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The motivation is trying to define the path integral, where at some point we get the integral $$ Z=\int e^{iS}Dx $$ which is then taken to imaginary time $$ Z_E=\int e^{-S_E}Dx $$ such that $Z_E$ can be calculated and then analytically continued back to real-time.

The bare bones of what I am wondering is if we have the integral $$ I=\int e^{ix^2 t}dx $$ can this be understood or evaluated without taking $t\mapsto -i\tau$ (or equivalently $x\mapsto z-i\epsilon$)

Qmechanic
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Toby Peterken
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  • The function $e^{ix^2 t}$ is not absolute integrable, and it needs some kind "regularization". Changing from a real $t$ to a complex variable makes the integral convergent inside the half plane and the real variable integral then can be interpreted as being a limit to a generalized "function". – hyportnex Nov 13 '23 at 18:16
  • See this Phys.SE answer by Jack. – Qmechanic Nov 13 '23 at 18:29

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