KMOC Formalism in Quantum-Classical Dynamics

• KMOC Formalism is a method that translates on-shell quantum amplitudes into classical observables such as momentum impulses, waveforms, and radiated energy.
• It employs a master formula integrating virtual and real kernels via generalized and reverse unitarity techniques to achieve precise predictions in high-energy and strong-field regimes.
• The formalism ensures cancellation of super-classical divergences and establishes equivalence with worldline and EFT approaches across gravitational and gauge theories.

The Kosower–Maybee–O’Connell (KMOC) formalism is a methodology for extracting classical observables—such as momenta, waveforms, impulses, spins, and radiated fluxes—directly from quantum field theoretical amplitudes, specifically S-matrix elements, in gauge and gravitational theories. It provides a systematic prescription to relate on-shell scattering amplitudes to classical dynamics at leading and subleading orders in the coupling, post-Minkowskian (PM), and post-Newtonian (PN) expansions, enabling ab initio calculations of classical observables relevant for two-body high-energy and strong-field interactions, including the emission of radiation and radiation reaction in general relativity and Yang-Mills theory (Herrmann et al., 2021, Damgaard et al., 2023, Capatti et al., 14 Dec 2024).

1. Foundational Principles of the KMOC Formalism

The KMOC formalism is built upon the observation that classical observables can be recast as expectation values of quantum operators in carefully constructed in and out states. For initial states comprising two massive (possibly spinning) particles prepared in sharply peaked coherent wavepackets, the difference in expectation value of any operator $\mathbb O$ after the S-matrix evolution yields the physical change:

$$\Delta O = \langle \text{out} | \mathbb O | \text{out} \rangle - \langle \text{in} | \mathbb O | \text{in} \rangle = \langle \text{in} | S^\dagger \mathbb O S - \mathbb O | \text{in} \rangle.$$

Expanding $S=1+iT$ and exploiting unitarity, this can be decomposed into "virtual" and "real" terms:

$$\Delta O = i\langle [\mathbb O, T] \rangle + \langle T^\dagger [\mathbb O, T] \rangle.$$

Classical observables are properly isolated by taking the $\hbar \to 0$ limit after organizing all powers of $\hbar$, preserving only those terms that survive. The reduction to explicitly classical integrands occurs before integration, ensuring only physically relevant contributions remain (Herrmann et al., 2021, Bern et al., 2021, Alessio et al., 3 Jun 2025).

2. The Master Formula and General Structure

The key output of the formalism is the KMOC master formula for the change in any classical observable $O$:

$$\Delta O = \int \hat{\mathrm d}^D q~\hat\delta(-2p_1 \cdot q)\,\hat\delta(2p_2 \cdot q)\, e^{i b \cdot q} \left[ \mathcal I_O^{(v)}(p_i, q) + \mathcal I_O^{(r)}(p_i, q) \right].$$

Here, $\mathcal I_O^{(v)}$ is the "virtual" kernel constructed from loop-level amplitudes, and $\mathcal I_O^{(r)}$ is the "real" kernel built from unitarity cuts (real emission). In practical computations, these kernels are expressed as integrals or sums over products of on-shell amplitudes with appropriate measurement insertions, such as momenta or angular momentum, decorating the cut lines (Herrmann et al., 2021, Cordero et al., 2022). The impact-parameter phase $e^{i b \cdot q}$ ensures localization in transverse space, and the delta functions enforce eikonal kinematics.

The conservative and radiative sectors are separated by restricting the set of intermediate on-shell states included in the sum over cuts: the conservative sector involves solely massive legs, while inclusion of massless radiative modes accesses radiation-reaction and emission effects.

3. Kernel Construction: Generalized Unitarity and Reverse Unitarity Methods

The computation of classical observables at a given PM order requires the systematic derivation of the relevant loop integrands using modern amplitude techniques:

• Generalized Unitarity: The relevant two-body scattering amplitude is decomposed into a sum over cubic graphs with unknown numerators. These are completely determined by evaluating a spanning set of unitarity cuts (openings along on-shell internal lines) and matching to the product of tree-level amplitudes. For radiative observables, three-particle cuts (two heavy lines and one massless line) encode the radiation channels.
• Reverse Unitarity: Multi-particle phase-space integrals arising from cut diagrams are mapped back to loop integrals by rewriting delta functions as differences of propagators, e.g.,

$$2\pi \delta(k^2) \longleftrightarrow \frac{1}{k^2 - i0} - \frac{1}{k^2 + i0}.$$

This allows a unified reduction of all integrals—virtual and real—to a canonical set of master integrals, which can be evaluated using integration-by-parts (IBP) identities, differential equations, and method of regions (Herrmann et al., 2021, Herrmann et al., 2021).

• Soft (Classical) Expansion: All loop momentum integrals are expanded in the small transfer limit ($q, \ell \sim \mathcal O(\hbar)$), and propagators are linearized (e.g., $1/[(k + p)^2 - m^2] \to 1/(2u \cdot k)$).

All super-classical ($1/\hbar^n$, $n>0$) divergences cancel identically between the virtual and real contributions, a structure established to hold order by order in perturbation theory (Capatti et al., 14 Dec 2024, Sinha, 24 Jan 2025).

4. Applications: Conservative and Radiative Observables

The KMOC formalism enables first-principles computation of a wide range of classical observables:

• Momentum Impulse and Scattering Angle: Setting $\mathbb O$ to the momentum operator yields the net impulse on each body. The explicit form, after integrating over the kernel and phase-space, gives the impulse in terms of simple transcendental functions of the Lorentz factor $\sigma = p_1 \cdot p_2/(m_1 m_2)$ and the impact parameter $b$. The scattering angle is extracted from the transverse impulse via algebraic or stationary-phase relations (Herrmann et al., 2021, Herrmann et al., 2021).
• Radiated Momentum and Energy: Taking the operator to be the total radiated graviton momentum computes the gravitational-wave recoil (bremsstrahlung). The leading nontrivial result appears at $\mathcal O(G^3)$ and depends on the full velocity dependence, with Fourier representations allowing analytic continuation into bound or elliptic orbits.
• Waveforms and Angular Momentum Loss: Insertion of field creation/annihilation operators or angular momentum operators accesses the waveform at infinity and radiated angular momentum, with explicit formulae involving phase-space integrals of five-point on-shell amplitudes (Angelis et al., 2023, Alessio et al., 3 Jun 2025).
• Spin Effects: By extending the formalism to wavepackets (asymptotic states) carrying arbitrary spin using coherent states and suitable operators (e.g., the Pauli–Lubanski vector or corresponding angular momentum tensors), one computes both linear and nonlinear spin contributions to the impulse and radiation. The KMOC approach accommodates all spin orders, with higher multipole effects incorporated directly at the amplitude level (Gatica, 2 Dec 2024, Cordero et al., 2022).
• Gauge, Color, and Memory Observables: The formalism naturally extends to Yang–Mills theory, enabling computation of both momentum ("linear impulse") and color ("color impulse") observables, and through soft expansions, yields electromagnetic and gravitational memory effects, matching the universal soft theorems at all orders (Cruz et al., 2021, A. et al., 2022, Akhtar et al., 21 Nov 2025).

5. Exponential/Coherent-State Representation and Dirac Brackets

A significant refinement introduces an exponential (coherent-state) representation of the S-matrix:

$S = \exp\Big(\frac{i}{\hbar} N \Big),$

where the Hermitian operator $N$ organizes (after projection) the entire series of classical commutators for any observable $O$ as nested brackets:

$$\Delta O = S^\dagger O S - O = \sum_{n=1}^\infty \frac{(-i)^n}{\hbar^n n!} [N,[\ldots[N,O]\ldots]].$$

In the classical limit, commutators become Dirac brackets, with constraints enforcing on-shellness and spin-supplementary conditions in phase space. This formalism avoids explicit computation of cut diagrams in higher orders, enabling a compact, gauge-invariant, and causal formulation of observables—including all fluxes and waveforms—in terms of a minimal set of on-shell kernels ($\mathcal K^{\rm cl}$) (Alessio et al., 3 Jun 2025).

6. Equivalence with Worldline and Effective Field Theory Approaches

The KMOC formalism is explicitly equivalent to worldline quantum field theory and post-Minkowskian EFT approaches:

• The in-in (closed-time-path, Schwinger-Keldysh) structure of KMOC exactly matches the doubled-worldline/retarded-propagator framework of worldline EFT, with the integration kernels and observable formulas mapping one-to-one at the integrand level (Damgaard et al., 2023, Capatti et al., 14 Dec 2024).
• Schwinger parameterization of both virtual and cut propagators localizes the combination of amplitudes onto a "quantum worldline," with time-ordering and causality (retarded prescription) emerging as a sum over forests and theta functions in proper time.
• All divergences (including super-classical $1/\hbar^n$) cancel locally among equivalence classes of diagrams; only retarded propagators survive, enforcing causality and matching the worldline diagrammatics.

This equivalence enables transfer of modern amplitude and unitarity technology into the worldline framework and vice versa, unlocking new computational strategies for higher-loop and high-multipole observables.

7. Soft Theorems and Smoothness of the Classical Limit

Constraining the structure of radiation and inclusive observables, the KMOC formalism is manifestly compatible with classical soft theorems:

• Soft constraints: The multi-moment expansions of the radiated field at each loop order obey infinite tower constraints, enforcing that all super-classical contributions cancel and the $\hbar \to 0$ limit is smooth and free of divergence (Bautista et al., 2021, Sinha, 24 Jan 2025).
• Nonperturbative memory: The KMOC formula for soft radiation extends beyond large-impact-parameter perturbative regime to arbitrary scattering, reproducing electromagnetic and gravitational memory in terms of universal soft factors alone, independent of the hard process (Akhtar et al., 21 Nov 2025).
• All-orders proof: A formal proof establishes that all negative powers of $\hbar$ cancel in any nested commutators for a broad class of classical observables, and that only $\hbar^0$ pieces contribute to the final result (Sinha, 24 Jan 2025).

As a result, the entire machinery is algorithmically robust, streamlining the extraction of classical radiation and memory effects from quantum amplitudes without spurious divergences or ambiguities.

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References:

• Classical limit, kernel construction, master formula, analytic results: (Herrmann et al., 2021, Herrmann et al., 2021, Bern et al., 2021).
• Spin and multipole effects: (Cordero et al., 2022, Gatica, 2 Dec 2024, Angelis et al., 2023).
• Exponential S-matrix and Dirac brackets: (Damgaard et al., 2023, Alessio et al., 3 Jun 2025).
• Equivalence with worldline and PM/EFT: (Damgaard et al., 2023, Capatti et al., 14 Dec 2024).
• Soft theorems, classical limit smoothness, inclusion of radiation/memory: (Bautista et al., 2021, Sinha, 24 Jan 2025, Akhtar et al., 21 Nov 2025).
• Yang–Mills, color impulse, classical field theory extension: (Cruz et al., 2021).

For explicit implementation details, analytic expressions, and worked examples in both gauge theory and gravity (with and without spin, through two loops and higher), see the cited sources.