Functional Schrödinger equation for scalar QED

We present an approximation scheme which, starting from the full functional Schrödinger equation for scalar QED, leads to the approximation of quantum field theory in a classical electromagnetic background. We solve the functional Schrödinger equation in this background exactly for evolving vacuum states. We show how well-known effects such as pair creation in a constant external electric field and dispersion are recovered in this framework. We give explicit results for the phase of the wave functional in two-dimensional QED. The next order of approximation leads to expressions for QED corrections to the external-field Schrödinger equation and for back-reaction effects of the matter field on the classical background. We discuss these terms and give explicit expressions for special cases. Contact is also made to a description in terms of Feynman diagrams. We finally comment on the analogous situation in quantum gravity.