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

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thatboi
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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.
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. Towards the end of the first lecture for the Qiskit Global Summer School 2025, Foundations of Quantum Mechanics, Olivia Lanes (Global Lead, Content and Education IBM) stated... Source: https://www.physicsforums.com/insights/quantum-entanglement-is-a-kinematic-fact-not-a-dynamical-effect/ by @RUTA
If we release an electron around a positively charged sphere, the initial state of electron is a linear combination of Hydrogen-like states. According to quantum mechanics, evolution of time would not change this initial state because the potential is time independent. However, classically we expect the electron to collide with the sphere. So, it seems that the quantum and classics predict different behaviours!
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