Amplitude–Hamiltonian–EOB Dictionaries

An on-shell amplitude and a two-body Hamiltonian must give the same invariant scattering angle $\chi(E,J)$ when they describe the same conservative scattering problem at fixed center-of-mass energy, angular momentum, masses, and spin data. This is a strong check, but the angle fixes the radial action only up to a function of energy; time-delay or other dynamical information may also be needed. Equality of a momentum-space kernel with a coordinate-space potential is neither necessary nor sufficient: state normalization, classical scaling, Born iterations, canonical transformations, and the EOB energy map all intervene before the invariant comparison.

Helpful background. The Einstein Equation supplies the gravitational interaction; Weak Metric Perturbations and Gauge Freedom supplies weak-field extraction; Presymplectic Potentials and Currents supplies canonical equivalence; Computational Relativity supplies numerical benchmarks; Gravitational Radiation and Waveforms supplies radiative observables; and Relativistic Two-Body Dynamics supplies the PN, PM, self-force, and EOB regimes joined here.

The invariant chain from amplitude to EOB dynamics

For nonspinning conservative scattering, use

$$M=m_1+m_2,\qquad \mu=\frac{m_1m_2}{M},\qquad \nu=\frac{\mu}{M},$$

together with total center-of-mass energy $E$ and angular momentum $J=b p_\infty$. The translation has five logically distinct stages:

Stage	Object	Convention that must be recorded	Invariant check
On-shell theory	Classical part of $\mathcal M(E,\mathbf q)$	External-state normalization, momentum-transfer sign, powers of $\hbar$, and conservative prescription	Impulse and momentum conservation
Potential matching	Iteration-subtracted kernel $\widetilde V(\mathbf q,\mathbf p)$	Relativistic factor $4E_1E_2$, infrared subtraction, and Born iterations	Tree-level Newtonian potential
Canonical dynamics	A Hamiltonian $H(\mathbf r,\mathbf p)$	Canonical gauge and allowed generating functions	The same $\chi(E,J)$
Hamilton–Jacobi bridge	Regulated radial action $W(E,J)$	Turning point, long-range subtraction, branch, and boundary conditions	$\chi=-\pi-\partial_JW$ for the full action; $\chi=-\partial_JW_{\rm int}$ after subtracting the free action
EOB representation	$H_{\rm eff}$ and its potentials	Real–effective energy map and EOB radial gauge	Scattering and bound invariants agree

The first row is already an observable construction rather than a reading of a Feynman graph. Kosower, Maybee, and O’Connell define the impulse from $S^\dagger P^\mu S-P^\mu$, restore $\hbar$, and show explicitly how the terms required by unitarity combine in the classical limit Kosower, Maybee, and O’Connell 2019, §§2–4, pp. 3–31, especially Eqs. (3.16), (4.29), and (4.30), PDF. Their next-to-leading example also demonstrates why cut-box and iteration contributions cannot be discarded graph by graph Kosower, Maybee, and O’Connell 2019, §5.2, pp. 34–45, PDF.

Normalization and iteration subtraction

Take relativistic one-particle states normalized by

$$\langle\mathbf p'\mid\mathbf p\rangle =2E_{\mathbf p}(2\pi)^3\delta^3(\mathbf p-\mathbf p').$$

In a common potential-matching convention,

$$\widetilde V(\mathbf q,\mathbf p) =-\frac{\mathcal M_{\rm sub}(\mathbf q,\mathbf p)} {4E_1E_2}.$$

This equation is a convention, not an invariant. Its sign must be checked by recovering attraction, and its energy factor follows from translating relativistic to nonrelativistic normalization. Kälin and Porto state the same $4E_1E_2$ conversion and formulate an “impetus” relation between the Fourier-transformed amplitude and the center-of-mass momentum Kälin and Porto 2020, §4.1, pp. 11–13, Eqs. (4.2)–(4.12), PDF.

At one loop and beyond, $\mathcal M_{\rm sub}$ is not simply the classical-looking part of the full amplitude. Solving a Hamiltonian iterates lower-order interactions automatically. The amplitude contains the corresponding Born iterations, so those pieces must be removed during matching or retained consistently in an exponentiated/eikonal observable. Otherwise the same long-range propagation is counted once in the kernel and again in the equations of motion. A useful diagnostic is that apparently “superclassical” inverse powers of $\hbar$ or imaginary terms cancel against iteration and cut contributions in a complete conservative observable; their isolated presence is not a new classical potential.

The subtraction depends on the chosen representation, whereas the final angle does not. This is why a disagreement between two published potentials should first be translated to $\chi(E,J)$ before being called a physical discrepancy.

Hamiltonian and radial-action equivalence

Given a local, rotationally invariant Hamiltonian, solve

$$H(r,p_r,J)=E$$

for $p_r(E,J;r)$. At a large radial cutoff $R$, define the full scattering action and then take its angular-momentum derivative:

$$W_R(E,J)=2\int_{r_{\min}}^R p_r(E,J;r)\,dr, \qquad \chi(E,J)=-\pi-\lim_{R\to\infty}\frac{\partial W_R}{\partial J}.$$

The full action contains a free linear divergence and, for a long-range $1/r$ force, a logarithmic divergence. The free radial action has $\partial_JW_{R,0}\to-\pi$. Subtracting it therefore changes the formula to $\chi=-\partial_JW_{\rm int}$, with no additional $-\pi$. The remaining logarithm also needs a specified subtraction scale if an action itself is quoted. In the spinless problem its divergent coefficient is independent of $J$, so it drops out of the angle. An arbitrary finite function $C(E)$ also drops out; it can affect an energy derivative such as a regularized time delay. Use identical subtraction conventions for that comparison.

In an isotropic canonical gauge, where the energy equation gives a function $p^2(r,E)=p_r^2+J^2/r^2$ independent of $J$, the angle becomes

$$\chi(E,J) =-\pi+2J\int_{r_{\min}}^\infty \frac{dr}{r^2\sqrt{p^2(r,E)-J^2/r^2}}.$$

The latter is the explicit Hamiltonian-to-angle dictionary Kälin and Porto 2020, §2.1, pp. 6–7, Eqs. (2.1)–(2.5), PDF. A radial action also permits analytic continuation to bound quantities such as periastron advance, provided the dynamics is conservative and local and the relevant branch structure is controlled Kälin and Porto 2020, §5, pp. 17–30, PDF.

Canonical nonuniqueness is harmless at this level. For a time-independent change of canonical coordinates $z'=z+\{z,g\}+O(g^2)$, the same Hamiltonian expressed in the new coordinates is

$$H'(z')=H(z')-\{H,g\}(z')+O(g^2).$$

This is the pullback sign: $H'(z')=H(z)$ and the inverse coordinate map has the minus sign. Hamilton’s principal function changes by endpoint terms. If the transformation approaches the identity and its action boundary term vanishes on the incoming and outgoing asymptotic states, $W$ and $\chi$ are unchanged. Otherwise the definitions of the asymptotic canonical variables and boundary terms must also be transformed; comparing the old $J$ with the new $J$ without that map is an inconsistent test.

The EOB energy dictionary

For the usual nonspinning EOB representation, define $\widehat H_{\rm eff}=H_{\rm eff}/\mu$. The real energy is

$$H_{\rm real} =M\sqrt{1+2\nu\left(\widehat H_{\rm eff}-1\right)}.$$

The quadratic map is part of the dictionary. Buonanno and Damour originally obtained it by matching real and effective action variables and energy levels Buonanno and Damour 1999, §§II–IV, pp. 3–12, PDF. Damour later showed that equality of the real and effective scattering functions at 1PM selects the same map at arbitrary velocity Damour 2016, §§II and VI–VIII, pp. 2–3 and 7–11, PDF.

The effective Hamiltonian may be expressed through potentials such as $A(u;\nu)$, $\bar D(u;\nu)$, and momentum-dependent $Q$. These potentials are useful coordinates on the dynamics, not observables. The robust amplitude-to-EOB comparison is:

$$\mathcal M_{\rm classical} \longrightarrow \chi_{\rm amp}(E,J) \stackrel{?}{=} \chi_{\rm EOB}(E,J),$$

after iteration subtraction and the energy map. If the equality holds, one may choose an EOB gauge and solve for potentials. Going directly from an amplitude coefficient to a named EOB potential without displaying these transformations hides the very ambiguities the dictionary is meant to control.

Boundaries of the dictionary

The local conservative chain does not automatically include radiation reaction. Once energy and angular momentum are radiated, the outgoing values differ from the incoming ones; a scattering angle must specify which values enter its conservative baseline and how radiative corrections are defined. Amplitude-based radiated momentum and the local reaction force can be compared through total momentum conservation, but not by inserting a five-point radiation amplitude into a conservative potential.

Hereditary tails also introduce time-nonlocal conservative terms. An analytic continuation from scattering to bound motion must then retain the nonlocal kernel or use a demonstrably equivalent localized representation. Spin requires a spin supplementary condition and a map between quantum spin states and classical spin variables. Finite-size operators require object-specific Wilson coefficients. Each extension can still be tested with invariant actions or angles, but the minimal table above is no longer complete.

Several limits do not commute operationally. The classical limit must be taken after restoring $\hbar$ to momenta, impact parameters, measures, and couplings; loop order is not the same as power of $\hbar$. The small-transfer limit must not be used before Born iterations are separated. The weak-deflection PM expansion and the low-velocity limit are nonuniform, as shown on PN–PM matching. Finally, analytic continuation to a bound orbit assumes the conservative branch has been identified before taking the binding energy through zero.

Solved tree-level dictionary

In the normalization above, suppose the nonrelativistic limit of the classical tree amplitude is

$$\mathcal M_{\rm tree}(\mathbf q) =\frac{16\pi Gm_1^2m_2^2}{\mathbf q^2}.$$

Derive the coordinate-space potential and its leading weak-scattering angle. Then explain whether adding a decaying canonical transformation can change the answer.

Solution

At low velocity $E_i=m_i+O(v^2)$, so

$$\widetilde V(\mathbf q) =-\frac{\mathcal M_{\rm tree}}{4m_1m_2} =-\frac{4\pi Gm_1m_2}{\mathbf q^2}.$$

Using

$$\int\frac{d^3q}{(2\pi)^3} \frac{e^{i\mathbf q\cdot\mathbf r}}{\mathbf q^2} =\frac{1}{4\pi r},$$

one obtains

$$V(r)=-\frac{Gm_1m_2}{r}.$$

For asymptotic relative speed $v_\infty$, the Newtonian potential by itself gives the exact hyperbolic angle

$$\chi_{\rm N} =2\arctan\!\left(\frac{GM}{b v_\infty^2}\right) =\frac{2GM}{b v_\infty^2} +O(G^3).$$

The $O(G^3)$ remainder here is only the next iteration of the Newtonian $1/r$ potential. The nonrelativistic tree amplitude fixes the leading low-velocity term displayed here, not all velocity dependence at 1PM. The full velocity-dependent tree amplitude gives $\chi_{\mathrm{1PM}}(E,J)+O(G^2)$; its relativistic corrections already occur at order $G$.

This verifies the sign, the factor $4E_1E_2$, and the mass normalization. A canonical transformation can rearrange terms in $H$ while preserving this $\chi$ if the transformation approaches the identity at both asymptotic endpoints and its action boundary term vanishes. The value of the generating function alone does not establish these conditions. A factor-of-four discrepancy in the leading angle would instead point to external-state normalization or an iteration error.

The same invariant-first logic underlies the chapter’s consistency tests and separates a genuine physical disagreement from two valid canonical representations.

References

• Buonanno, Alessandra, and Thibault Damour. “Effective One-Body Approach to General Relativistic Two-Body Dynamics.” Physical Review D 59, 084006 (1999). DOI. Open PDF.
• Damour, Thibault. “Gravitational Scattering, Post-Minkowskian Approximation and Effective One-Body Theory.” Physical Review D 94, 104015 (2016). DOI. Open PDF.
• Kälin, Gregor, and Rafael A. Porto. “From Boundary Data to Bound States.” Journal of High Energy Physics 2020, 072 (2020). DOI. Open PDF.
• Kosower, David A., Ben Maybee, and Donal O’Connell. “Amplitudes, Observables, and Classical Scattering.” Journal of High Energy Physics 2019, 137 (2019). DOI. Open PDF.