  We study the QED bound-state problem in a light-front hamiltonian approach.
Starting with a bare cutoff QED Hamiltonian, $H_{_{B}}$, with matrix elements
between free states of drastically different energies removed, we perform a
similarity transformation that removes the matrix elements between free states
with energy differences between the bare cutoff, $\Lambda$, and effective
cutoff, $\lam$ ($\lam < \Lam$). This generates effective interactions in the
renormalized Hamiltonian, $H_{_{R}}$. These effective interactions are derived
to order $\alpha$ in this work, with $\alpha \ll 1$. $H_{_{R}}$ is renormalized
by requiring it to satisfy coupling coherence. A nonrelativistic limit of the
theory is taken, and the resulting Hamiltonian is studied using bound-state
perturbation theory (BSPT). The effective cutoff, $\lam^2$, is fixed, and the
limit, $0 \longleftarrow m^2 \alpha^2\ll \lam^2 \ll m^2 \alpha \longrightarrow
\infty$, is taken. This upper bound on $\lam^2$ places the effects of
low-energy (energy transfer below $\lam$) emission in the effective
interactions in the $| e {\overline e} > $ sector. This lower bound on $\lam^2$
insures that the nonperturbative scale of interest is not removed by the
similarity transformation. As an explicit example of the general formalism
introduced, we show that the Hamiltonian renormalized to $O(\alpha)$ reproduces
the exact spectrum of spin splittings, with degeneracies dictated by rotational
symmetry, for the ground state through $O(\alpha^4)$. The entire calculation is
performed analytically, and gives the well known singlet-triplet ground state
spin splitting of positronium, $7/6 \alpha^2 Ryd$. We discuss remaining
corrections other than the spin splittings and how they can be treated in
calculating the spectrum with higher precision.
