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.
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.
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.
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.
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.
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.
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.
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.