  The specific features of nonlinear pair production and radiation processes in
an ultratsrong rotating electric field are investigated, taking into account
that this field models the antinodes of counterpropagating laser beams. It is
shown that a particle in a rotating electric field acquires an effective mass
which depends on its momentum absolute value as well as on its direction with
respect to the field plane. This phenomenon has an impact on the nonlinear
Breit-Wheeler and nonlinear Compton processes. The spectra of the produced
pairs in the first case, and the emitted photon in the second case, are shown
to bear signatures of the effective mass. In the first case, the threshold for
pair production by a $\gamma$-photon in the presence of this field varies
according to the photon propagation direction. In the second case, varying the
energy of the incoming electron allows for the measurement of the momentum
dependence of the effective mass. Two corresponding experimental setups are
suggested.
