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The Casmir force is explained by wave exclusion: Two parallel plates create a reduction in the possible modes of vacuum oscillations between them, reducing the energy of the vacuum. Since we define vacuum far from any plates as "zero" energy, a reduction of energy creates a negative energy. However, these sources also state that sometimes the regions in the vacuum gets a positive energy (depending on the geometry of the conductors). If so, how would "exclusion" create extra energy in a region?

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This is an interesting question. In 3 spatial dimensions, one may compare different topologies like paralles plates, infinite cylinder, and sphere. The first 2 toplogies have the same sign for the Casimir energy, while, for the sphere, the sign is different.

A short and violent answer would be "Shut up and calculate", but this is not quite satisfactory.

The Casimir energy for the infinite cylinder, while negative, is smaller (in absolute value) than the Casimir energy of the parallel plates, (with same specific length). So, we have a progression :

parallel plates -> infinite cylinder -> sphere

If we try to find the differences, my feeling is that we have to look at the concentation of the modes in some points. The concentration is null for parallel plates, it is 1-dimensional for the cylinder (a line of concentration points), and there is a unique concentration point for the sphere. It seems that, more the concentration of the modes is important is some point, more the total energy becomes positive.

Of course, it is a feeling, it would be interesting if someone would be able to bring a mathematical proof of that.

Note that the logic is qualitatively the same for the bosons modes and the fermions modes, while the numeric results are different.

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  • $\begingroup$ What are "concentration points"? $\endgroup$ Oct 4 '13 at 13:00
  • $\begingroup$ Well, if I imagine radial modes for the cylinder and the sphere, there is a point where these modes crosses each other, the center of circles for cylinder, and the center of the sphere. So, I called these points "concentration" points. My idea was that this "concentration" could be the reason of appearance of a positive energy, but I have no mathematical proof of this. $\endgroup$
    – Trimok
    Oct 4 '13 at 14:42
  • $\begingroup$ Could it be some form of resonance? $\endgroup$ Oct 5 '13 at 12:32
  • $\begingroup$ Well, I was thinking about a kind of repulsive force due to the fact that the radial modes crosses at some point... I don't know if this corresponds to your notion of resonance. $\endgroup$
    – Trimok
    Oct 5 '13 at 12:43

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