Confusions on QED renormalization 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:


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*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?

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

This depends on what order in perturbation theory you are working to. The corrections to the field strength ($Z$) will be proportional to $\alpha$ so if you are only interested in the leading contribution to any process, this does not need to be taken into account. In general, however, all the corrections need to be included if you are working to higher order. 


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

This is a very good question, and not trivial to see. The fact that this actually works (that a finite number of counterterms can cancel all infinities that arise at any order in perturbation theory) is what it means to be a renormalizable theory. This is something that has to be proved (and for QED or the standard model this has been proved). 
