My previous calculations below for the free photon field without any j.A coupling to electric charges assumes zero vacuum coherence. Putting in the virtual electron-positron pairs is equivalent to one virtual quantum of negative energy per transverse polarized mode. This is because of the Pauli exclusion principle's anti-commutation rules for the Dirac electron field. The electron has spin 1/2, the photon spin 1, neglecting supersymmetry is OK since there is no evidence for it at all, therefore, the longitudinally polarized virtual photon mode is not compensated for by the virtual electron-positron pairs.
In general with the macro-quantum vacuum coherence, the net virtual photon + virtual electron positron, zero point energy density is positive ~ (hc/2a^4)(1 - a^3|Vacuum Coherence|^2), which is adjusted to ZERO in the non-exotic vacuum without any dark energy and any dark matter. The latter two are simply exotic vacuum phases where
a^3|Vacuum Coherence|^2 < 1
a^3|Vacuum Coherence|^2 > 1
Ken Shoulders EVOs are stabilized by DARK ENERGY core that holds the unbalanced charge -Ne together via
V(r) = +(Ne)^2/mr + c^2/\zpfr^2
Note that the gradients of these two terms are of opposite sign allowing dynamical stability. This also explains the stability of a single spatially-extended electron as a Bohm hidden variable possibly allowing Vigier's tight atomic states as a new form of atomic energy investigated by Maric and Dragic in Beograd. We need to add the rotation term of course.
/\zpf = (8piG*/c^4)(ZPE Density) > 0
G* is the effective gravity constant at the small scale ~ 10^-5 - 10^-3 cm of the EVO, which may, or may not be, Newton's G. That is an empirical question.
N(h/mc)^2 = (Space-Warp Factor)4pi(Observed EVO Radius)^2
Where Space-Warp Factor << 1, i.e. non-Euclidean 3D space geometry for the mesoscopic EVO is my prediction.
On Sep 16, 2004, at 3:31 AM, Jack Sarfatti wrote:
In 3D case for a little box of volume xyz inside a big box of VOLUME XYX
The total free virtual photon ZPE in the little box is, with short-wave cutoff a:
The total free virtual photon ZPE in the BIG BOX outside the little box is
Note that in both cases the total 3D free virtual photon positive ZPE density is the same, i.e. ~ hc/2a^4, on both inside and outside of the little box inside the BIG BOX and is - the NEGATIIVE pressure, i.e. w = -1 not Puthoff's wrong value w = +1/3 that he uses in his most recent paper about Ken Shoulders EVO data.
What is the total ZPF force at the walls of the little box?
It is obviously ZERO!
For example, the total ZPF force perpendicular to the wall x cancels from both sides.
Take -(d/dx) of total ZPE inside + outside wall x. The -x term outside the wall in BIG BOX cancels the x term inside the box when you take the gradient!
This works same in any number of dimensions, also if you use spherical shell method with Pythagorean theorem, same final answer, i.e. r^3 and R^3 - r^3 terms, the radial gradients also cancel.
On Sep 15, 2004, at 7:09 PM, Jack Sarfatti wrote:
typo-corrected 2nd draft
If you posit that momentum space is a CONTINUUM metric space and assume FLAT Euclidean geometry with the Pythagorean theorem, then
1. Virtual photon ZPE on a 1D line of length L with short wave cutoff "a", L >> a, with standing wave modes ~ (L)^-1/2sin(kx)
Total ZPE per polarization ~ pi(hc/4a)(L/a) = (pi/2)(ZPE of shortest wave virtual photon)(Number of cells of phase space)
1D ZPE PRESSURE = STRING TENSION = -(d/dL)(Total ZPE) = pi(hc/4a^2) = constant = -ZPE LINE DENSITY, i.e. w = -1
2. Virtual photon ZPE on a 2D lattice of area A with short wave cutoff "a", A >> a^2, with standing wave modes ~(L)^-1sin(kx)sin(k'y)
Total ZPE per polarization ~ pi(hc/4a)(A/a^2) = (pi/2)(ZPE of shortest wave virtual photon)(Number of cells of phase space)
2D ZPE PRESSURE = Membrane Tension = -(d/dA)(Total ZPE) = pi(hc/4a^3) = constant = -ZPE Membrane DENSITY, i.e. w = -1
3. Virtual photon ZPE on a 3D volume V with short wave cutoff "a", V >> a^3, with standing wave modes ~ (L)^-3/2sin(kx)sin(k'y)sin(k"z)
Total ZPE per polarization ~ pi(hc/4a)(V/a^3) = (pi/2)(ZPE of shortest wave virtual photon)(Number of cells of phase space)
3D ZPE PRESSURE = -(d/dV)(Total ZPE) = pi(hc/4a^3) = constant = -ZPE DENSITY, i.e. w = -1
In all three cases, no contribution of virtual photons to any vacuum pressure difference that can be attributed to a VdW "Casimir force".
The effect of EFFECTIVE dimension is in how the "pressure" scales with short-wave cutoff. One could also imagine FRACTALS with a non-integer power of the short-wave cutoff.