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", L >> a, 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 LINE 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 "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.

## Wednesday, September 15, 2004

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