Why is the Electron Self-Energy Important in QFT Theory?

[ndung200790]

Please teach me this:
In QTF theory book of Peskin&Schroder they say:Call 1PI the one particle irreducible diagram,so Fourier two-point correlation function can write:
i/(gamma(vector).p-m0) +i/(gamma.p-m0)(1PI/(gamma.p-m0) +i/(gamma.p-m0)square(1PI/(gamma.p-m0))...=i/(gamma.p-m0-1PI).
They say that we can sum the left hand side to have the right hand side because the left is a geometric serie.But I don't understand why we can sum while we don't know 1PI/(gamma.p-m0) being smaller than 1 or not.Another worrying is that the calculation is done before the making renormalization and QED being asymtotic theory.
Thank you very much in advanced.

 

[A. Neumaier]

Please teach me this:
In QFT theory book of Peskin&Schroder they say:Call 1PI the one particle irreducible diagram,so Fourier two-point correlation function can write:
i/(gamma(vector).p-m0) +i/(gamma.p-m0)(1PI/(gamma.p-m0) +i/(gamma.p-m0)square(1PI/(gamma.p-m0))...=i/(gamma.p-m0-1PI).
They say that we can sum the left hand side to have the right hand side because the left is a geometric series. But I don't understand why we can sum while we don't know 1PI/(gamma.p-m0) being smaller than 1 or not.Another worrying is that the calculation is done before the making renormalization and QED being asymptotic theory.

Using 1+z+...+z^n+... = 1/(1-z) without bothering about convergence is called formal manipulation. Note that the formula is valid in the field of formal power series in z. Quantum field theory is full of such formal manipulations. After sufficiently skilful such manipulations and approximations one usually arrives at something (called ''renormalized'') where one can insert a particular value of z and get meaningful results. This can be justified in many cases by means of expansions of analytic functions, so one hopes that it also gives valid results when such justifications are too hard to come by (as in QED).
The proof is then by comparison with experiment rather than by rigorous mathematics.

 

[ndung200790]

I am profound grateful the kind helping of Prf.A Neumaier