why can we renormalize some theories but others not?..in fact what wo7uld happen if we apply renormalization group method,or Feynmann,s renormalization program to quatnum gravity (non renormalizable theories). I also found that http://arXiv.org/abs/hep-th/9507067 they claim to have solved the problem of renormalization..is that true?.
Any theory has parameters, numbers that define the theory. In quantum field theories, when you set up the equations, some of these parameters take on infinite values. Renormalization is a legitimate technique for modifying the theory temporarily so that the parameters stay finite ("regularization"), and then at the end of the development removing the modifications in such a way that the infinities don't appear in the physics ("renormalization"). It is very important that you only have to do this a finite number of times, for a finite number of parameters. Theories for which this is true are called renormalizable. Examples are quantum electrodynamics and the standard model. Theories for which an infinite number of parameters would have to be renormalized are called unrenormalizable, and they can't be developed with present methods as finite theories. Gravity turns out to be unrenormalizable. Every time you try to renormalize one parameter, another crops up.
and could the divergences in a non-renormalizable theory be removed by using Renormalization group method and its equations?....another question for any lagrangian where i could find information about renormalization group method.
As far as I understand it, there are two ways of looking at non-renormalizable theories. The first one is what Self-Adjoint explained. The other one, using renormalisation group stuff, is different. In that view, non-renormalizable terms are the small remnants from another theory at high energies. In fact, the coefficients of non-renormalizable terms, when going from high to low energies, flow to 0, so they are called, in this picture, irrelevant terms. People say that, because gravity is non-renormalizable, it is irrelevant, and indeed, the coupling, at low energies, of gravity is extremely small. The point is that you should consider that relevant terms (renormalizable ones) are essentially "uncoupled" from the large energy scale at which new physics comes in ; that's why, after regularization, we can take the limit to infinite energies and still have a finite answer. In non-renormalizable theories, results DO depend on the energy cutoff you introduce. That's normal, because they represent the tiny remnants at low energies of important interactions at high energies when new theories come in, so we cannot arbitrary change that scale without an impact on the low energy behaviour. So, you can say that gravity is a catastrophy, or you can say that it is a small remnant of another theory at high energies. cheers, Patrick.
And just to add to what Patrick said, renormalization group methods are not instead of renormalization, they are in addition to it. For example RG methods don't renormalize QCD but they do show that confinement follows from renormalization.
There is something about non-renormalizable theories what bothers me. In fact, I'm not sure whether, if you restrict yourself to a given order, say, second order perturbation, can you do loop diagrams, or are you screwed any way ? I mean, I don't know if you get already an infinitude of counterterms at a given order, or whether at each order, you have to add more, but a finite number each time, of counter terms. cheers, Patrick.
I think you only need to add more terms at each order. From the one theory that I'm aware of ('naive' IVB theory of the weak interaction), the divergences get worse order by order (ie they diverge as x^{kn}, where n is the order is k is constant).