An Effective Description of the Instability of Coherent States of Gravitons in String Theory

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

In this article, the authors consider that a dS space can be described by a coherent state of gravitons, as suggested in JCAP2017(06)028. Later, they use the Steepest Entropy Ascent Quantum Thermodynamics proposal to explain the evolution of the state. In this context, they study the system's stability and conclude that the dS space is unstable. In this way, the results of the article favor the swampland conjecture.

 

The results are interesting if specific hypotheses are accepted, and they should be publishable. However, the authors should make the following changes:

1. The authors should elaborate further on the point that a coherent state of gravitons can be considered as a representation of dS space. This assumption should be more thoroughly justified.

 

2)On page 4, line 157, the authors use a non-standard definition of von Neumann entropy. They should explain why this is permissible.

 

3)In equation (7), there is a parameter \beta that is identified with the inverse temperature. However, since it does not appear in equations (4), (5), and (6). The authors should explain its origin in (7). For example, is it related to \tau_D in some way?

 

4)On page 11, there is an inconsistency between lines 377 and 389, and there is a notational error: \tau_D = t_D.

After implementing these suggested revisions, I am confident that the article would be suitable for publication in UNIVERSE.

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

This paper considers the proposal that de Sitter space can be realized in superstring theory as a coherent state of gravitons, and tries to show that such a state is unstable.  This is an interesting proposal, but there are a number of issues, both conceptual and technical, that the authors should address before this article can be published.

The main conceptual issue which deserves some discussion is why a thermodynamic approach like SEAQT is appropriate for the universe as a whole.  The examples provided where SEAQT has been applied successfully involve interactions of a quantum system with its environment, but it's not clear what this should mean when the quantum system under consideration is the entire universe.  A brief explanation of this point would be helpful.

In section 2 there seems to be a discrepancy between the definition (4) of the dissipation operator and the calculation (7).  Specifically, they seem to differ by a factor of $-1/\tau_D$.  In equation (11), the right-hand side should presumably involve the projector $\hat{B}$ onto the support of the density matrix.  In (16), what choice of polarization tensor $\epsilon_{\mu\nu}$ is meant to correspond to de Sitter space?  What are the values of $k$ and $\tilde{k}$?  Presumably to get de Sitter one actually needs a superposition of such states with different values of $k$ and perhaps different ($k$-dependent) polarization tensors, no?  This needs at least some comment.

In section 3 there seems to be a sign error, first appearing in (22), which propagates throughout the calculations and risks invalidating some of the conclusions.  In (22), the coefficients $-\delta_i \log\delta_i$ in the second term are correct, but the coefficient in the first term should be $-(1-\epsilon) \log(1-\epsilon)$ which is approximately $+\epsilon$.  In other words, the sign of the first term seems to be wrong.  This sign error carries forward in equations (23)-(27) and as a flip in the overall sign in (29)-(32).  It also seems to indicate that (33) is not a correct analysis.  These details must be sorted out before the article is published.

In section 3.2 the solution (35), (36) is somewhat confusing.  What are the bounds in the integral in (36), and how does that make (35) a solution?  Also, how are the constants $k_i$ determined?

The Appendix also has some problems that must be addressed.  The algebra (A40) is not correct for superstring fermionic mode operators; they should be anticommutators with no factor of $m$ in the first line.  The coherent state (A41) is not the same as the one written earlier in (16).  Which is correct?  Somewhat more seriously, the Campbell identity used in (A45) is incorrect; the nested commutators should involve multiple instances of $\hat{C}+\hat{S}$ and only one instance of $\hat{C}$.  In (A46) and (A47) there appear to be an extra pair of oscillators written that shouldn't be there, and even removing them the result in (A47) doesn't look correct.  In (A48) the polarization tensors seem to suddenly have $n$ dependence, and it's not clear what the star means or why it has appeared going from (A45) to (A48).  The operators in (A49) and (A50) have not all been defined.  These several problems need to be fixed.

This work has at its core an interesting idea, and the results would be of some interest.  However the execution and attention to detail must be improved.

Author Response

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Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The authors have comprehensively addressed all recommended changes in this revised version. Nonetheless, Section 3.1 contains multiple errors that must be rectified for this manuscript to be suitable for publication.

a) Specifically, I contend that Equation (34) harbors a sign error as Equation (32) exhibits an overall negative sign.

b) Furthermore,  solution (35) is erroneous. Even if Equation (35) were corrected, the perturbation would still decay over time, contradicting the authors' growth assertion. This implies system stability. However, rectifying the sign error in Equation (34) would yield a solution demonstrating the perturbation's growth over time, thereby validating the presented results.

 

Provided the authors address this critical error, the article is well-suited for publication in Universe.

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

I would like to thank the authors for engaging with my recommendations.  I think they have improved the draft substantially.  There are, however, two areas where I feel that they have still not resolved my original issues.

First, in the equation defining their coherent state (now (18)), the authors have a traceless symmetric constant tensor epsilon, so their coherent state depends on epsilon.  On the other hand, they claim that this state should represent de Sitter space.  de Sitter space depends only on one dimensionful parameter (the cosmological constant), not on a choice of traceless symmetric tensor.  So which traceless symmetric tensor should be chosen if we want de Sitter space?  I suspect that the correct answer is that (18) does not strictly represent de Sitter space, but may (see my other objection below) represent some coherent state of gravitons.  Presumably to actually get de Sitter space, a complicated combination of graviton coherent states would be required.  For this reason I don't really think that the computations directly support any conclusions about the stability of de Sitter space, but only conclusions about the stability of this class of coherent states.

Secondly, there are still confusions and mistakes in the Appendix of the paper.  As emphasized in (A46), a massless graviton state in string theory is given by epsilon_{mu nu} S_0^{mu nu} |0>.  So why wouldn't a coherent state of gravitons be given by exp[ epsilon_{mu nu} S_0^{mu nu} ] |0>, i.e. why are the other S_n's being used at all?  This deserves some comment.  However even if this point is conceded, there are still some errors in the subsequent calculations.  In the BCH formula (A50), the sign in the third term on the right-hand side is wrong: the term with an iterated commutator of k copies of C+S and one copy of C should come with a coefficient (-1)^k / k!; for k = 0 and k = 1 this matches the written result (after reversing the order in the k = 1 commutator), but for k = 2 it gives the wrong sign.  The third line in (A51) should have a coefficient of -2 instead of +4.  (A52) should also have a coefficient of -2 on the right-hand side, but is otherwise correct, but it does not imply (A53).  In fact one can explicitly compute that [C,S] = -2 Tr(epsilon^2) zeta(0) I + 2 (epsilon^2)_{mu nu} sum_{n=0}^infinity (psi^mu_{-n-1/2} psi^nu_{n+1/2} + tilde{psi}^mu_{-n-1/2} tilde{psi}^nu_{n+1/2}).  In particular the summation is not of the form F(C), and does not commute with either C or S.  I don't believe that (A54) then holds, and so I don't think the conclusion (A55) is correct either.  Finally, I'm not really sure why (A55) is the correct test for whether |Phi_B> is a coherent state, since I would not have called C as the annihilation operator for gravitons.  Wouldn't that be just something like C_0?  Again, why are all the higher modes playing a role here?

The authors still need to clarify some of these issues.

Author Response

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Author Response File: Author Response.pdf

Round 3

Reviewer 1 Report

Comments and Suggestions for Authors

I am pleased to recommend this paper for publication in Universe.