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What is the connection between causality and wick rotation?

I came across implication of this connection multiple times but can't find a rigorous explanation.

For example in the answer to Wick rotation and the arrow of time it is said that

"(...) Wick rotation in the complex plane is only possible given the analyticity properties of the causal Green functions."

Could someone expand on this?

The question and answer below are also related. https://www.physicsoverflow.org/16852/wick-rotation-in-field-theory-rigorous-justification But it actually only talks about wick rotation in QM, making the claim that its extension to QFT should be trivial, which seems to be contradictory with causality playing an important role in the validity of wick rotation.

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  • $\begingroup$ There is obvious connection in terms of poles of the propagators/Greens functions. Rotation is impossible unless one pole is above the real line (time axis), and the other is below it. $\endgroup$ – Darkseid Nov 14 '17 at 5:21
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    $\begingroup$ @Darkseid, what does this mean in terms of causality? Yes wick rotating is equivalent to picking the Feynman propagator, which gives time ordered correlators. Where does causality come into play here? More specifically where do we see relativistic causality, i.e. the commuting of observables for spacelike separations. I think the statement is that this somehow plays a crucial role in the validity of Wick rotation in a relativistic theory. $\endgroup$ – Kvothe Nov 17 '17 at 14:04

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