Dual representation for the generating functional of the Feynman path-integral

The generating functional for scalar theories admits a representation which is dual with respect to the one introduced by Schwinger, interchanging the role of the free and interacting terms. It maps $\int V(\delta_J)$ and $J\Delta J$ to $\delta_{\phi_c}\Delta\delta_{\phi_c}$ and $\int V(\phi_c)$, respectively, with $\phi_c=\int J\Delta$ and $\Delta$ the Feynman propagator. Comparing the Schwinger representation with its dual version one gets a little known relation that we prove to be a particular case of a more general operatorial relation. We then derive a new representation of the generating functional $T[\phi_c]=W[J]$ expressed in terms of covariant derivatives acting on 1 $$T[\phi_c] = {N\over N_0} \exp(-U_0[\phi_c])\exp\Big(-\int V({\cal D}_{\phi_c}^-)\Big) \cdot 1$$ where ${\cal D}_{\phi}^{\pm}(x)=\mp\Delta{\delta\over\delta\phi}(x)+\phi(x)$. The dual representation, which is deeply related to the Hermite polynomials, is the key to express the generating functional associated to a sum of potentials in terms of factorized generating functionals. This is applied to renormalization, leading to a factorization of the counterterms of the interaction. We investigate the structure of the functional generator for normal ordered potentials and derive an infinite set of relations in the case of the potential ${\lambda\over n!}:\phi^n:$. Such relations are explicitly derived by using the Fa\`a di Bruno formula. This also yields the explicit expression of the generating functional of connected Green's functions.