Evidence of Time Evolution in Quantum Gravity
Round 1
Reviewer 1 Report
The article deals with the problem of time in quantum gravity. The article addresses a problem of great interest to the community. Below are my main objections to the manuscript.
1) Authors should discuss the idea of gauge and quantum gravity described in the first paragraph. How is this compatible with the Schrödinger equation?
2) This brings me to the next point. The introduction of time using an invariant equation for Galilean transformations must be incompatible with the Lorentz symmetry present in Lagrangian (1). This symmetry is not lost even for the simplification proposed by the authors.
3) Perhaps more details, on how eq. (4) leads to eq. (3), are necessary.
4) The Hamiltonian must be associated with the energy of the field, but the problem of gravitational energy is one of the oldest problems in General Relativity. The null constraint should prohibit time evolution. It seems to me that the authors must discuss this apparent contradiction.
For the above, I do not recommend publishing the article in its present form.
Author Response
Response to Reviewer 1 Comments
Point 1: Authors should discuss the idea of gauge and quantum gravity described in the first paragraph. How is this compatible with the Schr\"{o}dinger equation?
Response 1: The idea of compatibility of the Wheeler-DeWitt and Schrödinger equations in their connection with the gauge invariance and quantum gravity is discussed in Ref. [8], and we added the corresponding comment as the footnote in Page 1.
Point 2: This brings me to the next point. The introduction of time using an invariant equation for Galilean transformations must be incompatible with the Lorentz symmetry present in Lagrangian (1). This symmetry is not lost even for the simplification proposed by the authors.
Response 2: The authors, nevertheless, have to note that the example with the Lorentz symmetry given by the reviewer seems not very relevant, because, in the minisuperspace model considered in the paper, the variables do not depend on the spatial coordinates. That is, a(η) depending on the time η after the Lorentz transformation should depend on the spatial coordinates, as well. It is beyond of the minisuperspace model. It seems a more suitable example could be a bosonic string in the Minkowsky space-time [12], which could also be described by the Schrödinger equation. But even for this case, the Lorentz invariance is not a good example, because of the Schrödinger equation is written w.r.t. the time τ variable parameterizing the string surface Xμ(τ,σ), rather than in terms of the time X0 of external space. Concerning our paper, still more reasonable to say about the time reparametrization invariance [12]. This invariance is violated if one imposes the gauge condition. As a result, the Schrödinger equation arises in the methods A,E. If one quantizes the system before imposing the gauge condition (methods B,C,D, the WDW equation arises. After that, the gauge condition could be set in the scalar product (methods B,C) or at a level of the operator equations of motion (method D). This note was added to the last paragraph of the introduction. In this regard, let us remind that even in the quantum electrodynamics (QED) with the secondary quantized Hamiltonian, the Schrödinger equation is also used in the form of interaction representation, however, although the mathematical apparatus, as well as intermediate calculations, are not Lorentz-invariant, the final results turn out to be Lorentz-invariant there.
Point 3: Perhaps more details, on how eq. (4) leads to eq. (3), are necessary.
Response 3: It is written in the present version of the paper: "Variation of (4) on πφ and pa gives πφ=φ′a2/N and pa=a′/N, respectively. After substituting these values into (4) the Eq. (3) is recovered." Also, we have added more references, where the extended Hamiltonian systems are discussed, including the original Dirac papers.
Point 4: The Hamiltonian must be associated with the energy of the field, but the problem of gravitational energy is
one of the oldest problems in General Relativity. The null constraint should prohibit time evolution. It seems to me that the authors must discuss this apparent contradiction.
Response 4: The corresponding comments were added to the extended version of the conclusion.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
In this work, the authors consider the quantization of FRW gravity with a minimally coupled free scalar field. Time is introduced following reference 11, by imposing a gauge fixing corresponding to the solution for the scale factor of the classical equations with conformal time. There are four approaches: A) Schrödinger equation with hamiltonian following from the phase space reduction, the scalar product is the one of L2 (ref. 11). B) Wheeler-DeWitt equation with quantization following ref. 16 and time by gauge fixing, hermiticity is implemented by a sort of regularization. C) The scalar product of point B is written by means of supplementary integrals over Grassmann variables. D) This point is similar to the previous one, but now the operators are in a so called quasi-Heisenberg picture. E) The scalar product is implemented in an extended phase space with the delta function of the constraint and Fadeev-Popov determinant of the gauge fixing written as integrals in a Grassmann space. The gauge is fixed in two ways, with corresponding Hamiltonians. All the cases are based on the Hamiltonian constraint and the gauge fixing of the time reparametrization invariance (11). A comparison of the mean values for the scale factor and momentum operators is made. It is stated that in the simplest cases, lineal and quadratic, they coincide, but for higher powers, there are differences. The paper is interesting and gives new insights on the time interpretation in quantum cosmology.
I have the following observations:
I recommend the introduction of a description of the work in the introduction, to allow the reader to have a perspective before reading it.
Throughout the work, many of the results are given or mentioned without the computations or the corresponding expressions. Details should be given so that the reader can quickly convince himself. In particular, the differences among mean values should be explicitly shown.
The scalar product (17) seems to be in general time dependent, even if for the wave function (16) it is time independent. I recommend a discussion on it.
In the first paragraph of section 3.2 there are several, somewhat loose, statements regarding Wheeler-DeWitt equation which seem to be quotations, but there are not references. In order to avoid confusions, it would be useful to give references supporting these statements. Otherwise, it should be given a more detailed discussion. Moreover, regarding time dependent mean values in quantum cosmology there are previous works, which should be quoted, e.g. C. Ramirez and V. Vazquez-Baez, Phys. Rev. D93, 043505, 2016.
It is stated that the scalar product (24) and (25) is taken from reference 16. It does not seem obvious that it is so, it should be argued.
Taking into account the previous comments, I recommend the publication of this paper, after the observations are considered.
Author Response
Response to Reviewer 2 Comments
Point 1:I recommend the introduction of a description of the work in the introduction, to allow the reader to have a perspective before reading it.
Response 1: It is a very reasonable recommendation, and we added the corresponding comments to the last paragraph of the introduction.
Point 2: Throughout the work, many of the results are given or mentioned without the computations or the corresponding expressions. Details should be given so that the reader can quickly convince himself. In particular, the differences among mean values should be explicitly shown.
Response 2: We have introduced the references to the supplemental material
(Wolfram Mathematica notebooks) to the text of the paper.
Point 3: The scalar product (17) seems to be in general time-dependent, even if for the wave function (16) it is time-independent. I recommend a discussion on it.
Response 3: The phrase is added after the formula (21): "The time dependence in (21) arises from two sources: the time-dependent wave function in (17) and a= √2|k|η ."
Point 4: In the first paragraph of section 3.2 there are several, somewhat loose, statements regarding Wheeler-DeWitt equation which seem to be quotations, but there are not references. In order to avoid confusions, it would be useful to give references supporting these statements. Otherwise, it should be given a more detailed discussion. Moreover, regarding time dependent mean values in
quantum cosmology there are previous works, which should be quoted, e.g. C. Ramirez and V. Vazquez-Baez, Phys. Rev. D93,
043505, 2016.
Response 4: We have rewritten this paragraph more gently, and the footnote is added: "One has to note that the methods considered are not the exclusive methods describing the quantum evolution of the universe. For instance, one could take a scale factor or a scalar field [25] as the ``time variable.''"
Point 5: It is stated that the scalar product (24) and (25) is taken from reference 16 (in the present version [26]). It does not seem obvious that it is so, it should be argued.
Response 5: In the present version of the paper, we have added more references [18,26-28] where scalar products are discussed or mentioned. Then, we write more concretely ``...see the last formula of Ref.[26]...''. The paper [26] is a rigorous mathematical paper, and we sure that if a reader will be able to read it thoroughly, she acquires a sufficient knowledge about the scalar products for the Klein-Gordon equation.
Author Response File: Author Response.pdf
Reviewer 3 Report
The authors study the problem of time using an action containing both gravitational and scalar degrees of freedom. The latter is actually crucial. As has been shown in the context of loop quantum gravity, when gravity is coupled with matter, the latter can be used to define a "clock" unambiguously. The authors perform some computations that show how this idea works in practice.
The manuscript is well written, references and calculations are complete. I thus think that it deserves publication on the journal universe in the current form.
Author Response
Response to Reviewer 3 Comments
Point 1: As has been shown in the context of loop quantum gravity, when gravity is coupled with matter, the latter can be used to define a "clock" unambiguously.
Response 1: We added the footnote 3: "One has to note that the methods considered are not the exclusive methods describing the quantum evolution of the universe. For instance, one could take a scale factor or a scalar field [25] as the ``time variable.''"
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
The article is in a better shape now. The authors have cleared my previous points, thus I recommend this article for publication.
Reviewer 2 Report
The authors have considered satisfactorily the comments and recommendations.
