Geometry of variational methods: dynamics of closed quantum systems

1-A thorough exposition of the geometry of variational methods in quantum mechanics with a focus on the difference between Kähler and non-Kähler manifolds of variational wave functions.

2-Simple examples help illustrate basic concepts.

1-The paper is long, maybe unnecessarily long, and not always concise.

2-It is not always clear what the original contribution of this work are.

I broadly agree with the first report on this paper. The manuscripts provides a link to, and an exposition of, mathematical concepts that may be useful for various applications of variational methods in studies of closed quantum systems. The paper attempts to give a detailed and comprehensive discussion of these methods, in a way accessible to a relatively broad expertise. For these reason I believe the paper could have impact and should be considered for publication in SciPost Physics.

As a consequence of the aim to be comprehensive, maybe, the paper is quite long. I think that in principle the length is not a problem, but for such a long paper to be more likely to have an impact, one would have wanted the writing to be a bit more concise and focussed. For example, there are several sentences and paragraphs that discuss what will be done in the next section or in some other part of the paper, without actually adding anything to the story at that point. This has the effect on the reader (at least this reader) that they loose focus and it sometimes becomes hard to come back to the paper and find the information one is after.

I do not think I will insist on the paper being shortened, but I would encourage the authors to consider attempting to focus the writing a little bit. Since how that would be done is to a large extend matter of style, I don't think I should give specific instructions. What would be useful for a reader that maybe wants to use the result of the paper rather than reading every word, is to have a more detailed guide to the reader for where and how to find the key results. I also find that the summary and discussion could be used more efficiently to help the reader understand the key points of the paper. At the moment the summary reads more like the introduction (especially due to the chose of tense in some parts) and doesn't actually summarise very much. It may be useful to extend this section to ease the use of the paper. Again, I would probably not insist on these changes, as they are to some extend a matter of style, but I feel it would improve the readability of the paper.

Towards the end of the paper there is a discussion of the Generalized Gaussian states, and it is said that one of the main motivation of this work is to understand the nontrivial manifolds that arise from these states. There is also several references to earlier work on these states and it is not always clear what is fully new, what is described in a new way and what was discussed before. It would be useful to clarify this.

In addition to this I give a list of more detailed points for consideration in the Requested changes section.

Some of these points are more remarks than requested changes.

1-There are a few places, such as in the last sentence of paragraph 2 on page 2, which reads "Recently such methods have been used..." and then gives examples, where references are partially or completely missing. In this particular paragraph, for example, reference should be give to these recent works that are mentioned. Another example is start of IIB.3 "A standard approach..." Such a phrase seems to suggest that a reference would be appropriate.

2-First paragraph of section II reads "Readers may skip directly to section III." I suppose that this is an example of a guid to the reader I mentioned in my main report, except that here it is not useful. What readers may skip directly and for what reason should they do that?

3-I find the first four paragraphs of section III to be quite repetitive. Maybe one place to shorten?

4-There are several places where there is an additional article or a missing article. For example in second paragraph of III.A: "...where $|\psi(z)\rangle \in \mathcal{H}$ is a holomorphic in $z^\mu$..."

5-The definition of $\omega_{\mu\nu} $ given just below Eq. (8) and that in footnote 2, do not seem consistent. They seem to differ by an $i$.

6-Below Eq. (20) a $|\bar{v}_\mu\rangle$ is used. I may have missed it, but I don't think the bar notation has been defined.

7-In the paragraph below Example 2, should $\mathcal{P}(\mathcal{P})$ maybe be $\mathcal{P}(\mathcal{H})$?

8-Above Eq. (26) where the authors state that some things can be shown to be equivalent, is this a known result (and therefore maybe needing a reference) or is it something they can show but don't think is needed to expand on in the paper?

9-At the start of III should $\psi$ be $|\psi\rangle$.

10-Not all figures and tables seem to be referenced in the main text (for example Fig. 2). It would make sense to refer to the figures in the correct place. Also, I feel that sometimes the captions could be more detailed, as often not everything that is shown on the figures is defined or explained (for example caption to Fig. 3).

11-Should definition 1 read "...with inverse $\Omega^{\mu\nu}$..."?

12-Is there an superfluous dot at the end of Eq. (41)?

13-Start of IV: "Given a system $\mathcal{H}$" Here the concept of a system is being mixed with Hilbert space, since $\mathcal{H}$ is everywhere else used for that.

14-First sentence in the paragraph after Eq. (87): "In the case of the Lagrangian action principle...takes the following form." Nothing follows this, only the next sentence "A similar derivation..." What form does it take?

15-Last paragraph on page 16. "full Hilbert space, i.e., $\mathcal{M} = \mathcal{P}(\mathcal{H})$." Wasn't $\mathcal{P}(\mathcal{H})$ the projected Hilbert space?

16-Above proposition 5, again a sentence that ends in "take the following form" without any form following.

17-Below Eq. (98), should it be $\check{J}$ that "clearly satisfies $\check{J}^2=-1$?

18-IVB.2 starts with "A common alternative is..." A common alternative to what?

19-Above Eq. (19) the authors use $\mathcal{F}^i$ while elsewhere it is $\mathcal{F}^\mu$.

20-At the top of page 32, should the "associated tangent vector" be $|\delta\Gamma\rangle$ instead of $|\Delta\Gamma\rangle$?

21-In Eq. (265) one of these matrices should probably be $\delta\Gamma_2$.

22-In example 20, should some of the $q$ and $p$ have subscript 2 instead of 1. And also, is there a reason the order in $\xi$ is $q,p,q,p$ instead of $q,q,p,p$ as earlier in the paper.

23-In Eq. (292) would it make sense to show explicitly that the derivative is taken at $x=0$?

24-Proposition 13 says "The restricted Kähler structures are..." The restricted Kähler structures of what? It would be preferable to make the propositions self-contained.

25-In proposition 14, is is needed to write that $\mathcal{G}$ is a Lie group?

26-End of example 23. What balance between the properties of $\mathcal{M}_\phi$ and its dimension needs to be struck and for what purpose?

27-Example 24. "The representation of the representation on $\mathcal{H}_-$"?

28-First sentence in Example 25 seems unnecessary since the whole section has been discussing this.

29-Appendix A.3 starts with "In many areas of physics, ...". Then there is a heading and discussion goes somewhere else. The connection of this first statement to what follows is not clear.

30-In couple of places there are additional articles "we a reference", "we a corresponding" ...