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Comments to the question (v2): In (non-relativistic) quantum mechanics, time is a parameter (as opposed to a selfadjoint operator), cf. e.g. this Phys.SE post and links therein. In the phase space path integral $$K(q_f,t_f;q_i,t_i) ~\equiv~\langle q_f,t_f \mid q_i,t_i\rangle ~=~\int_{q(t_i)=q_i}^{q(t_f)=q_f} \!{\cal D}q~ {\cal D}p~ ... 1 I) Let us here phrase the problem in the context of some position operator \hat{q} of QM for simplicity. The generalization to QFT can formally be achieved by replacing the position operator \hat{q} with a quantum field \hat{\psi}({\bf x}). We know that the overlap with Minkowski (M) signature is given as a path integral ... 1 Hints to the question (v2): First of all, one should realize the abuse of notation in eq. (6.69) of Ref. 1 where \phi is used in two meanings: both as an external parameter and as internal integration/dummy variable. It is more properly written as$$ \widehat {Z}[\phi]~=~\frac{{e}^{iS[\phi]}}{{\cal N}}, \qquad {\cal N}~:=~\int\!{\cal D}\phi~e^{iS[\phi]} ...