Why is the coupling factor constant in QFT?

[Haorong Wu]

TL;DR Summary

Why the coupling factor is constant in QFT?

In Peskin's textbook, the coupling factor is given by a constant in the interacting field theory. The scattering matrix $S$ is given by the time-evolution operator, $\exp(-iHt)$, in the limit of very large t, i.e., $t\rightarrow \infty$, as expressed in Eq. (4.71).

In my mind, the scattering process is very short. Before and after the scattering process, the coupling factor should vanish, and only in the short interaction time, the coupling factor is nontrivial. Also, the coupling factor should be time-dependent.

Therefore, I do not understand why the scattering matrix is given by letting the coupling factor be a constant and letting the interaction time be infinity.

Thanks in advance.

 

[Demystifier]

QFT in particle physics deals with fundamental coupling factors such as the charge $e$ of the electron, which is obviously constant. Time dependent coupling factors often appear in effective Hamiltonians, but not in fundamental ones. Also note that time dependent coupling factor means that energy is not conserved during the interaction, which cannot be right in a closed system. In scattering, the interaction diminishes not by vanishing of the coupling factor, but by the increase of the distance between the particles.