On Functional Hamilton–Jacobi and Schrödinger Equations and Functional Renormalization Group

Round 1

Reviewer 1 Report

Report on the paper
On Functional Hamilton–Jacobi and Schrödinger Equations and Functional Renormalization Group
by Mikhail G. Ivanov, Alexey E. Kalugin, Anna A. Ogarkova and Stanislav L. Ogarkov

The aim of the paper is to study some aspects of functional Hamilton–Jacobi and Schrödinger equations.
The equations are formulated in D-dimensional coordinate and abstract (formal) spaces and a new “holographic” scalar field is introduced,
as the amplitude of delta-field or constant-field configurations.

The authors depict well the history of the subject and the connections with their study.
The content of the paper is interesting, relating new constructions to some types of known ones related to the field.
The authors compare in some examples their proposed approach, finding improvements and formulate new problems, for example concerning the construction of functional
functional renormalization group (FRG) flow equations.
In conclusion, we recommend the publication.

Author Response

Dear Reviewers, we are of a great appreciation for such detailed Reports on our manuscript. In accordance with these Reports, we have made the corrections in our paper.

Into the Report of the second Reviewer, we have added the comments of the third Reviewer using additional notes (stickers). This file is “Reviewers’ Reports (Old Version of the Paper)”. In the new version of our paper, we highlighted in green all the corrections that we made to the paper. The new paper file is “Functional HJ Paper (Symmetry)”. Please find your comments in the file “Reviewers’ Reports (Old Version of the Paper)” and our answers in the file “Functional HJ Paper (Symmetry)”. We assumed that the highlighting of responses in the new version of the paper will be convenient for you. If a line-by-line listing of changes is required, please let us know about it.

We sincerely hope that we have given detailed answers to all your comments.

Yours sincerely,
Stanislav L. Ogarkov and co-authors

Author Response File: Author Response.pdf

Reviewer 2 Report

This paper is a comprehensive analysis of key functional equations in broad areas of physics.  It is rigorous, carefully written, and very mathematically oriented.  The paper reflects significant expertise in functional analysis in many areas of physics. Experts in the field will probably have no trouble following the paper and will appreciate it.  For the rest of us, it is a bit of a challenge to follow. And my only comments are to consider including a few clarifying sentences, observations, comments about the physical phenomena, to help us follow and understand the significance of the equations.  For example, numerous approximations are made, and it might be helpful to explain their significance.  It might be helpful to occasionally translate from the condensed notation to more conventional notation just to remind the reader what it means.   

Comments for author File: Comments.pdf

Author Response

Dear Reviewers, we are of a great appreciation for such detailed Reports on our manuscript. In accordance with these Reports, we have made the corrections in our paper.

Into the Report of the second Reviewer, we have added the comments of the third Reviewer using additional notes (stickers). This file is “Reviewers’ Reports (Old Version of the Paper)”. In the new version of our paper, we highlighted in green all the corrections that we made to the paper. The new paper file is “Functional HJ Paper (Symmetry)”. Please find your comments in the file “Reviewers’ Reports (Old Version of the Paper)” and our answers in the file “Functional HJ Paper (Symmetry)”. We assumed that the highlighting of responses in the new version of the paper will be convenient for you. If a line-by-line listing of changes is required, please let us know about it.

We sincerely hope that we have given detailed answers to all your comments.

Yours sincerely,
Stanislav L. Ogarkov and co-authors

Author Response File: Author Response.pdf

Reviewer 3 Report

The article is equation intensive but difficult to read as a review and new results are not clearly mentioned. 

Comments for author File: Comments.docx

Author Response

Dear Reviewers, we are of a great appreciation for such detailed Reports on our manuscript. In accordance with these Reports, we have made the corrections in our paper.

Into the Report of the second Reviewer, we have added the comments of the third Reviewer using additional notes (stickers). This file is “Reviewers’ Reports (Old Version of the Paper)”. In the new version of our paper, we highlighted in green all the corrections that we made to the paper. The new paper file is “Functional HJ Paper (Symmetry)”. Please find your comments in the file “Reviewers’ Reports (Old Version of the Paper)” and our answers in the file “Functional HJ Paper (Symmetry)”. We assumed that the highlighting of responses in the new version of the paper will be convenient for you. If a line-by-line listing of changes is required, please let us know about it.

We sincerely hope that we have given detailed answers to all your comments.

Yours sincerely,
Stanislav L. Ogarkov and co-authors

Author Response File: Author Response.pdf

Round 2

Reviewer 3 Report

The paper is now much improved due to mainly referee2's detailed comments. The introduction has described the new results of the paper. I have some suggestions which the authors might consider,

(i) The assumption that the odd -order GF are zero is rather restrictive as the interesting scalar potential \phi^3 + \phi^4 which the authors mention are excluded in this discussion. The authors should try to generalize their results or at least comment on how their results would change if they use the odd-order terms in their Taylor expansions.

(ii) In the discussion about the semiclassical limit, in lines 84-86 in the introduction, the result in equation (90,91) is implied, then that should be interpreted as a classical limit. If there is a linear order in $\hbar$ hidden somewhere, then that should be mentioned, otherwise the semi-classical regime starts at order $\hbar^2$.

(iii) In equation (147) G(\Lambda)=1/\Lambda^{2n}. If I take the $\Lambda->0$ limit of this in equation (144) I am getting a $G(k)/0$, this equation must be corrected or if this $G(\Lambda)$ is not applicable to (144) then that should be clearly mentioned.  

(iv) I was expecting some further discussions on open quantum field systems, as this has new results. I would suggest that the authors refer to the paper 

A. Baidya et al `Renormalisation in open quantum field theory: Part-1:Scalar field Theory JHEP (2017) 204.

(v) Also as algebraic quantum field theory is defined in terms of GF, the authors should mention the use of their formalism in the above. 

Author Response

Dear Reviewer 3, we are of a great appreciation for your Report on our manuscript. In accordance with this Report, we have made the corrections in our paper. Into the new version of the paper we have added your comments using additional notes (stickers) as well as we highlighted in green all the corrections that we made to the paper. The new paper file is “Functional HJ Paper (Symmetry)”. Please find your comments and our answers in this file. We assumed that the highlighting of responses in the new version of the paper will be convenient for you. If a line-by-line listing of changes is required, please let us know about it. We sincerely hope that we have given detailed answers to all your comments.

Author Response File: Author Response.pdf