Infrared-Dressed Scattering States and Boundary Dictionaries

Bare Fock states are not exact asymptotic states of a theory with massless gravitons. Every hard particle carries a long-range field, virtual soft exchange makes exclusive Fock amplitudes infrared singular, and states in different soft-charge sectors can have vanishing overlap. A flat-space boundary dictionary must therefore encode dressed states or explicitly inclusive observables, not treat the regulator-dependent Fock S-matrix as final data.

Required background. Dressed States and Infrared-Finite Scattering constructs the asymptotic states, and BMS, Memory, and Soft Sectors as Holographic Data identifies their charge sectors.

Helpful background. Bloch–Nordsieck and KLN Cancellation supplies the inclusive alternative, while Eikonal Approximation and Wilson Lines organizes universal soft exchange.

Why Fock scattering fails

For a hard external momentum $p$, the leading soft-graviton coupling has the eikonal form

$$S_p(k,\varepsilon) =\frac{\kappa}{2}\frac{p^\mu p^\nu\varepsilon_{\mu\nu}(k)}{p\cdot k}, \qquad \kappa^2=32\pi G.$$

Integrating unresolved momenta $\lambda<|\mathbf k|<\Lambda$ produces logarithms of the infrared regulator $\lambda$. Virtual corrections exponentiate schematically as

$$\mathcal M_{\rm Fock}(\lambda) =e^{-B\log(\Lambda/\lambda)}\mathcal M_{\rm hard},$$

with a process-dependent positive quadratic form $B$ after the physical kinematics are fixed. The exclusive amplitude can vanish as $\lambda\to0$, even though suitably inclusive probabilities or dressed matrix elements are finite.

Coherent gravitational dressing

A Faddeev–Kulish-type state attaches a coherent soft cloud to each hard particle,

$$|p\rangle_{\rm FK} =\exp\!\left[ \int_{\lambda}^{\Lambda_s}\frac{d^3\mathbf k}{(2\pi)^3 2|\mathbf k|} \left(f_p^{\mu\nu}(k)a^\dagger_{\mu\nu}(k) -\overline{f_p^{\mu\nu}(k)}a_{\mu\nu}(k)\right) \right]|p\rangle,$$

where the leading part of $f_p^{\mu\nu}$ reproduces $\kappa p^\mu p^\nu/(2p\cdot k)$ and additional terms enforce the chosen gauge and asymptotic conditions. Soft-cloud overlaps cancel the universal virtual divergence. The hard scattering kernel remains, while the dressing records the long-range field and BMS charge.

Inclusive and dressed prescriptions answer different operational questions. An inclusive rate sums over unresolved radiation below a detector resolution; a dressed amplitude is a matrix element between coherent asymptotic sectors. Their finite pieces and state interpretation need not agree term by term.

First application. Dress a two-body gravitational scattering state, cancel the leading soft divergence in a simple amplitude, and record the associated asymptotic charge sector. Use the same soft cutoff and eikonal kernel in the virtual factor and coherent-state overlap, verify cancellation before sending $\lambda$ to zero, and compute the soft charge carried by the resulting cloud.

Boundary-state implications

The dressing is not unique. Radiation-free Coulombic clouds, line-like dressings, and dressings differing by finite soft radiation can represent different observables or states in the same hard sector. Large-gauge charges divide the asymptotic space into superselection sectors; local hard operators alone do not move consistently between them. A celestial transform applied before infrared completion merely transforms the divergent object.

Adversarial control. Remove $\lambda$ from the virtual Fock amplitude before combining it with real emission or dressing. The amplitude vanishes or diverges and cannot define a boundary inner product. Next change the finite part of the dressing while holding the hard momenta fixed; any proposed boundary operator that ignores the induced soft-charge or memory change is incomplete.

Evidence ceiling

Coherent dressing and inclusive cancellation control universal perturbative infrared divergences. They do not uniquely choose a gravitational asymptotic Hilbert space, settle black-hole sectors, or prove that a celestial or Carrollian representation supplies intrinsic boundary dynamics. A complete proposal must carry the dressing and its charges through every dictionary map.

An infrared-finite perturbative gravity S-matrix built from coherent asymptotic dressings is constructed by Ware, Saotome, and Akhoury 2013; this does not by itself supply a complete boundary dual or all superselection sectors.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Kulish, P. P., and L. D. Faddeev. “Asymptotic Conditions and Infrared Divergences in Quantum Electrodynamics.” Theoretical and Mathematical Physics 4 (1970): 745–757. DOI.
• Ware, Jonathan, Ryo Saotome, and Ratindranath Akhoury. “Construction of an Asymptotic S Matrix for Perturbative Quantum Gravity.” Journal of High Energy Physics 2013, no. 10 (2013): 159. DOI; Open PDF.
• Weinberg, Steven. “Infrared Photons and Gravitons.” Physical Review 140 (1965): B516–B524. DOI.