I understand the logic behind the Wick rotation by considering an imaginary time and in this way achieving an Euclidean-type metric. However, I am trying to understand this in a deeper way. Why precisely performing a rotation in the complex plane by moving to the $x$-axis to the $y$-axis has such an impact on the description of the space. I assume it has something to do with the reinterpretation of the hyperbolic geometry but I cannot clearly see this connection.
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