Classical Physics from Quantum Amplitudes

Loop order and classicality are not the same classification. In a scattering problem with massive bodies, one-loop amplitudes contain both classical post-Minkowskian terms and genuine quantum corrections. The classical pieces arise from long-range, nonanalytic momentum regions enhanced by large masses and small transfer. Extracting them requires restored powers of $\hbar$, subtraction of iterations, and a gauge-invariant observable such as a scattering angle or impulse.

Required background. Flat-Spacetime Quantum Field Theory supplies amplitudes and unitarity cuts. Gravitational Actions and Boundary Terms supplies the gravitational vertices.

Helpful background. Einstein Equations supplies post-Minkowskian observables, Quantum Information supplies unitarity language, and Quantum Gravity supplies the approximation hierarchy.

Restore Planck’s constant before classifying

The path-integral phase is $S/\hbar$. A propagator, vertex, loop measure, and external-state normalization carry compensating powers of $\hbar$. For elementary particles with masses held fixed, the usual loop expansion tracks quantum corrections. A classical two-body limit is different: the orbital action

$$J\sim bp$$

must satisfy $J/\hbar\gg1$, while momentum transfer is small compared with the hard masses. Factors of mass and inverse transfer can compensate the explicit loop power of $\hbar$. Counting the loop alone therefore mislabels some terms.

A reliable classification restores $\hbar$ in $q$, external wave packets, couplings, and Born normalization, then holds the classical impact parameter, velocities, and masses fixed as $\hbar\to0$. Terms that survive belong to the classical observable; terms proportional to positive powers of $\hbar$ are quantum. More singular terms can represent iterations of lower-order classical dynamics and must be removed or exponentiated rather than counted twice. The explicit one-loop analysis in Holstein and Donoghue 2004, §§3–5 shows how a mass-enhanced square-root nonanalyticity loses the nominal loop power of $\hbar$, while the logarithmic term retains it.

Post-Minkowskian hierarchy

The post-Minkowskian expansion orders powers of $G$ without assuming slow velocity. For conservative two-body scattering,

Order	Amplitude topology carrying new information	Typical long-range structure
First post-Minkowskian	Tree-level one-graviton exchange	Newtonian or relativistic $1/q^2$ pole
Second post-Minkowskian	One-loop triangles and boxes after iteration subtraction	Classical $1/\lvert\mathbf q\rvert$ term plus quantum logarithms
Third post-Minkowskian	Two-loop topologies and lower-order iterations	New classical nonanalytic terms and radiation-sensitive pieces

Thus a one-loop triangle can yield an $O(G^2)$ classical correction. A logarithm at the same loop order can yield a quantum $1/r^3$ tail. The distinction follows from mass, transfer, and $\hbar$ scaling, not the diagram’s loop label.

Modern unitarity methods build the relevant loop integrands from products of on-shell trees and then isolate the classical regions. Bern and collaborators used amplitudes to obtain a third-post-Minkowskian conservative two-body Hamiltonian from two-loop information Bern et al. 2019, pp. 1–6.

From an amplitude to an observable

The amplitude itself contains gauge-invariant on-shell information, but an intermediate potential or Hamiltonian depends on canonical variables. Three common routes are:

1. match the amplitude to a nonrelativistic EFT and subtract iterated lower-order potentials;
2. transform to impact-parameter space, extract the eikonal phase, and differentiate it to obtain the scattering angle;
3. compute the expectation value of momentum transfer to obtain the impulse, including real emission when required.

All routes must agree on a common classical observable after canonical transformations and convention changes. At second order, box diagrams contain an iteration of the first-order potential. Leaving that piece in a newly matched potential double counts classical propagation. Its imaginary part is also tied to unitarity and the eikonal phase rather than dissipation by itself.

For radiation, virtual loops are insufficient. Real on-shell gravitons carry energy and momentum, and inclusive observables combine real and virtual infrared contributions. Conservative and dissipative sectors should be separated by boundary conditions or time-symmetry prescriptions, not by deleting every imaginary term.

Method-of-regions checks

Let hard momenta scale with the masses and transfer $q$ be small. Loop momentum regions can be hard, soft, potential, or radiation-like. Classical long-distance terms arise from regions in which internal massive lines are nearly on shell and massless propagation remains nonanalytic in $q$. Hard regions expand analytically and match onto local coefficients.

Adverse checks include changing loop parametrization, integral basis, covariant gauge, and matching scheme; the final scattering angle or impulse must remain fixed. Taking the mass ratio $m/M\to0$ at fixed probe velocity tests the test-body limit against timelike geodesic scattering while retaining $bm/\hbar\gg1$. A strictly massless particle instead requires a separate null-probe limit with its energy specified; simply setting $m=0$ in a massive nonrelativistic expansion is not valid. Expanding at low velocity tests agreement with post-Newtonian results. Sending $q$ toward a heavy threshold invalidates the long-range expansion.

The approximation also requires impact parameter larger than the strong-gravity scale, wave packets narrow enough to define classical trajectories but broad enough in position, and an order-by-order treatment of radiation reaction. Large deflection, capture, or black-hole formation lies outside a fixed low-order post-Minkowskian calculation.

Exercises

Exercise. A one-loop small-transfer amplitude contains $A m/\lvert\mathbf q\rvert+B\log(\mathbf q^2/\mu^2)$ after lower-order iterations are subtracted. Which term can contribute to a classical second-post-Minkowskian potential, and which is the leading quantum-type tail? Give their radial powers.

Solution

The mass-enhanced $m/\lvert\mathbf q\rvert$ term survives the classical scaling and transforms as $m/r^2$, so it contributes to the classical second-post-Minkowskian interaction. The logarithm transforms as $1/r^3$ away from contact terms and carries the quantum correction once $\hbar$ is restored. Both originate at one loop; their different mass, momentum, and $\hbar$ scaling—not loop order—separates their classical and quantum roles.

References

• Bern, Zvi, et al. “Scattering Amplitudes and the Conservative Hamiltonian for Binary Systems at Third Post-Minkowskian Order.” Physical Review Letters 122 (2019): 201603. DOI.
• Holstein, Barry R., and John F. Donoghue. “Classical Physics and Quantum Loops.” Physical Review Letters 93 (2004): 201602. DOI. Open manuscript.