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The Wick's rotation $W$ facilitates dealing with integrals in the Minkowski space by rotating time into the Euclidean space. As this rotation in time is performed within integrals, one can view that as a change of integral's variables, and subsequently, the integrated results in the Minkowski and Euclidean space are the same.

On the other hand, spatial rotations $\cal R$ can be viewed as transformations that, depending on the properties of our system of interest with Hamiltonian $H$, may play the role of symmetries such that $${\cal R} H {\cal R}^{-1} = H.$$

My questions are

  1. Can we consider the Wick's rotation combined with spatial rotation as a spacetime transformation for a time-dependent $H$? i.e., writing $$(W{\cal R}) H (W{\cal R})^{-1}=H~? $$

  2. If yes, can you suggest a time-dependent Hamiltonian in which the Wick's rotation is its symmetry?

Sharing further insights regarding this issue is very appreciated.

Qmechanic
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Shasa
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  • Wick rotation, as I know it, is a specific trick we can do to shift the contour of integrals. How do you intend for it to act on your $H$ operator? [Note that time evolution using $H$ in Minkowski time and in Euclidean time looks very different, in real-time you get phases, in euclidean time you get exponential suppression of states] – QCD_IS_GOOD Jun 22 '22 at 21:16
  • One possibility is relaxing the Hermiticity of $H$. With complex spectrum, a dynamical phase and a decay of state may be interchangeable. – Shasa Jun 23 '22 at 14:41

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