How to demonstrate the asymptotic charateristic of perturbative QTF Theo?

[ndung200790]

Please teach me this:
Why the perturbative QTF Theory is an asymptotic theory,because in the asymptotic expansion,the error at N order is a ''infinity small'' of order the (N+1)th term (meaning O(term(N+1)).So I wonder why we know the error at order N in perturbative QTF Theory is of approximation of (N+1)th term in the series.
Thank you very much in advanced.

 

[A. Neumaier]

Please teach me this:
Why the perturbative QTF Theory is an asymptotic theory,because in the asymptotic expansion,the error at N order is a ''infinity small'' of order the (N+1)th term (meaning O(term(N+1)).So I wonder why we know the error at order N in perturbative QTF Theory is of approximation of (N+1)th term in the series.

It is known in special cases. In general, one just hopes. The only guarantee is that
''if the expansion parameter is sufficiently small'' the error is of the first neglected order.
But usually there are no guarantees at all that for the expansion parameter of interest, this will be the case.

 

[Quantumjump]

Please teach me this:
Why the perturbative QTF Theory is an asymptotic theory,because in the asymptotic expansion,the error at N order is a ''infinity small'' of order the (N+1)th term (meaning O(term(N+1)).So I wonder why we know the error at order N in perturbative QTF Theory is of approximation of (N+1)th term in the series.
Thank you very much in advanced.

You must do the calculation for the integral describing each type of process you find within a specific quantum field theory. For example, in quantum electrodynamics, you set up all the kinds of Feynman diagram that correspond to a S-matrix perturbated above tree level: this shows you that you have fundamentally 5 types of different processes only. Three of them only are a problem: vertex correction, photon self-energy and electron self-energy: they all give integrals which diverges. What does it mean exactly? Well, when you calculate the taylor expansion, you find out that what is supposedly giving you smalller and smaller terms give you terms that are becoming extremely huge and that explode to infinity. Renormalization is the process by which we mathematically get rid off that problem. It is very well defined mathematically, and has a physical interpretaion: basically, it is equivalent to the rescaling of wavefunction renormalization appearing in the propagators involved (fermionic/electromagntic), as well as a rescaling of the mass of partices involved.

 

[ndung200790]

I mean the ''asymptotic'' (or the divergent series) after renormalization for S-matrix.