⚛️ quantum physics

Dispersion interaction between two bodies of arbitrary shape

This paper derives a general framework for calculating Casimir dispersion forces between two arbitrarily shaped bodies by utilizing vacuum tensor Green's functions to establish Lippmann-Schwinger-type equations for electromagnetic field correlations, which are then applied to the Maxwell stress tensor and Lorentz force formula via electric and magnetic dipole excitations.

Original author: Michael V. Davidovich

Published 2026-11-10

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CC BY 4.0

Original author: Michael V. Davidovich

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the author. For technical accuracy, refer to the original paper. Read full disclaimer

In the invisible world of quantum physics, empty space is never truly empty. Even in a perfect vacuum, fields of energy flicker with a restless, jittery motion known as zero-point fluctuations. These are not mere theoretical ideas but physical realities that exert a tangible pressure. When two objects are placed close together, these fluctuations can push or pull them, creating a force known as the Casimir effect. For decades, scientists have understood how this force behaves between simple, flat surfaces, like two parallel mirrors. However, the real world is rarely so simple. Most objects are curved, irregular, or oddly shaped, and calculating the force between such complex bodies has remained a stubborn mathematical challenge. Understanding these forces is crucial for the future of nanotechnology, where tiny machines operate at scales where this invisible pressure can dominate over gravity or friction.

A researcher at Saratov National Research State University has now developed a new theoretical framework to calculate these forces for objects of any shape. Instead of relying on approximations that only work for flat layers or simple spheres, the team derived a set of general equations based on the fundamental laws of how electromagnetic fields interact with matter. The core of their approach involves treating the bodies as collections of tiny, fluctuating sources—essentially imagining the material as being made up of countless microscopic electric and magnetic dipoles that are constantly jiggling due to heat and quantum effects. By tracking how the fields generated by these jiggling sources in one body interact with the other, the researchers establish the mathematical foundation to determine the total force without needing to simplify the shape of the objects, though obtaining explicit numerical results for specific configurations requires further work.

The study derives a set of equations that link the random fluctuations of electric currents inside the materials to the resulting electromagnetic fields. These fields are then used to calculate the stress, or pressure, exerted on the surface of the bodies. The author demonstrates that this framework applies to two bodies of arbitrary shape, which could be spheres, cubes, or anything in between, placed in a thermal environment. The analysis accounts for both the random thermal vibrations caused by temperature and the inherent quantum fluctuations of the vacuum. A key finding is that the force depends entirely on the distance between the bodies and their specific geometric shapes, as well as their material properties like how they respond to electric and magnetic fields.

To make the calculations manageable, the researcher uses a technique that expresses the complex fields in terms of simpler "Green's functions," which act as a map showing how a disturbance at one point travels through space to another point. By solving these maps, the team can determine the correlations between the fields at different points on the surfaces of the two bodies. These correlations are then fed into formulas that describe the flow of momentum, allowing the total force to be calculated by integrating the pressure over the entire surface of the object. The paper confirms that this approach is consistent with the conservation of momentum, ensuring that the force pushing one body is equal and opposite to the force pushing the other.

The results show that this framework is flexible enough to handle not just simple shapes but also complex, anisotropic materials where properties change depending on the direction, and even bodies with internal variations in their composition. While the paper focuses on the theoretical framework, it notes that for very simple shapes like flat plates, the results align with known formulas. For more complex shapes, explicit ratios require the introduction of a specific body configuration, a task the author indicates needs to be considered in a separate paper. The author also touches on the role of temperature, suggesting that for very hot objects or those separated by large distances, thermal radiation might become the dominant factor over the quantum vacuum fluctuations. This new framework offers a powerful tool for physicists and engineers who need to predict how tiny, irregularly shaped objects will behave when they are brought close together, bridging the gap between idealized theory and the messy reality of complex geometries.

Technical Summary: Dispersion Interaction Between Two Bodies of Arbitrary Shape

Problem Statement
The paper addresses the theoretical challenge of calculating Casimir (dispersion) forces between two macroscopic bodies of arbitrary shape and configuration. While the phenomenon was introduced eighty years ago, existing analytical results are predominantly limited to plane-layered structures (utilizing the Lifshitz formula) or specific geometries like spheres and cylinders. For bodies with curved or arbitrary surfaces, current methods rely on approximations such as the Proximity Force Approximation (PFA) or complex scattering matrix theories that often involve perturbations. A significant difficulty lies in constructing tensor Green's functions (GFs) for structures with discontinuous coefficients and complex boundary conditions, particularly within the framework of macroscopic quantum electrodynamics. The paper notes that while several papers describe interactions for complex shapes like cylinders and balls, explicit results for bodies of arbitrary shape have not yet been obtained. This work aims to derive a general formalism to address this gap.

Methodology
The author proposes a new approach based on the vacuum tensor Green's functions of electrodynamics and the correlation of electromagnetic fields. The methodology proceeds through the following steps:

Field Representation and Sources: The macroscopic bodies are described by their permittivity ( $\hat{\epsilon}$ ) and permeability ( $\hat{\mu}$ ). The interaction is modeled by introducing fluctuating polarization current densities (electric and magnetic) within the bodies, which act as sources for the electromagnetic field. These fluctuations are linked to thermal radiation and zero-point vacuum fluctuations via the Rytov-Levin concept.
Integral Equations: The electromagnetic fields are expressed using Lippmann-Schwinger-type integral equations (LSIE). These equations relate the total field to the incident field (thermal or vacuum) and the induced polarization currents, mediated by the vacuum tensor Green's functions.
Correlation Functions: To determine the force, the paper derives correlation relations (CR) for the electromagnetic fields. These correlations are obtained by solving the LSIEs and are expressed in terms of the correlations of the fluctuating current densities. The spectral power density of these fluctuations is related to the Bose-Einstein function and the temperature of the bodies.
Force Calculation: Two distinct methods are employed to calculate the force:
- Maxwell Stress Tensor (MST): The force is determined by integrating the Maxwell stress tensor over the surface of one body. The tensor components are constructed from the field correlations ( $\langle E E^* \rangle$ and $\langle H H^* \rangle$ ).
- Lorentz Force: The force is calculated by integrating the Lorentz force density acting on the polarization currents and surface charges of the first body due to the field generated by the second body. The paper argues this approach is preferable as it is applicable in any medium where the magnetic induction vector $\mathbf{B}$ is defined.
Regularization: The derivation addresses the divergence of the zero-point energy integral. The paper suggests subtracting the limit value of the Green's function at high frequencies (which corresponds to the infinite energy of the vacuum independent of body configuration) to ensure the convergence of the frequency integrals.

Key Contributions and Results

General Formalism: The paper derives Lippmann-Schwinger-type equations for electromagnetic field correlations applicable to bodies of arbitrary shape. The author explicitly states that while the general formalism and integral equations are established, obtaining explicit numerical ratios requires the introduction of a specific body configuration, as explicit results for arbitrary shapes have not yet been obtained in the literature.
Dual Approach: It establishes the equivalence of calculating dispersion forces via the Maxwell stress tensor and the Lorentz force formula, demonstrating that both stem from the conservation of momentum of the electromagnetic field and matter.
Dipole Excitation Model: The analysis utilizes the excitation of bodies by electric and magnetic dipoles to formulate the correlation relations. The paper provides specific integral equations (Eqs. 4, 5, 26) and correlation formulas (Eqs. 17, 19, 20) that link field correlations to the Green's functions of the vacuum.
Handling Divergence: A specific regularization technique is proposed to handle the infinite energy of the quantum vacuum by subtracting the high-frequency limit of the Green's function, ensuring the resulting force depends only on the body configuration.
Extension to Complex Media: The framework is shown to be generalizable to anisotropic, inhomogeneous, and dispersive media, as well as bodies at different temperatures, though the paper focuses primarily on the isothermal case for simplicity.

Significance and Claims
The paper claims to provide a rigorous theoretical foundation for calculating Casimir forces in complex geometries where previous results were either unavailable or relied on approximations. By formulating the problem through vacuum tensor Green's functions and correlation relations, the author offers a method that can, in principle, be applied to any arbitrary configuration.

The author notes that while the general formalism is established, obtaining explicit numerical ratios requires specifying the body configuration. The paper suggests that the simplest explicit results would be for flat screens (e.g., graphene sheets) or finite layers, which would be the subject of future work. The approach is presented as a step toward a more complete description of dispersion interactions in macroscopic quantum electrodynamics, particularly for bodies where the proximity force approximation is insufficient. The work is supported by the Ministry of Education and Science of the Russian Federation.

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