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1) The statement that general relativity (GR) is not renormalizable - is it a statement only about the quantization of GR or is it non-renormalizable also as a classical field theory?

2) More basically - is there a notion of renormalization group (RG) flow and renormalization vs. non-renormalization in classical field theory? I assume that the answer should be yes, since it seems to me that also a classical theory could have a range of validity (in energy / length scales) and that outside the range we would need additional terms or different values of constants in the Lagrangian. Also, we know that in statistical mechanics we speak about RG flow.

3) If the answer to question 2 is yes, then how is the running of constants formalized/calculated? Do we get different results of the running when using the classical vs the quantum theory of the field?

4) And if the answer to question 2 is no, then what is the fundamental difference between the quantum theory and the classical regarding to RG flow?

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  • $\begingroup$ Mass renormalization has to be performed in classical electromagnetism if we try to consider point particles coupled to the EM field. However, the results obtained are not satisfactory since there are still unphysical effects such as the existence of "runaway solutions" (solutions whose acceleration diverges at infinity). Anyways, this is an example of classical renormalization ;-) $\endgroup$ – yuggib Jan 8 '15 at 8:00
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Renormalization group is basically a tool which shows us how our theory responds to the scale transformations. Since a classical theory is completely spawned by the action, this action should be invariant under the RG flow. It means that the classical RG flow is completely determined by the scale dimensions of physical quantities which can be derived from the action.

So yes, there is an analogue of the usual RG flow. Even more: the classical RG flow is used in the quantum theory when we try to determine whether the theory is renormalizable or not. We usually approximate the quantum RG flow by the classical RG flow (which gives its leading behaviour) and therefore classify couplings by their classical dimension.

Couplings with negative dimension ($m^{-d},\,d>0$) correspond to non-renormalizable interactions. This very situation occurs when determining the dimension of the gravitational constant $G$ (try this, it is actually extremely easy).

Couplings with positive dimension ($m^{d},\,d>0$) correspond to super-renormalizable interactions. They tend to vanish on short distances.

And finally, couplings with zero dimension (marginal couplings) are the most interesting ones. Since the leading (classical) behaviour of the RG flow vanishes, we are required to determine (at least to the first order of perturbation theory) the behaviour of the quantum corrections.

But there is a but: despite the fact that the gravitational coupling blows at short distances in the classical theory, it is not a problem since we don't use perturbative methods. But in perturbative QFT the disaster becomes when $G$ approaches one (~ the Planck scale), which indicates the meltdown of the perturbative approach.

UPD: note that gravity is only perturbatively non-renormalizable. It can still be that there is a complete quantum version of GR. For example, the 2+1 gravity was recently found to be exactly solvable despite that it had been considered non-renormalizable for a long time.

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