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Symplectic reduction of Yang-Mills theory with boundaries: from superselection sectors to edge modes, and back
by A. Riello
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Submission summary
| Authors (as registered SciPost users): | Aldo Riello |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202102_00016v1 (pdf) |
| Date submitted: | 2021-02-10 18:43 |
| Submitted by: | Riello, Aldo |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
I develop a theory of symplectic reduction that applies to bounded regions in electromagnetism and Yang-Mills theory. In this theory gauge-covariant superselection sectors for the electric flux through the boundary of the region play a central role: within such sectors, there exists a natural, canonically defined, symplectic structure for the reduced Yang-Mills theory. This symplectic structure does not require the inclusion of any new degrees of freedom. In the non-Abelian case, it also supports a family of Hamiltonian vector fields, which I call ``flux rotations,'' generated by smeared, Poisson-non-commutative, electric fluxes. Since the action of flux rotations affects the total energy of the system, I argue that flux rotations fail to be dynamical symmetries of Yang-Mills theory restricted to a region. I also consider the possibility of defining a symplectic structure on the union of all superselection sectors. This in turn requires including additional boundary degrees of freedom aka ``edge modes.'' However, I argue that a commonly used phase space extension by edge modes is inherently ambiguous and gauge-breaking.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2021-5-8 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202102_00016v1, delivered 2021-05-08, doi: 10.21468/SciPost.Report.2886
Report
Strengths:
This is very clearly written, and well worked out, account of symplectic reduction for YM theories, in the context of boundaries. It extends (?and completes?) an impressive line of work by the author and others over some five years. It discusses well, and convincingly, the relation to other approaches and constructions, especially by Friedel and co authors (beginning, so far as I know, with Donnelly and Friedel in 2016). This is the topic of Section 5.
Weaknesses:
1: About content: I think the only weakness is that I would like the Section 5 discussion of the relation to Friedel and co authors to say some more details about the relation to their more recent papers, which are cited as [27] to [29] on p. 23 but not discussed In the present draft, the only real point of reference is Donnelly and Friedel in 2016 (the author’s [26]).
2: About language. There are slight lapses of English throughout; but one can always reconstruct the meaning. Examples at the beginning and end are as follows.
Sec 1.1, paragraph 3, line 1: say: perspective, what is responsible for this ETC
Sec 1.1, paragraph 4, line 3: say: start with the second task
Sec 1.2, paragraph 2, line 2: say: concentrate one’s efforts
Sec 1.2, last paragraph (p.3), line 1: say: The fixing of a (covariant) superselection sector ETC
Sec 1.3, paragraph 3, last line: say: denoting by s^i
Sec 1.3, last paragraph (page 4), lines 3 to 5: about the introductory intuitive statement of superselection.
(A): line 3 to 4: you say: quantum states cannot be stirred …any physical operation: This is too loose (if true!). I would just say the obvious, like: 0 matrix elements between the sectors, all observables block diagonalized. And
(B): line 5: you say: theory factorizes … sectors labelled ETC
Again too loose.
So I suggest instead, you should say: and the theory’s states are (decompose into) statistical mixtures of states in sectors labelled ETC.
And at end:
Sec 5.8, last paragraph (page 31), line 1: replace: At the light of this: by: In the light of this
Sec 5.9, paragraph 2(page 31), line 4: replace: but by far non-unique: with EITHER: but very non-unique: OR: but far from unique.
Section 6, last line, p. 32: should say: mismatches to be a neat example
The author may find it best to get an academic native speaker of English to go through and fix these minor matters of language.
Also: Souriau has an A in his name; so not: Sourieu!
I would certainly recommend publication, subject to minor revisions
Report #1 by Nicholas Teh (Referee 1) on 2021-4-22 (Invited Report)
- Cite as: Nicholas Teh, Report on arXiv:scipost_202102_00016v1, delivered 2021-04-22, doi: 10.21468/SciPost.Report.2827
Strengths
1.This paper presents a genuinely novel point of view on how to identify quasilocal degrees of freedom for Yang-Mills gauge theory, thus completing a project initiated by Gomes and Riello in a series of interesting investigations into the field space geometry of gauge theory in the presence of spatial boundaries. This identification is done by the introduction of covariant superselection sectors and an additional KKS (canonical) symplectic form in each sector.
2. The presentation is clear throughout: the calculations are reasonably explicit so the reader can check them, and when they are not, multiple references are given. The geometrical foundations of the theory are clearly explained.
Weaknesses
1. The greatest weakness of the paper is that, despite calling the procedure "symplectic reduction", it is unclear how this procedure relates to the usual symplectic geometry constructions of reduction that involve a moment map. One simple example that it might be helpful for the author to briefly comment on for the reader is how this form of symplectic reduction relates to the construction given in Atiyah and Bott's "The Moment Map and Equivariant Cohomology" (which results in a finite-dimensional moduli space) -- this would also help establish a link between standard equivariant constructions and the equivariant symplectic geometry introduced by the author.
2. The geometrical formalism used in this paper (sans field space connection) is generally introduced in the context of the covariant phase space approach: it would be nice if the author could briefly explain how the spatial constructions in this paper relate to a fully covariant approach.
3. In criticizing the approach of Donnelly and Freidel, I would recommend that the author update his criticism to deal with the more developed point of view on edge modes offered in https://arxiv.org/pdf/2006.12527.pdf
Report
I would recommend publication in this journal subject to minor corrections.
Requested changes
I would ask that the author briefly comment on points (1), (2) and (3) of the Weaknesses section in this paper.