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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~ ...


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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 ...


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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]} ...



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