Can we demontrate the convergence of perturbation quantum field theory?

[ndung200790]

Please teach me this:
Can we demontrate the convergence of perturbation series of quantum field theory(Feymann
diagrams) after making the renormalizing procedure? If we can't demontrate that,why we still consider the perturbative method using in quantum field theory being useful and believable theory?
Thank you very much in advance.

 

[ndung200790]

Sorry,I have had an wrong spelling:"demontrate",I mean "demonstrate".I am Vietnamese

 

[A. Neumaier]

Please teach me this:
Can we demontrate the convergence of perturbation series of quantum field theory(Feymann
diagrams) after making the renormalizing procedure? If we can't demontrate that,why we still consider the perturbative method using in quantum field theory being useful and believable theory?

The renormalized perturbation series of QED is most likely divergent (and corresponding series in certain simpler l2-dimensional theories are provably divergent). But divergent asymptotic series often give useful approximations, and in case of QED even very accurate ones!

 

[ndung200790]

Please explain for me what is divergent asymtotic series.It is seem to me that being convergent series,the higher power in series,the more acurate approximation.But in devergent series,the higher power, the less acurate approximation.

 

[sheaf]

This reference contains some discussion, including a brief statement of Dyson's intuitive argument:

http://arxiv.org/abs/hep-ph/0508017"

 

[A. Neumaier]

Please explain for me what is divergent asymptotic series.It is seem to me that being convergent series,the higher power in series,the more accurate approximation. But in divergent series, the higher power, the less accurate the approximation.

Only for sufficiently high power -- there is a difference between the limit and the approximation quality of a few terms, truncating at a fixed power. For QED, the basic arguments for the divergence of the perturbation series are given in

F.J. Dyson,
Divergence of perturbation theory in quantum electrodynamics,
Phys. Rev. 85 (1952), 613--632.

Things are explained in detail in the entry ''Summing divergent series'' of Chapter B4 of my theoretical physics FAQ at http://arnold-neumaier.at/physfaq/physics-faq.html