How do we Wick rotate the Maxwell $U(1)$ gauge theory's field strength, say in 3 space and 1 time dimensions? Suppose we start with a Lorentz signature with coordinates $(x_0, x_1, x_2, x_3)$, then we define $$ x_0= -i x_4 $$ to rotate to $(x_4, x_1, x_2, x_3)$. I wanted to ask the Wick rotations of the followings from the Lorentz to the Euclidean signature? (The question may be trickier than you thought, because we need to worry about the relation between $F_{j0}$ and $F_{j4}$, where $j=1,2,3$.

Precisely I have four subquestions on Wick rotations from the Lorentz to the Euclidean signature:

  1. How to Wick rotate $ F_{\mu\nu}=(\partial_\mu A_\nu -\partial_\nu A_\mu )$?

  2. How to Wick rotate $F=\frac{1}{2} F_{\mu\nu} (dx^\mu \wedge dx^\nu)$?

  3. How to Wick rotate $F \wedge * F =\frac{1}{2} F_{\mu\nu} F^{\mu\nu} d^4 x $?

  4. How to Wick rotate $F \wedge F= \epsilon^{{\mu\nu}{m n} } F_{\mu\nu} F_{m n} d^4 x$?

p.s. Here I use the fact $(*F)_{\mu_1\mu_2\dots \mu_{n-p}}=\frac{1}{p!}\epsilon^{\nu_1 \dots \nu_p}_{\mu_1\mu_2\dots \mu_{n-p}} F_{\nu_1 \dots \nu_p}$, so $(*F)_{\mu_1\mu_2}=\frac{1}{2!}\epsilon^{\nu_1 \nu_2}_{\mu_1\mu_2} F_{\nu_1 \nu_2}$,


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