Euclideanizing Action: Refs & Fermionic/Spin-1 YM Actions

[TriTertButoxy]

Is there a good reference on the procedure of Euclideanizing the action? In particular, giving a detailed account of Wick rotating in this context. I can't seem to figure out if they are supposed to give precisely the same answer (even numerically), but just that one converges better, or if they are two different quantities (just one related to the other).

Also, is there a standard way of Euclideanizing the fermionic action and the spin-1 Yang Mills action? There seem to be subtleties which seems very poorly motivated. For example, multiplying $\sigma^0$ and $A^0$ by the imaginary unit, $i$.

 

[RedX]

I don't understand Euclideanizing the action either, so I'm bumping this!

Does it even make sense to integrate over imaginary time? How would you go about doing this? - finding the anti-derivative of the integral, and plugging in imaginary +- infinity via the fundamental theorem of calculus?

Sometimes Euclideanizing the action is called a Wick rotation, which is confusing because there is this other Wick rotation used in momentum space which is definitely sound.

As for multiplying $$A^0$$ by the imaginary unit 'i', that would kind of make sense, since $$A^0$$ transforms like time. However, shouldn't it be '-i', and not '+i', since the relation is $$\tau=it$$ where $$\tau$$ is the Euclidean time.