# Is Wick rotation invariant under proper conformal transformations?

Is Wick rotation invariant under proper conformal transformations? Why or why not?

Does Wick rotation apply to conformal field theories? $(1-i\epsilon )T$ is not invariant under proper conformal transformations.

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It is invariant! $\epsilon$ is a positive infinitesimal number - but when it comes to its magnitude, only the sign and the fact that it's infinitesimal matter. If you rescale it by a positive factor of $k$, it is still the same $\epsilon$. This $i\epsilon$ is only there to calculate the limits from the right side. – Luboš Motl Jan 2 '13 at 11:55
I asked about proper conformal transformations, not dilatations. – Hubla Jan 2 '13 at 12:10
The question you're asking is somewhat confusing. Conformal transformations are defined with respect to some manifold $M$ and metric $g_{ij}$ on that manifold. If you change the metric, you change the conformal group, but in this flat-space case of Minkowski vs. Euclidean, the structure of the group doesn't change (you have the same SCT's, but in one case the inner product in its expression involves $\eta_{\mu \nu}$, in the other one it's the Kronecker delta). In that sense the conformal group respects Wick rotations. – Vibert Jan 2 '13 at 21:33