Measures of Distance in Quantum Mechanics

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

                 Review   report  on  paper  universe-2775560

This article is interesting and important. I am not against its publication in  Universe. However, I have two significant comments.

1) The article is formatted carelessly.

This applies to formulae  (2.3), (2.6) where the sign of the integral is missing. The formulae (4.2), (4.5), (4.6) and etc. are also carelessly formatted.

The authors must carefully check the entire text and correct everything; otherwise, there can be no talk of any publication of this article.

2) Since the article is about the  measures of distance in the general relativity  and quantum mechanics, i.e.  essentially about the approach to the quantum theory of gravity, it was correct to add a part connecting the authors’ results with the measure in the corresponding partition function (that is) for quantum gravity. In this case, an extensive study is certainly not required, but this part of the work should at least be indicated.

Author Response

1. In the processing of the file appeared those errors (missing symbols for integrals, sums, etc) that were not in the original file. In fact, the article looks sloppy, as the Reviewer rightly pointed out. The right file is attached.
2. The title of the paper has been changed to make the content more precise: "Measures of distance in quantum mechanics" 
3. In the "Conclusions" were added explanations of the Reviewer's comments.
4. For the convenience of Reviewers, all changes resulting from comments are written in italics. 

 

 

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

This article evaluates information metric for two quantum mechanical systems, a free particle and a harmonic oscillator, and computes relative entropy of two states of a scalar field. While the derivations are straightforward, the results might be interesting to researchers working on foundations of quantum mechanics, so I recommend this article for publication. However, the title is somewhat misleading: while the authors derive the information metrics in quantum mechanics, no discussion of GR is presented beyond review of very basic geometric facts in section 2. Therefore, a more appropriate title for this article is "Measures of distance in quantum mechanics". I recommend this article for publication in Universe once the title is changed.

Author Response

1. As suggested by the Reviewer, the title has been changed: "Measures of distance in quantum mechanics"
2. For the convenience of Reviewers, all changes resulting from comments are written in italics.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

see comments

Comments for author File: Comments.pdf

Comments on the Quality of English Language

English language is OK

Author Response

1. Response to REMARKS 

In the processing of the file appeared those errors (missing symbols for integrals, sums, etc) that were not in the original file. In fact, the article looks sloppy, as the Reviewer rightly pointed out. The right file is attached.

2. Responses to ISSUES

    #5 - On the issue of combining gravity with quantum mechanics, my opinion that this is a distant endeavor does not change. I base my conviction on two reasons: i) the lack of experimental facts that could give some guidance of the direction of the theory, ii) despite so many efforts over the years, the theory of quantum gravity does not exist. These two arguments are correlated. The first attempts to undertake quantization of gravity date back to the 1930s and were carried out by L. Rosenfeld. More on this topic in [1, 2]. No doubt, much progress has been made since then. But many problems remain unsolved further. According to the one of authors (PG), the most advanced approach to connecting gravity and other fields at the quantum level was superstring/M-theory. However, due to the impossibility of determining a single ground state and the lack of any experimental facts confirming the truth of the theory's predictions, this theory can be sent back to "beautiful theories that have been killed by facts".
    #6 - As for the inner product in Hilbert space, I have added explanations in the text. 
    #7 - Of course, the Reviewer is right that there are other approaches to quantum gravity. In addition to canonical quantization based on geometrodynamics, there are covariant quantization, loop gravity, which is some variation of geometrodynamics. All these approaches are equivalent and even must be so! Otherwise, the "theory" (which does not yet exist) would be inconsistent from its origin.
    Let me point out to the Reviewer that the WDW equation is "defined" on superspace and not on 4-dimensional space-time. This superspace is the space of all metrics on 3-dimensional hypesurface modulo 3-dimensional diffemorphisms. However, due to the extreme complexity of this equation, it remains poorly understood, but it can be applied to problems in cosmology and for black holes. The origin of the "problem of time" is in the interpretation of spacetime in general relativity; spacetime is the analogue of a particle trajectory in mechanics; so after quantization spacetime disappears in the same way as the trajectory disappears -- only space remains. It is the "problem of time" (e. g. [3]-[6]) In [3] question is asked by Mason: "Why doesn't the search for an internal time go against the spirit of general covariance?"  I will quote Kuchar's entire reply in full: "If you search for an intrinsic time that satisfies the criterion of general covariance, i. e., that is a space-time scalar, you probably fail. There is not any that is' a local functional of the metric. I feel it is highly unlikely that you will find one that is nonlocal. The search for an intrinsic time thus seems to go against the spirit of general covariance. I would say that the general covariance is somehow hidden under the surface of superspace: There are geometrically privileged variables, like' the curvature scalars in a given curved space-time, which can be reconstructed from the phase-space data. Such variables can be used as privileged coordinates"  Such variables were found in [7] using the the Wheeler-DeWitt approach to quantum cosmology, where superspace is reduced to only a few degrees of freedom (minisuperspace). As the result the time was obtained, called "WKB time". Thus, there is hope that the WDW equation is a bridge (at least at a formal level) from quantum gravity to quantum field theory in curved spacetime that (and, in particular, time) emerged as an approximate concept.
    [1] C. Kiefer, Quantum gravity-unfinished revolution [arXiv:2302.13047]
    [2] G. Peruzzi, A. Rocci Tales from the prehistory of Quantum Gravity, The European Physical Journal H, Vol. 43, pages 185--241, (2018) [arXiv:1802.08878]
    [3] K. V. Kuchar, The Problem of Time in Canonical Quantization, in A. Ashtekar & J. Stachel (eds.), Conceptual Problems of Quantum Gravity. Birkhauser (1991).
    [4] C. J. Isham, Canonical Quantum Gravity and the Problem of Time, [arXiv:gr-qc/9210011]
    [5] K. V. Kuchar, Time and interpretations of quantum gravity, Int. J. of Modern Physics D20, pp. 3-86 (2011)
    [6] E. Anderson, The Problem of Time in Quantum Gravity, in V.R. Frignanni (edi.) Classical and Quantum Gravity: Theory, Analysis and Applications, 2012, ch.4 [arXiv:1009.2157]
    [7] C. Kiefer, H. D. Zeh, Arrow of time in a recollapsing quantum universe, Phys.Rev. D51 (1995) 4145-4153, [arXiv:gr-qc/9402036].

3. For the convenience of Reviewers, all changes resulting from comments are written in italics.
    

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

I see that the work has been corrected (file universe-2775560-coverletter.pdf). I accept the authors' arguments on pages  2, 3, 18.  I believe that the work can be published in present  form.

Author Response

The authors thank the Reviewer for his/her remarks. 

Reviewer 2 Report

Comments and Suggestions for Authors

I thank the authors for making a requested change. I recommend this paper for publication.

Author Response

The authors thank the Reviewer for his/her remark. 

Reviewer 3 Report

Comments and Suggestions for Authors

see comments

Comments for author File: Comments.pdf

Author Response

The authors thank the Reviewer for her/his remarks, which have been included in the text.

 

Author Response File: Author Response.pdf