I apologize if my question seems a little half-baked. I was wondering if while working with a QFT, one can make transitions from imaginary time to real time and thereby changing the underlying manifold from Euclidean to Minkowski (and vice versa), do we know if this mapping also perseveres the invariants. More precisely, if my QFT is a projective unitary representation of (for instance) group $SO(3,1)$, then if I make an imaginary-time transformation, do I end up with a projective unitary representation of $SO(4)$? Also do we know what is the relation between $SO(3,1)$ and $SO(4)$? Are they isomorphic?
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