Résumé
We propose an improvement of standard quantum perturbation theory around the vacuum or some other classical solution, in order to incorporate from the beginning quantum mechanical effects in a non-trivial way into functional methods. This improvement is obtained by introducing an a priori
arbitrary quadratic kinetic term in the action. At a given order in perturbation, the answer depends on the kernel of this quadratic term. Equations of motion are then obtained by imposing that the answer be stationary with respect to this kernel. In particular lowest order cases, one recovers mean-field theory, variational, Hartree and Hartree-Fock results, for which this method can be seen as providing a systematic improvement. It also provides an approach to bound states, like positronium and the hydrogen atom, which, although present in weak coupling, cannot appear in a standard functional treatment of the classical field theory. Finally, contrary to most previous attempts at variational calculations in continuum quantum field theories, this approach is compatible with renormalization and the usual cancellation of infinities.