are vectors, the dot is the scalar product. By this expression one could also compute higher terms in delta r. It seems that you used 4th order in eq.(25) but neglected 3rd order at other places.

The total expression for E is

Horst

Am 28.11.2017 um 10:35 schrieb EMyrone:

Thanks again, agreed about the typo. A check by computer algebra of this hand calculation would be essential, in my opinion, because things get complicated. If errors are fund by computer, the papers will be corrected and reposted. One of the important things to note is that the gradient operator is defined by Eq. (16) of Note 393(4). The basic axiom is that r is replaced by r + delta r in all physics. More generally, angles are also changed in the same way in all physics. However, for the Coulonb law and dipole fields it is sufficient to consider the replacement of r by r + delta r. This is the procedure used in the highly accurate Lamb shift theory.

To: EMyrone
Sent: 27/11/2017 20:48:59 GMT Standard Time
Subj: Re: 393(6): Shivering Electric Dipole Field in the Presence of the Vacuum

Is there a typo in eq.(10) ? should it read <delta X^2> instead of <X^2> , etc. ?
I hast still to check the rest of the note.

Horst

Am 22.11.2017 um 12:09 schrieb EMyrone:

This is Eq. (27) to first order in x of the binomial expansion of previous notes. The zitterbewegung theory results in a very rich structure and a completely new subject: the zitterbewegung or shivering theory of macroscopic electrodynamics in the presence of the vacuum. This theory is ideal for computer algebra because the calculations quickly become laborious. My hand calculations for UFT393 should be checked as usual by computer algebra, and graphed, when Horst returns from vacation. The effect of the vacuum is very intricate, and the vacuum is ubiquitous and ever present. Note carefully that this is the same theory as used to calculate the Lamb shift with great accuracy. So there is great confidence in the theory.