Causal structure in spin-foams

0- the issue of causality in spin foam models is very important, and too little explored in the literature, so this paper is a welcome contribution in this respect

1-the presentation is very good in terms of clarity and structure, very pedagogic and very well written

2-the technical improvements of the construction of causal spin foam amplitudes, and in particular the EPRL model, provided by the authors, are valuable

3- the paper adds several elements of precision in the definition of causal structures in simplicial gravity, and the relevant discussion is very well done

1-In the end, the original contribution of this paper to the literature is not too substantial. The technical improvements in the construction of causal spin foam models seem rather marginal; the definition of causal structures in simplicial gravity is precise but not new, and the overall strategy for defining causal spin foam models or identifying causal elements in existing ones is the same that had been proposed earlier.

2-if the paper is to be understood (also) as an introduction to the topic of causality in spin foams or as a review of the existing work, it is definitely lacking in giving a proper account of how the current knowledge has been achieved, in explaining the connection to other facts about quantum gravity (e.g. those explained in the early QG papers by Teitelboim et al.) and in citing relevant work.

3-the outstanding open issues concerning causality in spin foam models have to do with the use of the "causal models" and their physical interpretation, as independent (but of course related) constructions from the orientation-independent ones, again in the spirit of the formal QG path integral constructions or of the QFT analogues. These issues are not even touched by the authors, and in some points their presentation is on the contrary quite confusing, if not misleading, concerning these aspects.

This paper presents an overview of how causality can be identified at the level of the simplicial complexes and geometries underlying spin foam models and how the latter can be modified to give "causal spin foam amplitudes". In particular, a specific construction of a causal version of the EPRL amplitudes is presented.

I have several comments about this paper.

a. the basic definitions of discrete bare causality and time-orientability, for the simplicial complex, the translation in terms of dual 1-skeleton, and the Lorentzian simplicial geometry in terms of Regge calculus, i.e. sections II,III and IV, are basically like in [6], only slightly more detailed in what concerns the global Lorentzian properties of the simplicial complex and its geometry, and about boundary data. This should be made more clear.

b. in fact, the broader idea could be traced back to the formalism of quantum causal histories, developed in F. Markopoulou, L. Smolin, gr-qc/9702025 [gr-qc]; F. Markopoulou, L.Smolin, gr-qc/9712067 [gr-qc]; F. Markopoulou, hep-th/9904009 [hep-th]; E. Hawkins, F. Markopoulou, H. Sahlamnn, hep-th/0302111 [hep-th]; they should be cited

c. also for what concerns the restriction of spin foam amplitudes to causal configurations, i.e. the construction of causal spin foam amplitudes, the strategy outlined at the end of page 7 is exactly the one argued for and employed in [6], where the relation to Teitelboim's construction of the QG causal propagator was also discussed, and then later in [7]. The authors are possibly improving on this latter construction, although not in a very dramatic way, but the overall strategy, and procedure is basically the same. Also this should be made clear when introducing the general procedure, rather than simply mentioning it when comparing the BC and EPRL construction on page 12.

d. it should be made much more evident that the dual 1-skeleton of the simplicial complex, when interpreted in causal terms, contains closed timelike loops. Transitive closure (which gives poset from dual 1-skeleton) implies contraction of some dual faces (the closed timelike loops), but it is important to note that this contraction changes the combinatorics, thus the duality with the simplicial complex (it amounts to a specific coarse graining of the simplicial complex). It would also change the spin foam amplitudes, in fact. The role of closed timelike loops in causal spin foam models has been studied, from a quantum information perspective, in E. Livine, D. Terno, gr-qc/0611135 [gr-qc], which should be cited.

e. the discussion of the PR model is focusing exclusively on the spin representation of the model; this is surprising, since the orientation independence of the amplitudes, the connection to the discrete gravity path integral, which makes clear where this orientation-independence comes from, and the geometric interpretation of the same amplitudes, is much clearer in the equivalent expression as lattice BF theory, or, indeed, 3d lattice gravity written in Lie algebra and group variables. The spin foam representation is simply the result of a change of variables, which can be performed explicitly and straightforwardly in both directions (this is the exact counterpart of writing down the path integral for a particle on the 3-sphere in terms of spherical harmonics). From this point of view, there is no need to go to the semiclassical limit to see the connection to discrete 3d gravity, as implied by the authors. In fact the same is true for any spin foam model, although the corresponding lattice gravity path integral is more involved (as it should be) in the 4d case. This has been shown in several papers, for example in M. Finocchiaro, D. Oriti, 1812.03550 [gr-qc]
In particular, it is true for the BC model (whose lattice gravity expression has been introduced in V. Bonzom, E. Livine, 0812.3456 [gr-qc]), and for the EPRL model. This shows that the reliance on the semi-classical approximation to give a "causal"interpretation to the terms appearing in the spin foam amplitudes is not needed.

f. From this point of view, the strategy applied by the authors to impose a causality restriction on the amplitudes, and by the other authors before them including the authors of [6,7], is rather artificial. Indeed, it basically amounts to forcing the amplitudes in the spin representation to take a "path integral-like" form, with the appearance of the exponential of an action, and then restrict the corresponding sum over opposite orientations for the wedge sub-amplitudes., as one would naturally do in a quantum gravity path integral following Teitelboim. Given that the path integral expression is already available for the same amplitudes, without any artificial splitting and in the classical variables from the underlying phase space, it would seem much more natural to perform the causal restriction in this expression. In fact, this was done for the Ponzano-Regge model coupled to point particles, in D. Oriti, T. Tlas, gr-qc/0608116 [gr-qc], where the resulting appearance of the causal propagator for the point particles is also shown.
The authors could explain why the focus on the spin representation instead, since their strategy is actually motivated and phrased in a path integral perspective (by the way, one can again do the same for the point particle on the 3-sphere, and construct different versions of the path integral, the causal propagator as well as the orientation-independent Hadamard propagator; both can be expanded in spherical harmonics; the expression in spherical harmonics of the causal propagator does not coincide with the "causal restriction of the Hadamard propagator written in spherical harmonics". Even insisting for some reason on applying the restriction in the spin representation, the authors fail to explain why their procedure would be more convenient that using the integral expression of the 6j-symbol and restricting that, which was the procedure in [6] in the 4d case. Using this integral expression, one sees immediately the terms which will then be selected by the semiclassical limit as forming the discrete gravity path integral, which the case also in the 4d case, for both BC and EPRL model.

g. the authors are of course free to regard the "causal spin foam models" as simply nicely identified components of the original spin foam models, rather than new models, and their usefulness as limited to signaling out underlying causal properties. However, this is rather unnatural from the QG path integral perspective and quite unnatural also from the point of view of particle propagators or QFT 2-point functions, where causal and orientation-independent constructions are used alongside one another, and on equal footing (in fact, the Feynman propagator is clearly playing a more prominent role in QFT), since they correspond to different "observables" or, more generally, answers to different questions one can pose to the theory. In any case, causal models seem to be actually put forward as new models also by the authors, when they suggest that they can "cure" issues of the non-causal ones. If this was not the case, i.e. if one did not work with causal models as "the correct models", by dropping all non-causal configurations, they could not cure anything, since one would still be working with the non-causal ones, in the end.

h. I am puzzled by the discussion of the relation with the construction in [7]; in particular, the answer to the question posed in 1. on page 13 seems obviously "yes". It is the same analysis by Teitelboim, that shows that, as the only difference between the two sectors of solutions of the simplicity constraints (the third being the degenerate configurations) is indeed the orientation of the resulting bivectors, and this is exactly what is interpreted in causal terms by the authors to motivate their restriction. When the Immirzi parameter is present the difference between gravitational and topological sectors is basically irrelevant, for what concerns us here. This is also confirmed by the BC case, and by the fact that the simplicity constraints that are actually imposed in the construction of BC and EPRL models are in fact the linear ones, which are stronger and select away the topological sectors.

i. I am in fact even more puzzled by the authors mentioning the "cosine problem". Their own analysis of causality and of the gravitational path integral, discrete or continuous, should have made clear to them that there is no cosine problem at all. Spin foam amplitudes which are intended to define the projector onto solutions of the Hamiltonian constraint of canonical quantum gravity must be orientation independent, and thus sum over opposite orientations also in the semiclassical limit (unless the configurations with opposite orientation are suppressed by boundary conditions/states). The appearance of the cosine in the asymptotic is a confirmation that they do what they are supposed to do, at least in this regard. By the way, this is also confirmed by the 3d Ponzano-Regge model and the way the causal restrictions break symmetries that underly the topological nature of the model and, from the canonical perspective, prevent the imposition of the canonical constraints.
Causal spin foam amplitudes correspond to a different path integral construction, which is possibly equally valid and possibly useful also from the canonical perspective, but do not define such imposition of the Hamiltonian constraint. They are defining something like a "Green function" for the Hamiltonian constraint, which is in fact more natural from a Lagrangian path integral perspective, but less easy to interpret from a canonical one. This was discussed in the early spin foam literature, but it is also very clear already in the old papers by Teitelboim. In fact, the appearance of the cosine from a sum over opposite orientations in the (formal) QG path integral in the continuum, and an explanation of why this is actually needed to have a path integral that satisfies the canonical constraints, is explained in J. Halliwell, J. Hartle, Phys.Rev.D 43 (1991) 1170-1194 (section IVA, see in particular the end of page 1181) from the Hamiltonian perspective, relating it in particular to the known difference between diffeomorphisms and canonical transformations, thus adding an important layer to the interpretation.

- in the end, the technical improvements of the spin foam construction provided by the authors seem rather marginal to me, compared to the existing literature, although valuable, and their contribution to the overall strategy is limited to some added element of precision in the definition of causal structures in simplicial gravity, although this can be appreciated. Thus the original contribution of this paper is not too substantial.

- the presentation is very good in terms of clarity and structure, very pedagogic and very well written; however, if it is to be understood as an introduction to the topic of causality in spin foams or as a review of the existing work, it is definitely lacking in giving a proper account of how the current knowledge has been achieved, in explaining the connection to other facts about quantum gravity (e.g. those explained in the early QG papers by Teitelboim et al.) and in citing relevant work.

- finally, technical improvements in specific spin foam constructions are welcome. However, the outstanding open issues concerning causality in spin foam models have to do with the use of the "causal models" and their physical interpretation, as independent (but of course related) constructions from the orientation-independent ones, again in the spirit of the formal QG path integral constructions or of the QFT analogues. These issues are not even touched by the authors, and in some points their presentation is on the contrary quite confusing (if not misleading), concerning these aspects.

Given the above, and while I appreciate very much several aspects of this paper, and its original contribution, I cannot recommend publication in the present form. I recommend instead a profound revision, and then, in case, resubmission.

The authors should revise substantially the paper, taking into account points a-i in the report.