A coupling prescription for post-Newtonian corrections in Quantum Mechanics

I wish to thank the authors for their effort in going through the vast amount of comments in my previous report, and for their considerate taking so many of them into account in revising the paper. In my opinion, the article has benefitted a lot from these changes.

However, a few minor points still remain which I would like clarified before publication—most of them due to me being imprecise in my previous report, for which I apologise. (I will refer to the numbering from the first report and the authors’ answer):

1. I understand that you use the expressions from section 3.3, in the explicit case of spherical coordinates, in the Kerr example. However, the only equations in section 3.3 that are ‘explicit expressions in spherical coordinates’ are the coordinate transformation formula (3.39), the explicit metric components directly before (3.40), and the metric determinant (3.41). Therefore, in my opinion the formulation that ‘the methods […] can be generalised to arbitrary coordinates’ is quite misleading, since there is nothing coordinate-specific about the *method* used here: having read this sentence, the reader might wonder what about the methods would still need generalisation, except for the concrete form of the metric components and connection coefficients. Hence, I would still strongly appreciate some rewording of this section.

2. I realised that in my previous report I had a typo in the scalar density transformation behaviour, which you included in the paper as eq. (A.45): the exponent should be $+1/2$, such that the rescaled wave functions are really scalar densities of positive weight (not negative). I apologise for this typo. I also realised that the notation I used in this formula is inconsistent with your notation in the rest of this appendix: while you use the presence or absence of a prime on the symbol $\hat\Psi$ to denote the coordinate chart with respect to which functions are expressed, I used it to denote which local frame (in our case chart-induced) is used to represent the scalar density by a scalar function—i.e. for that purpose for which you use the subscript "Schw" or "Iso". To make the notation consistent with the rest of the paper, I suggest to write the transformation behaviour (A.45) in the form
\[\hat\Psi_x(x) \to \hat\Psi'_{x'}(x') = (\det(\partial x^i/\partial x'^j))^{1/2} \hat\Psi'_x(x').\]
Since $x'$ denotes the isotropic and $x$ the Schwarzschild coordinates, from this we obtain concretely
\[\hat\Psi'_\text{Iso}(r') = \left(\frac{r^2}{r'^2} \frac{\partial r}{\partial r'}\right)^{1/2} \hat\Psi'_\text{Schw}(x'),\]
which rearranged gives your (A.46).

I also suggest to add the fact that $\Psi$ is a scalar only under *spatial* diffeos not just above (A.23), but also in the introductory paragraph of section A.2, i.e. in the 8th line.

4. Here I also had a typo in my previous report: of course, it is not ref. [52] (old number, now [15]) that deals with low-energy quantum systems under gravity, but [51] (now [46]). I would be grateful if you could correct this oversight of mine and cite not the former, but the latter in the amended place in the beginning.

7., third point: Of course you are right that the general form (1.1) of the Schrödinger equation describes any quantum system. My point was a different one: in the sentence preceding (1.1) you speak explicitly of a wave function *on $\mathbb R^3$*, i.e. that of a system with configuration space $\mathbb R^3$. That’s why I thought that the formulation ‘the quantum mechanical wave function’ sounded a little too generic, and hence suggested to add ‘of a point particle’ to explicitly state the system under consideration from the very beginning. I still think this would be a usefull small clarification, but perhaps I am overly pedantic here—so feel free to ignore this point.

8., third point: I apologise, I automatically thought of the equation as being a spacetime-covariantised one (since the result does not depend on the choice of spacetime affine connection, as long as it's torsion-free and compatible with $\tau$ and $h$).

We thank the authors for addressing the feedback provided in the review process. We deem the updated submission to be well-suited for publication.