Wick rotation and analytic continuation

[wnvl2]

In quantum mechanics and quantum field theory, Wick rotation is often used to transform oscillatory integrals into exponentially convergent ones by replacing real time with imaginary time. The result is then analytically continued back to real time.

For example, a time-evolution amplitude contains a factor $e^{-iH\Delta t/\hbar}$, which leads to oscillatory integrals in path-integral formulations. After Wick rotation, these often become much easier to evaluate.

My question is: under what precise mathematical conditions is this procedure justified?

Is Wick rotation merely a computational trick, or does it rely on additional assumptions about analyticity, convergence and the singularities of the integrand? How can we be sure that analytic continuation back to real time recovers the original physical result?

I am particularly interested in references that explain the mathematical foundations of these methods, including examples where Wick rotation fails or requires special care.

 

[renormalize]

Is Wick rotation merely a computational trick, or does it rely on additional assumptions about analyticity, convergence and the singularities of the integrand? How can we be sure that analytic continuation back to real time recovers the original physical result?

Have you looked here: https://en.wikipedia.org/wiki/Wick_rotation and the references therein? In particular, that link states:

 

[wnvl2]

No. Thankyou for the reference. In the meantime I have als a simple answer from AI:
The main conditions are:

1. Analyticity: The integrand must be analytic in the region through which the contour is deformed, apart from singularities that are treated separately.
2. Vanishing arc contribution: The integral over the connecting contour at infinity must vanish. For example, this can be verified for the Fresnel integral $\int_{-\infty}^{\infty} e^{it^2},dt$ by rotating the contour by $45^\circ$.
3. Convergence: The rotated integral must converge or have a well-defined interpretation. In quantum field theory, Wick rotation can transform oscillatory factors $e^{iS_M}$ into exponentially damped factors $(e^{-S_E})$, provided the relevant conditions hold.
4. Correct treatment of poles: Singularities must not be crossed without accounting for their contributions. In Feynman propagators, the $(i\varepsilon)$-prescription determines how the poles are positioned relative to the integration contour.