Can we attain a disire approxomation for asymptotic perturbative QTF Theo?

[ndung200790]

Please teach me this(please forgive me if stupid question):
Can we attain an arbitrary small error(e.g calculating S-matrix) for an asymptotic perturbative QTF Theory.Because if we can not,I think that is problematic.But with asymptotic perturbative theory,in general the series does not converge,so the error may not tend to zero.
Thank you very much in advanced.

 

[vanhees71]

It depends on the theory. For QED up to now we have a phantatic "convergence", i.e., doing high-order perturbative calculations leadds to a very good accuracy for observed quantities (in comparison to experiment). For asymptotic series, one should stop at the order of the calculation where an n-th order correction (in whatever approximation scheme you work; in QFT usually you work in the number of loops, i.e., orders of [itex]\hbar[/tex]) becomes larger than the correction at (n-1)-th order.

If it comes to QCD, which is much more complicated due to the self interaction of the gluons, we usually observe a very bad "convergence" of the series. E.g., if you calculate the pressure of a quark-gluon plasma, it alternates like crazy between the different orders. Thus perturbation theory for this (in principle observable quantity!) becomes ill defined right away. The solution for this problem are clever resummation schemes. In this case one has to use the 2PI formalism of quantum field theory in combination with the hard-thermal loop (HTL) approximation which leads to the notion of massive quasi particles making up the pressure of the QGP. This method is a pretty good approximation for temperatures above two to three times the critical temperature, as can be shown by comparing to lattice-QCD calculations.