Quantum Gravity Effective Action Provides Entropy of the Universe

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

I was invited to review the manuscript "Quantum Gravity Effective Action Provides Entropy of The Universe". Building on previous work on renormalizable quantum gravity [15-19] and the BRST conformal invariance [27-33], the author shows that the action of the theory is equal to the entropy of the Universe. More precisely, using a simplified model that allows analytical calculations, the author uses the action (1) to compute the entropy of the Universe, thus obtaining eq. (6). By comparing the result with the entropy density of the Universe (as given in ref. [1]), the author finds what should be the parameters of the quantum gravity under study if eq. (1) holds. 

The calculations seem to be correct, but I am not convinced that this is a strong evidence that quantum gravity provides the entropy of the universe. It is not a surprise that one can set the free parameters of a theory in order to make two quantities equal. I am also concerned about the citations in the article. 17 out of the 42 references are papers written by the author of the article under review. Aren't there other people working in this quantum gravity theory? If the answer is yes, this is not reflected by the list of references. If the answer is no, then I would be very concerned about the interest to the readers.

Therefore, the major points I would like the author to address are:

1- Expand the discussion about the matching between the action and the entropy of the universe. Explain in detail why the calculations constitute strong evidence that quantum gravity provides the entropy of the universe. 

2- Concerning the list of references, are there other people working on this quantum gravity theory? Are the proper references cited?  

I also would like the author to address the following minor points:

3 - All equations should be numbered;

4 - All variables and parameters should be defined as soon as they appear in the text. For instance, \Gamma_{QG} is not defined in (1), \mathcal{L}_M and R are not defined in the bottom of page 2, and so on.

5 - How is the numerical solution discussed in the end of Section 3 implemented? I think the  details should be explained in the article.

Overall, the calculations seem to be correct and the article is sufficiently well written. On the other hand, I believe that the article will be of limited interest and will produce low impact in the field. Nevertheless, I think that the community working on these types of quantum gravity theories might benefit from the work. Therefore, I recommend the paper for publication after the author addresses the points I have raised.    

 

Author Response

Thank you for your sincere report. I added a number of explanations throughout. Please check the following amendments.

 

The problem behind the entropy of the universe is that Einstein's theory of gravity cannot explain effectively the entropy of the universe without introducing an unknown scalar field as its source. So, I add a sentence to Introduction to point out this fact. One of the aims of this paper is to explain the entropy using the gravitational field that follows the principles of gravity and quantum field theory. Unfortunately, since it is not yet possible to solve strong coupling dynamics completely, I have introduced free parameters to describe it.

 

Answers to the points you made are as follows:

 

1 Add explanations that the effective action of quantum gravity gives entropy, in two paragraphs containing equations (1) and (2) with additional references and a footnote. The path integral over the gravitational field represents the sum over the states of spacetime, so it represents the entropy for the state of the universe. That is transferred to matter at the spacetime phase transition, yielding the entropy of the current universe. To explain this process more clearly, add the energy conservation equation (9) and sentences to explain it. Furthermore, add Footnote 7 in relation to this equation and a paragraph below equation (13).

 

2 Most of the quantum gravity approaches known in recent years aim to eliminate ghosts completely. To do this, many either break diffeomorphism invariance, sacrifice renormalizability, or deny the existence of the gravitational field itself, at the Planck scale. This has been a trend continued from the mid-1980s to the present, and unfortunately, the trend has not changed. I have turned my back on this and conducted research to show that diffeomorphism invariance and renormalizability are guiding principles even in the trans-Planckian world. Here, I am dealing with the problem of entropy of the universe as a good example derived from the principles.

 

3 All equations are numbered.

 

4 According to your suggestions, add explanations for various symbols.

 

5 Footnote 8 indicating the specific calculation method is added.

 

In addition, add a number of sentences and footnotes to various places. In particular, add a paragraph starting with “The ghost” in Introduction to explain about ghost modes and a paragraph starting with “Since” in Section 3 to explain conformal mode dominance. The first paragraph in Introduction is modified. Footnote 1 is added. In third-to-last paragraph in Introduction, some sentences and Footnote 2 are added. Minor corrections are made here and there.

 

Reviewer 2 Report

Comments and Suggestions for Authors

The renormalizible quadratic gravity is an interesting approach to formulate a quantum gravity theory, but the main problem is the classical action, which is of the fourth order in time derivatives. Hence the classical Cauchy problem requires the knowledge of the initial values of the 2nd and 3rd order time derivatives, so that the classical trajectories can be radically diferent from the standard classical trajectories. This means that one has to insure that in the relevant time interval the deviations are negligible, a task which is difficult to realize in general case, and also very unlikely to be the case.

Beside this general criticism of the approach used by the author, there are more specific points which have to be cleared. First, the author should give the details of how he obtains the effective action (2) from the classical action I plus the Rigert action. Second, the formula (6) is puzzling, since there is no classical entropy contribution, so that the author should clarify this. Then the author talks about the ghost modes, and it is not clear to what he is referring to. He should give a definition of the ghost modes.

As far as the remarks about the Hamiltonian in the conclussions are concerned, they are not correct, because the Hamiltonian in general relativity is defined with respect to a time variable, otherwise we would have a static universe. The Hamiltonian constraint is zero, but this does not imply that the Hamiltonian is zero. If the author thinks about the energy, it may not be conserved, if the Hamiltonian depends on the time variable. Hence he must change the remarks about the Hamiltonian of a gravitational system in the conclusions.

Author Response

Thank you for your sincere report. I would like to answer your questions and criticisms.

 

The inflationary solution is a stable solution, and it converges to this solution even if the initial conditions are changed significantly. Therefore, although the uncertainty caused by the initial conditions that you pointed out is of course a problem, I believe that it can be dealt with　positively. Regarding this, Footnote 6 is added.

 

The definition of effective action follows the usual renormalizable quantum field theory such as QCD. A description about the Weyl part of the effective action is added in the second paragraph with Footnote 5 from the bottom in Section 2.

 

To explain why the Weyl part does not contribute to the entropy, add the paragraph starting with “Since” in Section 3. Furthermore, added the energy conservation equation (9) to explain that almost no matter exists during the inflation period. At the same time, add explanations of the process by which matter is produced during the phase transition, in the paragraph containing (9), Footnote 7, and also a paragraph below equation (13).

 

More descriptions about ghost modes are added in the paragraphs starting with “However” and “The ghost” in the middle of Introduction. According to this revision, Appendix A is slightly modified.

 

Equation (1), including its explanations with additional references, is added as an equation showing that the Hamiltonian vanishes exactly when gravity is quantized. This equation will be broken if some operation breaking diffeomorphism invariance, such as introducing an ultraviolet cutoff, is carried out upon quantization, and such an operation is always required in non-renormalizable quantum gravity. Renormalizability means that we can take a continuum limit that makes the cutoff infinite properly.

 

The way of thinking about the energy and momentum conservation law is also different from the normal one. Here, we capture it as an RG invariant. In addition, changes in spacetime are expressed as changes within the system while the energy of the entire system remains zero. Equation (9) expresses this fact.

 

In addition, the first paragraph in Introduction is modified. Footnote 1 is added. In third-to-last paragraph in Introduction, some sentences and Footnote 2 are added. Footnote 8 is added in Section 3 to clarify the numerical calculation method, and minor corrections are made here and there. Minor corrections are made here and there.

 

Reviewer 3 Report

Comments and Suggestions for Authors

In this paper, the author considers a particular theory of gravity with higher derivative  corrections and, after some assumptions, calculates the entropy of the universe by semiclassically solving the equation of motion for the conformal mode of the metric tensor.

The main concern is about the ground on which this result is built, and it is necessary to support some strong statements with more references that shows that there is a general consensus of the scientific community around them (so not only self citations). In particular:

1) Lines 31-32: motivate better why , when gravity is quantized, the energy momentum tensor has to vanish. Is this statement supported by other authors? What are the predictions of other quantum theories of gravity, such as string theory?

2) Lines 76-80: motivate better why the theory proposed is renormalizable, why in this case an infinite tower higher derivative terms can be avoided, differently from examples from other theories of quantum gravity such as string theory or f(R) theories.

3) Lines 123-124: why is most of the entropy carried by the conformal mode?

4) Lines 144:147: describe more explicitly the process for which the quantum gravity energy is transferred to the matter fields (which has not been considered in the action so far). Is the author assuming an initial non-zero energy and entropy before the big bang? Can this statement be supported?

If the author cannot support these statements with more references which are accepted by the quantum gravity community, then it must be clearly pointed out that there is no general consensus about them. 

 

Author Response

Thank you for your sincere report. I would like to answer your concerns in turn as follows.

 

1 Although it is widely known that the energy-momentum tensor vanishes in a gravitational system, this can only be shown as an identity when gravity is quantized. As a formula to express this, add equation (1), and then add literature. However, the SD equation only works in renormalizable quantum field theory, so you would not see it in any other approach.

 

Related to your first and last comments, research on quantum gravity from the late 1980s to the present has mostly been developed based on the premise that gravity is not renormalizable. String theory, f(R) gravity, asymptotic safety gravity, etc. are all like that. Behind the denial of renormalizability is the ghost problem. Indeed, that remains a problem when employing the traditional graviton picture, as pointed out in Introduction and Appendix. What I have shown is that the appearance changes when we adopt a more realistic perturbation expansion method that is closer to an image of the Planck scale world. So, another theme of this paper is to help you understand that "ghosts are unphysical, but exist". The goal of this research is to demonstrate that the guiding principles of gravity and quantum field theory, diffeomorphism invariance and renormalizability, are keys to solving many problems related to gravity. Here is taken up entropy of the universe, as one of the issues related.

 

2 The reason why the higher derivative terms do not appear is because the perturbation expansion is performed with a dimensionless coupling constant. A sentence emphasizing that the coupling constant is dimensionless is added in the paragraph starting with “The problem” in Introduction.

 

3 The predominance of the conformal mode is due to the asymptotic background freedom, which indicates that when the coupling constant is small, a configuration in which the Weyl action disappears is selected. In the revised version, the part that you pointed out is rewritten and a new paragraph starting with “Since” in Section 3 is added.

 

4 To answer your concerns, add the energy conservation equation (9) and sentences to explain it. Furthermore, Footnote 7 is added in relation to this equation.

 

In addition, the first paragraph in Introduction is modified. Footnote 1 is added. In third-to-last paragraph in Introduction, some sentences and Footnote 2 are added. Footnote 8 is added in Section 3 to clarify the numerical calculation method. Minor corrections are made here and there.

 

Reviewer 4 Report

Comments and Suggestions for Authors

Please see attached report

Comments for author File: Comments.pdf

Comments on the Quality of English Language

Please see attached report.

Author Response

Thank you for your detailed report. I would like to answer your concerns.

 

In the revised version, rewrite the first paragraph of Introduction. To answer your concerns, add the Schwinger-Dyson (SD) equation (1), the energy conservation equation (9), and sentences and footnotes to explain them.

 

First, quantum gravity is defined by the path integral over the gravitational field. So, the partition function represents sum over spacetime states, and the effective action is given by the logarithm of it. The identity that the total energy-momentum tensor vanishes is expressed as an integral equation as shown in equation (1), where “total” means all fields coupled to gravity, including gravity itself. This is the SD equation applied for the gravitational field, and the reason why this equation holds is exactly because the integration by the gravitational field is performed. As a description of the SD equation, also add a sentence containing Footnote 4 at the end of the paragraph with equation (4).

 

This equation will be broken if some operation breaking diffeomorphism invariance, such as introducing an ultraviolet cutoff, is carried out upon quantization, and such an operation is always required in non-renormalizable quantum gravity. Renormalizability means that we can take a continuum limit that makes the cutoff infinite properly.

 

It is clear from the definition that the effective action represents the entropy of spacetime states. Since the Hamiltonian of the system disappears, the effective action gives only entropy. To understand this fact, it may be helpful to think of the statistical mechanics formula. As you pointed out, effective action in ordinary quantum field theory corresponds to free energy. However, as emphasized in Introduction and so on, the quantum gravity theory discussed here starts by discarding the particle picture propagating in a specific spacetime. This picture is not appropriate to describe spacetime that itself fluctuates quantum mechanically. Hence, since there are no ordinary particles, including gravitons, in quantum spacetime, the usual concept of temperature does not exist there. Regarding this, Footnote 1 is added.

 

During the inflation period, the de Sitter solution is given as an average value of spacetime fluctuations. A thermal state emerges after spacetime fluctuations are transferred to matter at the spacetime phase transition. The so-called temperature is a quantity defined when particles exist, but not a quantity defined by gravity. What should be noted here is that equation (2) holds true even without the concept of temperature.

 

The energy conservation equation (9) indicates that almost no matter exists during the inflation period, and also explain the process by which matter is produced at the phase transition.

 

If you want to get a statistical mechanical image of the quantum gravity, consider lattice quantum gravity using the dynamical triangulation method. The partition function is represented by the sum over random 4-volumes made up of 4-simplices, which is a four-dimensional version of the random surface approach to two-dimensional quantum gravity. Its statistical behavior appears to be in the same universal class as the quantum gravity with conformal invariance. Please see Footnote 2 with additional reference.

 

As for your comment about pure states, here I consider a state in which the Hamiltonian disappears, not a state in which entropy disappears. I am simply stating that if all modes are positive definite, entropy vanishes as a result.

 

More descriptions about ghost modes are added in the paragraphs starting with “However” and “The ghost” in the middle of Introduction. According to this revision, Appendix A is slightly modified.

 

The answer to the question about singularity is as follows. In quantum field theory, physical objects are given by solutions whose action is finite, such as an instanton or soliton solution. Solutions where the action diverges are excluded as unphysical. In fact, the Riemann tensor of the Schwarzschild solution has a value and diverges at the origin, whereas the Ricci tensor disappears. Therefore, the Einstein-Hilbert action composed of the Ricci scalar is zero, that is, finite. In other words, from a field-theoretical standpoint, the singularity cannot be claimed to be unphysical in Einstein's theory of gravity. Adding the square of the Riemann tensor to the action solves this problem. Of course, this statement does not mean that objects like black holes do not exist, but that the vicinity of their centers should be subject to changes due to quantum gravity.

 

Lastly, one of the conclusions of this paper is summarized in the statement "In any case, quantum gravity can generate sufficient entropy" written at the end of Section 3. In fact, some of the parameters are treated as free parameters, so there is no point in pursuing that the values match exactly. Here, I do not take into account the entropy of black holes, but I also do not consider that its amount exceeds others, including dark matter, and there is no proof that it does. As mentioned at the beginning of Introduction, in the framework of Einstein's theory of gravity, entropy generation cannot be achieved without introducing an unknown scalar field. This is one of the reasons why I say a mystery. One of the purposes of this paper is to derive the vast entropy from the known field ruled by the guiding principles of gravity and quantum field theory, diffeomorphism invariance and renormalizability.

 

Regarding what I said above, the revised version introduces equations (1) and (9) and adds related sentences in various places. In addition, more footnotes are added. Please confirm them.

 

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

The main problem with this paper is the definition of the entropy. The entropy in QFT is defined as the temperature derivative of the free energy. The free energy is the temperature times the logarithm of the partition function. Hence

S = k log Z + kT d(log Z)/dT.

The author used only the first term, so that his calculation is wrong.

Note that the partition function is given by the Euclidean path integral for a 3d space with periodic imaginary time, whose period is the inverse temperature, i.e.

Z(T) = \int Df(x) exp( - S[f(x), T]/hbar ).

In the saddle point approximation we get

Z(T) ~ exp(-S[f_0(x), T]/hbar) (det(S''(f_0(x), T)))^{-1/2}

where f_0(x) is a solution of the classical Eucledean action S equations of motion, so that

log Z ~ - S_0 (T)/hbar - Tr log (S''_0 (T)),

where S_0 (T) = S[f_0(x),T].

The author here takes the first term only, as the most dominant. He calls    S_0 (T) the "effective action", which is a wrong terminology. However, he still did not clarify why the action (3) is given by the action (6), and also he did not indicate the classical solution he used for the saddle point approximation.

Author Response

Quantum gravity is defined by the path integral over the gravitational field, and the partition function represents the summation of all possible spacetime configurations. Furthermore, since the Hamiltonian vanishes, it represents a pure sum of spacetime states, and its logarithm is nothing but a Boltzmann's statistical entropy.

 

The second term in the entropy expression shown in your report gives the expectation value of energy, and the vanishing of the Hamiltonian just implies that it disappears. However, the concept of thermal equilibrium does not exist exactly because there are no particle states in the world of quantum gravity that we are considering here, so I have not emphasized it previously.

 

The reason why I brought up the relationship between effective action and free energy in Footnote 1 was to answer questions from other reviewers. So, I apologize if the footnote has confused you. This research begins by discarding the ordinary particle picture, as mentioned in third paragraph from the end of Introduction. All particles, including gravitons, are born after the spacetime phase transition, and then a usual temperature can be defined. The quantum gravity entropy is then inherited by thermodynamic entropy while changing there. Accordingly, Footnote 1 and related parts have been slightly modified.

 

If you want to get a statistical mechanical image of the quantum gravity, consider lattice quantum gravity using the dynamical triangulation method. The partition function is represented by the sum over random 4-volumes made up of 4-simplices, which is a four-dimensional version of the random surface approach to two-dimensional quantum gravity. Its statistical behavior appears to be in the same universal class as the quantum gravity with conformal invariance. Please see Footnote 2 with additional reference.

 

In addition, add Appendix B, a number of sentences in Conclusion, and a reference [45] following correspondence with another reviewer.

 

Reviewer 3 Report

Comments and Suggestions for Authors I thank the author for the answer and the changes implemented in the paper.

I'm still confused by some of the claims of the author.

Regarding answer 1), most of the recent literature tried to avoid ghosts since they lead to problems with stability and unitarity. These ghost can in principle be avoided using appropriate boundary conditions, see

https://arxiv.org/pdf/1105.5632.pdf


On the other hand, the author appeals to a BRST conformal invariance to remove the ghosts from the spectrum. Since this is the ground on which this work is based on, and since these might not be concepts which are familiar to the rest of the community, it would be a good idea to add an appendix were a clear review on how the ghosts are eliminated is given, possibly with some details for the case of interest.



Regarding answer 4), I think it is still not very clear yet if at the beginning of the time evolution the universe was in a thermal state, which is necessary in order to define thermodynamics quantities such as the entropy. If this is the case, then what I would expect is that the time evolution of this thermal state is given by a path integral defined on a Schwinger-Keldysh contour in the complex plane, which is not what the author considered.
If the starting point is not a thermal state, then the universe can still evolve into a thermal state after some coarse graining if the theory is ergodic, then the author should describe how entropy is calculated from this procedure and prove the assumptions.

Let me also add that this is the expectation for a theory which respects unitarity and does not have ghosts, the author should also address how ghosts affect this calculation.

 

Author Response

Thank you for your honest report. I would like to answer to your questions as follows.

 

Answer to the first half:

As you say, this work is different from the consensus of many researchers. I have recognized that there are a lot of criticism for that. In this paper, since the existence of ghost modes is essential when discussing the entropy of the universe, I have dared to mention the ghost issue more deeply than usual.

 

Employing the weak-field approximation, i.e., the graviton picture is nothing more than reducing the gravitational system to a system in special relativity. It is an ordinary quantum-mechanical system, in which case it is not taken into account that the Hamiltonian vanishes, and states that appear there become objects of observation, thus ghost modes must be erasable. This method only applies to local particle worlds.

 

On the other hand, when considering the very spacetime state, ghost modes play an essential role. The existence of the Friedmann universe and black holes are due to the ghost mode that makes the Einstein-Hilbert action unbounded below. The ghost modes are necessary in the trans-Planckian world to remove singularities that Einstein gravity could not do, to vanish the total Hamiltonian while providing entropy, and for the inflationary solution to exist as well.

 

As in your report, it may be an idea to introduce boundaries and remove ghost modes by appropriately imposing boundary conditions, but here we are considering a universe in which there are no boundaries leading to the unknown outside world, which is also a condition for the SD equation (1) to hold. Regarding this, add Footnote 9 and a reference [45].

 

In addition, add a more specific explanation as Appendix B about the problem with the weak-field approximation and how it is solved by the BRST conformal invariance. Some sentences regarding this are also added in Conclusion.

 

 

Answer to the second half:

First of all, it should be pointed out that the traditional quantization method deals with gravitons as quanta, but this theory begins by discarding such a particle picture as inadequate for describing the trans-Planckian physics, as mentioned in third paragraph from the end of Introduction. A state in which particles are propagating will appear after the spacetime phase transition occurs. The energy conservation equation (9) shows that matter hardly exists during the inflation period, but is generated at the phase transition point.

 

Since there are no particles in the state where spacetime itself is quantized, there is no concept of temperature there, and thus such a state cannot be called a thermal equilibrium state, nor a non-equilibrium state. The state that the Hamiltonian vanishes is a state in which only entropy exists. The effective action in this case directly expresses a statistical entropy counting the number of states. Accordingly, Footnote 1 and related parts have been slightly modified.

 

In order to describe the process leading to the realization of a thermal equilibrium state while following matter production during the phase transition, it is necessary to solve the strong coupling dynamics. Although it is difficult to do that, it is natural to think that a thermal equilibrium state is realized after the phase transition, and that entropy is inherited into that state.

 

 

Regarding the last comment:

As mentioned in the first half, the ghost modes contribute to the dynamics of the universe. The existence of ghost modes in Einstein gravity is also responsible for the instability of the Friedmann universe, that is, the growth of fluctuations after the big bang resulting in the large-scale structure of the current universe. Conversely, I have demonstrated that spacetime fluctuations reduce in amplitude during the inflation period. The reduced fluctuations will give the initial value of the Friedmann universe. This stability is due to the fact that the fourth-derivative action is positive definite, even though it contains ghosts as sub-modes. In this way, these various ghost modes play a decisive role when considering the universe as a whole. We should pay more attention to this fact. Regarding to this, add a number of sentences in Conclusion.

 

Reviewer 4 Report

Comments and Suggestions for Authors

No comments for the authors:  Please see comments for the editors

Comments on the Quality of English Language

Not applicable

Author Response

Additionally, the following modifications have been made:

Add Appendix B according to the suggestions from another reviewer. In addition, slightly modify Introduction and add a number of sentences in Conclusion following correspondences with other reviewers.

 

Round 3

Reviewer 2 Report

Comments and Suggestions for Authors

The author has made an effort to improve the presentation of the manuscript, but he should correct the equation (2) or justify it as an approximation, since the correct relation between the entrophy S and the Eucledean effective action Gamma is given by

S = log Z + T d(log Z)/dT ,

where T is the temperature, and Z is the partition function. Note that 1/T is the radius of the compact imaginary time dimension.

Also the relationship (5) is not correct, since the exact effective action satisfies an integro-differential equation (see, for example, the book on path integrals by Kleinart) so that (5) is the one-loop approximation. The author should indicate this. In the zero-loop (classical) approximation, one obtains

log Z = - Gamma,

and therefore

S = - Gamma - T d(Gamma)/dT .

However, the author should explain why one can ignore the second term.

Also, it is not the Hamiltonian that vanishes in quantum gravity, but it is the Hamiltonian constraint, and these two are not the same, and vanishing of the Hamiltonian constraint does not imply vanishing of a Hamiltonian (note that there may be more than one Hamiltonian, which depends on the choice of the time variable).

Author Response

I would like to answer your criticism.

 

First of all, here the energy-momentum tensor is defined by the variation with respect to the gravitational field, and the SD equation (1) is derived using this fact. In other words, the disappearance of the Hamiltonian can be expressed as the equation of motion for the gravitational field. The SD equation is a general field-theory formula that holds true even involving quantum corrections. On the other hand, what is generally called the Hamiltonian constraint is actually discussed in classical theory. The two are the same in the sense that they are derived as a consequence of diffeomorphism invariance. However, the latter is not based on renormalizability, while it is essential in the SD equation, so they cannot be compared with.

 

The formula (5) for the partition function is a general expression that holds true for all orders, and the Wess-Zumino (WZ) action includes not only 1-loop but also those that appear at higher loops. The Riegert action (6) is a quantity that is mainly discussed at the 1-loop level, but the replacement of the coupling constant t with the running coupling constant \bar{t} done in the paper is exactly achieved by incorporating such higher order WZ actions. For the relationship between the WZ actions and running coupling constants at higher orders, please see reference [42]. In this paper, I have described the strong-coupling dynamics by modeling contributions from higher-order corrections, so I have not gone into the theoretical details. Regarding this, add a new Footnote 5 and also add sentences to Footnote 6.

 

Once again, in a world where spacetime itself is fluctuating greatly, there are no so-called particles, and thus there is no usual concept of temperature. The normal relationship between effective action and free energy is defined in the world of special relativity, in which the Hamiltonian eigenstate with a positive eigenvalue is considered. However, in the world where gravity fluctuates significantly, the eigenvalue becomes zero, as shown in (1). Footnote 1 helps us with understanding, but strictly speaking it does not give the correct answer. The inability to define temperature and the vanishing of the Hamiltonian are complementary. The partition function of quantum gravity is defined as purely counting the number of spacetime states, and the statistical entropy is given by its logarithm. Regarding this, add a sentence in Footnote 1.

 

As mentioned in Conclusion, if you use the weak-field (graviton) approximation to reduce the gravitational system to a system of special relativity, you can treat it as a system whose temperature can be defined as you are thinking. However, although this approach is effective for describing a local particle world that does not cover the entire universe, it is not proper for describing the trans-Planckian world where the entire spacetime is thought to fluctuate significantly. The starting point of this theory is to discard the particle picture and quantize spacetime itself. In that world, the concept of temperature no longer exists.

 

Please reconsider your report. I believe that reconciling quantum mechanics and gravity requires a deep understanding of what it means for the Hamiltonian to vanish.

 

Finally, add references in Appendix B.

 

Reviewer 3 Report

Comments and Suggestions for Authors

We thank the author for all the modifications implemented in the paper.

Author Response

Thank you for reading the manuscript. The correspondence with you has been very helpful.