Wick rotation and contour integral

In summary, the conversation discusses the use of a contour integral in the complex plane and the conditions required for performing Wick rotation. If the integrand vanishes fast enough as the absolute value of q goes to infinity, the contour can be rotated counterclockwise by 90 degrees without passing over any poles. However, the fast vanishing condition is necessary for Wick rotation, but does not guarantee that the rotation will not pass over any poles. The rotation passing through no poles is another condition that must be satisfied.
  • #1
koolmodee
51
0
We have an integral over q from -[tex]\infty[/tex] to +[tex]\infty[/tex] as a contour integral in the complex q plane. If the integrand vanishes fast enough as the absolute value of q goes to infinity, we can rotate this contour counterclockwise by 90 degrees, so that it runs from -i[tex]\infty[/tex] to +i[tex]\infty[/tex].
In making this, the contour does not pass over any poles. Why?

When I make a contour integral, the big semi-circle that I use in the upper half has the -x+iy pole in it. How does the fast vanishing of the integral and the rotation make the integral not pass over the pole? I don't see it.

thank you
 
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  • #2
koolmodee said:
We have an integral over q from -[tex]\infty[/tex] to +[tex]\infty[/tex] as a contour integral in the complex q plane. If the integrand vanishes fast enough as the absolute value of q goes to infinity, we can rotate this contour counterclockwise by 90 degrees, so that it runs from -i[tex]\infty[/tex] to +i[tex]\infty[/tex].
In making this, the contour does not pass over any poles. Why?

When I make a contour integral, the big semi-circle that I use in the upper half has the -x+iy pole in it. How does the fast vanishing of the integral and the rotation make the integral not pass over the pole? I don't see it.

thank you
I think the fast vanishing of the integral is a required condition for performing Wick rotation. Only when the fast vanishing boundary condition is satisfied, one can perform the Wick rotation.
The condition that the rotation of the contour does not sweep through any poles is not a consequence of fast vanishing condition. Fast vanishing does not imply that the rotation sweeps no poles. Correctly speaking, the rotation passing through no poles is another condition one needs in order to be able to perform Wick rotation.
 
  • #3
thanks ismaili!

I misread something in my book. It says the rotation does not pass over no pole ( in the specific integral given). I misunderstood that as after the rotation there are no poles in the contour.

But why is the fast vanishing of the integral necessary for Wick rotation?
 
  • #4
koolmodee said:
thanks ismaili!

I misread something in my book. It says the rotation does not pass over no pole ( in the specific integral given). I misunderstood that as after the rotation there are no poles in the contour.

But why is the fast vanishing of the integral necessary for Wick rotation?
If the integral vanishes fast toward the infinity, the big semicircles contribute nothing so that the contour integral from [tex]-\infty[/tex] to [tex]\infty[/tex] in the real axis can be analytic continuated to the imaginary axis.
(This is my understanding, everyone is welcome to correct me or supplement.)
 
  • #5
Makes sense.

thanks again
 

1. What is Wick rotation and how is it related to contour integration?

Wick rotation is a mathematical technique used to simplify certain integrals in quantum field theory. It involves rotating the real time axis in the complex plane to the imaginary axis. This transformation allows for the use of contour integration, which is a powerful tool for evaluating complex integrals.

2. Why is Wick rotation necessary in quantum field theory?

In quantum field theory, certain integrals can become divergent when evaluated on the real time axis. Wick rotation allows for the use of contour integration, which helps to regularize these divergences and make the integrals mathematically well-defined.

3. How does Wick rotation affect the physical interpretation of a theory?

Wick rotation does not change the physical predictions of a theory, but it can make the calculations mathematically simpler. In particular, it can help to simplify integrals involving imaginary time, which can be easier to work with than integrals involving real time.

4. What is the relationship between Wick rotation and the Minkowski spacetime metric?

Wick rotation is closely related to the Minkowski spacetime metric, which describes the geometry of spacetime in special relativity. When performing a Wick rotation, the time component of the metric is multiplied by a factor of i, transforming it from a positive to a negative sign. This allows for the use of Euclidean geometry and contour integration techniques.

5. Are there any limitations or drawbacks to using Wick rotation and contour integration?

While Wick rotation and contour integration can be extremely useful tools in simplifying integrals in quantum field theory, they are not always applicable. In some cases, the integrals may not converge after Wick rotation, or the contour needed for integration may not exist. Additionally, Wick rotation can only be used in certain types of theories and does not work for all physical systems.

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