Large-time evolution of an electron in photon bath

The problem of infrared divergence of the effective electromagnetic field produced by elementary charges is revisited using the model of an electron freely evolving in a photon bath. It is shown that for any finite travel time, the effective field of the electron is infrared-finite, and that in each order of perturbation theory the radiative contributions grow without bound in the large-time limit. Using the Schwinger-Keldysh formalism, factorization of divergent contributions in multi-loop diagrams is proved, and summation of the resulting infinite series is performed. It is demonstrated that the effective electromagnetic field of the electron vanishes in the large-time limit, and that this vanishing respects the total charge conservation and the Gauss law. It is concluded that the physical meaning of infrared singularity in the effective field is the existence of a peculiar irreversible spreading of electric charges, caused by their interaction with the photon field. This spreading exists in vacuum as well as at finite temperature, and shows itself in a damping of the off-diagonal elements of the momentum-space density matrix of electron, but does not affect its momentum probability distribution. It precludes preparation of spatially localized particle states at finite times by operating with free particle states in the remote past. Relationship of the obtained results to the Bloch-Nordsieck theorem is established, and discussed from the standpoint of measurability of the electromagnetic field. The effect of irreversible spreading on the electron diffraction in the classic two-slit experiment is determined, and is shown to be detectable in principle by modern devices already at room temperature.
26 pages, 11 figures