  We consider a non-relativistic electron interacting with a classical magnetic
field pointing along the $x_3$-axis and with a quantized electromagnetic field.
The system is translation invariant in the $x_3$-direction and we consider the
reduced Hamiltonian $H(P_3)$ associated with the total momentum $P_3$ along the
$x_3$-axis. For a fixed momentum $P_3$ sufficiently small, we prove that
$H(P_3)$ has a ground state in the Fock representation if and only if
$E'(P_3)=0$, where $P_3 \mapsto E'(P_3)$ is the derivative of the map $P_3
\mapsto E(P_3) = \inf \sigma (H(P_3))$. If $E'(P_3) \neq 0$, we obtain the
existence of a ground state in a non-Fock representation. This result holds for
sufficiently small values of the coupling constant.
