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Long-range magnetic interaction within quantum electrodynamics formalism

This paper utilizes the S-matrix formalism of quantum electrodynamics to derive a long-range magnetic interaction potential between atoms that extends classical predictions by accounting for state-dependent deviations, specifically calculating dispersion coefficients for hydrogen s-states and demonstrating the method's applicability to hydrogen-antihydrogen systems.

Original authors: D. Solovyev, T. Zalialiutdinov, A. Anikin, A. Bobylev, P. Kvasov, J. J. Lopez-Rodriguez, A. Moshkin, D. Zinenko, D. Glazov

Published 2026-07-27

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CC BY 4.0

Original authors: D. Solovyev, T. Zalialiutdinov, A. Anikin, A. Bobylev, P. Kvasov, J. J. Lopez-Rodriguez, A. Moshkin, D. Zinenko, D. Glazov

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 authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine two invisible dancers floating in a vast, empty ballroom. They aren't touching, and they aren't holding hands, yet they somehow "know" what the other is doing. In the world of physics, this is the story of how atoms talk to each other across empty space. Usually, we think of atoms as tiny, neutral balls that only bump into each other when they get very close. But even when they are far apart, they whisper to one another through invisible forces. The most famous of these whispers is the "van der Waals" force, a kind of magnetic handshake that happens because the atoms' internal electric charges wiggle and create tiny, fleeting ripples in the space between them. Scientists have known about these electric whispers for a long time. But there is another, quieter kind of whisper: the magnetic one. This happens because electrons inside atoms spin like tiny tops, creating their own miniature magnetic fields. While we know how these magnetic fields work when atoms are close, the rules for how they behave when atoms are very far apart have been a bit fuzzy. Understanding this is crucial because it helps us predict how atoms stick together to form molecules, or how they might behave in extreme environments, like the edge of a black hole or in a lab trying to create antimatter.

This paper dives deep into that quiet magnetic whisper using the most rigorous rulebook physics has: Quantum Electrodynamics (QED). Think of QED as the ultimate instruction manual for how light and matter interact. The authors, a team of physicists from Russia, decided to use this manual to calculate exactly how two atoms interact magnetically when they are far apart, without changing their internal states. They didn't just guess; they used a mathematical tool called the "S-matrix" to track the exchange of a single photon (a particle of light) between two atoms. Their main finding is that they successfully derived a formula for this long-range magnetic force that matches what classical physics predicted, but with a twist: they showed that the rules change depending on the specific "state" of the atoms.

Here is the fun part: the authors discovered that if you have a pair of normal atoms (like two hydrogen atoms), they push and pull in a certain way. But if you swap one of them for its evil twin, an "antihydrogen" atom (which is made of antimatter), the magnetic interaction flips its sign. It's like if two magnets usually repel each other, but if you turn one of them inside out, they suddenly attract. This isn't just a theoretical curiosity; the paper suggests that this attraction could be a secret door to understanding why we don't see much antimatter in the universe today. The authors calculated that at room temperature, the "heat" of the environment (thermal radiation) can actually make this magnetic attraction stronger, potentially causing hydrogen and antihydrogen to crash into each other and annihilate much faster than we thought.

The paper also looked at what happens when atoms are in a special "hyperfine" state, where their internal spins are slightly different. In this case, the interaction doesn't just fade away quickly like a normal magnet; it stretches out much further, following a different mathematical rule (falling off as $1/R$ instead of $1/R^3$ ). While this effect is tiny and hard to measure with current tools, the authors show that it is a real, quantum phenomenon with no classical equivalent. They even calculated the numbers for hydrogen atoms: at a distance of 10 atomic units, the magnetic energy is about $87.6$ MHz, and at the distance where hydrogen molecules usually sit, it matches known spin-spin corrections.

Crucially, the authors are careful not to claim they have solved the mystery of antimatter's disappearance. They suggest that their findings could explain why antimatter is rare, because higher temperatures might speed up the annihilation process between matter and antimatter. They emphasize that this is a "qualitative explanation" and that the imaginary parts of their equations (which represent decay or energy loss) need more study. They didn't measure this in a lab; they derived it mathematically from first principles. So, while the paper doesn't give us a new engine or a cure for a disease, it gives us a clearer, more precise map of the invisible magnetic landscape between atoms, showing us that even in the emptiness of space, the dance of matter and antimatter is far more complex and fascinating than we previously imagined.

Technical Summary: Long-range magnetic interaction within quantum electrodynamics formalism

Problem Statement
The investigation of long-range interactions between atoms is a well-established field, traditionally described by fourth-order perturbation theory for dipole-dipole (van der Waals) interactions. While the classical electrodynamics expression for magnetic dipole-dipole interaction is known, its rigorous derivation from first principles within Quantum Electrodynamics (QED) for large interatomic distances remains a specific focus. Furthermore, the behavior of these interactions under finite temperature conditions and the potential for deviations from classical predictions based on atomic state structures (specifically hyperfine structure) require further theoretical clarification. The paper addresses the derivation of the magnetic interaction potential between two one-electron atoms at large distances, considering both zero and finite temperatures, and examines how the asymptotic behavior depends on the specific atomic states involved.

Methodology
The authors employ the S-matrix formalism within QED perturbation theory to analyze the interaction between two atoms (denoted A and B). The analysis is conducted under the assumption of one-photon exchange, which is the lowest-order contribution for magnetic interactions when the dominant electric dipole-dipole term vanishes at large distances.

Key methodological steps include:

Formalism: The calculation utilizes the S-matrix element for one-photon exchange, incorporating both direct and exchange Feynman diagrams to account for the permutation symmetry of identical atoms.
Photon Propagator: The photon propagator is expressed in the temporal (Weyl) gauge and includes contributions from both zero-temperature ( $D^0_{\mu\nu}$ ) and finite-temperature ( $D^\beta_{\mu\nu}$ ) environments, the latter accounting for the Planck frequency distribution of thermal photons.
Approximations: The derivation employs the adiabatic approximation, assuming electronic motion is much faster than the variation of the internuclear separation. The inter-electronic distance is expanded via a Taylor series around the internuclear distance $R$ .
State Considerations: The study considers two scenarios:
1. Unchanged States ( $\varepsilon_{aa'} = 0$ ): The atoms remain in their initial states, leading to a static magnetic interaction.
2. Hyperfine Transitions ( $\varepsilon_{aa'} = \Delta_{HFS} \neq 0$ ): The interaction involves transitions between hyperfine sublevels, where the energy difference corresponds to the hyperfine splitting.
Operator Reduction: The Dirac $\alpha$ -matrices are related to the magnetic moment operator $\mu = e[\mathbf{r} \times \boldsymbol{\alpha}]/2$ to extract the magnetic dipole-dipole interaction terms from the general amplitude.

Key Contributions and Results

Derivation of Magnetic Potential: The authors derive an expression for the magnetic interaction potential that agrees with the well-known classical result for the spin-spin (J-coupling) interaction in the limit of unchanged atomic states. The derived potential for two identical atoms in the same state is:
$\Delta E^{(m)}_{AB} = \frac{\langle (\boldsymbol{\mu}_A \boldsymbol{\mu}_B) \rangle}{R^3} - \frac{3\langle (\boldsymbol{\mu}_A \mathbf{R}) (\boldsymbol{\mu}_B \mathbf{R}) \rangle}{R^5}$
This result is obtained rigorously from QED without relying on phenomenological generalizations.
Finite Temperature Effects: The paper derives the thermal contribution to the magnetic interaction. Unlike the zero-temperature case which scales as $1/R^3$ , the thermally induced interaction for unchanged states scales as $1/R^2$ (with an imaginary component). The authors note that for pairs like hydrogen-antihydrogen, the sign of the magnetic interaction is opposite to that of ordinary matter due to the charge of the constituent particles.
Hyperfine Structure and Asymptotic Behavior: A significant finding is that when considering transitions between hyperfine states ( $\varepsilon_{aa'} = \Delta_{HFS} \neq 0$ ), the asymptotic behavior of the interaction potential deviates from the classical $1/R^3$ prediction.
- The real and imaginary parts of the interaction energy for hyperfine transitions scale as $1/R$ .
- Specifically, the dispersion coefficient for the $1s$ state of hydrogen shows a change in scaling from $1/R^3$ to $1/R$ .
- The thermally induced contribution for hyperfine transitions is enhanced by the Planck distribution function $n_\beta(|\Delta_{HFS}|)$ , making the imaginary part of the correction four orders of magnitude larger than the non-perturbed counterpart, though still small at current experimental precision.
Hydrogen-Antihydrogen System: The formalism is applied to the hydrogen-antihydrogen system. The authors demonstrate that the sign of the interaction potential reverses compared to the hydrogen-hydrogen system. This sign reversal affects both the real part (interaction energy) and the imaginary part (decay/annihilation rates).

Significance and Claims
The paper claims that the QED formalism provides a rigorous foundation for understanding long-range magnetic interactions, extending beyond classical electrodynamics by explicitly treating atomic energy level structures.

Deviation from Classical Predictions: The study highlights that the asymptotic behavior of the interaction potential is not universally $1/R^3$ ; it depends on the atomic states. Specifically, interactions involving hyperfine transitions exhibit a $1/R$ scaling, a deviation from the standard van der Waals or static magnetic dipole behavior.
Matter-Antimatter Comparison: The derived expressions allow for a straightforward comparative analysis of long-range interactions between matter and antimatter. The sign reversal in the hydrogen-antihydrogen system suggests distinct interaction dynamics compared to ordinary matter.
Thermal Annihilation: The authors interpret the positive imaginary part of the interaction potential in the context of the hydrogen-antihydrogen system as a thermally induced decay channel. They suggest that higher temperatures could stimulate annihilation between hydrogen and antihydrogen atoms, offering a qualitative explanation for the absence of antimatter in the universe, though they explicitly state this is a qualitative explanation requiring further detailed study.
Precision Spectroscopy: The work notes that while these long-range magnetic interactions are generally negligible at large distances (e.g., micrometers), they could become relevant in high-precision measurements of atomic transitions (such as the $1s-2s$ transition in hydrogen) if the atomic density is high or if the system involves symmetric/antisymmetric diatomic states where exchange contributions do not cancel.

The authors conclude that while the magnetic long-range interaction is suppressed by factors of the fine-structure constant and temperature-dependent coefficients, the rigorous QED treatment reveals nuances in asymptotic behavior and thermal effects that are not captured by conventional phenomenological approaches.

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