The logic of LSZ, counterterms and field strength renormalization

Question

I'm reading Coleman's lecture notes on Quantum Field Theory(World Scientific Press).

Coleman first showed that[Chapter 13, equation (13.2)], in a theory with an adiabatically turned on&off interaction and counterterms. That is $H = H_0 + f(t; \Delta, T) H_I$, where $f(t)$ is an adiabatic function. Counterterms are needed to ensure conditions such as $\left<0|S|0\right>$(Chap 9 & 10). The scattering amplitude in the theory can be written as: $$ \left<k_3, k_4 | S - 1 | k_1, k_2\right> = \prod_{r = 1}^{4} (-i)(k_r^2 - \mu^2) \tilde{G}^{(4)}(-k_3, -k_4, k_1, k_2) $$ where $\tilde{G}^{(4)}(-k_3, -k_4, k_1, k_2)$ is the sum of all Feynman diagram with four external lines' propagator included. In this theory, there is no field redefinition.

Then the LSZ reduction is proved: $$ \left<k_3, k_4 | S - 1 | k_1, k_2\right> = \prod_{r = 1}^{4} (-i)(k_r^2 - \mu^2) \tilde{G}^{'(4)}(-k_3, -k_4, k_1, k_2) $$ where $\tilde{G}^{'(4)}(-k_3, -k_4, k_1, k_2)$ is the Fourier transform of the 4-point renormalized Green function in the physical theory($H = H_0 + H_I$, without adiabatic function) $$ G^{'(4)}(x_1, x_2, x_3, x_4) = \left<0|T(\phi'(x_1)\phi'(x_2)\phi'(x_3)\phi'(x_4)|0\right> $$ and $\phi'$ is the renormalized field s.t. $\left<0|\phi'(x)|0\right> = 0$ and $\left<k|\phi'(x)|0\right> = e^{ikx}$.

I'm a little confused about these two equations, since they are both right. Can someone point out the difference between these two equations and the underlying logic behind LSZ reduction.

This post imported from StackExchange Physics at 2024-11-01 12:03 (UTC), posted by SE-user Jason Chen

Comments

In the first formula everything is "bare" including fields. The counterterms serve to fix this "drawback". This fixing is reduced to renormalization of everything.