The Casimir force at high temperature

Abstract

Abstract. -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(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 = βπhc/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 ofh and c)

Casimir energies: Temperature dependence, dispersion, and anomalies

“…In particular, the lack of a thermodynamic inconsistency has been conclusively demonstrated [33,34], by showing that the free energy for a Casimir system made from real metal plates with impurities has a quadratic temperature dependence at low temperature. Further evidence for the validity of the notion of excluding the TE zero mode for metals comes from the recent work of Buenzli and Martin [35], corroborating earlier work by these authors and others [36,37], who show from a microscopic viewpoint that the high-temperature behavior of the Casimir force is half that of an ideal metal, a rather conclusive demonstration that the TE zero mode is not present. 1 Our purpose with the present paper is not to study the temperature corrections in further detail.…”

Present status of controversies regarding the thermal Casimir force

“…At large separations or high temperatures the Drude model approach predicts 1/2 the magnitude of the Casimir free energy and force than between ideal metals. (The same prediction was obtained using the model of a conducting wall by the classical Coulomb fluid [16] or 'nonrelativistic quantum electrodynamics' [17].) This prediction is in contradiction with the classical limit based on Kirchhoff's law [18].…”

“…Note that the approximation was made only at ω = 0, whereas in the limit ω → 0, which is of special importance for our problem, equations (17) and (19) remain equivalent.…”

“…We emphasize that the dispersion relations (19) with the Leontovich impedance have both photonic and plasmonic solutions. It is not admissible, however, to preserve k ⊥ = 0 in the impedance function Z ⊥ (ω, k ⊥ ), and to consider ω → 0 because this violates the dispersion relations (17). The Matsubara frequencies and the transverse momentum become independent only after application of the argument theorem.…”

Casimir force on real materials—the slab and cavity geometry

“…Moreover, we note the connection between our approach and the statistical mechanical approach of Buenzli and Martin [69]. These authors computed the force between two quantum plasma slabs within the framework of non-relativistic quantum electrodynamics including quantum and thermal fluctuations of both matter and field.…”