Effective Action for a Statistical System with a Field dependent Wave Function

We compute the first order correction in $\hbar$ to the field dependent wave function in Statistical Field Theory. These corrections are evaluated by several usual methods. We limit ourselves to a one dimensional model in order to avoid the usual difficulties with the UV divergences that are not relevant for our purposes. The main result of the paper is that the various methods yield different corrections to the wave function. Moreover, we give arguments to show that the perturbative integration of the exact renormalization group provides the right result.