Long-time behavior of an electron interacting with a quantized radiation field

概要

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The long-time behavior of an electron coupled to a quantized radiation field is discussed in the ground state and in equilibrium states at finite temperatures. The electron is not confined in an external potential. The model used is a d-dimensional extension of a standard model (the Pauli–Fierz model) in nonrelativistic quantum electrodynamics: The electron moves in R^d (d=>2) and the radiation field is over R^d. Further, the energy function ω of one free photon is taken to be a general one. In defining the interaction part of the Hamiltonian of the model, an ultraviolet cutoff is introduced for photon momenta with a cutoff function ρ-hat and the dipole approximation is used. It is proved that at each finite temperature T>0, the mean-square displacement of the electron behaves like (kBTd/m)t^2 as time t tends to infinity, where kB is the Boltzmann constant and m>0 is a renormalized mass of the electron which should be identified with the observed mass of the electron. The long-time asymptotics of the mean square displacement of the electron in the ground state is different from that at the finite temperatures; it depends on the space dimension d and on the infrared behavior of ω and ρ-hat.