Physical interpretations of the generating functions $Z[J]$ and $W[J]$ (or $E[J]$)

Question

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 $$Z[J]=\exp[-iE[J]]=\int D\phi\exp[i\int d^4x(\mathcal{L}(\phi)+J(x)\phi(x))] \tag{11.43}$$ From this, how can we arrive at or understand the following statements in Peskin and Schroeder (page 365, eqn. 11.43):

(i) "The RHS of the equation above is the functional integral representation of the amplitude $\langle\Omega|e^{-iHT}|\Omega\rangle$, where $T$ is the time extent of functional integration, in presence of the source $J$."

(ii) "$E[J]$ is just the vacuum energy as a function of the external source $J$."

Answer 1

By definition, $H|\Omega\rangle = E|\Omega\rangle$, so that $\langle \Omega |e^{-i T H}|\Omega\rangle=e^{-i T E}$. The presence of source terms in the Hamiltonian does not change anything about that.

The RHS of the equation 11.43 is just the functional integral rewriting of $\langle \Omega| e^{-i T H}|\Omega\rangle$.