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This formula for the effective action is valid if (i) you are interested in a slowly varying field (i.e. in the low energy/momenta modes) and (ii) if the effective action not singular in this expansion. If the expansion is well behaved, then you always have the right to do the expansion, which will obviously be valid only for fields varying slowly enough. ...


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 $\langle \Omega| e^{-i T H}|\Omega\rangle$.

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