Wick rotation and Minkowski/Euclidean space

[Pablo88]

Hello, I was wondering why in integrals such as
$$\int d^4k F(k^2)$$
where $$k^2 = (k^0)^2 - |\vec{k}|^2$$ ranges from -∞ to ∞, once the Wick-rotation is performed, we have $$-k^2_E = -(k^0_E)^2 - |\vec{k}_E|^2$$ which lies in the (-∞,0) interval ... So, the contribution which lies in the part where $$k^2 >0$$ it seems that fades away. This function would have even poles. It is still a bit counter-intuitive for me that this part just fades away.

I guess we could shift the space-variables as $$\vec{k} \rightarrow \vec{k+k_0}$$ so this always lays in the euclidean space so to say. Anyway I am not sure, if this shift is always allowed.

Asides, things get more annoying for me if we have other four-vectors in the function, appart that the loop variable, which are Minkowskian, i.e. $$F((k+q)^2,(k-p)^2)$$ where $$q^2 = m^2, \ p^2 = M^2$$ (both are positive quantities) ...

I find this kind of examples in triangle loops diagrams where the function F(k,q,..) is the form factor, and I this form factor is supposed to be the in the Spacelike or Euclidean region ...

Thanks in advance!

 

[eljose79]

Hello,

Thank you for your question. The reason for this behavior is due to the Wick rotation, which is a mathematical tool used to simplify certain integrals in quantum field theory. When performing a Wick rotation, we are essentially transforming the integral from Minkowski space (with a Lorentzian metric) to Euclidean space (with a Euclidean metric). This is done by rotating the time coordinate from the Minkowski signature (t, x, y, z) to the Euclidean signature (it, x, y, z). This rotation has the effect of changing the sign of the time component, which is why we have -k^2_E in the Euclidean integral.

Now, in the Euclidean space, the allowed range of k_E is from -∞ to ∞, which corresponds to a range of k^2_E from 0 to ∞. This is why the contribution from the region where k^2 > 0 seems to fade away - it is now included in the integral over k_E. This is a consequence of the Wick rotation, and it is a perfectly valid mathematical manipulation.

As for your concerns about shifting the space-variables, this is indeed allowed as long as the integral remains convergent. However, it is not always necessary to do so, as the Wick rotation takes care of this issue.

In the case of other four-vectors in the function, the same principles apply. The Wick rotation will transform all the four-vectors in the function to Euclidean space, and the integral will be over Euclidean variables. As for the form factor, it is indeed supposed to be in the Euclidean region, as this is where the Wick rotation takes us.

I hope this helps to clarify your doubts. If you have any further questions, please feel free to ask. Happy researching!