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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.

2000mg Haigo
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  • Hi 2000mg Haigo, The standard Wick rotation usually has the convention $t_E=it_M$. Are you following a reference? – Qmechanic Jul 26 '22 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. – 2000mg Haigo Jul 26 '22 at 12:06

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  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.

  2. OP is also right that holomorphic/complex differentiation of the fields is a too strong/restrictive notion for Wick-rotation.

  3. For scalar fields, see e.g. this related Phys.SE post.

Qmechanic
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  • thank you, yes, I stumbled upon these references by now, I think I am more concerned with why these transformation laws arise, since the argument of imposing O(4) invariance on the Lagrangian after the wick rotation, from which these laws are derived then, seemed a little bit ad hoc to me... – 2000mg Haigo Jul 26 '22 at 12:09
  • Related post by OP: https://physics.stackexchange.com/q/720158/2451 – Qmechanic Jul 28 '22 at 07:43