Why General Relativity is non-renormalizable quantum theory?

[tatanacevedo]

Hello

I have not understand this.

 

[radium]

Because the simplest term you can write down in gravity (the Einstein Hilbert action in curved space time) is the Ricci scalar which allows you to see that the scaling dimension of Newton's constant in 4d is -2. This means it is irrelevant under the RG flow and flow to strong coupling at high energies. If you tried to renormalize, you would see that you could no get rid of divergences with a finite number of counter terms.

However, when theories are non-renormalizable they can viewed as an effective theory at the appropriate energy scale, so even though Einstein gravity is nonrenormalizable, it works just fine at low energy scales. We just don't know what the full theory is.

Another example of a nonrenormalizable theory is the four fermi theory. However, it turns out this is just an effective theory which has a UV completion. So it is not really nonrenormalizable if you consider that it contains other things at high energies.

 

[RockyMarciano]

Note also that quantum field theories in general are not renormalizable either. They have to meet certain stringent conditions to be renormalizable that can be summarized as having a reduction from infinite dimensional group symmetries to their finite dimensional closed subgroups. Then similarly to what was commented in the previous post one obtains a low energy efective theory that works fine like the standard model of particle physics , so the parallelism with the case in GR is clear.