1 4 Ju n 20 05 The Casimir force at high temperature

– The standard expression of the high-temperature Casimir force between perfect conductors is obtained by imposing macroscopic boundary conditions on the electromagnetic field at metallic interfaces. This force is twice larger than that computed in microscopic classical models allowing for charge fluctuations inside the conductors. We present a direct computation of the force between two quantum plasma slabs in the framework of non relativistic quantum electrodynamics including quantum and thermal fluctuations of both matter and field. In the semi-classical regime, the asymptotic force at large slab separation is identical to that found in the above purely classical models, which is therefore the right result. We conclude that when calculating the Casimir force at non-zero temperature, fluctuations inside the conductors can not be ignored. Casimir showed in 1948 [1] that the zero-point energy of the quantum electromagnetic field generates an attractive force between two perfectly conducting metallic plates at distance d and zero temperature. In his calculation, the microscopic structure of the conductors is not taken into account. The latter are merely treated as macroscopic boundary conditions for the electromagnetic fields requiring the vanishing of the tangential electric field. This geometrical constraint modifies the field eigenmodes depending on d. The d-dependence of the modified zero-point energy is the source of the well known Casimir force f vac (d) = − π 2 ¯ hc 240 d 4 (1) (¯ h denotes Planck's constant, c the speed of light). The generalisation of Casimir's calculation to thermalized fields was given some years later in [2, 3], see [4] for a recent account. When the temperature T is different from zero, one can form the dimensionless parameter α = βπ¯ hc/d (the ratio of the thermal wave length of the photon to the conductors separation; β is the inverse temperature). A large value of α (low temperature, short separation) characterizes the quantum regime whereas a small value of α (high temperature, large separation) yields a purely classical asymptotic result (independent of ¯ h and c) f = − ζ(3) 4πβd 3 + O(e −b/α), α → 0, b > 0 (2)