Semiclassical Casimir energies at finite temperature

We study the dependence on the temperature T of Casimir effects for a range of systems, and, in particular, for a pair of ideal parallel conductors, l₁×l₂, separated by a vacuum to a distance l₃ apart, with l₁≫l₃ and l₂≫l₃. We find the Helmholtz free energy A^T combining Matsubara’s formalism in which the temperature T appears as a periodic Euclidean fourth dimension of circumference l_T=ħc/k_BT, with the semiclassical periodic orbital approximation of Gutzwiller. The approximation was shown to be exact for parallel plates at T=0. By inspecting the known semiclassical results for the Casimir energy at T=0 in two cases of a rectangular parallelepiped, (l₁≫l₃ and l₂≫l₃, and l₁≫l₂ and l₁≫l₃), one is led to guess at the expression for A^T of two ideal parallel conductors without performing any calculation. The result, A^T=-(2ħcl₁l₂l₃/π²)∑_n3=1^∞∑_nT=-∞^∞L^-4(n₃,n_T), where L(n₃,n_T)=[(2n₃l₃)²+(n_Tl_T)²]^1/2 is the length of a classical periodic path on a two-dimensional cylinder section, has been derived previously but never via periodic paths. At T=0 the semiclassical approach provides a finite and systematic approximation scheme in terms of classical paths, which is useful when the normal modes of the cavity cannot be determined either explicitly or implicitly. Slightly extending the domain of applicability of Gutzwiller’s semiclassical periodic-orbit approach, we here evaluate the free energy at T>0 in terms of periodic classical paths in a four-dimensional cavity, which is the tensor product of the original cavity and a circle. The validity of this approach is at present restricted to particular systems.