Electromagnetic waves near perfect conductors. II. Casimir effect

Abstract

The Casimir free energy of the electromagnetic field in regions bounded by thin perfect conductors with arbitrary smooth shapes is studied. This free energy is expressed as a convergent multiple scattering expansion, in which the wave is damped between scatterings taking place on conductors. The result depends neither on the large box needed to render the spectrum discrete, nor on the shape of the high frequency cut-off corresponding to slightly imperfect conductivity; this establishes the universal character of quantum electro-magnetic forces on perfectly conducting thin objects. General expansions are derived for the Casimir free energy in the low and high temperature limits. They exhibit a dependence on the topology of the conducting surfaces. At low temperature, the Casimir entropy vanishes as T 3 for simply connected conductors, and as T for multiply connected ones. At high temperature, the free energy has the semiclassical behavior − C T log( T h ̵ cQ ) . The constant C = (128φ) −1ʃdθ( 3 R 1 2 + 3 R 2 2 + 3 R 1 R 2 )−n may be interpreted as the number of additional modes of finite frequency created by introducing the conducting surface. It depends on the topology of the surface through its genus n, and on its local curvature radii R 1, R 2. The average wave number Q depends on the shape of the surface and its size. The symmetry between low and high temperatures is recovered for parallel plates. While a plane foil is stable against Casimir stresses at zero temperature, such stresses would tend at finite temperature to create wrinkles of dimensions larger thn 2.9 h ̵ c T . The same wrinkling effect exists for an arbitrary conductor at high enough temperature. The presence of a curved conducting surface transfers the free energy of radiation from the concave to the convex side at zero temperature, and does the converse at high enough temperatures. Wedges may be considered for slightly imperfect conductors; the electro-magnetic energy is lowered at finite temperature by creation of wedges, and also at zero temperature by creation of multiple wedges as in a honeycomb structure. The formalism is also applied to evaluate Van der Waals forces and torques at arbitrary temperature between remote conductors, and to show that the Casimir energy of a cylinder at zero temperature vanishes to lowest order in the multiple scattering expansion. Finally, the sphere is used as a test for studying convergence of the multiple scattering expansions. Using radial and transverse combinations of vector spherical harmonics yields a simple expression for the electromagnetic Green functions. This expression is recovered by summation of the multiple scattering expansion; the convergence domain, in the plane kR = x + iy, of the expansion includes a neighborhood of the origin, as well as the whole region y >0.0348 |x| 1 3 . The numerical convergence of the Casimir free energy is rapid, the lowest order term already yields a result with less than 7% error.