# What are Tom Banks's arguments against a QFT of quantum gravity? [closed]

https://arxiv.org/abs/1007.4001

Pages 1-9

What are his arguments here against a QFT of QG, can someone provide a simpler explanation of these? And to what specific theories of QG do these arguments apply to?

• It's currently unclear what exactly this question is asking without clicking on the link you provided. To make questions more accessible and guard against link rot, please include all relevant information, such as the explanation of notation or specific terminology used, in your question. Aug 17, 2018 at 10:56

Gravity is not renormalizable in four dimensions. The entire idea of formulating gravity as QFT in four dimensions rests on the fact that whether one can actually find an interacting fixed point in UV such that based on experiments, one can fine-tune finite (and small) number of parameters to define a consistent QFT. This possibility was put forward by Weinberg in the 1970s. This is now known as 'asymptotic safety'. If gravity is indeed asymptotically safe, then one expects a fixed point in the UV (small scales) around which the theory behaves like a conformal field theory (CFT). For a CFT, one expects that entropy will go as, $\sim S ∼ E^{(d-1)/d}$ [See https://arxiv.org/abs/0709.3555]. In GR, the entropy goes as, $\sim S ∼ E^{(d-2)/(d-3)}$. They are different scalings for any integer $d$. The author notes - "It, therefore, follows that the large energy asymptotics of the density of states in a theory of gravity in asymptotically flat spacetime is not that of any conformal field theory, and therefore, it is not a renormalizable quantum field theory". This paper mentions that this result was initially discussed by Banks and Aharony in https://arxiv.org/abs/hep-th/9812237. This is the summary (simplest) of Banks's argument.
• They are different scalings for any (integer) $d$, not only for $d=4$. Aug 16, 2018 at 2:21
• The point 1) is related to the fact that Einstein-Hilbert action contains second derivative of $g_{\mu\nu}$, to define $T_{\mu\nu}$, you need to drop a boundary term and that spoils gauge invariance and hence cannot define conserved stress-energy tensor. See for ex. P. A. M. Dirac, General Theory of Relativity, Ch 31 (the entire book is just 69 pages long, probably the shortest GR book by page count) I would rather keep mum on other points and let other people answer better than me since I don't understand well enough to explain. Aug 16, 2018 at 17:13