  We carry out a Dirac sea reinterpretation of a discretized version of the
Hamiltonian of quantum electrodynamics (QED), and analyze the perturbed vacuum
in the continuum limit. We argue that if certain operators can be shown to be
the self-adjoint, the perturbed vacuum will have solutions that converge nicely
in this limit to states of the infinitesimal subspace of the Hilbert space
spanned by momentum eigenstates whose momenta sum to 0. This convergence allows
us to analyze what effect the perturbed vacuum has on the physical states of
the system. We then show that the interaction and the interference between the
perturbed vacuum and the physical states actually vanish in the continuum
limit, due to the fact that the states of the perturbed vacuum only live in
this vanishingly small part of the momentum space. We furthermore show that
this allows us to remove the terms that perturb the vacuum from the
Hamiltonian, without changing the dynamics of the physical states, thus
yielding us a well-defined Hamiltonian for the full theory of QED.
