Quantum electrodynamics on a lattice: A Hamiltonian variational approach to the physics of the weak-coupling region

We develop and apply a Hamiltonian variational approach to the study of quantum electrodynamics formulated on a spatial lattice in both 2 + 1 and 3 + 1 dimensions. Two lattice versions of QED are considered: a noncompact version which reproduces the physics of continuum QED, and a compact version constructed in correspondence with lattice formulations of non-Abelian theories. Our focus is on photon dynamics with charged particles treated in the static limit. We are especially interested in the nonperturbative aspects of the solutions in the weak-coupling region in order to clarify and establish aspects of confinement. In particular we find, in accord with Polyakov, that the compact QED leads to linear confinement for any nonvanishing coupling, no matter how small, in 2 + 1 dimensions, but that a phase transition to an unconfined phase for sufficiently weak couplings occurs in 3 + 1 dimensions. We identify and describe the causes of confinement.