Review of standard phase space calculations of zero point energy.

Spectral density in 3D space is from a spherical shell in momentum space.

4pip^2dp/h^3 = 3D Spectral Density

Where for the zero point virtual photon the energy is pc/2.

Therefore, integrate from p = 0 to p(max) to get

Total virtual photon zero point energy is in volume V

ZPE(3D) = Integral of (pc/2)4pip^2dp/h^3V = (4pic/8)p^4(max)V/h^3 = (pi/2)(hc/a)(V/a^3)

where p(max) = h/a

a = short-wave cutoff

This assumes Euclidean geometry continuum in 3D momentum space with basically ignoring oddly shaped cavities.

Same kind of calculation in 2D with an annulus in 2D momentum space gives (ignoring factors of 2, pi etc)

2pipdp/h^2 = 2D Spectral Density

ZPE(2D) ~ (hc/a)(A/a^2)

And in 1D along a line

dp/h = 1D Spectral Density

ZPE(1D) ~ (hc/a)(L/a)

In 1D one can do an exact calculation using finite series Sum of n from 1 to N = (1/2)N(N+1), but when L/a >> 1 you get essentially the same result as using the continuum integration.

Next go back to 3D with ZERO VACUUM COHERENCE.

3D Pressure = -d(Internal Energy)/dV

In this case

3D ZPF Pressure = -(pi/2)(hc/a^4) Independent of V! Also the pressure is negative!

w = Pressure/(Energy Density) = -1 not Hal Puthoff's "w = + 1/3" which is true only for real photons not virtual ZPF photons.

If you have a little box of volume v = xyz inside a large BOX of volume V = XYZ, then the total ZPE inside the little box is in 3D

ZPE(v) ~ (hc/a^4)v

The total ZPE inside the BIG outer BOX is

ZPE(V - v) ~ (hc/a^4)(V - v)

Note the LINEAR additive rule

ZPE(v) + ZPE(V - v) = ZPE(V)

Similarly in 2D for ZPE(A) and in 1D for ZPE(L).

Back to 3D what is the total ZPF force perpendicular to the wall x of the little box?

It is obviously

Fx(ZPF) = -dZPE(v)/dx - dZPE(V - v)/dx = -yz(hc/a^4)(1 - 1) = 0

That is, with ZERO VACUUM COHERENCE there is NEVER any NET purely FREE virtual photon force on any wall. Therefore, you can never explain the actual Casimir force as some kind of free virtual photon ZPF pressure differential on a pair of electrically UNCHARGED conducting parallel plates as Hal Puthoff, and others, have repeatedly done in popular physics articles. The Casimir force is a Van der Waals electrostatic force and in no way can ever tap the completely random free virtual ZPF photons neglecting the actual QED coupling j.A.

Similarly in 2D and 1D. The number of effective space dimensions in conditions of different symmetries on the boundary conditionns makes no essential difference to this general conclusion.

## Thursday, September 16, 2004

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