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Starting in the Minkowski formulation, the Feynman $i\varepsilon$-prescription is just the first infinitesimal angle $\theta=\varepsilon$ of a Wick rotation

$$ t(\theta) = e^{i\theta} t_M, \qquad \theta~\in~[0,\frac{\pi}{2}], \qquad t(\theta\!=\!0)~=~t_M, \qquad t(\theta\!=\!\frac{\pi}{2})~=~it_M=t_E, $$

in the complex $t$-plane to the Euclidean formulation. Heuristically, on physical grounds, no poles & branch cuts are expected after the first infinitesimal rotation $\theta=\varepsilon$, so this in turn is equivalent to the full $\theta=\frac{\pi}{2}$ Wick rotation.

NB: Wick rotation of spinors is subtle, cf. e.g. arXiv:hep-th/9611043 & this Phys.SE post.

Starting in the Minkowski formulation, the Feynman $i\varepsilon$-prescription is just the first infinitesimal angle $\theta=\varepsilon$ of a Wick rotation

$$\begin{align} t(\theta) ~=~& e^{i\theta} t_M, \qquad \theta~\in~[0,\frac{\pi}{2}], \cr t(\theta\!=\!0)~=~&t_M, \qquad t(\theta\!=\!\frac{\pi}{2})~=~it_M=t_E, \end{align}$$

in the complex $t$-plane to the Euclidean formulation. Heuristically, on physical grounds, no poles & branch cuts are expected after the first infinitesimal rotation $\theta=\varepsilon$, so this in turn is equivalent to the full $\theta=\frac{\pi}{2}$ Wick rotation.

NB: Wick rotation of spinors is subtle, cf. e.g. arXiv:hep-th/9611043 & this Phys.SE post.

The Feynman $i\varepsilon$-prescription is just the first infinitesimal angle of a $e^{i\pi/2}$ Wick rotation in the complex $t$-plane. Heuristically, on physical grounds, no poles & branch cuts are expected after the first infinitesimal rotation, so this in turn is equivalent to the full $e^{i\pi/2}$ Wick rotation.

NB: Wick rotation of spinors is subtle, cf. e.g. arXiv:hep-th/9611043 & this Phys.SE post.

Starting in the Minkowski formulation, the Feynman $i\varepsilon$-prescription is just the first infinitesimal angle $\theta=\varepsilon$ of a Wick rotation

$$ t(\theta) = e^{i\theta} t_M, \qquad \theta~\in~[0,\frac{\pi}{2}], \qquad t(\theta\!=\!0)~=~t_M, \qquad t(\theta\!=\!\frac{\pi}{2})~=~it_M=t_E, $$

in the complex $t$-plane to the Euclidean formulation. Heuristically, on physical grounds, no poles & branch cuts are expected after the first infinitesimal rotation $\theta=\varepsilon$, so this in turn is equivalent to the full $\theta=\frac{\pi}{2}$ Wick rotation.

NB: Wick rotation of spinors is subtle, cf. e.g. arXiv:hep-th/9611043 & this Phys.SE post.

The Feynman $i\varepsilon$-prescription is just the first infinitesimal angle of a $e^{i\pi/2}$ Wick rotation in the complex $t$-plane. Heuristically, on physical grounds, no poles are expected after the first infinitesimal rotation, so this in turn is equivalent to the full $e^{i\pi/2}$ Wick rotation.

NB: Wick rotation of spinors are subtle, cf. e.g. arXiv:hep-th/9611043 & this Phys.SE post.

The Feynman $i\varepsilon$-prescription is just the first infinitesimal angle of a $e^{i\pi/2}$ Wick rotation in the complex $t$-plane. Heuristically, on physical grounds, no poles & branch cuts are expected after the first infinitesimal rotation, so this in turn is equivalent to the full $e^{i\pi/2}$ Wick rotation.

NB: Wick rotation of spinors is subtle, cf. e.g. arXiv:hep-th/9611043 & this Phys.SE post.