## Saturday, September 18, 2004

Let d be the separation between the flat uncharged plates.

The observed Casimir VdW force is ~ A/d^4 ~ 0.016/d^4 dynes per square cm. Do a Google search.

I have calculated the FREE VIRTUAL PHOTON ZPF attractive force in this same situation to be

- (1/2)(hc/d^2)(A/a^2)

A is area of the plates and a is the short-wave cutoff.

Therefore, unless Hal & Co can point out a specific error in the precise detailed phase space calculation below, all the articles and text books that attribute the Casimir force to a pressure differential of purely free virtual photons between the inside and the outside of the plates are simply wrong.

In fact the free virtual photon ZPF force must be exactly ZERO in the non-exotic vacuum limit where the real formula is

ZPF Free Virtual Photon Force (parallel plates) = - (1/2)(hc/d^2)(A/a^2)[1 - a^3|Vacuum Coherence|^2] = 0

=/= Casimir VdW Force ~ 0.016/d^4 dynes per square cm

The precise calculation of the ZPF Free Virtual Photon Force is short-wave cutoff dependent from simple phase space calculations. The error Milonni and other Pundits make in their text books is to do the phase space integrations incorrectly to get the spurious result ~ 1/d^4 for the virtual photons. That is they use the spherically symmetric phase space integral in an inappropriate situation that does not match the non-spherically symmetric boundary conditions!

On Sep 18, 2004, at 6:50 AM, Jack Sarfatti wrote:

On Sep 18, 2004, at 2:57 AM, RKiehn2352@aol.com wrote:

Dear Jack
I find it miraculous that you admit to a mistake (re ZPF).  (grin)

Milonni and Hal Puthoff appear to have made even bigger ones!
My today's correction to my yesterday's correction now agrees in form, i.e. -x^-2 attraction (mod dimensionless coefficient that one must use QED with j.A to get) with A. Zee's calculation in "Quantum Field Theory in a Nutshell" and as I recall disagrees with Milonni's in "The Quantum Vacuum". I have former book with me but not the latter here in Southern California. See below

*
There may be another way to interpret Ken Shoulders EVO's.
**
Ken put together a meeting (financed by Church's Fried Chicken) a long time ago in the SAF bay area.  Attendants: Shoulders, Aharonov, Puthoff, Kiehn.
*
Since then,I have always been interested in Ken's experiments.
**
Last week I attended euromech448 conference on Vortex Dynamics and Field interactions.
Crowdy from Imperial college gave an interesting talk on exact COMPACT vortex solutions to the Euler equations in terms of complex variable theory.  The key idea here is COMPACT structures of reasonable lifetimes.
The numerical people had been experimenting with "Patch Distributions" of vorticity (curl V) combined with vortex line "1-D singularities" or strings.  The key result is they got numeric results that seemed to mimic compact vortex structures (like hurricanes).  Now Crowdy comes along and finds exact families of analytic solutions.  Neat.
*
That evening I woke up with the idea of how to generalize Crowdy's ideas beyond 2 dimensions, and express the ideas in terms of exterior differential forms.  The technique combines "2-D singularities" or branes with patches of charge-current densities to form COMPACT charge structures.   The combination of the singularities and the patches can be used to define a deformable boundary of a COMPACT  topological cohererent structure (blobs) -- the higher dimensional analogue of the COMPACT vortex topologically coherent structures in fluids.  This of course does not require metric.
*
I immediately thought of Ken's experiments.
*
You might find interest in
and
http://www.cartan.pair.com/ebooks.htm
regards,

RMK

More corrections. Puthoff's argument is still not correct. It's easy to
make a mistake of a - sign. Appealing to text books is not good enough.
They may all be wrong about these ZPE calculations because the rules of
QED conflict with the rules of GR and one cannot really use tricks like
normal ordering of the photon creation and destruction operators.
Milonni's book, for example, completely ignores GR. The discovery of
dark energy shows one cannot ignore GR even at small scales.

On waking next morning at 5AM I realized we better take another look.

Given the asymmetric boundary conditions of parallel plates, we need to
be careful about how to do the integration in momentum space. We cannot
use spherical coordinates in momentum space. We must use Cartesian
rectangular coordinates, i.e. for the spectral density

dpxdpydpz not 4pip^2dp

Let the parallel plates be separated by x along the x-axis.

Between the plates, the limits are from px(min) = h/2x to px(max) = h/a.

However, in the plane of the plates the limits are from 0 to p(max) =
h/a.

How do we do the virtual photon ZPE integral?

That integral must split into longitudinal and transverse parts.

The longitudinal FREE VIRTUAL PHOTON ZPE integral BETWEEN THE PLATES is

ZPE(x) = (1/2)c[px(max)^2 - px(min)^2]py(max)pz(max)xYZ/h^3

= (1/2)c[(h/a)^2 - (h/x)^2](h/a)^2xYZ/h^3

= (1/2)(h/a)^2(h/a)^2xYZ/h^3 - (1/2)c(h/x)^2(h/a)^2xYZ/h^3

= (1/2)(hc/a)(xYZ/a^3) - (1/2)(hc/x)(YZ/a^2)

The ZPF force from this term is

-(d/dx)ZPE(x) = -(1/2)(hc/a^2)(YZ/a^2) - (1/2)(hc/x^2)(YZ/a^2)

= -(1/2)(hc/a^4)YZ - (1/2)(hc/x^2)(YZ/a^2)

The longitudinal FREE VIRTUAL PHOTON ZPE integral OUTSIDE THE PLATES is

ZPE(X-x) = (1/2)c[px(max)^2 - px(min)^2]py(max)pz(max)(X-x)YZ/h^3

= (1/2)c[(h/a)^2 - (h/(X-x))^2](h/a)^2(X-x)xYZ/h^3

= (1/2)(hc/a)((X-x)YZ/a^3) - (1/2)(hc/(X-x))(YZ/a^2)

The ZPF force from this term is from -(d/dx)ZPE(X-x) =

+(1/2)(hc/a^2)(YZ/a^2) + (hc/2)(YZ/a^2)(d/dx)(1/(X-x))

(d/dx)(1/(X-x)) = +(X - x)^-2

Next look at the transverse free virtual photon integral ZPE(xYZ).

ZPE(xYZ) = (c/2)xYZ[(px(max) - px(min)](Integral of 2pip^2dp from 0 to
p(max))/h^3

= (hc/2)[(1/a) - (1/x)]xYZ(Integral of 2pip^2dp from 0 to p(max))/h^2

= (hc/2)[(1/a) - (1/x)]xYZ(2pi/3)(1/a)^3

This term gives a net attractive force along x of -(hc/2a^4)YZ.

ZPE((X-x)YZ) = (hc/2)[(1/a) - (1/(X-x))](X-x)YZ(Integral of 2pip^2dp
from 0 to p(max))/h^2

= (hc/2)[(1/a) - (1/(X-x))](X-x)YZ(2pi/3)(1/a)^3

This term gives a net repulsive force along x of +(hc/2a^4)YZ.

Therefore, as expected, we need only look at the longitudinal integral
for these non-spherically symmetric parallel plate boundary conditions.

Area of plates A = YZ.

Thus the NET ZPF force along x at the plate located at x is

-(1/2)(hc/a^4)A - (1/2)(hc/x^2)(A/a^2) +(1/2)(hc/a^2)(A/a^2) +
(hc/2)(A/a^2)(d/dx)(1/(X-x))

- (1/2)(hc/x^2)(A/a^2) + (hc/2)(A/a^2)(1/(X-x)^2)

Assume x << X. As X --> infinity we can ignore the second term from
outside the plates.

This is a net attractive force ~ -1/x^2 not - 1/x^4, but it is too weak
to contain unbalanced charges in the EVO.