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arXiv:2606.26946

On the simple derivation of the Casimir effect

Hai-Chau Nguyen, Matthias Kleinmann

quant-phphysics.ed-phSubmitted Jun 25, 2026• Announced Jun 26, 2026
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Abstract

The Casimir effect in its simplest form describes the attraction of two parallel conducting plates at close distance due to the vacuum fluctuation of the electromagnetic field. Its derivation can be found in many introductory works on quantum optics. Here we return to the original paper by Casimir and find subtle nuances in his derivation that are worth discussing to give a complete picture of a mathematically sound derivation of the effect.

Comments
5 pages, 1 figure. Comments are welcome!

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The Casimir effect in its simplest form describes the attraction of two parallel conducting plates at close distance due to the vacuum fluctuation of the electromagnetic field. Its derivation can be found in many introductory works on quantum optics. Here we return to the original paper by Casimir and find subtle nuances in his derivation that are worth discussing to give a complete picture of a mathematically sound derivation of the effect.
In previous works Suppes and de Barros used a pure particle model to derive interference effects, where individual photons have well-defined trajectories, and hence no wave properties. In the present paper we extend that description to account for the Casimir effect. We consider that the linear momentum \\sum\\frac{1}{2}\\hbar {\\bf k} of the vacuum state in quantum electrodynamics corresponds to the linear momentum of virtual photons. The Casimir effect, in the cases of two parallel plates and the solid ball, is explained in terms of the pressure caused by the photons. Contrary to quantum electrodynamics, we assume a finite number of virtual photons.
Abstract The electromagnetic Casimir effect manifests as the interaction between uncharged conducting objects that are placed in a vacuum. More generally, the Casimir-like effect denotes an induced interaction between external bodies in a fluctuating medium. We study the Casimir-like interaction between two impurities embedded in a weakly interacting one-dimensional Bose gas. We develop a theory based on the Gross–Pitaevskii equation that accounts for the effect of quantum fluctuations. At small separations, the induced interaction between the impurities decays exponentially with the distance. This is a classical result that can be understood using the mean-field Gross–Pitaevskii equation. We find that at larger distances, the induced interaction crosses over into a power law dependence due to the quantum fluctuations. We obtain an analytic expression for the interaction that interpolates between the two limiting behaviors. The obtained result does not require any regularization.
We investigate the Casimir effect in rainbow space-time, focusing on leading-order corrections to the Casimir energy and force. Starting with the scalar field Lagrangian in rainbow space-time, with parallel plates introduced through delta-function potentials, we find the corresponding energy-momentum tensor. We obtain the vacuum expectation value of this energy-momentum tensor by expressing it as a quadratic operator acting on the Green's function. By solving the Euler-Lagrange equation of a scalar field in rainbow space-time, we obtain the Green's function solutions. Employing these Green's function solutions in the vacuum expectation value of the energy-momentum tensor, we obtain the modified Casimir energy and Casimir force expressions in rainbow space-time. We study the variation of the deformed Casimir force and energy with the distance between the plates for different choices of rainbow functions. Our results show that for two choices of rainbow functions, the absolute value of the Casimir energy and force is decreasing or increasing, whereas for one specific choice of rainbow functions, it remains the same as the standard result in Minkowski space-time. Comparing our result with experimentally measured value of Casimir force, we obtain the bound on the rainbow parameter dependent terms to be of the order of 10^-24.
The quantum geometric tensor has been shown to bound the gap, optical absorption, and dielectric susceptibilities of materials. Here we derive new quantum-geometric bounds on the magnitude and sign of the Casimir force between two-dimensional plates in the long-distance limit. These bounds limit the previously attributed benefit of increasing the plate's Chern number to maximize repulsion, and give a quantum geometric origin to the stronger attractive force of metallic plates, regardless of their Chern number. These bounds allow us to infer that flat Chern bands that saturate geometric bounds, including Landau levels and moiré flat bands, enlarge the window where Casimir repulsion exists and bring the repulsive crossover to smaller, more experimentally relevant distances. We derive estimates for material platforms such as twisted MoTe2_22​. Our work shows that quantum-geometric bounds constrain repulsive Casimir forces beyond previously known theorems, and suggests new optimization strategies to observe repulsion.
The interference pattern of coherent electrons is effected by coupling to the quantized electromagnetic field. The amplitudes of the interference maxima are changed by a factor which depends upon a double line integral of the photon two-point function around the closed path of the electrons. The interference pattern is sensitive to shifts in the vacuum fluctuations in regions from which the electrons are excluded. Thus this effect combines aspects of both the Casimir and the Aharonov-Bohm effects. The coupling to the quantized electromagnetic field tends to decrease the amplitude of the interference oscillations, and hence is a form of decoherence. The contributions due to photon emission and to vacuum fluctuations may be separately identified. It is to be expected that photon emission leads to decoherence, as it can reveal which path an electron takes. It is less obvious that vacuum fluctuations also can cause decoherence. What is directly observable is a shift in the fluctuations due, for example, to the presence of a conducting plate. In the case of electrons moving parallel to conducting boundaries, the dominant decohering influence is that of the vacuum fluctuations. The shift in the interference amplitudes can be of the order of a few percent, so experimental verification of this effect may be possible. The possibility of using this effect to probe the interior of matter, e.g., to determine the electrical conductivity of a rod by means of electrons encircling it is discussed. (Presented at the Conference on Fundamental Problems in Quantum Theory, University of Maryland, Baltimore County, June 18-22, 1994.)
It is hypothesized that, within Bose-Einstein condensates, contaminants will form a potential that effects the energy state of a condensate. While assuming a system governed by the Gross-Pitaevskii equation, contaminants are modelled as boundary conditions for the wave function of the condensate. It is then found that the energy of the system depends directly upon the distance between contaminants. Energy is minimized as two particles either come together or move apart depending on the nature of the condensate. This is due to the presence of induced standing waves in the condensate between two contaminants, similar to the attractive effect caused by standing electromagnetic waves in a vacuum, the Casimir effect. Quantum calculations are also done to determine the expected strength of the "contaminant in condensate" effect.