Classical Limit of Quantum Scattering Amplitudes

Scattering amplitudes are quantum transition data, but the long-range motion of two compact bodies is classical. The bridge between them is not obtained by merely erasing loops or setting $\hbar$ to zero inside an amplitude. One must specify a localized incoming state, restore the powers of $\hbar$, isolate the soft momentum-transfer region, and form an observable such as an impulse or scattering angle. Only that complete object has a controlled classical limit.

Required background. Events, Intervals, and Minkowski Geometry supplies timelike and spacelike four-vector geometry; Weak Metric Perturbations and Gauge Freedom supplies linearized graviton exchange and its gauge freedom. Helpful background. The Einstein Equation fixes the gravitational normalization; Presymplectic Potentials and Currents clarifies why invariant observables are preferable to representation-dependent potentials; Relativistic Two-Body Dynamics locates the amplitude method among PN, PM, self-force, and effective-one-body descriptions.

Long-distance kinematics and the ℏ limit

Consider elastic scattering of two massive bodies,

$$p_1+p_2\longrightarrow p'_1+p'_2, \qquad q^\mu=p_1^{\prime\mu}-p_1^\mu,$$

with $p_A^2=p_A^{\prime 2}=-m_A^2$ in the site convention $(-,+,+,+)$. The invariant relative boost is

$$\sigma\equiv-\frac{p_1\cdot p_2}{m_1m_2}\geq 1.$$

For a weak encounter, $q^\mu$ is spacelike and small compared with $p_A^\mu$. The impact parameter $b^\mu$ is chosen in the transverse plane, $b\cdot p_1=b\cdot p_2=0$, and $b^2\gt 0$. These definitions describe asymptotic data; no coordinate-space potential has yet been introduced.

To expose the classical scaling, temporarily restore $\hbar$ while keeping $c=1$. Hard momenta $p_A^\mu$, masses $m_A$, the macroscopic impact parameter $b^\mu$, and the classical spins are held fixed. Momentum transfer and emitted massless momenta instead scale as wavenumbers:

$$q^\mu=\hbar\,\bar q^\mu, \qquad k^\mu=\hbar\,\bar k^\mu, \qquad \hbar\longrightarrow0$$

with $\bar q^\mu$ and $\bar k^\mu$ fixed. Here $q$ is the soft integration variable inside the perturbative observable construction; it must not be identified with the final, finite classical impulse. Likewise $k$ is the momentum of one emitted quantum. A classical radiation field has a large occupation number and can carry finite total energy as each quantum becomes soft. In the normalization used by Kosower, Maybee, and O’Connell, the restored coupling is $\kappa=\sqrt{32\pi G/\hbar}$, so an $n$-point, $L$-loop amplitude carries the explicit coupling scaling $\hbar^{1-n/2-L}$ before soft momentum powers are counted. Phase-space measures, wave packets, and cut terms supply further powers. The accounting is given in Kosower, Maybee, and O’Connell 2019, § 2, pp. 3–5, and §§ 3–4, pp. 5–31, PDF.

Localized particles require wave packets. Their Compton lengths $\ell_{c,A}=\hbar/m_A$, packet widths $\ell_{w,A}$, and impact parameter must obey a window such as

$$\ell_{c,A}\ll \ell_{w,A}\ll \sqrt{b^2}.$$

The first inequality suppresses intrinsically quantum position uncertainty; the second makes the classical trajectory and impact parameter sharp. A plane wave has no localized impact parameter, so a plane-wave cross section cannot simply be relabeled as the classical encounter.

Long-range terms in momentum space

Return to $\hbar=1$ for the Fourier-transform formulas in this section:

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

Then nonanalytic behavior at $\mathbf q^2=0$ diagnoses propagation over macroscopic distance:

$$\frac{1}{\mathbf q^2}\longleftrightarrow\frac{1}{4\pi r}, \qquad \frac{1}{\lvert\mathbf q\rvert}\longleftrightarrow\frac{1}{2\pi^2r^2},$$

while $\log(\mathbf q^2/\mu^2)$ produces a $1/r^3$ tail away from the origin, up to a convention-dependent contact term. A polynomial in $\mathbf q^2$ transforms into delta functions and their derivatives and therefore cannot affect separated point particles. This is why the classical pieces at successive PM orders are associated with increasingly subtle nonanalytic structures.

Nonanalyticity alone, however, does not distinguish classical from quantum physics. A loop can contain both, and quantum massless propagation can also be nonanalytic. The reliable classifier combines the small-$q$ hierarchy with restored $\hbar$ counting. In the center-of-momentum frame, large orbital angular momentum gives

$$J\sim \lvert\mathbf p\rvert\,b\gg\hbar, \qquad \frac{\lvert\mathbf q\rvert}{\lvert\mathbf p\rvert} \sim \frac{\hbar}{J}\ll1.$$

The second estimate applies to the soft variable with typical wave number $\lvert\bar q\rvert\sim1/b$. It is not the scattering angle: $\lvert\Delta\mathbf p_{\rm cl}\rvert/\lvert\mathbf p\rvert$ can remain finite as $\hbar\to0$, with smallness controlled by $G$, energy, and impact parameter. In the eikonal construction the resummed phase produces precisely that finite transfer at its saddle.

Loop order is therefore not synonymous with quantum order. Massive propagators can approach their mass shells inside a loop, producing classical long-range terms. Hard loop regions instead describe short-distance quantum corrections. The momentum regions are developed in Bern et al. 2019, § 2, pp. 5–12, PDF.

Classical spin has its own scaling. A fixed quantum spin label $s$ gives an angular momentum of order $\hbar s$ and vanishes as $\hbar\to0$. A macroscopic spinning body instead requires $s\to\infty$ with $S\sim\hbar s$ fixed. The on-shell spin representation must then be matched to a chosen classical spin tensor, supplementary condition, and multipole normalization; this matching is carried out on Spin, Multipoles, and Massive On-Shell States.

From transition data to a classical observable

Let $S=1+iT$, and define the invariant amplitude by

$$\langle f\lvert T\rvert i\rangle =(2\pi)^4\delta^{(4)}(p_f-p_i)\,\mathcal M_{fi},$$

with one-particle states normalized as

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

For a normalized two-packet state $\lvert\Psi\rangle$, the impulse on body 1 is the observable

$$\Delta p_1^\mu = \langle\Psi\lvert S^\dagger P_1^\mu S-P_1^\mu \rvert\Psi\rangle.$$

Expanding the operator gives

$$S^\dagger P_1^\mu S-P_1^\mu = i(P_1^\mu T-T^\dagger P_1^\mu) +T^\dagger P_1^\mu T.$$

The term quadratic in $T$ is not optional. At the first order where loop amplitudes enter, it cancels pieces required by unitarity and combines with real emission when radiation is allowed. Omitting it can leave an apparently divergent “super-classical” term and violate momentum conservation. The observable formulation and its classical wave-packet limit are summarized in Kosower, Monteiro, and O’Connell 2022, §2, pp. 3–8, PDF.

An alternative conservative route matches the amplitude to an effective potential or Hamiltonian. That intermediate object depends on field variables, coordinates, canonical transformations, and the chosen external-state normalization. At loop order it also contains repeated propagation generated by lower-order interactions. Those Born iterations must be subtracted before identifying genuinely new higher-PM information. By contrast, the scattering angle $\chi(E,J)$, the asymptotic impulse, and the radiated four-momentum are invariant once their asymptotic frame and sign conventions are fixed.

Infrared phases illustrate the distinction. A long-range interaction makes the plane-wave amplitude infrared singular, and repeated tree exchange exponentiates into a phase. The phase is real physical information about long-range propagation but is not by itself a new conservative potential coefficient. It cancels or reorganizes when one takes the logarithm appropriate to the eikonal, performs effective-theory matching, or constructs the full impulse. At sufficiently high order, potential and radiation regions overlap and conservative tail effects appear; then a potential-region calculation alone is not complete.

Checks that discriminate a classical contribution

A candidate term should pass several independent tests.

First, its final observable must be $\hbar$-independent after the soft scaling and all iteration or cut terms are included. Second, elastic on-shell kinematics implies

$$2p_1\cdot q+q^2=0, \qquad -2p_2\cdot q+q^2=0,$$

so the leading classical transfer is transverse to both incoming momenta. Third, the impulse must satisfy total momentum balance; without radiation, $\Delta p_1^\mu+\Delta p_2^\mu=0$. Fourth, the spinless answer must be symmetric under exchanging the two bodies. Finally, the slow-motion limit must recover the Newtonian potential and deflection, while the exact relativistic dependence should be expressible through invariants such as $\sigma$, $E$, and $J$.

These tests are stronger than inspecting a diagram’s topology or counting loops. They ask whether the complete construction produces a finite, normalized, invariant piece of classical motion.

Common pitfalls

Taking the limit of an isolated amplitude. Negative powers of $\hbar$ in an amplitude need not signal a divergence of an observable. The state, measure, soft transfer, and unitarity terms belong to the same limit.

Calling every nonanalytic loop term classical. Nonanalyticity signals long distance, not classicality. Restored $\hbar$ scaling and a check against an invariant observable are still required.

Equating a potential with an observable. Potentials related by canonical or field transformations can encode the same scattering angle. Compare invariant boundary data before declaring disagreement.

Exercise

Show that a tree amplitude whose nonrelativistic matching gives

$$\widetilde V(\mathbf q) =-\frac{4\pi Gm_1m_2}{\mathbf q^2}$$

reproduces Newtonian gravity, and explain why a term proportional to $\mathbf q^2$ does not alter the force at nonzero separation.

Solution

Using the stated Fourier convention and

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

one obtains $V(r)=-Gm_1m_2/r$. Therefore $\mathbf F=-\boldsymbol\nabla V=-Gm_1m_2\,\mathbf r/r^3$. A polynomial term $C\mathbf q^2$ transforms into $-C\nabla^2\delta^{(3)}(\mathbf r)$, which has support only at $\mathbf r=0$. It may encode a contact interaction or a choice of local operator, but it cannot change the long-range force between separated bodies.

The next page applies these rules to a concrete one-graviton exchange and extracts the leading invariant deflection: Tree Amplitudes and Leading Gravitational Deflection.

References

• Bern, Zvi, Clifford Cheung, Radu Roiban, Chia-Hsien Shen, Mikhail P. Solon, and Mao Zeng, “Black Hole Binary Dynamics from the Double Copy and Effective Theory”, Journal of High Energy Physics 2019, 206 (2019); Open PDF.
• Kosower, David A., Ben Maybee, and Donal O’Connell, “Amplitudes, Observables, and Classical Scattering”, Journal of High Energy Physics 2019, 137 (2019); Open PDF.
• Kosower, David A., Ricardo Monteiro, and Donal O’Connell, “The SAGEX Review on Scattering Amplitudes, Chapter 14: Classical Gravity from Scattering Amplitudes”, Journal of Physics A: Mathematical and Theoretical 55, 443015 (2022); Open PDF.