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arXiv:2608.16619

On Casimir force between two real metallic plates

Michael V. Davidovich

quant-phSubmitted Aug 17, 2026• Announced Aug 18, 2026
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Abstract

The corrections to Casimir pressure (force) on two plates of real metal described by the dielectric constant of the Drude, Drude-Lorentz, Drude-Smith and plasma models at the finite temperature and conductivity are considered. We consider the contour integral lost in the Lifshitz formula and discuss inconsistencies for the corrections to the Casimir force given in the literature. Numerical and analytical results in the near and far zones are presented. The two first-order correction obtained due to the actual dielectric constant of the metal coincided with the literature data.

Comments
49 pages, 4 figures

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•Jan 1, 2005•14 citations
The corrections to Casimir pressure (force) on two plates of real metal described by the dielectric constant of the Drude, Drude-Lorentz, Drude-Smith and plasma models at the finite temperature and conductivity are considered. We consider the contour integral lost in the Lifshitz formula and discuss inconsistencies for the corrections to the Casimir force given in the literature. Numerical and analytical results in the near and far zones are presented. The two first-order correction obtained due to the actual dielectric constant of the metal coincided with the literature data.
We consider the Casimir force in configurations with magnetic metal plates and analyze the reasons why the predictions of the Lifshitz theory using the dielectric permittivity of the Drude model are inconsistent with the measurement data. For this purpose, the contributions of the electromagnetic waves with the transverse magnetic and transverse electric polarizations to the Casimir force are computed using the Lifshitz theory expressed in terms of the pure imaginary Matsubara frequencies. Furthermore, the fractions of the evanescent and propagating waves in these contributions are found using an equivalent formulation of the Lifshitz theory along the real frequency axis. All computations are performed for Au-Ni and Ni-Ni plates using the Drude model and the experimentally consistent plasma model over the separation region from 0.5 to 6~mum, where the total force value is determined by conduction electrons. It is shown that the transverse magnetic contribution to the Casimir force does not depend on the used model of the dielectric permittivity, so that the total difference between the predictions of the Lifshitz theory using the Drude model and the measurement data is determined by the transverse electric contribution. In doing so, as opposed to the case of nonmagnetic metals, both fractions of the evanescent and propagating waves in this contribution depend on the model of the dielectric permittivity used in computations, whereas the magnetic properties of the plate metal influence the Casimir force solely through the fraction of propagating waves in the transverse electric contribution. The issue of a more adequate theoretical description of the electromagnetic response of magnetic metals is discussed.
Based on the van Kampen-Schram method, a formula for the Casimir force between two plane-layered structures is obtained, which includes the sum of the Matsubara frequencies and a contour integral. It is shown that the contour integral is not taken into account in the Lifshitz formula. Taking it into account gives a small contribution at high temperatures when the Lifshitz formula is approximately correct. However, at low temperatures, its contribution is significant, and the Lifshitz formula for the finite temperature does not transform into the Lifshitz formula at zero temperature. An example of interaction of metal plates is considered.1
We develop a finite-temperature field-theoretical description of the Casimir interaction between two perfectly conducting boundaries with the intervening gap filled by a two-component quantum Boltzmann plasma embedded in a homogeneous dielectric background. The quantum motion of the charged particles is represented by imaginary-time worldlines, while the electromagnetic field is treated in the Gaussian approximation. The resulting partition function contains distinct longitudinal and transverse response contributions and an additional quantum-induced source term with no classical counterpart. In the classical limit the theory reduces to the usual Casimir-screening picture: the static longitudinal mode is Debye screened, whereas the transverse electromagnetic sector is unaffected by classical mobile charges. Finite thermal de Broglie wavelengths modify the nonzero Matsubara response and produce the leading quantum correction to the force. In the static long-wavelength limit, the longitudinal dielectric function has, through the leading quantum correction, the same expansion structure as the Lindhard response of a degenerate electron gas under the correspondence kTF↔κDk_{TF}\leftrightarrowκ_DkTF​↔κD​ and kF↔λdB−1k_F\leftrightarrowλ_{dB}^{-1}kF​↔λdB−1​, while the static transverse response reproduces the leading high-temperature Landau orbital diamagnetic susceptibility.
The Casimir force due to thermal fluctuations (or the pseudo-Casimir force) was previously calculated for a perfect Bose gas in the slab geometry for various boundary conditions. The Casimir pressure due to quantum fluctuations in a weakly-interacting dilute Bose–Einstein condensate (BEC) confined to a parallel plate geometry was recently calculated for Dirichlet boundary conditions. In this paper we calculate the Casimir energy and pressure due to quantum fluctuations in a zero-temperature homogeneous weakly-interacting dilute BEC confined to a parallel plate geometry with periodic boundary conditions and include higher-order corrections to the leading-order result which we refer to as Bogoliubov corrections. The leading-order term is identified as the Casimir energy of a massless scalar field moving with a wave velocity equal to the speed of sound in the BEC. We then obtain the leading-order Casimir pressure in a general three-dimensional rectangular cavity of arbitrary length and obtain the finite-size correction to the parallel plate scenario.