How-to Compute EPRL Spin Foam Amplitudes
Round 1
Reviewer 1 Report
The authors aim to provide a guide for the computation of spin foam amplitudes, which are a central element in loop quantum gravity. There appears to be a gap between the mostly mathematical literature focussing on abstract concepts, and the cutting-edge computational literature, which the authors try to fill. In the manuscript, within a concrete example the authors first explain the basic building blocks of the triangulation/2-complex, then relate it to the amplitude, and finally give some concrete numbers that the reader can compare to.
In my opinion, the paper is indeed much needed and fills an important gap in the literature. I however also think that the manuscript in its current form falls somewhat short of achieving this goal. My main concerns are the following:
1) To me it is not clear who exactly this guide is aimed towards/what background knowledge the reader is supposed to have. Especially the introduction, but also some intermediate paragraphs in the non-numerical part are *highly* technical/mathematical, and likely only comprehensible to readers who already have considerable expertise in LQG. A concrete example is the beginning of section 3 - there are a lot of buzzwords (topological BF theory, EPRL model, \gamma-simple unitary representation, state sum model) with zero explanation, but in the end their relevance for the guide itself is questionable. I would thus suggest that the authors carefully review what level of mathematical detail is strictly necessary for the paper, and then give more explanation on the essential concepts.
Let me also say that this is in stark contrast to the numerical part in section 6, which is very accessible and quite a pleasure to read.
2) I find it enormously confusing that at no point in the whole paper, it is explained explicitly what amplitude will be computed (in terms of a concrete formula), and what physical significance it carries. This is in a sense related to point 1) - the reader is assumed to know what an EPRL amplitude is, but for such a reader the value of the how-to is questionable. I also think that some explanation on where the concrete example discussed in the manuscript comes from/why it is interesting would be beneficial.
3) It took me quite a while to understand the structure of the paper - the sections are interweaved with the different steps of the guide, and it is not clear where each step ends. This should be presented more clearly.
4) I find the general order/logic of presentation somewhat confusing, and to me the authors seem to jump back and forth. The general starting point is the graph in fig. 1. At step (I), they then start with the object in eq. (1) which falls from the sky, but only really in eq. (6) it is somewhat explained what it represents, and that it actually comes from the object in eq. (4). Next the general example is transformed into the graphical language in eq. (10), after which one goes back to one of the four building blocks, only to eventually come back to the final result.
If these points are addressed by the authors, I believe that the guide can indeed fill the mentioned gap in the literature, making it an important contribution to the field of LQG.
Besides these major points, the authors should also consider the following minor comments:
a) In the first paragraph of section 2, the authors write the sentence “The generalization is straightforward.”. As a reader who wants to learn a topic, I find such sentences extremely demotivating if I do not immediately know how to generalise. I would suggest to instead point to the relevant literature.
b) In eq. (10) there are also building blocks of the form of eq. (6) but with only one blue line. To me it was not entirely clear what this corresponds to in terms of building blocks.
c) The authors explain the well-known method of series acceleration (concretely the Aitken delta-squared process) in quite some detail. In my opinion the authors should state more explicitly that this is nothing new/something they invented, and potentially cite some relevant literature on the theory behind series acceleration.
Author Response
Please see the attachment.
We hope that we can upload the new draft version soon (the system does not allow to do that before we replied to each referee).
All the best.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
This paper presents a pedagogical introduction to spin foams and vertices. It is a welcome addition to the literature since as the authors point out, the barriers of entry to the field are high.
I would urge the authors to expand section 2. It seems to assume too much from the reader. What is the connection between what is done and the path integral? What is the connection with loops and spin networks? The simplicity constraints are just mentioned without explaining what they are or come from. It will improved the readability of the article.
There seems to be an issue with the references. Sometimes they appear correctly with a number like [4], sometimes the raw LaTex seems to appear, at least in the version that got to me.
Author Response
Please see the attachment.
We hope that we can upload the new draft version soon (the system does not allow to do that before we replied to each referee).
All the best.
Author Response File: Author Response.pdf
Reviewer 3 Report
I find this paper useful in providing a guide to spin foam amplitude calculations in a certain model of loop quantum gravity. However, there are some deficiencies:
- The EPRL-FK model certainly deserves at least a brief description before the issue of amplitudes is addressed. In particular, the term BF Lorentzian theory is not even defined.
- List of literature is missing in the paper; this is probably a technical defect.
Author Response
Please see the attachment.
We hope that we can upload the new draft version soon (the system does not allow to do that before we replied to each referee).
All the best.
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
The authors have addressed all of my criticism, and the new version of the manuscript has been improved significantly. I can now recommend publication of the work.
Reviewer 3 Report
The authors have revised the paper according to my comments, and I can recommend it for publication.
