When developing quantum electrodynamics in the 1930s, Max Born, Werner Heisenberg, Pascual Jordan, and Paul Dirac discovered that in perturbative calculations many integrals were divergent.

The divergences appear in calculations involving Feynman diagrams with closed loops of virtual particles in them.

Renormalization problem is to extract from the divergent integrals the convergent part that has the physical meaning and there are examples when it works!

Some mathematically rigorous "renormalizations" of divergent integrals is developed in the "principal value of divergent integral" and in the theory of distributions..

You may read more in the article Renormalization-Wikie.

Hello Gro,

Would the integrals be considered "improper"?

http://en.wikipedia.org/wiki/Improper_integral

Ok, I think I get it now (yes they are improper).

Do you know what the specific problems are in physics which create these divergent integrals?

Maybe helpful to look on Dirk Kreimer's papers on renormalization, related to Hopf algebras and the Feynman graph operads.

Dear J. Inman, I hate to be so immodest, but you will find a short introduction to the core of the problem in http://arxiv.org/pdf/0810.2173.pdf . Especially understand the first page! The problem of renormalization has been discussed in a clear mathematical setting by Epstein and Glaser in 1973; however, their works are tough to read. Kreimer's papers mentioned above are based on the findings of Epstein and Glaser as well.

One would compare the situation as in Lagrange and Hamilton formalisms, the latter may be considered only as a formal development of the Lagrange one. The Hopf algebra and operadic approach in Kreimer's approach is in similar situation - as an effective mathematical reformulation of renormalization process, which shares light on the mathematical tools needed for further developments and understanding in quantum, the deep space-time, domains.

Renormalizable theories mean that the infinities arising from loop contributions can be cancelled by adding a finite number of counter-terms to the (bare) lagrangian of the theory. Nonrenormalizable means that infinite terms should be added to the theory to cancel the infinities. This characteristic is related to the dimensionality of the terms in the lagrangian of the theory. For instance, gravity and extra dimensional theories are non renormalizable, whereas four dimensional Yang-Mills theories are typically renormalizable.

Renormalization is in general is one of the paradoxical aspects of quantum mechanics. Quantum computations give rise to infinities which however can be cancelled by adding new contributions. Magically, at the end of the day, you obtain the correct results with astonishing agreement with experiments. This is commonly interpreted as an evidence of a cut-off scale in the validity of our theories. More recently, integrable theories, such as twistors and holography, allow to calculate quantum observable in particular cases in terms of geometrical considerations, without the necessity of cancellations of infinities.

@Dolce: It seems to lead to paradoxical aspects if one is going to take the representation theory (especially the efforts for an analytic representation of perturbation theory) too serious... :-)

It is indeed not clear that the appearance of infinities is a fundamental flaw in QED. It rather can be seen as the consequence of the fact that the infinite series of terms (Feynman digrams) is not apparently convergent. (Proof: It is not invariant to using -e instead of e for the electron charge) The arrangement of terms selected by Feynman creates infinities. It is like the series 0 = 1-1+1-1+1-1 + ... that does not satisfy the Cauchy criterium for convergent series. A rearrangement to 1+1+1+1+1+1 ...-1-1-1-1 will deliver infinity - infinity which you struggle to deal with.

This could be only overcome by using non-perturbative approaches.

The renormalization problems have nothing to do with quantum theory. The best way to understand them is to consider the same problem in condensed matter theory. Imagine you have an atomic theory, but no experiment which allows to measure the size of the atoms. So you have observables for large distances, and continuous large distance approximations from the atomic theory. Given this background, you can compute the fundamental parameters of the corresponding atomic model. But this computation depends on the unknown value of the critical atomic distance.

Whatever the dependence of the hypothetical fundamental parameters on the critical distance, it would not be strange at all (at least for a proponent of an atomic theory) if some of these parameters become infinite in the limit of zero for the critical distance.

So these infinities are problematic only for those who think that existing theory has to be true down to zero distance. A strange assumption, but, strangely, quite popular. The reason is IMHO spacetime metaphysics. The metric of GR is identified with "spacetime" and relativistic symmetry assumed to be fundamental. What would happen with this interpretation if there is a different theory below a critical distance? A new, different theory has, probably, a different symmetry group. And the metric will be some continuous field which approximates something different. but hardly remains "spacetime".