Quantum Field Theory: Evaluating Integrals on Page 27

In summary: Here, M is the maximum length of the interval [a,b] in which |h(z)|<M.Is ML estimate maximum likelihood estimate? How can ML estimate be used here since it is about probability?When I said ML-estimate I mean the following:Suppose C is a piecewise smooth curve. If h(z) is continuous function on C then\displaystyle{\left|\int_{C} h(z)\, dz\right| \leq \int_{C}|h(z)|\, |dz|}.and if C has length L and |h(z)|\leq M on C
  • #1
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I don't understand how Peskin & Schroeder can evaluate the integral on page 27 by having the real axis wrapping around branch cuts just like that. The picture of the contours are on page 28.
 
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  • #2
I think what they have done is simply completed a loop (like a key-hole contour) but the arc/circular bit dies away as your variable go to infinity so effectively the flat/horizontal bit is same as the two vertical bits (by Cauchy theorem... as no poles inside loop)

describing the loop: first bit is the original bit the flat/horizontal (-R,+R) bit with R eventually taken to infinity, then to complete the loop you need to add a 1/4 of an arc going from +R to +iR, then comes down to avoid the branch cut, go around the pole and goes up again before arch back from +iR to -R.
 
  • #3
mjsd said:
I think what they have done is simply completed a loop (like a key-hole contour) but the arc/circular bit dies away as your variable go to infinity so effectively the flat/horizontal bit is same as the two vertical bits (by Cauchy theorem... as no poles inside loop)

describing the loop: first bit is the original bit the flat/horizontal (-R,+R) bit with R eventually taken to infinity, then to complete the loop you need to add a 1/4 of an arc going from +R to +iR, then comes down to avoid the branch cut, go around the pole and goes up again before arch back from +iR to -R.

Why would the arc or circular bit dies away as the variable goes infinity?
 
  • #4
I haven't check this particular example and see if it does goes away... but it usually does and that's why we close the contour in the first place...by the way, I did say "I think"...perhaps you can check that... to prove that you need to look at your integrand and see what happen when R becomes large (ie. when the integration variable expressed in polar form becomes large). Sometimes Jordon's lemma or ML-estimate maybe used to help.
 
  • #5
mjsd said:
I haven't check this particular example and see if it does goes away... but it usually does and that's why we close the contour in the first place...by the way, I did say "I think"...perhaps you can check that... to prove that you need to look at your integrand and see what happen when R becomes large (ie. when the integration variable expressed in polar form becomes large). Sometimes Jordon's lemma or ML-estimate maybe used to help.

I looked up on Jordan's lemma, and yeah the integrand of the semicircular path (excluding the real axis) tends to zero as R goes infinity.
OOOooo contour integrals are so interesting !
Thank you!

Is ML estimate maximum likelihood estimate? How can ML estimate be used here since it is about probability?
 
  • #6
when I said ML-estimate I mean the following:
Suppose C is a piecewise smooth curve. If [tex]h(z)[/tex] is continuous function on C then
[tex]\displaystyle{\left|\int_{C} h(z)\, dz\right| \leq
\int_{C}|h(z)|\, |dz|}.[/tex]
and if C has length L and [tex]|h(z)|\leq M[/tex] on C then
[tex]\displaystyle{\left|\int_{C} h(z)\, dz\right| \leq ML}[/tex]
 

1. What is Quantum Field Theory?

Quantum Field Theory (QFT) is a theoretical framework that combines the principles of quantum mechanics and special relativity to describe the behavior of subatomic particles and their interactions.

2. What is the purpose of evaluating integrals in QFT?

In QFT, integrals are used to calculate the probability amplitudes of different particle interactions. By evaluating these integrals, we can determine the likelihood of certain particle interactions occurring.

3. How is the evaluation of integrals different in QFT compared to classical mechanics?

In classical mechanics, integrals are computed over a continuous range of values. In QFT, the integrals are computed over a discrete set of values, which corresponds to the discrete energy levels of particles in a quantum system.

4. What challenges are faced when evaluating integrals in QFT?

One of the main challenges in evaluating integrals in QFT is dealing with infinities that arise in the calculations. These infinities can be dealt with using various techniques such as renormalization.

5. How are Feynman diagrams used in QFT to evaluate integrals?

Feynman diagrams are graphical representations of mathematical expressions that describe the behavior of particles in QFT. They are used to simplify the evaluation of integrals and make predictions about particle interactions.

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