Dear Editor,

We thank you for the rapid and efficient handling of our manuscript. We greatly appreciate the work of the Referees who carefully read our manuscript and gave us numerous constructive suggestions for improving its clarity and focus. We found all recommendations entirely reasonable, and have extensively revised the manuscript accordingly.

Hereafter we respond point-by-point to the Referees’ comments.

=========================== REFEREE 1 [Anonymous] ===========================

The Referee's report appears based on generic arguments regarding existing textbooks which are not reflected in the actual content of these texts.

The Referee writes:

<< As far as I can see the paper does not contain any new material that has not already be covered in the textbooks by, for instance, Kleinert, Negele&Orland, and Stoof et al. to name just a few. >>

On the contrary, most of the material we cover is not presented in the books cited. Section 3 discusses aspects which have been debated in the recent literature, such as the issues related to change of variables and the Hubbard-Stratonovich transformation, which are not treated in the books cited. Sections 5 and 6 discuss both the Bose gas and the BCS considering finite-range interactions, which are not treated in the books cited. Furthermore, none of the books cited provides an explicit comparison between the Hamiltonian approach and the path integral; typically, they present one or the other.

<< Moreover, instead of properly referencing this literature the authors seem to imply that these subtleties are treated here for the first time, which is just not fair in my opinion and does not do justice to the existing textbooks that have treated all the topics of this review >>

Relevant textbooks [1-8] are cited repeatedly throughout the text where appropriate. We are certainly not claiming to discuss these subtleties or anything fundamentally new for the first time; this is reflected in the fact that we have submitted our work to a journal whose scope is to publish "didactic material", not research articles. However, we believe that these notes provide a more detailed treatment than that offered in the cited textbooks, which for obvious reasons cannot dwell too much on these details. In addition, as remarked above, much of the material we present is not treated at all in these textbooks.

<< existing textbooks [...] have clearly explained the problems involved. >>

The fact that these issues are not fully explained in textbooks is demonstrated by the publication, in recent years, of research articles on these issues [9-12, 14, 15]. And it is reflected also in the reports by the other two Referees.

=========================== REFEREE 2 [Prof. Rancon] ===========================

We are grateful to the Referee for his precise and detailed assessment of our manuscript and for his suggestions, which allowed us to correct some errors and improve the overall clarity and focus of the work. We anticipate that all of their suggestions have been carefully implemented in the new version.

The Referee writes:

Report

<< The manuscript discusses pedagogically the subtleties that arise when dealing with coherent state path integrals. These include the issue of non-linear changes of variables, which can lead to wrong results when not performed properly; the fact that Hubbard-Stratonovich fields are white noise and should be handled properly in functional determinants; the problem of convergence factors in Matsubara sum. The first two issues have been the subject of some debates in the recent literature, and could deserve being the subject of a review or of some pedagogical notes. This is much more debatable concerning the last issue, since it is well-known and discussed in many textbooks. >>

We concede that the opportunity of treating the "problem of convergence factors in Matsubara sum" may be "debatable". So let us express our opinion. While it is true that this problem is covered in most textbooks, and thus nothing fundamentally new or different can be said, we believe that textbooks (not through fault, but by necessity) do not devote the necessary space to this issue so that students approaching the path integral for the first time can learn how to calculate thermodynamic potentials without feeling like something fishy is going on behind the scenes, reading middle-derivation that "we forgot to consider some commutation rules", or coming across rather airy arguments to erase spurious contributions to the energy, and so on. Moreover, these issues are typically discussed in footnotes or "optional paragraphs" written in small print, see e.g. the books by Dupuis or Stoof, which are still among the clearest in their treatment. In our experience as lecturers and students, sentences such as

"Note that we do not obtain the correct value of the constant term in the rhs of (7.253) because the Nambu formalism mixes creation and annihilation fields" (Dupuis, p.511)

or long "chains of justifications" such as

"the sum ... comes from interchanging the fermionic spin-down fields and the corresponding equal-time limiting procedure in order to write the above action in matrix form. This was more extensively explained in Sect. 11.4." (Stoof, Section 12.7, p. 286) -> "where the second term on the right-hand side is explained by the discussion from Example 11.1." (Stoof, Section 12.4, p.249) -> "In the last step of (11.39), we had to add a factor ... to compensate for the change of limits, as follows from Example 7.2". (Stoof, Example 11.1, p.249) -> ...

do not make it easy for the beginner to learn with confidence.

Our goal is to make the mechanism as transparent as possible. With this new version of the manuscript, greatly improved thanks to the inputs of the Referee, we believe we are moving well in this direction.

<< Furthermore, the manuscript is somewhat unfocused, since additional textbook material unrelated to coherent state path integrals are discussed at length (sec. 5.1.3, 5.2, 6.3, App A and B). These sections should be removed. >>

Sections 5.1.3, 6.3, Appendix A and B have been removed. However, we believe that the discussion of finite-range interactions is useful and should not be omitted, especially since it is not typically covered in textbooks. To this end, we have structured Section 5 differently, directly addressing the finite-range case, similarly to what we did in Section 6. In both cases, the standard results for zero-range interaction are briefly presented as a special case of the results we discussed.

<< Sometimes, the authors don't seem to quite know the level of pedagogy they aim for. For instance, they want to start at a very basic level, but do not give the canonical (anti)-commutation relations of the operators, which seems strange. >>

We understand the Referee's point. Consequently, Section 2 has been greatly expanded to include many more of these details.

<< 1- I don't quite agree with the discussion of Q_k(eps) around Eq. 2.4. In Eq. 2.4, all terms of order greater than 1 in eps are neglected, not just the ones in Q_k(eps). See the (correct) discussion of the functional determinant, which is computed with an error of order 1/M^2~eps^2. >>

We recognize that this discussion was somewhat unclear. In the new version of the manuscript, we present an improved discussion [around Eq. (2.7)] based on a different manipulation of the double series. In our opinion, this goes straight to the point in a much cleaner way, and should ease the Referee's concern.

<< 2- page 12, I do not understand what the discussion of mean-field vs saddle point brings as it is written. Furthermore, I have an issue with the calculation done in Eq. 3.41: there, N is NOT a HS field behaving as a white noise, so there is no reason why there should be a correction term g/2 in the exponent exp(beta(mu+g/2-Ng). Furthermore, Eq. 3.39 is ambiguous: what is S_MF in that equation? [...] My suggestion is that, after the calculations done on that page are corrected/clarified, a discussion of the difference between MF and SP would be interesting, and, it seems to me, new. This is a question left open in the conclusion of Ref. 14 that would be interesting to address here. >>

We realized that this discussion was partially flawed and thank the Referee for pointing this out. We rewrote from scratch the discussion, and put it in the separate subsection 3.3.2.

<< 3- In the fermionic case (p.13), there does not seem to be an Ito correction term. Why is it so? It is unfortunate that the authors do not discuss the discretized version to explain why it appears to be so different from the bosonic case. This is again a problem that has not been addressed in the literature and deserves a pedagogical discussion here. >>

The difference with respect to the bosonic case originates in the fact that in the fermionic case there are two different HS fields, which are not auto-correlated. This is now discussed in Section 3.4, where we also provide a "physical picture" of the HS fields as "force carriers" that allow us to resolve a contact interaction between four fermionic fields as the exchange of HS fields (in particular, phi_1 - phi_2 mixing) between two pairs of fermionic fields. Such physical interpretation of the HS transformation is also provided below Eq. (3.32) for the bosonic case. We believe that this should make the discussion, and the white-noise nature of HS fields, clearer and more concrete.

<< 4- When dealing with convergence factors for Matsubara sums, the authors do not do much better than what most textbooks usually do: they drop the convergence factors that appear at the discretized level, to add them back when necessary, with a sign only chosen to reproduce the correct result. This is sloppy and goes against the advertised pedagogical purpose of these notes. >>

In the new version of the manuscript, we have been careful in keeping the convergence factors throughout the calculations. Furthermore, we have rewritten the beginning of Section 4 (p. 16-17) and extended the discussion of Section 4.2 to make it very clear that the "sign" of convergence factors is not arbitrarily "chosen to reproduce the correct result", but it is dictated by the underlying time-ordering. Elaborating on the method adopted by Stoof, in Section 4.2 we explicitly show that one can effectively change the time ordering and with that the signs of the convergence factors, so that the same result if obtained in the end.

<< The sentence "‘reg’ reminds us that the divergent summation should be properly regularized. Here ‘properly’ means, of course, that the result should give back Eq. (4.7)" is in this respect particularly damaging. >>

The Referee's point is understandable. We changed the tone of our approach to this point, and invite the Referee to consider the new discussions around Eq. (4.22) and (4.36). The calculation in Eq. (4.25)-(4.28) is also a new addition. We believe this modifications go in the direction implied by their comment.

<< Similarly, as it is written, Sec. 5 and 6 are superfluous, as the handling of convergence factors is not done better than in textbooks (it is just a rehash of, e.g., chapter 7 of Dupuis'). For instance, the sentence "The treatment we have presented, which is based on the careful introduction of convergence factors motivated by the implicit time ordering of the path integral" is misleading, since the convergence factors have disappeared in Eq. 5.37, and are added by hand in 5.44 to obtain the correct result. If they were kept throughout the calculation, there would be at least a pedagogical interest. >>

In the new version of the manuscript, we have been careful in keeping the convergence factors throughout the calculations, and we hope the Referee will find that satisfactory. Again, we are not doing anything fundamentally different from what is done, e.g., by Dupuis and others. However, we are doing it less hastily, and hopefully this will be useful to students. Moreover, in our opinion there are several other aspects of pedagogical value that are unique to Sec. 5-6 of these notes: - Explicit comparison of the Hamiltonian and path-integral approaches - Treatment of finite-range interactions - Comparison of two alternative time-orderings, which explicitly shows how the form of convergence factors is dictated by the underlying time-ordering.

Requested changes

The Referee asks:

<< 1 - remove sec. 5.1.3, 5.2, 6.3, App A and B >>

Sec. 5.1.3, 6.3, App. A and B have been removed. Sec. 5.2 has been expanded to replace and generalize the discussion of Sec. 5.1, as described in our comments above.

<< 2- Sec. 2: the canonical (anti)-commutation relations should be given. The fact that fermionic operators anticommute with Grassmann variables should be discussed. The latter should be defined. >>

We have given the canonical (anti)-commutation relations, defined Grassmann variables, and discussed their relevant properties.

<< 3- Eq. 3.23: H(N) should be defined right away. It is implicit, and can very well be confusing (since it looks very much like \hat H, and could imply that it is \hat H with \hat N replaced by N, which it is not!) >>

H(N) is now defined explicitly.

<< 4- Bosonic and fermionic Matsubara frequencies should be defined around eq. 4.2, not in Sec. 4.2 >>

Matsubara frequencies were already defined in Eq. (4.1), just above Eq. (4.2), and it is the same in the new version.

<< 5- after eq. 4.16: The number N can be set by taking the limit beta->infty, which unbigously set it to 0. >>

We agree. We added this comment after Eq. (4.16).

<< 6- below Eq. 5.2, L is not defined, and the dilute limit should read a_s^D N/L^D<<1. >>

We agree. L is now explicitly defined as the linear size of the system and we corrected the typo on the dilute limit.

<< 7- p. 31: I do not understand the equation i pi lim_{delta->0^+} Sum_n e^{-i omega_n delta}=0, nor its connection to Eq. 4.27 >>

Indeed this is not a rigorous equality, but rather a regularization condition. We now state this more clearly after Eq. (4.31), which is referenced also after Eq. (6.35).

=========================== REFEREE 3 [Prof. Tempere] ===========================

We thank very much the Referee for his positive evaluation and useful comments.

The Referee writes:

<< Firstly, it was necessary for me to remind myself that Grassman variables not only anticommute among each other, but also anticommute with their operators. That appears when having the annihilation operator act on the right hand side of 2.14, in order to obtain 2.15a. It might be useful to remind other readers of this as well. >>

Section 2 has been expanded to include many more of these details, including canonical commutation and anti-commutation relations, the definition and the properties of Grassmann variables, the expressions for the trace using coherent states, the expressions of complex and Grasmann Gaussian integrals.

<< A step that may deserve a bit more attention is the one going from (3.20), the second-quantized version of the Hamiltonian, to (3.22), the Euclidean action. In the last term of (3.22). In relation to the problem identified in the section below, one could wonder why there is (a a)^2 to start with, rather than (a a)(a* a-1). Usually, the problem is formulated the other way around, with a quantization scheme (for example Weyl quantization) specifying how to assign operators to phase space variables. Since there are different choices possible, I think it would be clarifying if the authors gave more explanation at this point. >>

The Referee is correct in pointing out that different "orderings" (e.g. normal ordering, anti-normal, Weyl ordering, ...) are possible when defining the discretized path integral, which lead to different discretized Euclidean actions representing the same quantum Hamiltonian. All orderings are physical equivalent in the sense that if one computes the partition function in discrete time with the chosen discretization, and at the end takes the limit dt->0, they will obtain the same result, which is the exact partition function of the original quantum Hamiltonian. However, among the possible path integrals/actions, the normal-ordered action is is distinguished in that it admits a safe, direct continuum interpretation. In other words, only the normal-ordered action can be evaluated directly as a path integral in the continuum, without the need to add correction terms related to the discretization rule. In the present example, the normal-ordered form of the interaction Hamiltonian is (a a a a). The fact that the normal-ordered form of the Hamiltonian must be used is now stated more explicitly in Section 2; it is also implicit in the definition (2.11) of the continuous-time Euclidean action, since H(a*, a) = < a | \hat H | a >, and only if all creation operators are to the left and all annihilation operators are to the right one is able to evaluate explicitly this expectation value. This comment on the different orderings has been added on p.5.

<< Another interesting detail that could be commented on is the \omega=0 frequency term for bosons. When using the \omega_(-n) = - \omega_n symmetry as in (4.22) care must be taken for that special term that is the only one not doubled by this symmetry. It would be nice if the authors could add a discussion of that detail as errors there also lead in an additional term not unlike the one they address already. >>

We have modified Section 4.2, see in particular Eq. (4.22) of the new manuscript, to make this point clearer.

Summary of the main changes (for details on some particular changes, see the response to the Referees):

Sections 5.1.3, 6.3, Appendix A and B of the original manuscript have been removed.

Section 2:

• Added canonical commutation and anti-commutation relations, the expressions for the trace using coherent states, the expressions of complex and Grasmann Gaussian integrals
• Rewrote and clarified the discussion around Eq. (2.4) [Eq. (2.7) in the new version]
• Defined Grassmann variables and their properties

Section 3.3.2: Completely rewritten

Section 3.4: Expanded and improved clarity

Section 4.2: Expanded and improved clarity

Section 5: Considered directly finite-range interactions

Section 5.2:

• Clarified the origin of convergence factors
• Added discussion of alternative time-ordering

Section 6.2:

• Clarified the origin of convergence factors
• Added discussion of alternative time-ordering

Some typos have been corrected