Covariant gaussian approximation. I. formalism

The variational Gaussian approximation is generalized to the time-dependent approach capable of giving time-dependent Greens functions. This covariant Gaussian approximation is represented, in full analogy with the classical approximation, as an initial truncation of the Dyson-Schwinger equations followed by functional differentiation of the effective action. Intuitively simple Schrödinger and Heisenberg pictures of the approximation are also discussed.