Asymptotic Completeness for Compton Scattering

Abstract

Scattering in a model of a massive quantum-mechanical particle, an "electron", interacting with massless, relativistic bosons, "photons", is studied. The interaction term in the Hamiltonian of our model describes emission and absorption of "photons" by the "electron"; but "electron-positron" pair production is suppressed. An ultraviolet cutoff and an (arbitrarily small, but fixed) infrared cutoff are imposed on the interaction term. In a range of energies where the propagation speed of the dressed "electron" is strictly smaller than the speed of light, unitarity of the scattering matrix is proven, provided the coupling constant is small enough; (asymptotic completeness of Compton scattering). The proof combines a construction of dressed one-electron states with propagation estimates for the "electron" and the "photons".

The dressed nonrelativistic electron in a magnetic field

“…Therefore it follows from (A.3) and (A.7) that φ n , H mod,σ (P 3 )φ n = φ n ,Γ(j R ) ⋆Ȟ mod,σ (P 3 )Γ(j R )φ n + o(R 0 ) (A.8) uniformly in n (cf. [21] and [16]). Denoting by P Ω∞ the orthogonal projection on the vacuum of delocalized photons, we have using (A.6) φ n ,Γ(j R ) ⋆Ȟ mod,σ (P 3 )Γ(j R )φ n ≥ E mod,σ (P 3 ) + σ 2 − σ 2 φ n ,Γ(j R ) ⋆ (I ⊗ I ⊗ P Ω∞ )Γ(j R )φ n .…”

“…This result is not surprising but the complete proof is long. Actually it follows by mimicking [20,21,16] (see also [34]) and, here, we only give a sketch of the proof. In this appendix we are faced with the lack of smoothness of the ǫ µ (k)'s which define vector fields on spheres |k| =cst (see [33,23]).…”

“…Let ω mod (k) be the modified dispersion relation as defined in ( [21], section 5, hypothesis 3), i.e. : ω mod (k) is a smooth function satisfying…”

“…By using [16] (sections 2.13 and 2.14) and [21] (sections 2.5 and 2.6) we obtain thatȞ mod,σ (P 3 ) and H mod,σ (P 3 ) are almost conjugated byΓ(j R ), namely Γ(j R )H mod,σ (P 3 ) −Ȟ mod,σ (P 3 )Γ(j R ) = − g 2m √ 2T R (H)(x ′ )Γ(j R )…”

Non-Relativistic Matter and Quantized Radiation

“…The improved results discussed thereafter are important in the study of photon scattering at a free electron (Compton scattering) [16].…”