The S-Matrix Formulation

An S-matrix formulation is available when the complete physical states can be specified on asymptotic past and future boundaries and every channel carrying conserved information is included in the outgoing space. For a forming and fully evaporating black hole, this requires more than a late cloud of approximately thermal quanta: it requires asymptotic completeness, a well-defined infrared prescription, and no omitted remnant, horizon, or baby-universe sector.

Required background. Quantum Information supplies isometries, unitary maps, density operators, and trace-preserving channels used to state completeness.

Helpful background. Semiclassical Gravity supplies the evaporation regime; Quantum Fields in Curved Spacetime supplies asymptotic particle states and Hawking flux; Causality and Global Structure supplies null infinity, horizons, and complete slices.

The asymptotic map

In an asymptotically flat scattering problem, incoming states are prepared on past null infinity $\mathscr I^-$, with any massive particles entering through past timelike infinity, and outgoing states are read on $\mathscr I^+$ together with massive future sectors. After fixing gauge constraints, charges, and the infrared treatment, denote the physical spaces by $\mathcal H_{\mathrm{in}}$ and $\mathcal H_{\mathrm{out}}$. The scattering operator is

$$S:\mathcal H_{\mathrm{in}}\longrightarrow\mathcal H_{\mathrm{out}}, \qquad \mathcal A_{\beta\alpha}=\langle\beta,\mathrm{out}\rvert S\lvert\alpha,\mathrm{in}\rangle.$$

Probability conservation for every incoming state requires $S^\dagger S=I_{\mathrm{in}}$. This makes $S$ an isometry. Full unitarity also requires $SS^\dagger=I_{\mathrm{out}}$: every state counted as physically outgoing must lie in the image. In finite dimensions the two conditions imply equal dimensions; in field theory they encode completeness of the chosen asymptotic sectors.

The corresponding map on density operators is $\rho\mapsto S\rho S^\dagger$. It preserves trace, positivity, all spectral entropies, and overlaps. A merely inclusive probability rule can preserve total probability after summing over unobserved quanta while failing to define a unitary operator on the truncated hard-particle space. Thus “the probabilities add to one” is weaker than a complete unitary S-matrix.

Conditions imposed by evaporation

For collapse followed by complete evaporation, an S-matrix description assumes that an arbitrarily late exterior region is sufficient to encode the final state. The working conditions are:

• the geometry admits suitable asymptotic preparation and detection regions;
• the final space contains radiation, massive remnants if present, conserved-charge sectors, and whatever soft or dressed degrees of freedom the infrared theory requires;
• no independent state is lost through a lasting event horizon or singular boundary;
• time evolution connects the in- and out-algebras without an external bath that has been silently traced out;
• the approximation used to compute amplitudes controls the late-time endpoint well enough to define the out-space.

Hawking’s superscattering proposal denied precisely this factorization into $S(\cdot)S^\dagger$ when the interior terminates and only exterior radiation remains Hawking 1976, §§ II–IV. The modern unitary formulation instead asks for a complete microscopic amplitude whose subtle correlations purify the radiation. Harlow develops this scattering formulation and its quantum-information consequences while keeping it distinct from what leading semiclassical effective field theory computes Harlow 2016, §§ 5.1–5.4.

Completeness checks

Suppose one proposes amplitudes $S_{\beta\alpha}$ in an orthonormal basis. Two independent sums test the two sides of unitarity:

$$\sum_\beta S_{\beta\alpha}^{*}S_{\beta\alpha'}=\delta_{\alpha\alpha'}, \qquad \sum_\alpha S_{\beta\alpha}S_{\beta'\alpha}^{*}=\delta_{\beta\beta'}.$$

The first checks preservation of incoming inner products. The second detects outgoing states that the proposed dynamics never produces. If a stable remnant label $r$ exists, the first sum must run over $(\beta,r)$; summing over $\beta$ alone gives a reduced channel on radiation. Likewise, if a baby-universe label $u$ is included in the full theory but inaccessible asymptotically, then

$$\rho_{\mathrm{rad}}=\operatorname{Tr}_{u}\!\left(S\rho_{\mathrm{in}}S^\dagger\right)$$

need not evolve unitarily even though the enlarged map does. The apparent answer therefore changes with the declared out-algebra.

An eternal black hole supplies another discriminating limit. Radiation at one exterior infinity is not a complete future out-space when independent horizon or second-exterior data remain. In asymptotically anti-de Sitter spacetime, timelike boundary evolution can be unitary, but it is not literally the flat-space $\mathscr I^-\to\mathscr I^+$ S-matrix defined here. These cases do not refute unitarity; they show that the scattering question must fit the global structure.

Boundary of the statement

This page does not construct the microscopic black-hole amplitudes. It gives the criteria a proposed construction must satisfy. Generic scattering theory and infrared analysis lie with quantum field theory, fixed-background Hawking production lies with Quantum Fields in Curved Spacetime, and endpoint backreaction lies with Semiclassical Gravity. Explicit holographic or string constructions are separate. Within this scope, a unitary black-hole S-matrix exists only after the complete asymptotic states and every correlation-carrying sector are identified.

Exercise

Let $S\lvert0\rangle=\lvert0\rangle\lvert r_0\rangle$ and $S\lvert1\rangle=\lvert1\rangle\lvert r_1\rangle$, where the first factor is radiation and $\langle r_0\vert r_1\rangle=0$. Is the radiation map unitary?

Solution

The enlarged map is an isometry because the two output states are orthonormal. Tracing over the remnant removes the off-diagonal entry of an input superposition, so the radiation channel is dephasing and is not unitary. If the remnant states are part of the complete outgoing space, no information is lost; if the proposed S-matrix contains radiation alone, its out-space is incomplete. The example directly separates enlarged unitarity from asymptotic radiation unitarity.

References

• Harlow, Daniel. 2016. “Jerusalem Lectures on Black Holes and Quantum Information.” Reviews of Modern Physics 88: 015002. Open text.
• Hawking, Stephen W. 1976. “Breakdown of Predictability in Gravitational Collapse.” Physical Review D 14: 2460–2473. DOI.