I What Causes Ambiguity in Wick Rotation for Time-Ordered Correlators?

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Hi all,
I am reading the following post: https://physics.stackexchange.com/q...o-time-ordered-correlator-in-real-time-how-do
and was confused regarding the portion in the accepted answer where they made a distinction for the ##t>0## and ##t<0## case. I believe the ##e^{\pm i\pi}## factor comes from the negative sign in front of ##t^{2}## but cannot understand why there is an ambiguity in the final result.
 
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