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 at each order of perturbation theory the radiative contributions grow unboundedly with time. 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 found that despite the unbounded growth of individual contributions to the effective field, their sum is bounded, tending to zero in the limit of infinite travel time. 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 electromagnetic field. This spreading originates from the quantum electromagnetic fluctuations, rather than the electron–photon scattering, and exists in vacuum as well as at finite temperatures. It 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. This effect is discussed in terms of thermalization of the electron state, and the asymptotic growth of its quantum entropy is determined. Relationship of the obtained results to the Bloch–Nordsieck theorem is established and considered 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.

► Infrared finiteness of the effective electromagnetic field of a free electron is proved.
► Quantum radiative effects result in an irreversible spreading of free electrons.
► The irreversible spreading produces a detectable effect on the electron interference.