Semiclassical motion of dressed electrons

We consider an electron coupled to the quantized radiation field and subject to a slowly varying electrostatic potential. We establish that over sufficiently long times radiation effects are negligible and the dressed electron is governed by an effective one-particle Hamiltonian. In the proof only a few generic properties of the full Pauli-Fierz Hamiltonian H_PF enter. Most importantly, H_PF must have an isolated ground state band for |p| < p_c ≤ ∞ with p the total momentum and p_c indicating that the ground state band may terminate. This structure demands a local approximation theorem, in the sense that the one-particle approximation holds until the semiclassical dynamics violates |p| < p_c. Within this framework we prove an abstract Hilbert space theorem which uses no additional information on the Hamiltonian away from the band of interest. Our result is applicable to other time-dependent semiclassical problems. We discuss semiclassical distributions for the effective one-particle dynamics and show how they can be translated to the full dynamics by our results.