On Sep 11, 2004, at 2:14 PM, main_engineering wrote:

"Jack,

The ZPF energy density in any given interval of frequency, w2 > w1, is
proportional to the difference in the 4th power of these frequencies,

~(w2^4 - w1^4)

when you integrate the spectral energy density over that frequency interval
to get the energy density. The w^3 dependency of the spectral energy density
of the ZPF is required in order to retain Lorentz invariance. When you
integrate over the frequency interval, you get the above dependence over any
finite interval.

If you lower the low end cut-off (w1) by making your volume bigger, not only
does the energy content in that volume increase, but the overall energy
density is also increased by the difference of the 4th powers.

Therefore, when you said,

"Therefore the 1D Zero Point Energy Density is CONSTANT!"

This is absolutely Wrong in any 3D closed container. It is not constant. The
larger you make your volume, the larger the energy density of the ZPF
becomes in that volume and in all smaller volume elements contained therein.

main_engineering"


Your argument is false as is proved by direct calculation, which I here repeat with, I hope a clearer notation. The long-wave cutoff cancels out of the problem! Only the short-wave cutoff remains. You have made an elementary error.

Imagine a closed cylinder of length L made of the same kind of perfect uncharged conducting material used in the plates of actual Casimir force measurements, with a movable piston made of same kind of material inside that is separated by distance d from the left wall. We will introduce a short-wave cutoff "a". We can solve this problem completely for the standing wave electromagnetic field oscillator modes. We neglect electron-photon interactions. So does Puthoff. This is, of course an error because the real Casimir force is simply the integrated Van Der Waals force of mutually-induced electric dipole moments between the neutral molecules making the material of the plates. The correct explanation is in 7.6 of Peter Milonni's "The Quantum Vacuum" and in papers by Ian Peterson of Coventry University in UK. Even Milonni and others, maybe even Casimir, seem to have made an incorrect approximation for the FREE PHOTON FIELD, sans QED interaction that by a weird fluke, gives the same answer as the correct Van Der Waals explanation! I have not had time to go into that in detail. However, I can easily do the exact calculation that I give below. The key point for metric engineering warps, wormholes and EVO "micro-fusion" weapons is that the observed tiny Casimir force is irrelevant only "tapping" very weak electrostatic molecular energy, i.e. virtual photons exchanged between charges on the parts of the separated neutral molecules of the material!

We can ignore the transverse wave numbers. The problem is effectively 1D by symmetry. All that matters is the shift in d as the internal piston is moved.

The boundary conditions in the left partition A are for the wave number

k{n(A)} = n(A)(2pi/2d) = n(A)(pi/d)

n(A) = 1, 2, 3 ... nearest integer to d/a.

Similarly, for the right partition

k{n(B)} = n(B)(pi/(L - d)

n(B) = 1, 2, 3 ... nearest integer to (L - d)/a.

The mathematical error made in MANY text books seems to be on p. 66 of "Quantum Field Theory in a Nutshell" by A. Zee, where the entirely unnecessary fudge factor e^-cka is put in adhoc giving an infinite series n --> infinity with the FALSE result on the left side

ZPEnergy(d) ~ - hc(pi/2d)e^api/d/(e^api/d - 1)^2 eq. 18 p. 66

That WRONGLY gives the Casimir force as

-[ZPE(d) - ZPE(L - d)],d ~ pihc/24d^2 when L >> d

i.e. the right Van Der Waals answer for entirely the wrong physical reasons!

It is not enough to be right. One must be right for the right reasons!

Here is the EXACT calculation without the bogus exponential cutoff fudge factor that permits a completely unphysical sum over an infinite number of modes. The actual number of modes is finite and is easily exactly calculated. This seems to be the source of the widespread WRONG IDEA that the Casimir force works because of some kind of effective LOCAL spatial gradient in the completely random virtual photon TOTAL ENERGY caused by a difference in virtual photon pressures between the inside of the plates and the outside of the plates.

The EXACT calculation is:

1. Assume, as everyone does, that each k{n(A(B))} mode has ONE virtual ZPF photon of energy (1/2)hck{n(A9B))}

2. The finite series of integers

Sum from 1 to N of n = (1/2)N(N + 1)

Therefore,

3. Total EXTENSIVE ZPEnergy(A) = Sum from 1 to d/a (with small round-off error) (1/2)hck{n(A)}

= Sum from 1 to d/a of hc(pi/2d)n(A)

= hc(pi/4d)(d/a)[(d/a) + 1] = hc(pi/4a)[(d/a) + 1] ~ hc(pi/4a)(d/a) when d/a >> 1

(with small round-off error)

The ZPF FORCE from this TOTAL ZPF POTENTIAL ENERGY on the left partition is the negative gradient of this potential function that is LINEAR in the separation of the plates! Mind you that this is an EXACT calculation with an ignorable round off arithmetic error that reflects the fact that there are only a finite number of modes not an infinite number with a smooth exponential cutoff fudge factor that gives a misleading result!

Therefore, the ZPF force on the left side A is

F(A) = -hcpi/4a^2

Note that this force is independent of d the separation between the plates, but it does depend on the short-wave cutoff!

Next, I compute the ZPF force on the RHS B.

4. Total EXTENSIVE ZPEnergy(B) = Sum from 1 to (L - d)/a (with small round-off error) (1/2)hck{n(B)}

= Sum from 1 to (L - d)/a of hcn(B)pi/2(L - d)

= [hcpi/2(L - d)][(L - d)/a][((L - d)/a) + 1] = hc(pi/4a)[(L - d)/a) + 1] ~ hc(pi/4a^2)(L - d) when d/a >> 1

Note the - sign for d!

Therefore, again taking the negative gradient

F(B) = + hcpi/4a^2

Notice that the LONG-WAVE CUTOFF CANCELS OUT OF THE ZPF FORCE!

This is an elementary example of REGULARIZATION and FINITE RENORMALIZATION!

The ZPF FORCE on each side is CONSTANT but EQUAL IN MAGNITUDE AND OPPOSITE IN DIRECTION

Therefore, the TOTAL ZPF FORCE for FREE TOTALLY RANDOM VIRTUAL PHOTONS (i.e. ZERO VACUUM COHERENCE - another story) EXACTLY VANISHES!

F(A) + F(B) = 0

THE ZPF FORCE IS NOT THE OBSERVED CASIMIR FORCE! What is observed is the Van Der Waals force.

Puthoff & Co are completely off-base about that! So is NASA BPP.

Note that the ZPF total energy diverges to infinity as the short-wave cutoff -> 0 but, nevertheless the TOTAL ZPF FORCE on EACH PLATE is EXACTLY ZERO.

The hand waving mode-counting argument is BOGUS. Even Scientific American fell for it. I did as well until I did this calculation!