Quantization of Gravity and Finite Temperature Effects

Round 1

Reviewer 1 Report

The paper is dedicated to a general study of the quantization of gravity and related physical problems.

First, the authors discuss a 4D-covariant derivation for the physical state condition, exploiting the gauge-independent of the scattering amplitude.

The vacuum energy is then analyzed working in a finite temperature setup. This in turns gives some relevant results to address the problem of cosmological constants, by exploiting the role of some effects at finite temperature.

Finally, the author considers the possibility of asymptotic freedom in finite temperature quantum electrodynamics, arguing that this remains a reasonable possibility when attention is posed on the finite parts after divergences subtraction.

The problems considered are of wide interest for physics and the author's arguments are interesting.

I recommend acceptance in this form.

Author Response

Thanks very much for your review!

Reviewer 2 Report

This paper continues and extends the author's previous analysis of `foliation-based quantization' (FBQ). Its validity, therefore, depends upon the validity of the entire FBQ program. However, here, I do not make any judgement on this, and instead judge the present work in the context of existing studies of the FBQ method, which is now well established in the quantum gravity literature. In this context, it is a valuable contribution to the existing works and, therefore, is in principle worthy of publication in Particles. 

In particular, the present study shows that the physical state condition (PSC) of the theory, which was imposed in a non-covariant way in previous works, may be imposed covariantly by requiring the gauge independence of scattering amplitudes in a toy gravity-matter system, consisting of a flat spacetime background with a Higgs type potential. The ground state of the system is then analysed to 2-loop, extending the 1-loop analysis given previously in ref. [9] of the text. These results indicate that finite-temperature effects yield important low-energy (IR) contributions to the renormalized vacuum energy (the effective cosmological constant) backing up previous qualitative claims made in [9]. Finally, the FBQ method is applied to QED in Sec. 4, again using a toy model in flat space, rather than an arbitrary Einstein-Maxwell system. Based on these results, the author argues that QED may exhibit asymptotic freedom, once quantum gravitational corrections to the gauge couplings are taken into account. 

These results build on previous work and open up interesting new avenues for future research. Nonetheless, several outstanding issues must be addressed before final publication. These are listed below.  

Important scientific issues: 

(1) The procedure for fixing the physical states of the theory, outlined in Sec. 2.1, relies critically on the symmetries of the spacetime. Specifically, the author claims that the first class constraints of the theory generate ``translations in the $r$-direction''. However, earlier in the text (below Eq. (1)), it is explicitly stated that the $r$-coordinate refers to the radial direction in an asymptotically flat spacetime. Note that it is {\it impossible} to perform translations along the `radial direction' of any coordinate system, since this direction is not fixed. Such transformations, in fact, typically represent enlargements of the spatial hypersurface. How does the author respond to this point? 

(2) The author's results show that, in the FBQ scheme, the renormalised mass of the toy Higgs field, which gives rise to the effective cosmological constant, is of the same order as the temperature. This is fine for the present day universe but, naively, implies that the value of the cosmological `constant' was much greater in the past when the temperature of the universe was higher. Is this the case in his model? If so, how does this implication fit with existing observational constraints on the time-dependence of dark energy?

(3) The contributions to the effective cosmological constant corresponding to Fig. 2(b) generate a graviton mass, but what is its order of magnitude value in this scheme? How does it compare with the renormalized mass of the Higgs field? Most importantly, is this estimate compatible with current experimental bounds on the graviton mass, obtained from cosmological and astrophysical observations?

Minor issues: 

(1) The author's use of the terms `small' and `large' gauge symmetries is somewhat non-standard and the precise meanings of these terms are not made entirely clear in the existing text, which simply states that ``The `small' gauge transformation is redundancy of the the degrees of freedom whereas the large represents part of the moduli (a collection of vacua) of the theory.' This is insufficient and a more detailed explanation of these terms should be given in the introduction. 

(2) Without expanding this section to an unmanageable length, it would also be helpful to include a brief overview or summary of the background field method (BFM). 

(3) The text contains a large number of acronyms, e.g. FBQ, BFM, OPT, ADM, CC, PSC, LGT, PMS, etc. While this is largely unavoidable, it would be helpful to add an appendix with a glossary of terms. 

(4) Some terms and notations are introduced without being formally defined, e.g. 1PI, $\overline{MS}$. For clarity, all terms should be well defined when they are first introduced in the text. 

(5) Although the text is mostly well written and clear, a few of the sentences don't make sense or are ambiguous, e.g. ``(For convenience, we are viewing, at the moment, the large symmetry as the residual symmetry of the small and large symmetries combined.)''.

Author Response

“These results build on previous work and open up interesting new avenues for future research. Nonetheless, several outstanding issues must be addressed before final publication. These are listed below.”  

“Important scientific issues:”

“(1) The procedure for fixing the physical states of the theory, outlined in Sec. 2.1, relies critically on the symmetries of the spacetime. Specifically, the author claims that the first class constraints of the theory generate ``translations in the $r$-direction''. However, earlier in the text (below Eq. (1)), it is explicitly stated that the $r$-coordinate refers to the radial direction in an asymptotically flat spacetime. Note that it is {\it impossible} to perform translations along the `radial direction' of any coordinate system, since this direction is not fixed. Such transformations, in fact, typically represent enlargements of the spatial hypersurface. How does the author respond to this point?”

Although FBQ is not the same as the so-called radial quantization, it has a certain similarity. It is true that the translation will cause the hypersurface to get enlarged. However, this must be immaterial since the surface is located in the asymptotic region anyway.  There are indeed infinitely many directions depending on the angular coordinates. One may, however, just fix those angular coordinates. The reason for considering the radial (as opposed to time) direction is to remove all of the gauge redundancy in such a way that one still has dynamics. In other words, the hypersurface still contains the time-direction and thus dynamic. (I would like to invite the referee to take a look at Fig. 1 in [40] (https://arxiv.org/pdf/1807.11595.pdf).)

“(2) The author's results show that, in the FBQ scheme, the renormalized mass of the toy Higgs field, which gives rise to the effective cosmological constant, is of the same order as the temperature. This is fine for the present day universe but, naively, implies that the value of the cosmological `constant' was much greater in the past when the temperature of the universe was higher. Is this the case in his model? If so, how does this implication fit with existing observational constraints on the time-dependence of dark energy?”

Good point! Although one must repeat the `usual’ hydrodynamic-type analysis in the new setup of the manuscript, it is indeed expected that the CC was much bigger in the past. (As a matter of fact, a decaying vacuum energy was demonstrated in a zero-T setup in section 4.2 of [39] (https://arxiv.org/pdf/1606.08384.pdf). So perhaps such time-dependence may well be pretty generic, regardless of the presence of temperature.) This happens to be one of the issues that I am vigorously pursuing presently. Until completing such an endeavor, I am afraid that I cannot give any concrete statement. Nevertheless I certainly believe that there are issues to explore, including the issues raised by the Referee.

“(3) The contributions to the effective cosmological constant corresponding to Fig. 2(b) generate a graviton mass, but what is its order of magnitude value in this scheme? How does it compare with the renormalized mass of the Higgs field? Most importantly, is this estimate compatible with current experimental bounds on the graviton mass, obtained from cosmological and astrophysical observations?”

This is definitely an interesting issue to explore.  As a matter of fact, the first paper of mine that has mention of the possibility of graviton mass is [39] (https://arxiv.org/pdf/1606.08384.pdf),  (if I remember correctly). As stated in footnote 6 of [39], there have been some debates. In ref [40] therein (https://arxiv.org/pdf/gr-qc/0610054.pdf), this issue was examined, and the Compton length comparison seems to satisfy the bound (see the discussion around eq. (2.7)).

A bigger challenge should be the prediction of the present temperature-dependent CC on the graviton mass interpretation: for instance, if the interpretation of the CC terms as the graviton mass is correct, the temperature-dependent mass will imply that the graviton mass was much bigger in the early Universe.  Again, I believe that the issue deserves a work dedicated to itself. At this point I can just state that the present proposal will lead to cosmology substantially different from the standard one.   

"Minor issues: 

(1) The author's use of the terms `small' and `large' gauge symmetries is somewhat non-standard and the precise meanings of these terms are not made entirely clear in the existing text, which simply states that ``The `small' gauge transformation is redundancy of the the degrees of freedom whereas the large represents part of the moduli (a collection of vacua) of the theory.' This is insufficient and a more detailed explanation of these terms should be given in the introduction. "

These concepts are known, e.g., in the Seiberg-Witten theory. I have added [47,48] and referred to in p 2 and 5. For instance, it is stated in [47] that “In gauge theory it is important to make a distinction between small gauge transformations g which are those approaching the identity at spatial infinity and large gauge transformations which do not approach the identity at spatial infinity.” and in [48] that “We distinguish ”small” from ”large” gauge transformations because they have very different physical meaning. A ”small” gauge transformation connects two equivalent field configurations, while a ”large” gauge transformation corresponds to a global symmetry, as the translational symmetry, and therefore two field configurations related by a ”large” gauge transformation are physically inequivalent.”

"(2) Without expanding this section to an unmanageable length, it would also be helpful to include a brief overview or summary of the background field method (BFM). "

I believe what’s meant here is perhaps refined BFM. The refined background is based on eq. (16.1.17) of Weinberg’s QFT vol 2. The central idea is what’s given in section 3.1. (That’s really all there is to it.) At the technical level, more pedagogical details of the steps were given [37,38].

"(3) The text contains a large number of acronyms, e.g. FBQ, BFM, OPT, ADM, CC, PSC, LGT, PMS, etc. While this is largely unavoidable, it would be helpful to add an appendix with a glossary of terms."

Appendix A has now been added.

"(4) Some terms and notations are introduced without being formally defined, e.g. 1PI, $\overline{MS}$. For clarity, all terms should be well defined when they are first introduced in the text. "

Done

"(5) Although the text is mostly well written and clear, a few of the sentences don't make sense or are ambiguous, e.g. ``(For convenience, we are viewing, at the moment, the large symmetry as the residual symmetry of the small and large symmetries combined.).''

The sentence was put in analogy with string theory.  If I may use a string theory analogy again, it is stated in p 149 of Polchinski’s string theory vol. 1, “…parameters in the metric that cannot be removed by the symmetries, and symmetries that are not fixed by the choice of metric. The unfixed symmetries are known as the conformal Killing group (CKG).” To use an analogy with the string theory case, it is necessary to treat the small and large gauge symmetry as one umbrella of total gauge symmetry, hence the sentence.

Thanks very much for your suggestions!

Reviewer 3 Report

In this article, the author presents some additional advances to previously developed works (sequels), regarding to the renormalizability of gravity for certain physical states via foliation-based quantization, which leads to a contribution in the attempt to obtain a quantum theory of gravity. First, an alternative method (4D-Covariant) of derivation physical state condition is presented. Second, the author shows that there exists an optimal perturbation theory (OPT), which allows him to avoid (allows addressing a two-loop resolution) the problem of the cosmological constant (CC) when it is implemented. Finally, the author presents an analysis of an Einstein-Scalar system in the finite-temperature framework of quantized gravity, in order to analyze the possibility of asymptotic degrees of freedom in the theory of quantum electrodynamics (QED).

With regard to work in general, the paper explains in detail the background, the motivations, and the results of the analysis. The assumptions and the methodology are discussed in detail and allow for results to be reproduced. The bibliography captures the bulk of the current work. English and spelling are adequate.

For the reasons above, I suggest the paper for publication in the present form.

I only have a minor problem: In conclusion, what does he mean by "organic nature of spacetime"?. If you can explain it better and / or reference it or or better address the comment (I don't know if it refers to the "granular" nature of spacetime), fine, if not, then it seems better to remove it, 

Author Response

I have revised the part in question. Thanks very much for the review.