Time-Dependent 4D Quantum Harmonic Oscillator and Reacting Hydrogen Atom
Round 1
Reviewer 1 Report
-The definition of tau after eq. 5 is inconvenient, since gamma also is not defined.
-For the reviewer (fro me at least) it is difficult to check the correctness of all the mathematical expressions in the manuscript. The author have to check their formulas carefully.
Author Response
The authors grateful to the reviewer for comments, which were all taken into account in the new version of the article.
Reviewer 2 Report
The authors intend to explain some deep aspects of the physics of the Hydrogen atom by understanding some fundamental aspects of the symmetries involved on it. The paper could be interesting. However, before further consideration, I suggest the authors to consider the following:
1). Explain why do you consider a Harmonic oscillator initially if what you expect is to describe a hydrogen atom.
2). Eq. 6 looks inconsistent. Could you verify it?
3). The authors claim that they have found an S-matrix for the transitions in the hydrogen atom. How can we be sure about it?
4). Where is the S-matrix? Could you express it more explicitly? Why do you think that it should be related to the hydrogen atom?
5). Are the authors rather analyzing processes like laser emission or photon emission, photon absorption, etc? I think that this would be the only way to consider Harmonic oscillators.
6). How are the symmetries of the system involved in the mentioned transitions? Where did you specify it?
7). There are several typos. For example the words "was used was used" (repeated twice) appear on page 2. There are more typos, please verify.
8). If there is a S-matrix, and we have an initial and final state. Could we consider integrable systems here?
9). Can we perceive the process analyzed in the paper as a phase transition?
10). Add more references.
After the authors revise the paper in agreement with these comments, I would happily revise it again.
Author Response
The authors are grateful to the reviewer for useful comments.
Author Response File: 
Author Response.docx
Reviewer 3 Report
Point 1: The physical system under consideration is, supposedly, the ``reacting hydrogen atom'' but what, exactly, this term means is never concretely defined. It is not made clear in the Introduction and, as such, it is not at all clear to the reader what any of the formulae presented in the following sections actually represent.
Point 2: For example, on page 2, the hydrogen atom is described as ``a charged spinless particle in a Coulomb field''. But a hydrogen atom is charge-neutral, being a composite system (bound state) comprised of a negatively charged spin-1/2 electron and a positively charged spin-1/2 proton, with a Coulomb potential acting between them.
Point 3: However, it should be noted that no Coulomb fields appear anywhere in the mathematical analysis presented by the authors.
Point 4: Similarly, the analysis makes use of the so-called ``reference equation method'' and, in particular, ``low-dimensional reference equations'' but there is no exposition or summary of this method, or of the precise meaning of the term ``low-dimensional'' in this context.
Point 5: There is a great deal of non-standard notation, whose meaning is also very unclear. For example, x = \overline{x_1,x_4} \in \mathbb{R}^4$ in Eq. (1) and y = y(y_1,…, y_4) below Eq. (10). I think that the authors simply mean x = (x_1,…,x_4), y = (y_1,…,y_4), i.e., that x and y denote vectors in 4D Euclidean space, but this is not what they write.
Point 6: Similarly, together with the time-like parameter \tau, the set (y,\tau) are described, below Eq. (11), as denoting ``the new space-time continuum''. But there is no space-time here. There whole analysis is manifestly non-relativisitic and all time-like variables are, formally, treated as parameters (c-numbers).
Point 7: There are a number of typos and grammatical errors which, though easily corrected, render the paper even more difficult to read and interpret. These include but are not limited to the incorrect use of the articles ``a'' and ``the'', the misspelling of names in the text that do not match the list of references, and the use of other undefined terms.
Point 8: To cap off the confusion, the Conclusions section does not, in fact, contain a summary of the article's main results. It is therefore not clear which of the results presented in this work are truly new and which were already presented in similar previous analyses, listed in the references.
Author Response
The authors are grateful to the reviewer for useful comments.
Author Response File: Author Response.docx
Reviewer 4 Report
The main goal of this paper, in the authors' own words, is ``to construct a mathematically rigorous and consistent representation of the reacting hydrogen atom using the explicit and hidden symmetries of a four-dimensional isotropic oscillator''.
This is done by beginning with the equations of motion (EOM) of the 4D quantum harmonic oscillator (QHO), with an additional constraint, then performing a coordinate transformation that maps these to the EOM of the system under consideration, i.e., that of the ``reacting hydrogen atom''.
Presumably, this is intended as an extension of the analysis of the ``non-reacting hydrogen atom'', which was analysed in ref. [5], using similar techniques.
As such, the aim of the study is valid, and its results are potentially interesting, but there are a number of problems with the way the paper is currently formulated, which make it unsuitable for publication.
I outline the main problems below.
\begin{enumerate}
\item The physical system under consideration is, supposedly, the ``reacting hydrogen atom'' but what, exactly, this term means is never concretely defined.
It is not made clear in the Introduction and, as such, it is not at all clear to the reader what any of the formulae presented in the following sections actually represent.
For example, on page 2, the hydrogen atom is described as ``a charged spinless particle in a Coulomb field''.
But a hydrogen atom is charge-neutral, being a composite system (bound state) comprised of a negatively charged spin-1/2 electron and a positively charged spin-1/2 proton, with a Coulomb potential acting between them.
However, it should be noted that no Coulomb fields appear anywhere in the mathematical analysis presented by the authors.
\item Similarly, the analysis makes use of the so-called ``reference equation method'' and, in particular, ``low-dimensional reference equations'' but there is no exposition or summary of this method, or of the precise meaning of the term ``low-dimensional'' in this context.
\item There is a great deal of non-standard notation, whose meaning is also very unclear.
For example, ${\bf x} = \overline{x_1,x_4} \in \mathbb{R}^4$ in Eq. (1) and ${\bf y} = {\bf y}(y_1, \dots y_4)$ below Eq. (10).
I think that the authors simply mean ${\bf x} = (x_1,x_2,x_3,x_4)$, ${\bf y} = (y_1,y_2,y_3,y_4)$, i.e., that ${\bf x}$ and ${\bf y}$ denote vectors in 4D Euclidean space, but this is not what they write.
Similarly, together with the time-like parameter $\tau$, the set $\left\{{\bf y},\tau\right\}$ are described, below Eq. (11), as denoting ``the new space-time continuum''.
But there is no space-time here.
There whole analysis is manifestly non-relativisitic and all time-like variables are, formally, treated as parameters (c-numbers).
\item There are a number of typos and grammatical errors which, though easily corrected, render the paper even more difficult to read and interpret.
These include but are not limited to the incorrect use of the articles ``a'' and ``the'', the misspelling of names in the text that do not match the list of references, and the use of other undefined terms.
\item To cap off the confusion, the Conclusions section does not, in fact, contain a summary of the article's main results.
It is therefore not clear which of the results presented in this work are truly new and which were already presented in similar previous analyses, listed in the references.
\end{enumerate}
In summary, the paper, in its current form, is far too vague and unclear to warrant publication in Symmetry.
To be considered for publication, the authors must rewrite the article thoroughly, explaining clearly the physical system under consideration, and how it relates to the mathematical formulae given in the text.
The must clearly define all terminology and use use clear (and preferably standard) mathematical notation in their analysis.
Comments for author File: 
Comments.pdf
Author Response
The authors are grateful for the reviewer's comments, which where all taken into account in the new version of article.
Author Response File: Author Response.docx
Round 2
Reviewer 2 Report
The authors have replied to the referee's comments. However, as a minor arrangement, the authors should emphasize and mention those references where they claim that others have addressed the problem of the Hydrogen atom inside the framework of the Quantum Harmonic oscillator. It would be great if the authors dedicate one (or part of a) section to the Hydrogen atom and its connection with the QHO. After this minor arrangement, the paper can be published.
Author Response
Thanks to the reviewer for the important suggestion.
Author Response File: Author Response.docx
Reviewer 4 Report
The authors’ response to point 8 of my first report seems to have nothing to do with the criticism I made. The issue is whether or not Conclusions section of their paper contains a summary of the conclusions of their analysis, or not. I do not understand their reference to phase transitions in this context, which seems rather bizarre.
Also, I do not agree that point 6 is “irrelevant to the issue at hand”. The authors referee to a `space-time’ structure, where, to the best of my understanding, no such structure exists. The 4D in their analysis represent 4 spatial dimensions and time is everywhere treated as a parameter (c-number) as is always the case in non- relativistic mechanics (in any number of dimensions). Is my understanding correct, or have I misinterpreted their formalism?
Their remaining responses, and amendments to the text, may have adequately addressed my previous concerns, but the issues above must first be dealt with before the article can be recommended for publication.
Author Response
Many thanks to the reviewer for important comments.
Author Response File: Author Response.docx
