  Variational (Rayleigh-Ritz) methods are applied to local quantum field
theory. For scalar theories the wave functional is parametrized in the form of
a superposition of Gaussians and the expectation value of the Hamiltonian is
expressed in a form that can be minimized numerically. A scheme of successive
refinements of the superposition is proposed that may converge to the exact
functional. As an illustration, a simple numerical approximation for the
effective potential is worked out based on minimization with respect to five
variational parameters. A variational principle is formulated for the fermion
vacuum energy as a functional of the scalar fields to which the fermions are
coupled. The discussion in this paper is given for scalar and fermion
interactions in 1+1 dimensions. The extension to higher dimensions encounters a
more involved structure of ultraviolet divergences and is deferred to future
work.
