Ref. The Quantum Vacuum by Peter Milonni, Acad Press, 1994
2.7 The Casimir Effect pp 54 - 58
This calculation was plausible until the 1999 discovery of dark energy as most of the stuff of our mostly "virtual" (i.e. off mass shell) universe. The problem is that the virtual photon zero point energy density as well as all the other contributions like virtual electron-positron pairs are directly observable in principle because of the laws of gravity! This is precisely the cosmological constant paradox, which is to physics today, what the black body radiation paradox was 100 years ago.
Even if we take Milonni's calculation as is, it fails because it posits an infinite virtual photon zero point energy background. The universe cannot exist in that case. It is no longer a valid rule in The Game to pretend that this infinite energy can be simply "subtracted". Indeed one must put in a short-wave cutoff, but one can do it in a generally covariant way.
You need vacuum coherence to absorb the large but finite incoherent zero point fluctuation energy in the ordinary vacuum, which when disturbed, results in anomalous "dark energy" exotic vacuum of negative pressure or anomalous "dark matter" exotic vacuum of positive pressure. Both forms of exotic vacuum have w = (pressure/energy density) = -1. w = -1 because of the generally covariant laws of Einstein's gravity theory AKA "GR".
For example, look at Milonni's E(infinity) i.e. eq. (2.100) p.57
What you really have there is E(a) where a is the short-wave cutoff, but E(a) has the vacuum coherence factor
(1 - a^3|Vacuum Coherence|^2) = 0 under "normal conditions".
Look at Milonni's "spectral density" of quantum harmonic field oscillators (2.73) p.49 that comes from the more fundamental quantity
Mode Density ~ 4pip^2dp/h^3 = phase space factor per unit volume = = number of quantum harmonic field oscillators per unit volume
Because of spherical symmetry, take pc/2 as the common virtual photon energy of that spherical shell in momentum space to get
ZPE density per unit volume = (2pi)cp^3dp/h^3
In general, with finite asymmetric boundary conditions you need to use
ZPE Mode Density ~ dpxdpydpz/h^3 = number of quantum harmonic field oscillators per unit volume
i.e. as I showed yesterday, without the vacuum coherence factor that can be put in at end of the calculation.
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) =
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
= (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) +
- (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.
This FINITE WAY of mode counting does not agree with Milonni's Rube Goldberg sleight-of-hand shell game giving the observed Casimir force potential (2.108) p. 58
U(d) = -(pi^2hc/720d^3)A
from subtracting two infinities to get this finite difference.
It's the right empirical answer for the wrong reasons!
Milonni's 3.10 is also spurious because virtual photons do not propagate energy and momentum the way real photons do! Milonni here assumes w = +1/3, which is true for real photons, but not for virtual photons where w = -1. The only acceptable explanation for the Casimir force is indicated in Milonni's 3.11 on the Van der Waals forces. Of course the virtual photons play a vital role there but not as directly as they do in the first two models that are misleading because they are not consistent with the laws of Einstein's general relativity that reach down even to the microscale. It's not enough to get the right empirical answer. One must get it for the right reasons. Again up until the discovery of dark energy the choices between the three ways Milonni presents of looking at the orgin of the Casimir force was moot or degenerate. The discovery of dark energy "lifts the degeneracy" and only the retarded Casimir-Polder VdW potential eq. (3.91) p.105 ~ r^-7, which when integrated over many induced dipoles in both parallel plates allegedly gives the result close to (2.108) p. 58 is physically acceptable. See Ian Peterson's papers on this that the only energy you can get from the Casimir force is the small electrostatic induced dipole energy not the large virtual photon ZPE energy!
That is, zero point energy densities and their equal-in-magnitude but opposite-in-sign pressures STRONGLY WARP space-time geometry if not absorbed into the vacuum condensate! Upsetting that equilibrium is the key to the practical metric engineering of warp, wormhole and weapon that we see in the flying saucer evidence on the NIDS website.
Guv + /\zpfguv = 0
/\zpf = a^-2[1 - a^3|Vacuum Coherence|^2]
a = short-wave invariant cutoff (proper spacelike interval)
/\zpf is a "scalar"
ZPE density = t00(vac) = (c^4/8piG*)/\zpfgoo = (String Tension)(Micro-Quantum ZPE Curvature of Vacuum)goo
For example, in the Schwarzschild vacuum solution
goo = (1 - 2r*/r) when r* < r
But /\zpf is a function of r as well.