Cauchy theorem and Wick rotation

In summary, the conversation discusses a proof known as 'Wick rotation' in a textbook on QFT. The proof involves rotating the real axis into the imaginary axis and using the Cauchy theorem on the path-independence of integrals. The last step involves taking a finite closed path and letting it tend to infinity, checking that the integral at infinity tends to zero. The conversation also discusses the assumption that f(r)/g(r) is O(1/r2) or less and how the difference between the integrals between ±r on each axis tends to 0 as r tends to infinity. The conclusion is that the integrals between infinity are the same.
  • #1
parton
83
1

Homework Statement



Hi!

I have a little problem understanding a proof ('Wick rotation') which I found in a textbook on QFT.

Assume: f and g are polynomials with degree(g)-degree(f) >= 2

and f/g has no poles on the closed 1. and 3. quadrant.

proposition:

[tex] \int_{-\infty}^{+\infty} \mathrm{d}x \dfrac{f(x)}{g(x)} = i \int_{-\infty}^{+\infty} \mathrm{d}x \, \dfrac{f(ix)}{g(ix)} [/tex]

proof (just until the part I don't undestand):

We rotate the real axis into the imaginary axis (counter-clockwise). Because the rotating real axis does not hit poles of f/g we can write

[tex] \int_{-\infty}^{+\infty} \mathrm{d}x \dfrac{f(x)}{g(x)} = \int_{-i\infty}^{+i\infty} \mathrm{d}x \dfrac{f(x)}{g(x)} [/tex]

by the Cauchy theorem on the path-independence of integrals over holomorphic functions.


Could anyone explain this last step to me?

Homework Equations





The Attempt at a Solution



I heard about path-independence, but I thought we need two paths with identical initial and end point, but here it is obviously not the case... maybe I am missing something
 
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  • #2
hi parton! :smile:

yes, you're right, we need two paths with identical initial and end point (or one continuous closed path), but we can take a finite closed path, apply the Cauchy theorem, and then let the path tend to infinity, checking that the bit "at infinity" has an integral that tends to zero

i think the idea here is that you integrate between ±r along one axis, then round a quadrant of a circle of radius r, back along the other axis, and then another quadrant to close the curve

the theorem assumes f(r)/g(r) is O(1/r2) or less, and the integral along the quadrants is proportional to πr times f(r)/g(r), and so is O(1/r)

so the difference between the integrals between ±r on each axis -> 0 as r -> ∞, ie the integrals between ∞ are the same :wink:
 

Related to Cauchy theorem and Wick rotation

1. What is Cauchy's theorem and how does it relate to complex analysis?

Cauchy's theorem is a fundamental result in complex analysis that states that if a function is analytic within a closed contour, then the integral of that function over the contour is equal to zero. This theorem is important because it allows us to calculate complex integrals by using simpler methods such as the Cauchy integral formula.

2. What is the significance of Wick rotation in quantum field theory?

Wick rotation is a technique used in quantum field theory to convert the calculations of time-dependent systems into the calculations of imaginary time systems. This allows for easier calculations and the avoidance of divergent integrals that often arise in quantum field theory.

3. How does Wick rotation relate to Cauchy's theorem?

Wick rotation is based on the idea of rotating the contour of integration in the complex plane, which is similar to the idea behind Cauchy's theorem. By using Wick rotation, we can convert complex integrals into simpler ones that can be solved using Cauchy's theorem.

4. Can Wick rotation be applied to any function?

No, Wick rotation is only applicable to functions that are analytic in the complex plane. This means that the function must be differentiable at every point in the complex plane. If a function has singularities or poles, Wick rotation cannot be used.

5. Are there any limitations to using Wick rotation?

One limitation of Wick rotation is that it can only be used in certain regions of the complex plane where the function is analytic. Additionally, the results obtained through Wick rotation may not always be physically meaningful or applicable, so it is important to use caution when applying this technique.

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