⚛️ high-energy theory

Inclusive Radiation and Backreaction from the Phase-Space S-Matrix

This paper develops a phase-space S-matrix framework that unifies classical scattering observables—such as waveforms, impulses, and angular momentum—by partially Weyl-transforming the matter sector, revealing how nonlinear gravitational memory, radiation reaction, and static-field effects emerge from an inclusive coherent waveshape and its associated quantum geometric structure.

Original authors: Nathan Moynihan

Published 2026-08-24

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Nathan Moynihan

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

When two massive objects, like black holes or charged spheres, fly past one another at high speeds, they do not simply bounce off like billiard balls. They interact through invisible fields that stretch across space, exchanging energy and momentum in a complex dance of cause and effect. As they swing around each other, they accelerate, and this acceleration causes them to shed ripples in the fabric of space and time, or bursts of electromagnetic energy, depending on the forces at play. These ripples carry away information, leaving the objects with slightly different speeds and positions than they started with. Physicists have long sought a precise way to calculate these changes, known as scattering, to understand everything from the orbits of planets to the gravitational waves detected by observatories on Earth. The challenge lies in connecting the quantum rules that govern the tiniest particles with the smooth, deterministic laws of classical physics that describe these massive collisions. For decades, researchers have used a tool called the scattering matrix, or S-matrix, which acts as a mathematical ledger recording every possible outcome of a collision. However, this ledger is often cluttered with quantum noise and infinite possibilities, making it difficult to extract the clean, predictable signal of a classical event.

In this new work, a researcher at Queen Mary University of London has developed a fresh way to look at this ledger, one that separates the heavy, solid matter from the light, fleeting radiation it emits. The core idea is to treat the massive particles and the radiation they produce as two different kinds of things within the same mathematical framework. The massive particles are treated as fixed, countable objects moving along specific paths, while the radiation is allowed to exist in any amount, behaving like a fluid that can be measured as a whole. By applying a specific mathematical transformation to the massive particles but leaving the radiation in its original, quantum form, the author creates a new kind of map. This map, called a phase-space symbol, acts as a bridge. It allows physicists to calculate the final state of the radiation and the final position of the massive objects directly from the fundamental rules of the collision, without getting lost in the quantum details.

The paper reveals that this new map unifies several different measurements that physicists usually calculate separately. The shape of the radiation wave that flies out into the universe, the change in the speed of the colliding objects, and the shift in their final positions all emerge as different views of the same underlying object. Previously, these were often treated as separate additions to a calculation, but here they are shown to be different projections of a single, coherent structure. The author demonstrates that the radiation field can be described by a "waveshape," a specific pattern that tells us exactly what the classical field looks like. Crucially, this waveshape is not just a simple copy of the first burst of radiation; it includes contributions from more complex interactions where multiple particles are created and then reabsorbed in the calculation. This inclusion is necessary to explain a phenomenon known as nonlinear memory, where the passing of a gravitational wave leaves a permanent, lasting distortion in space that cannot be explained by simple, linear effects.

One of the most significant findings is how the paper handles the "backreaction," or the way the radiation pushes back on the objects that created it. The author shows that the change in the objects' motion is driven by a geometric property of the space of all possible radiation waves. This property, known as a Berry connection, acts like a subtle force that corrects the path of the objects. It explains why the objects end up in a slightly different position than they would if only the direct push of the radiation were considered. This correction accounts for both the immediate recoil from the radiation and the lingering effects of the static fields that surround the objects. The work explicitly argues against the idea that the final state of the radiation is always a simple, coherent wave. While the classical field we observe behaves like a coherent wave, the underlying quantum state is more complex and "squeezed," containing correlations that a simple wave cannot capture. However, for the purpose of predicting the deterministic motion of massive objects, the coherent wave description is sufficient and accurate.

The paper also establishes a new balance law for the universe. In a closed system, the total area of the phase space—a measure of the possible states of the system—should remain constant. The author shows that while the massive particles alone do not preserve this area when they emit radiation, the missing area is not lost. Instead, it flows into the radiation sector. The change in the geometry of the massive particles is exactly balanced by the change in the geometry of the radiation field. This provides a deeper understanding of how energy and momentum are conserved in the presence of radiation, showing that the apparent loss of order in the massive particles is simply a transfer of that order to the waves they emit. By deriving these results directly from the fundamental Dyson S-matrix, the work provides a rigorous foundation for calculating classical observables, ensuring that the predictions for gravitational waves and electromagnetic radiation are consistent with the underlying quantum theory.

The implications of this work extend to the study of gravitational waves, which are ripples in spacetime generated by colliding black holes. The paper explicitly derives how the nonlinear memory effect, a subtle but permanent shift in spacetime caused by the collision, arises from the interplay of different quantum amplitudes. This confirms that the memory effect is an inclusive observable, meaning it depends on the sum of all possible radiation outcomes, not just the most likely single outcome. The author illustrates this by calculating the static contributions to the position shift and angular momentum in both scalar quantum electrodynamics and gravity, showing that the new geometric approach correctly reproduces known classical results while offering a clearer path to higher-order calculations. The work does not claim to solve every problem in scattering theory, but it provides a robust and unified framework that clarifies how classical physics emerges from quantum mechanics, offering a reliable tool for future research into the dynamics of the universe.

Technical Summary: Inclusive Radiation and Backreaction from the Phase-Space S-Matrix

Problem Statement
Modern scattering amplitude approaches have successfully organized classical two-body scattering, particularly for gravitational wave physics, often utilizing eikonal formulations where the S-matrix is treated as a phase. However, extending these formulations to include radiative observables and radiation-reaction effects presents challenges. Specifically, there is a tension between the "exclusive" nature of scattering amplitudes (which organize radiation into fixed-multiplicity sectors) and the "inclusive" nature of classical deterministic waveforms (which are one-point functions). Recent work suggested that nonlinear gravitational memory might require going beyond simple coherent state descriptions, as the exact outgoing state is generally not coherent due to nonlinear interactions. Furthermore, the origin of certain phase-space translations (Bopp shifts) used to define eikonal phases beyond leading order was previously observational rather than derived from first principles.

Methodology
The author develops a phase-space formalism based on a partial Weyl transform of the Dyson S-matrix.

Partial Weyl Transform: The matter sector (massive particles) is mapped to an on-shell phase-space symbol, while the radiation sector (massless particles) remains operator-valued in the Fock space. This results in an operator-valued symbol $S_W(\tilde{z})$ acting on the radiation vacuum, where $\tilde{z}$ represents the midpoint phase-space variables (midpoint positions and momenta).
Star Products and Bopp Shifts: The composition of operators in the matter sector translates to a Moyal star product ( $\star_m$ ) of their symbols. The author demonstrates that the "half-way" or "barred" momenta variables, often used in amplitude calculations, naturally arise as the midpoint variables of the Weyl transform. The star product induces Bopp shifts, which route local momentum transfers through the ordered sequence of current insertions along the worldline.
Inclusive Projections: Classical observables are derived as specific projections of this symbol:
- The inclusive waveform (waveshape $\alpha_I$ ) is obtained by an inclusive one-point projection involving the annihilation operator.
- The phase-space displacement (impulse and position shift) is derived from the derivative of the symbol with respect to phase-space variables.
Endpoint Approximation: The author distinguishes between the full star-product structure (which retains local recoil history) and the "endpoint approximation," where local shifts are replaced by global midpoint variables. They argue that endpoint reduction must only be applied after the full radiative block and Bopp shifts have been evaluated to correctly capture radiation.

Key Contributions and Results

Unified Origin of Observables: The paper establishes that the inclusive waveform, the classical impulse, the position shift, and the angular momentum are different projections of the same partially Weyl-transformed S-matrix symbol. They are not independent structures appended to an eikonal but arise from the same underlying object.
Derivation of Bopp Shifts: The author shows that the translation operators used in previous eikonal formulations are a manifestation of the two-particle Bopp shift, a natural consequence of the star-product structure of operator composition in phase space.
Coherent Representative vs. Exclusive State: The paper clarifies the distinction between the exact exclusive outgoing state (which is generally non-coherent and contains squeezed states due to nonlinearities) and a coherent representative of the inclusive waveform.
- The exact state is not coherent; it contains higher-order connected correlators.
- However, a coherent state defined by the full inclusive waveshape $\alpha_I = \langle a \rangle$ correctly reproduces all deterministic classical observables (linear in the field).
- Higher exclusive kernels (e.g., two-radiation amplitudes) contribute to the inclusive waveshape through cuts (e.g., $5pt^* \times 6pt$ cuts) even though they are $\hbar$ -suppressed when fully resolved.
Nonlinear Gravitational Memory: The formalism explicitly derives the nonlinear gravitational memory term as an inclusive observable. It confirms that this effect arises from a cut involving a six-point amplitude (composed with a five-point amplitude) and cannot be obtained from the uncut exclusive one-radiation kernel alone. This supports the view that the coherent representative is valid for the one-point field, even if the full state is not coherent.
Quantum Geometry and Radiation Reaction: The dependence of the coherent waveshape on hard scattering data induces a quantum geometry on the space of waveforms.
- This geometry is characterized by a Berry connection ( $A_A$ ) and curvature ( $F_{AB}$ ).
- The Berry connection contributes directly to the phase-space displacement, providing a geometric interpretation of radiation reaction.
- The author derives a symplectic area balance law: the loss of symplectic area in the massive particle sector due to radiation reaction is exactly balanced by the symplectic area spanned by variations in the outgoing radiation waveform.
Static Contributions: The author explicitly calculates the leading static-field contributions to the position shift and field angular momentum in both scalar QED and gravity. They show that the Berry connection captures these static effects, which were previously difficult to reconcile with purely radiative eikonal proposals. In gravity, these contributions are shown to be frame-dependent (related to BMS supertranslations).

Significance
The paper claims to place the proposal of a coherent radiative extension to the eikonal (from Ref. [1]) on firmer ground by deriving it directly from the Dyson S-matrix. It resolves the apparent contradiction between the non-coherent nature of the exact exclusive state and the existence of a deterministic classical waveform. By identifying the inclusive waveshape as the relevant coherent displacement, the author shows that the coherent representative correctly captures classical observables, including nonlinear memory and static backreaction, without needing to reproduce the full noise or squeezing of the exclusive state.

Furthermore, the work provides a rigorous phase-space origin for Bopp shifts and introduces a quantum geometric framework (Berry connection) that unifies the description of radiation reaction and static field effects. The derivation of the symplectic area balance law offers a new perspective on radiation reaction as a Hamiltonian flow on an enlarged phase space that includes the radiative degrees of freedom.

The author notes that while the coherent representative is sufficient for deterministic classical observables, the exact exclusive state remains necessary for quantum observables such as graviton noise or correlations between detectors. The formalism is presented as a tool to explore these distinctions and to compute higher-order classical corrections systematically.

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