Confusions on QED renormalization

[Sven2009]

TL;DR Summary

why do people use free photon instead of correted one in QED process?

In many QFT textbooks, we usually see the calculations of vertex function, vacuum polarization and electron self-energy.

For example, one calculates the vacuum polarization to correct photon propagator $\langle{\Omega}|T\{A_{\mu}A_{\nu}\}|\Omega\rangle$, where $|\Omega\rangle$ is the ground state of an interaction Hamiltonian.

My questions are:

1. why do people use free photon propagator (and free electron propagator) in QED process instead of corrected one? You calculate those stuff, but you don't use them?

2. How can we guarantee that all infinites in any QED process can be absorbed by counterterms($\delta_m, Z_2...$ ect.)?

Thanks!

 

[vanhees71]

Ad 1) Of course you use them, but any diagram in perturbation theory consists of the free propagators and the point vertices. So in a higher-order calculation, involving loop diagrams, you calculate the proper vertex functions (i.e., the one-particle irreducible amputated diagrams) first. Also only these have to be "renormalized" to get sensible finite results. Then you use them to evaluate measurable quantities. The most simple that comes to my mind is the one-loop calculation of the anomalous magnetic moment of the electron.

Ad 2) That has been finally proven by the BPHZ formalism in the mid 1960ies (in application to non-Abelian gauge theories a bit later in 1971 by 't Hooft and Veltman), which completed Dyson's earlier proof by solving the issue with overlapping divergences.