**Repulsive Casimir Force II: Shape Effects**

**Abraham-Lorentz (+Casimir) model of electron:**

A spherical conducting shell of radius

R, with a uniform chargee ."Introductory remarks on quantum electrodynamics," H.B.G. Casimir, Physics

19, 846 (1956)

Casimir's "mousetrap" to catch the fine structure constant:

Balance the repulsive Coulomb energy, with the (presumed attractive) Casimir energy:

"Quantum Electromagnetic Zero-Point Energy of a Conducting Spherical Shell and the Casimir Model for a Charged Particle," T.H. Boyer, Phys. Rev.

174, 1764 (1968)

2A = - 0.09235is negative![also obtained by R. Balian and B. Duplantier (1977)]"Vacuum means of energy-momentum tensor of quantized fields on manifolds of different topology and geometry. I,"

S.G. Mamaev and N.N. Trunov, Sov. Phys. J. (USA)

22, 51 (1979)

The "Casimir energy" of a parallelopiped changes sign with aspect ratio:

Does this imply a repulisve force?

"Attractive Casimir forces in a closed geometry,"

M. P. Hertzberg, R. L. Jaffe, M. Kardar, and A. Scardicchio, Phys. Rev. Lett.

95, 250402 (2005)In the physically accessible geometry of a piston, the partition is always attracted to the closer side.

"Opposites Attract: A Theorem about the Casimir Force,"

O. Kenneth and I. Klich, Phys. Rev. Lett. 97, 160401 (2006) (two halfs of a cut sphere attract)

Contrained repulsion?

"Casimir repulsion between metallic objects in vacuum,"

M. Levin, A.P. McCauley, A.W. Rodriguez, M.T.H. Reid, S.G. Johnson, arXiv:1003.3487v2 (2010)