1
$\begingroup$

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.

$\endgroup$
2
  • $\begingroup$ Hi 2000mg Haigo, The standard Wick rotation usually has the convention $t_E=it_M$. Are you following a reference? $\endgroup$
    – Qmechanic
    Jul 26, 2022 at 11:35
  • $\begingroup$ 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. $\endgroup$ Jul 26, 2022 at 12:06

1 Answer 1

1
$\begingroup$
  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.

$\endgroup$
2
  • $\begingroup$ 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... $\endgroup$ Jul 26, 2022 at 12:09
  • $\begingroup$ Related post by OP: physics.stackexchange.com/q/720158/2451 $\endgroup$
    – Qmechanic
    Jul 28, 2022 at 7:43

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.