# Wick rotation of contravariant vector field holomorphic?

Suppose one has a contra variant vector field in Minkowski spacetime $$A^\mu : \mathbb{R}^{1+3} \to \mathbb{R}$$ for each component $$\mu$$ inside some path integral. I assume, that one can analytically continue also the function $$A^\mu$$ to $$\mathbb{C}\times \mathbb{R}^3$$, and this continuation is unique. Now, after wick rotation, i.e. choosing the time $$t \in i\mathbb{R}$$ with parametrisation $$t=i\tau$$, this results in $$x^0_E = i\tau = ix^0$$. Now often times it is claimed that also for this vector field $$A^0((i\tau, \vec{x})) = i A^0(\tau, \vec{x})$$ and $$A^j((i\tau, \vec{x})) = A^j(\tau, \vec{x})$$. How does the vector field look for arbitrary $$t \in \mathbb{C}$$?

Is the generalisation of that simply $$A^0((e^{i\phi}\tau, \vec{x})) = e^{i\phi} A^0(\tau, \vec{x})$$ and $$A^j((e^{i\phi}\tau, \vec{x})) = A^j(\tau, \vec{x})$$?

Because probing if such functions could be holomorphic simply fails, with the Cauchy Riemann conditions.

Thus I believe there must be another argument as to why this is so, for example regarding the contravariant nature of the vector.

• Hi 2000mg Haigo, The standard Wick rotation usually has the convention $t_E=it_M$. Are you following a reference? Jul 26, 2022 at 11:35
• no I am unfortunately not, I am rather left confused by this subject, particularly in the application to the path integral , but I changed it now, for good practice. Jul 26, 2022 at 12:06

1. OP is right. All 4-vectors (such as $$x^{\mu}$$ and $$A^{\mu}$$) transform in the same way under Wick-rotation, cf. e.g. this Phys.SE post.