Non renormalizability non predictiveness

[Jim Kata]

Ok, I know Goroff and Sagnotti proved that the two loop contribution to pure gravity is divergent, but I guess I don't understand non - renormalizability. Looking, at the BPHZ method it seems that you could subtract out the divergences at each level. Although, you need an infinite number of counter terms for the entire path integral. It seems to me that at each loop level it would only take a finite number of counter terms to cancel the divergences. What am I missing here?

 

[genneth]

Nothing. But if you need to subtract out an infinite number of independent terms than your theory has an infinite number of degrees of freedom, i.e. you would need to make an infinite number of observations to extract any useful prediction. This makes the theory less useful...

 

[mhill]

but although you need an infinite number of parameters could not try an 'ansatz' for the infinite quantities ? or if the problem of divergences is that integrals are divergent could not approximate these divergent integrals by divergent series and then use resummation methods to obtain finite values for these series ?? or at least assume that all the particles have a 'radius' proportional to their Compton wavelent $$\lambda = \hbar (mc)^{-1}$$ .

Also i am not sure but i think that at least for low energies we have only a finite number of divergences, in case you increase the energy scale then the infinite terms become relevant and spoil your prediction.

 

[nrqed]

Ok, I know Goroff and Sagnotti proved that the two loop contribution to pure gravity is divergent, but I guess I don't understand non - renormalizability. Looking, at the BPHZ method it seems that you could subtract out the divergences at each level. Although, you need an infinite number of counter terms for the entire path integral. It seems to me that at each loop level it would only take a finite number of counter terms to cancel the divergences. What am I missing here?

But the problem is that every time you bring in a new interaction that is required to patch up the divergences, you need to fix the finite piece as well! Sure, you can fix the divergent part of the bare coeffcient to insure that you result is finite but you are left with an arbitrary finite piece. How to fix this finite piece? By performing a new experiment. So you end up losing predictive power. If you can't make any prediction without requiring extra measurements to fix the constants of your theory, the theory is useless.

The way out is of course to treat the theory as an effective field theory.

 

[BenTheMan]

The way out is of course to treat the theory as an effective field theory.

This much should be apparent just from the fact that Newton's constant is dimensionful. Just as we don't expect Fermi Theory to hold in the infinite energy (zero length) limit, we shouldn't expect gravity to hold in that regime either.