Analytic approximations to hamiltonian lattice field theories. Pt. 1

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 calculabe. 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.). (orig.)