Covariant canonical formulations of classical field theories

The review is pedagogical with particular emphasis on the development of the field with rather exhaustive references.

Some technical concepts are sometimes mentioned without definition and the boundary between what is pedagogically described and what is only mentioned as a reference is not always sharp.

In this review the author gives a comprehensive description of several approaches to the covariant canonical formulation of relativistic field theories with constraints. It is often necessary to use the hamiltonian formulation of a theory to describe its Cauchy problem, its conserved charges, and eventually its quantization. The lack of Lorentz invariance and more generally diffeomorphism invariance is often a technical drawback that has lead mathematical physicists to define a covariant canonical formulation of classical field theory. This review describes these attempts with particular emphasis on the historical evolution of the field. The three approaches described in the review are the multiphase or multisymplectic approach, the covariant phase space approach and finally the variational bicomplexe that somehow encompasses the two other approaches. The review emphasizes that these constructions are all equivalent, including the Peierls bracket (a Lorentz invariant bracket defined such as to reproduce the commutator of causal quantum fields). I guess that this equivalence is only at the formal level usually required by physics analysis, but the mathematical objects may differ in a rigorous definition since the truncated phase space seems to be defined with the topology of a finite dimensional jet space whereas the covariant phase space is defined with the appropriate topology on the function space of solutions to the equations of motion, which may differ from the topology of the infinite jet space in the bicomplexe. The review is well written and can certainly provide an interesting reference in the literature that deserves to be published.

Nevertheless, I believe the presentation could perhaps be improved.

I found the introduction and in particular the overview rather hard to read. The first presentation of the different approaches could be more digest if the author was not putting so much technical details on the evolution of the field and on comparaisons of objects that are not yet defined in the review. I appreciate that the author is very precise about references, but it may help to postpone some of the historical points to section 1.4. Actually I have not found the precise definition of the jet bundle affine dual in the review whereas it is already mentioned twice in page 2. If it is only going to be defined as the extended multi-phase space, including both the canonical momentum vector field and the Hamiltonian as a separated variable, I believe that this more mathematical object could only be mentioned in section 2 with the appropriate reference. The description of line 2.b is also particularly cryptic. Since the author decides to enumerate the various approaches according to the way they appear in the flow diagram, the flow diagram should probably appear in the same subsection. The reference to Appendix C in the discussion of line 1 is not very clear, probably the author means that a review of the Hamiltonian formulation can be found in Appendix C, but I think this should be said in words, perhaps in a footnote.

This may be a personal opinion, but I was slightly disappointed that no more details would be given in section 3.3 about the Peierls bracket of conserved charges in general relativity, and similarly in 6.1.3 for corner charges in general relativity. These examples are particularly relevant to recent research developments in asymptotic symmetry and I think they would deserve to be included in the review.

Sometimes the author briefly mention something without defining it and nevertheless refers to it afterward just like if he did. This the case for the jet bundle affine dual in the introduction as mentioned above, but also for the Koszul-Tate differential mentioned in (6.15) and then referred to in (6.38). The Koszul-Tate complexe is mentioned three times in the review but never defined. I believe it would be appropriate to give the definition in (6.15) and maybe its relation to the antifield formalism of section 6.6.2.

I have not understood why the author needs to linearise in (6.96) to define the color charge in (6.100).

Despite these minor queries, I believe this review is of significant interest to the community and deserves to be published in Scipost.

1) I found the introduction and in particular the overview rather hard to read. The first presentation of the different approaches could be more digest if the author was not putting so much technical details on the evolution of the field and on comparaisons of objects that are not yet defined in the review. I appreciate that the author is very precise about references, but it may help to postpone some of the historical points to section 1.4.

2) Actually I have not found the precise definition of the jet bundle affine dual in the review whereas it is already mentioned twice in page 2. If it is only going to be defined as the extended multi-phase space, including both the canonical momentum vector field and the Hamiltonian as a separated variable, I believe that this more mathematical object could only be mentioned in section 2 with the appropriate reference.

2) The description of line 2.b is also particularly cryptic.

3) Since the author decides to enumerate the various approaches according to the way they appear in the flow diagram, the flow diagram should probably appear in the same subsection.

4) The reference to Appendix C in the discussion of line 1 is not very clear, probably the author means that a review of the Hamiltonian formulation can be found in Appendix C, but I think this should be said in words, perhaps in a footnote.

5) This may be a personal opinion, but I was slightly disappointed that no more details would be given in section 3.3 about the Peierls bracket of conserved charges in general relativity, and similarly in 6.1.3 for corner charges in general relativity. These examples are particularly relevant to recent research developments in asymptotic symmetry and I think they would deserve to be included in the review.

6) Sometimes the author briefly mention something without defining it and nevertheless refers to it afterward just like if he did. This the case for the jet bundle affine dual in the introduction as mentioned above, but also for the Koszul-Tate differential mentioned in (6.15) and then referred to in (6.38). The Koszul-Tate complexe is mentioned three times in the review but never defined. I believe it would be appropriate to give the definition in (6.15) and maybe its relation to the antifield formalism of section 6.6.2.

7) I have not understood why the author needs to linearise in (6.96) to define the color charge in (6.100). Could you explain why.

This is an interesting review article on the so-called covariant canonical approaches to classical field theories. The review is well-written and has the advantage of presenting various approaches scattered in existing literature in a unified picture with coherent notations. Because this is mainly a review article, I will not focus on possible criticisms of shortcoming of the various approaches described, but instead list several points which I think should be revised in order to improve the clarify of the manuscript. I recommend the manuscript for publication in SciPost, provided the following points are addressed and clarified:

- Generally speaking it could be useful to clarify the role of boundaries at various points throughout the manuscript. It seems to me that in most of the manuscript the author is considering that boundaries are absent, although they are clearly important for the discussions of e.g. section 6. The author could for example state more precisely in which section boundaries are present or not, and if they are absent, how the various results would extend in the case of non-trivial boundaries. For example, since the Peierls bracket discussed in section 3 relies on the use of Green functions, it seems that the latter would be strongly affected by the presence of boundaries. Similarly, in the middle of page 38 (paragraph ‘’From the invariance…’’) it is claimed that (4.39) and (4.45) are equivalent, but it is not clear whether this holds because there are no boundaries. It also does not seem that reference [111] cites at this point is the most appropriate reference for this statement. Similarly, in (4.12) for the proof of invariance of the symplectic 2-form under choice of \Sigma, the author is then seemingly introducing a boundary, but invoking the condition that J^\mu vanishes on the boundary to get the desired result. I agree that this is a possibility, but once again, I think that this part would benefit from a clearer discussion of the role of boundaries. It seems for example that references such as https://arxiv.org/abs/1906.08616 discuss the role of boundaries and relation with the Peierls bracket.

- in section 6.5 it is not clear to me what the author means by ‘’asymptotic symmetries’’. There is no notion of ‘’asymptotia’’ defined in this part, especially in the subsection about general relativity (where there is also no discussion on ‘’symmetries’’).

- the author could consider including a discussion on the so-called Weiss variational principle (https://arxiv.org/abs/1708.04489) which also gives rise to a covariant Hamiltonian.