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

Understand

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 ${\cal D}_{\phi}^{\pm}(x)=\mp\Delta{\delta\over\delta\phi}(x)+\phi(x)$.