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Wick rotation is formally described by the transformation $$t \mapsto it.$$ In many place it is stated more rigorously as an analytic continuation into imaginary time. I understand why we do it but now how we do it. When someone says analytic continuation I think taking a function defined in some domain in $\mathbb{C}^n$ and re-centering its power series to extend it (if possible) beyond its original domain.

I do not see how the above process is being applied when we Wick rotate. In particular, what is the function we are extending using analytic continuation? How is it guaranteed this can always be done?

I have taken a look at these questions:

but they do not get into details on how the analytic continuation is actually done, how it matches the definition of analytic continuation I described, what justifies that this can always be done, or (in the context of quantum field theory) what justifies that we can always recover the Minkowski field from the Euclidean one.

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  • $\begingroup$ The real line can be consider a "some domain in $\mathbb{C}$." So if your power series expansion is valid in a region of $\mathbb{C}$ that is not only the real line then you can analytically continue off the real line into the broader complex plane. In particular, you might be able to analytically continue all the way up the imaginary axis. In which case, you can just make the formal substitution $t \to it$ where $t$ is real. $\endgroup$
    – hft
    Commented Feb 16 at 4:44
  • $\begingroup$ @hft Thank you. And what is the function we are analytically continuing in this case, the field itself? $\endgroup$
    – CBBAM
    Commented Feb 16 at 4:47
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    $\begingroup$ It's whatever function of $t$ you are power series expanding. If that function is the "field," then yes. $\endgroup$
    – hft
    Commented Feb 16 at 4:48
  • $\begingroup$ But, more likely it is is some expectation value of fields, like a green's function $\endgroup$
    – hft
    Commented Feb 16 at 4:49
  • $\begingroup$ @hft What ensures that each field or green's function can be analytically continued to the whole imaginary axis in a unique way? $\endgroup$
    – CBBAM
    Commented Feb 16 at 4:51

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