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arXiv:2601.07661

On the Lifshitz formula of dispersion interaction

Michael V. Davidovich

quant-phSubmitted Jan 12, 2026• Announced Jan 13, 2026

Abstract

The Lifshitz formula and methods of its preparation in the literature are considered. It is shown that in Lifshitz's work itself, this formula is given without a consistent conclusion. Moreover, the approach to the conclusion proposed in this work does not allow us to obtain it. The most general conclusion of this formula can be the method proposed by Levin and Rytov, the variation method of Schwinger and the method proposed by Van Kampen and co-authors. The Levin and Rytov approach is applicable in principle to bodies of arbitrary shape if the diffraction loss fields for electric and magnetic dipoles are determined, while the Van Kampen approach is applicable to any plane-layered structure and is quite simple. It is enough to write down the dispersion equations of the plasmon-polaritone structure. The specific dispersion force for a number of structures is calculated based on the Van Kampen method. It is shown that at small gaps, the force (pressure) density changes the inverse fourth-degree dependence on the distance and practically ceases to depend on it at distances less than 1 nm. For thin identical plates, this density is proportional to the square of their thickness at such distances, but the dependence quickly becomes saturated and already at thicknesses of the order of 10 nm practically ceases to depend on it.

Comments: arXiv admin note: substantial text overlap with arXiv:2508.00025

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The Lifshitz formula and methods of its preparation in the literature are considered. It is shown that in Lifshitz's work itself, this formula is given without a consistent conclusion. Moreover, the approach to the conclusion proposed in this work does not allow us to obtain it. The most general conclusion of this formula can be the method proposed by Levin and Rytov, the variation method of Schwinger and the method proposed by Van Kampen and co-authors. The Levin and Rytov approach is applicable in principle to bodies of arbitrary shape if the diffraction loss fields for electric and magnetic dipoles are determined, while the Van Kampen approach is applicable to any plane-layered structure and is quite simple. It is enough to write down the dispersion equations of the plasmon-polaritone structure. The specific dispersion force for a number of structures is calculated based on the Van Kampen method. It is shown that at small gaps, the force (pressure) density changes the inverse fourth-degree dependence on the distance and practically ceases to depend on it at distances less than 1 nm. For thin identical plates, this density is proportional to the square of their thickness at such distances, but the dependence quickly becomes saturated and already at thicknesses of the order of 10 nm practically ceases to depend on it.

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

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.

With the use of second-order perturbation theory in the long-range interatomic interaction for the degenerate states of two Rydberg atoms we have obtained a general formula for the dependence of atomic interaction energy on the interatomic distance R in the presence of the Förster resonance. Inside of the ‘Förster sphere’ ( R < R F ) this dependence transforms to the formula for electric dipole interaction energy Δ E d − d = C 3 / R 3 and for R > R F it transforms to the formula for the van der Waals interaction energy Δ E VdW = − C 6 / R 6 . The van der Waals constant C 6 is represented as an expansion in terms of irreducible components which define the dependence on the interatomic axis orientation relative to the quantisation axis of projections M of the total angular momentum J . The numerical values of the irreducible components of tensor C 6 were calculated for rubidium atoms in the same Rydberg states | nlJM 〉 with large quantum numbers n . We present the calculated resonance interaction energy of two rubidium atoms in the states |43D 5/2 M 〉, whose total energy exceeds by only 8 MHz the total energy of one of the atoms in the state |45P 3/2 M 〉 and of the other in the state |41F 7/2 M 〉.

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.