Talk:Asymptotic Safety in quantum gravity
Overall, this is a very clear, well-written article, written by experts in the field. I have only a few suggestions for correction/clarification:
1. The sentence
- Due to the requirement of a UV fixed point the bare action is a prediction of the Asymptotic Safety program rather than an input.
will be clear to experts, but perhaps not to others. It's not quit "the bare action" that is the prediction, it's the detailed structure of the bare action. Perhaps this should be something like
- The requirement of a UV fixed point restricts the form of the bare action and the values of the bare coupling constants, which become predictions of the Asymptotic Safety program rather than inputs.
Also, the link from "bare action" is not helpful -- it leads to a general page on actions, not really relevant here.
2. The sentence
- The idea of a UV completion by means of a nontrivial fixed point was proposed even before in scalar field theory
is not quite standard English -- "before" should be "earlier".
3. The sentence
- The cutoff scale dependence of this functional is governed by a functional flow equation which contrary to earlier attempts can easily be applied in the presence of local gauge symmetries also.
is also not quite standard English -- "which contrary to earlier attempts can" should be "which, in constrast to earlier attempts, can"
4. Figure 1 shows one trajectory that leaves the UV critical surface. It might help if it had an arrow, and a more explicit mention in the caption.
5. In the section "Taking the UV limit," the term "scale" may be confusing to nonexperts -- it is not completely obvious whether it means "momentum scale" or "distance scale". (It means momentum scale, of course, but to nonexperts the term scale usually connotes distance.) This could be fixed by simply saying "momentum scale" the first time the term is used.
6. In the sentence
- Its critical exponents agree with the canonical mass dimensions of the corresponding operators which usually amounts to the trivial fixed point values g∗_α=0 for all α.
surely "all α" isn't quite right -- that would imply a vanishing action.
7. In the section Dynamical vs. background fields, the link to "fluctuation" goes to an essentially irrelevant page,
8. It might be worth expanding the section "The microscopic structure of spacetime" at least slightly. In particular, in addition to the arguments related to the spectral dimension, one can argue that at a UV fixed point fields necessarily acquire anomalous dimensions such that their total dimensions are those of a field theory in two spacetime dimensions (see, e.g., section 3 of Niedermaier, Class.Quant.Grav. 24 (2007) R171). This would obviously have to be phrased carefully, since it uses the word "dimension" in two very different senses in the same sentence.
I find the paper particularly well written and clear, also its length fits the scopes of Scholarpedia.
However there is a difficulty which, to my knowledge, has been overlooked in the literature on Asymptotic Safety and is not mentioned in this work, which has a relevant role in the general formal approach even if possibly much less relevant in the Asymptotic Safety analysis.
Since the paper should give a sound, however simplified, description of the method, the difficulty mentioned above must be discussed in the sections “The gravitational effective average action” and in the following one, “Exact functional renormalization group equation”.
Let us now come to the nature of the mentioned difficulty. As clearly stated in the section “Dynamical vs. background fields” any gauge (diff.) invariant theory in the presence of a background field is characterized by two independent invariance properties that correspond to “true” and “background” gauge transformations. The first invariance is accounted for by the Slavnov-Taylor identities, while the second one is controlled by the background Ward identity. In the present construction this second identity is guaranteed by the systematic recourse to covariant quantities while the first one is not, however both must be satisfied in order the theory to make sense. See e.g.
- L.F. Abbott, Marcus T. Grisaru, Robert K. Schaefer , The Background Field Method and the S Matrix , Nucl.Phys. B229 (1983) 372
and more pedagogical;
- C. Becchi and R. Collina, Further comments on the background field method and gauge invariant effective actions, Nucl.Phys. B562 (1999) 412-430.
Of course there is a problem in implementing the Slavnov-Taylor identities in the presence of , more or less, sharp cut-offs, indeed this requires the introduction of true non-invariant terms fine tuning the identities, the effect of this introduction on the renormalization group equations could even be irrelevant, at least in some approximation. I consider very difficult an exhaustive analysis of this impact and hence, in my opinion, in the present paper it should be sufficient, however necessary, mentioning that this effect should exist, in much the same way as that of truncated terms
Response to reviewer A
The authors would like to thank reviewer A for very valuable suggestions. Points 1 - 5 and point 7 have now been adopted as they stand. Concerning point 6, clearly not all couplings vanish at the Gaussian fixed point, but only the essential ones. As for point 8, the section "The microscopic structure of spacetime" has been expanded by a few comments on the effective/total spacetime dimension, including a remark on the resemblance to field theory in 2 dimensions.
Response to reviewer B
The authors would like to thank reviewer B for important comments on the different kinds of Ward identities. Indeed the background Ward identity is guaranteed by construction. However, as already stated, the true gauge transformations cannot be accounted for by the usual BRST Ward identities due to the inclusion of a cutoff term in the action. This cutoff term gives rise to a modification and leads to what we now call "generalized BRST Ward identities" in the revised article. Only for vanishing cutoff scale, k = 0, the standard identities are recovered. The generalized BRST Ward identities can be found in Reuter (1998) of the article's reference list. We agree that a serious check of these identities for the truncations considered so far is still lacking in the literature (for calculational reasons).
In order to implement these comments we slightly modified the article in three subsections (without going too much into technical details): The subsection "Dynamical vs. background fields" has been supplemented by some additional notes, and two further sentences have been added in "Exact functional renormalization group equation" and "Truncations of the theory space".