  We consider an electron, spin 1/2, minimally coupled to the quantized
radiation field in the nonrelativistic approximation, a situation defined by
the Pauli-Fierz Hamiltonian $H$. There is no external potential and $H$ fibers
according to the total momentum $p$. We prove that the ground state subspace of
$H$ with a total momentum $p$ is two-fold degenerate provided the charge $e$
and the total momentum $p$ are sufficiently small. We also establish that the
total angular momentum of the ground state subspace is $\pm1/2$ and study the
case of a confining external potential.
