In quantum field theory, the generator of all Green's functions $Z[J]$ and that of the connected Green's functions $E[J]$ are related as $$Z[J]=\exp[-iE[J]]=\int D\phi\exp[i\int d^4x(\mathcal{L}(\phi)+J(x)\phi(x))]$$$$Z[J]=\exp[-iE[J]]=\int D\phi\exp[i\int d^4x(\mathcal{L}(\phi)+J(x)\phi(x))] \tag{11.43}$$ From this, how can we arrive at or understand the following statements in Peskin and Schroeder (page 365, eqn. 11.43):
(i) The"The RHS of the equation above is the functional integral representation of the amplitude $\sim \langle\Omega|e^{-iHT}|\Omega\rangle$$\langle\Omega|e^{-iHT}|\Omega\rangle$, where T$T$ is the time extent of functioanlfunctional integration, in presence of the source $J$."
(ii) "$E[J]$ is just the vacuum energy as a function of the external source $J$."
