Physical Significance of Noether Symmetries
Round 1
Reviewer 1 Report
This survey, which specializes on Noether symmetries and its applications in different fields of mathematical physics and even of economics, has some elements that make it fascinating: the illustrative language and the putting together of such a (seemingly) different information is something that makes it original and useful.
I would recommend it to be published in Symmetry, MDPI, after some changes.
First of all, I personally felt sad to see that, while so many paragraphs were dedicated to Supersymmetry, Twistor Theory is completely missing from this discussion: even if the authors do not have a definite answer about any relationship between the two, I strongly recommend to include some paragraph, discussing this.
Secondly, a successful survey should include open questions: these questions should be enumerated and either grouped to the Paragraph/Section that they are linked, or they can all be listed in a final section.
Author Response
We have revised the paper according to the comments of Reviewer 1, and made the changes in the manuscript as the attached pdf file.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
This is a review paper. The authors have made efforts to be very pedagogical and detailed in many aspects in order to convey the fundamental concept of symmetries in physical systems from Classical and Quantum mechanics, classical and quantum Field theory and General Relativity, as well as in engineering and even economics. They also consider the content and application of the Noether theorem (Emmy Noether, 1915), corresponding to the symmetries. Noether theorem states that if the action integral (or equivalently, its density, the Lagrangian), which generates the equations of motion of the system, is invariant under some n-parametric Lie group, there exist n independent conserved quantities (conservation laws). This helps solving the equations, or at least leads to their reduction to simpler ones. Important symmetries discussed in the text are gauge symmetries (in Electrodynamics, in the Standard model), isometries, conformal (Weyl) symmetry (in General Relativity).
I have the following remarks.
- In section 4 the authors have chosen to illustrate the Lie symmetries and the Noether theorem on the example of classical electrodynamics. Here they mostly discuss the equations of motion in their historically first formulation in terms of the observable electric E and magnetic H fields unlike the standard canonical Lagrangian action formulation in terms of the vector potential Aµ , where the Noether symmetry becomes explicit. The authors discuss the latter formulation later in Section 6.5.3, therefore, the lengthy details in section 4 look unsuitable for the aim of the paper.
- There are several places in the text where the authors are referring to a book by one of them (reference [20]) for well-known expressions of fundamental quantities in General Relativity (e.g., the Christoffel symbols Гμνχ , the Riemann curvature tensor Rμνλχ ). It would be more appropriate to add at least some references to standard original textbooks on the subject (e.g., Weinberg’s).
After taking into account the above remarks, I shall recommend the paper for publication.
Author Response
We have revised the paper according to the comments of Reviewer 2, and made the changes in the manuscript as the attached pdf file.
Author Response File: Author Response.pdf
Reviewer 3 Report
I would start by commending the authors for the job well done. However, this paper is not suitable enough to be published in its current form, this journals template must be used. Moreover, I noticed that there is no theorem, convergence or stability proved in this paper, which I think is not agreed with the scope of this journal. However, I would suggest that the paper should be accepted with minor revision due to some of the
corrections I pointed out above and in order to raise the standard of this paper. The English need to be polished, punctuation mark needs to be administered after each equation.
Finally, I will be available for further revision of this paper.
Comments for author File: Comments.pdf
Author Response
We have revised the paper according to the comments of Reviewer 3, and made the changes in the manuscript as the attached pdf file.
Author Response File: Author Response.pdf
