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Covariant canonical formulations of classical field theories

by Francois Gieres

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Authors (as registered SciPost users): Francois Gieres
Submission information
Preprint Link:  (pdf)
Date submitted: 2023-07-10 17:48
Submitted by: Gieres, Francois
Submitted to: SciPost Physics Lecture Notes
Ontological classification
Academic field: Physics
  • Gravitation, Cosmology and Astroparticle Physics
  • High-Energy Physics - Theory
  • Mathematical Physics
Approach: Theoretical


We review in simple terms the covariant approaches to the canonical formulation of classical relativistic field theories (in particular gauge field theories and general relativity) and we discuss the relationships between these approaches as well as the relation with the standard (non-covariant) Hamiltonian formulation. Particular attention is paid to conservation laws (notably related to geometric symmetries) within the different approaches. Moreover, for each of these approaches, the impact of space-time boundaries is also addressed. To make the text accessible to a wider audience, we have included an outline of Poisson and symplectic geometry for both classical mechanics and field theory.

Author comments upon resubmission

I sincerely thank the two referees for their careful review and positive feedback
as well as for the detailed comments and insightful suggestions.
The latter led me to revise and improve several parts or points of the text.
I have also taken this opportunity to update and complete the references.
I hope that these additions, modifications and comments represent satisfactory answers to
all of the points raised by the referees.

List of changes

I first spell out the major changes made in the text:

- Points 1-4 of referee 2: To make the introductory part easier to read, I have reorganized the introduction
by postponing the historical evolution and synthetic overview to a new section 2 entitled "Pre-/overview of results
and historical evolution of the subject".
As emphasized there, the goal of this section is to try to convey already some of the basic concepts and ideas
(within their historical context) while postponing the details to the
later parts of the notes. Accordingly, I have added references for the
mathematical notions (that are mentioned here)
to the adequate equations or subsections
of the text where these notions are discussed.
These indications should help the reader to go right away to the technical details
if he wishes to do so in this overview part of the notes.
Concerning this part, I agree with referee 2 that the description of the multisymplectic approach is somewhat cryptic
from the mathematical point of view. In fact, I have tried to put forward the physical aspects
while referring to various extensive mathematical reviews which focus on the latter aspects.
Following the suggestion of referee 2, I have postponed the mention of the "twisted affine dual" to section 3 (footnote 3).

- Point 5 of referee 2: For the Peierls bracket, I have expanded the subsection "Geometric symmetries and conservation laws"
(now subsection 4.3) so as to elaborate on the given example and on the bracket of charges.
Moreover, I added some comments concerning the mathematical underpinnings (new subsection 4.4).
The definition and construction of gravitational charges is now discussed in some detail in the new subsections 7.7 and 7.8.

- Point 6 of referee 2: Since the Koszul-Tate differential was mentioned (but not defined) in several places of the notes,
I have included a new appendix B which provides a synthetic introduction to the various differentials in field space that are
considered in the main body of the text: the horizontal differential, the BRST differential, the Koszul-Tate differential
and the BV differential. Hopefully, this addition is also useful in its own.

- Referee 1 raises the interesting and important issue of boundaries.
Concerning the boundaries of space-time at infinity or at a finite distance, I have added a
new subsection 1.3 in the introduction. (Here, I have referred in particular to the quite recent review
"From asymptotic symmetries to the corner proposal" by L. Ciambelli: the introduction of this review
provides several hundred references which address the issue of boundary conditions and terms
as well as the relationship with theories in the bulk. Moreover, it elaborates in detail on the particular approach
of the corner proposal.)
In the main part of the text, I have followed the suggestion of referee 1 by adding
some comments on the issue of boundaries
within the different approaches that are considered in the notes (new subsections 3.5, 4.5, 5.5, 6.7 and 7.8
as well as an elaboration on boundary conditions and boundary actions in general relativity on pages 47-51).

- Since most of the recent literature on the covariant phase space approach to gravitational theories
is based on the work of R. Wald and his collaborators, I have added an introduction to this topic
while considering the recently given geometric reformulation of this work (new section 7.7).
This section also addresses the relationships with the results discussed in other parts of the notes.

Concerning some details raised by the referees:

- Point 7 of referee 2: As I have now spelled out in the text, eq.(7.97) represents the (complete) expansion
of the Lagrangian density with respect to the coupling
constant g viewed as a deformation parameter. Only the part L _0 describing the dynamics of a collection
of free Abelian fields is invariant under the given symmetry transformation (7.99) thus leading to the conservation of the
associated ``color charges'' (7.101). As noted after eq.(7.101), the latter have the same expression as (7.66)
for vanishing background charges. (Other surface charges as well as their algebra are discussed in the cited reference [242].)

- Referee 1 made a remark concerning the equivalence of different expressions for the symplectic two-form of gravity
(which are now numbered by (5.44) and (5.55)): as he suggested I added an earlier reference for this result (namely the work
of J. Lee and R. Wald) and I made more precise the role of the boundary term in the action.

- Referee 1 emphasized the role of boundaries and boundary conditions for establishing (in eq.(5.12))
the independence
of the symplectic two-form on the hypersurface Sigma over which one integrates.
The new subsection 7.8 now treats some basic aspects of manifolds with a boundary
notably following the work of D. Harlow and J. Q. Wu. The results obtained by these authors
concerning the equivalence of Poisson brackets and Peierls brackets are now mentioned in the new subsection 5.5.

- Concerning the remark of referee 1 on the subsection "Asymptotic symmetries" (now numbered as subsection 7.5):
At the beginning of this part, I have added a parenthesis concerning the ``asymptotia''.
Moreover, in the paragraph dealing with general relativity I added a parenthesis (before eq.(7.71))
to emphasize that the Killing vector fields (of the background metric) parametrize the asymptotic symmetries.

- Following the suggestion of referee 1, I have included some comments and relevant references to the
so-called Weiss variational principle (after equation (6.44) as well as on top of page 122).
Indeed, as it is explicitly pointed out in reference [234], this variational principle amounts to a derivation
of Noether's first theorem if the infinitesimal variations are viewed as infinitesimal symmetry transformations
of an invariant action functional. Thus it leads to the canonical energy-momentum tensor in field theory
(which is referred to as ``Hamiltonian Complex'' or ``Hamiltonian tensor'' in reference [234]).

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 2 on 2023-10-20 (Invited Report)


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


Does not discuss much quantization.


Let me thank the author for doing many corrections. I believe the introduction is clearer now, and I appreciated the additions in sections 4.3 to 4.5, 7.7 and Appendix B. I recommend the paper for publication.

Requested changes

Let me just mention a minor misprint above equation 4.15 were the text is strangely cut.

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Anonymous Report 1 on 2023-7-17 (Invited Report)


This revised version addresses in a constructive manner the various questions and comments which were raised, so I recommend it for publication.

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