Analytic approximations to hamiltonian lattice field theories: (I). Periodic gaussian model for U(1) theories

Periodic gaussian models are introduced for local and global U(1) invariant hamiltonian lattice field theories. The models coincide with standard lattice theories at weak coupling, but the leading non-perturbative contributions to wave functions and physical quantities are exactly calculable. Electric charges are confined and the mass gap is finite if correlations of an integer-valued magnetic field are of infinite range (d = 2 + 1 gauge model). Otherwise, for short-range correlations, the mass gap and the string tension vanish at weak coupling (QED, XY model, etc.)