LONG-TIME BEHAVIOR OF AN ELECTRON INTERACTING WITH A QUANTIZED RADIATION-FIELD

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 greater-than-or-equal-to 2) and the radiation field is over R(d). Further, the energy function omega 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 rho-triple-overdot 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 (kappa-B Td/m)t2 as time t tends to infinity, where kappa-B 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 omega and rho-triple-overdot.