Can one just change the notation of four vectors so as instead of having $$ X^{\mu} =(X^0, \vec{X})$$we define $$ X^{\mu}=(X^0,i\vec{X})?$$ This way we could use the Euclidean metric instead of $$g^{\mu\nu}=\text{diag}(1,-1,-1,-1).$$

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    $\begingroup$ Older books tend to do this I think (they put the $i$ in the time component though). As for why people swapped - another user might have opinions on this. $\endgroup$
    – jacob1729
    Jan 4, 2020 at 14:47
  • $\begingroup$ Why do you want to pretend that the metric is Euclidean, and spacetime is complex, when neither is true? (Yes, I know there are some reasons in QFT for doing this.) $\endgroup$
    – G. Smith
    Jan 4, 2020 at 14:52
  • $\begingroup$ @G.Smith why do you say that neither are true? If they are mathematically equivalent, they should both be ""true"", whatever true means $\endgroup$
    – user728261
    Jan 4, 2020 at 14:56
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    $\begingroup$ Here is my paraphrase of Misner Thorne Wheeler’s “Farewell to ict”: physics.stackexchange.com/a/327516/148184 $\endgroup$
    – robphy
    Jan 4, 2020 at 15:28
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    $\begingroup$ Does this answer your question? Minkowski's complex Euclidean space vs. the real pseudo-Euclidean version $\endgroup$
    – Kyle Kanos
    Jan 4, 2020 at 16:22