- #1

Jim Kata

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- TL;DR Summary
- Matter-free gravity should be locally conformally invariant. Conformal field theories should have a beta function equal to zero. Which should imply a finite theory.

In reading Weinberg volume 1 I learned gravity is not renormalizable by Dyson power counting. This means that it has an infinite number of free parameters, and such theories lose their predictive power at energies of the common mass scale. This being said, T Hooft and Veltman showed miraculously at the one-loop level pure gravity is renormalizable. However, about ten years later, Goroff and Sagnotti showed that gravity at the two-loop level is not renormalizable. That is to say, there is no way to introduce counterterms to cancel the ultraviolet divergences at the two-loop level. Is this the end of the story?

Einstein's field equations for pure gravity imply that the Ricci tensor is zero. The Ricci tensor is constructed out of the Riemann curvature tensor. The decomposition of the Riemann curvature tensor into irreducible representations of the Poincare group takes the form (2,0)+(0,2)+(1,1)+(0,0). The only parts of the Riemann curvature tensor itself not constructed from the Ricci tensor are the (2,0) and (0,2) parts which represent the Weyl curvature tensor. The Weyl curvature tensor is locally conformally invariant.

All of that is just a long-winded way of saying that the field equations for pure gravity are locally conformally invariant. The significance of this is quoting Wikipedia:

"Conformal symmetry is associated with the vanishing of the beta function. This can occur naturally if a coupling constant is attracted, by running, toward a

or to quote physicist Jean Zinn-Justin

"The scale dependence of a quantum field theory (QFT) is characterized by the way its coupling parameters depend on the energy scale of a given physical process. This energy dependence is described by the renormalization group and is encoded in the beta functions of the theory.

For a QFT to be scale-invariant, its coupling parameters must be independent of the energy scale, and this is indicated by the vanishing of the beta-functions of the theory. Such theories are also known as fixed points of the corresponding renormalization group flow."Jean-Zinn-Justin

The vanishing of the beta function ,

To quote Weinberg in volume 3 of his QFT books about N = 4 SYM.

" The beta function vanishes. This is therefore a finite theory with no renormalizations at all."

In summary, why have physicists not been able to prove asymptotic safety for pure gravity? They can prove it conformally invariant. They act as though conformal invariance guarantees

Einstein's field equations for pure gravity imply that the Ricci tensor is zero. The Ricci tensor is constructed out of the Riemann curvature tensor. The decomposition of the Riemann curvature tensor into irreducible representations of the Poincare group takes the form (2,0)+(0,2)+(1,1)+(0,0). The only parts of the Riemann curvature tensor itself not constructed from the Ricci tensor are the (2,0) and (0,2) parts which represent the Weyl curvature tensor. The Weyl curvature tensor is locally conformally invariant.

All of that is just a long-winded way of saying that the field equations for pure gravity are locally conformally invariant. The significance of this is quoting Wikipedia:

"Conformal symmetry is associated with the vanishing of the beta function. This can occur naturally if a coupling constant is attracted, by running, toward a

*fixed point*at which*β*(*g*) = 0. "or to quote physicist Jean Zinn-Justin

"The scale dependence of a quantum field theory (QFT) is characterized by the way its coupling parameters depend on the energy scale of a given physical process. This energy dependence is described by the renormalization group and is encoded in the beta functions of the theory.

For a QFT to be scale-invariant, its coupling parameters must be independent of the energy scale, and this is indicated by the vanishing of the beta-functions of the theory. Such theories are also known as fixed points of the corresponding renormalization group flow."Jean-Zinn-Justin

The vanishing of the beta function ,

*β*(*g*) = 0 , is synonymous with a theory being UV finite.To quote Weinberg in volume 3 of his QFT books about N = 4 SYM.

" The beta function vanishes. This is therefore a finite theory with no renormalizations at all."

In summary, why have physicists not been able to prove asymptotic safety for pure gravity? They can prove it conformally invariant. They act as though conformal invariance guarantees

*β*(*g*) = 0. So what am I missing? What is the problem?