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I am reading a famous paper by S.Hawking - "Quantum gravity and path integrals" https://doi.org/10.1103/PhysRevD.18.1747.

On the third page left column there is a statement, after the derivation of the relation between area and entropy $S = A /4$ :

The reason the classical solutions in gravity have intrinsic entropy whereas those in Yang-Mills or scalar field theories do not, is closely connected to the facts that the gravitational action is not scale invariant and that the gravitational field can have different topologies.

How does the existence of the intrinsic entropy follow form the non-scale invariance of the action and complicated topology?

Does he mean the scale invariance of the classical action? On the quantum level there is dimensional transmutation and the corresponding generation of the energy scale $\Lambda_{\text{QCD}}$.

What is more, Yang-Mills theories have instantons and the field configurations can have various topological charges. How does it conform with the above claim?

On the 5th page, when considering the number of eigenvalues of the operator $A$ in the action $$ I_2 = \frac{1}{2} \int d^{4}x \sqrt{g_0}\ \phi A \phi $$ he defines $N(\lambda)$ - number of eigenvalues less than $\lambda$, and the expansion $\lambda \rightarrow \infty$: $$ N(\lambda) = \sum_{n = 0} P_n \lambda^{2-n} $$ First two terms are divergent in the $\lambda \rightarrow \infty$ and one would like get rid of them. And then he considers the term $P_2$ :

In Yang-Mills theory or quantum electrodynamics (QED) the quantity corresponding to $P_2$ is proportional to the action of the field. This means that one can absorb the $\mu$ dependence into an effective coupling constant $g(\kappa)$ which depends on the scale $\kappa$ under consideration. If $P_2$, is positive, $g(\kappa)$ tends to zero logarithmically for short-length scales or high energies. This is known as asymptotic freedom.

In gravity, on the other hand, the dependence cannot be absorbed because $P_2$, is quadratic in the curvature whereas the usual action is linear.

How can one see that, the $P_2$ coefficient proportional to the action leads to possibility of putting the dependence on the scale $\mu$ into the running coupling constant?

Does the explicit dependence of the path integral on the scale $\mu$ imply non-renormalizibility of the action?

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