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arxiv: 2604.07334 · v1 · submitted 2026-04-08 · ✦ hep-th · gr-qc

Recognition: 1 theorem link

· Lean Theorem

The BEF Symplectic Form: A Lagrangian Perspective

Mohd Ali , Georg Stettinger
Authors on Pith no claims yet

Pith reviewed 2026-05-10 18:03 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords BEF symplectic formL-infinity Lagrangiancovariant phase spaceBarnich-Brandt symplectic formcorner termsgeneral relativitynon-local theories
0
0 comments X

The pith

The BEF symplectic form arises directly from an L∞-Lagrangian and coincides with the Barnich-Brandt construction for second-order theories.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that the BEF symplectic structure proposed for non-local theories can be obtained by applying the covariant phase space method to an L∞-Lagrangian. This establishes that the BEF form is equivalent to the Barnich-Brandt symplectic form whenever the equations of motion are at most second order. Consequently, the appearance of the canonical corner term in general relativity finds a natural explanation inside the BEF framework. The construction further reveals that the BEF structure encodes details of generic corner terms as well as some information on boundary conditions. A general formula for the Hamiltonian in L∞ theories is also given and demonstrated through examples.

Core claim

The BEF symplectic structure can be derived directly from an L∞-Lagrangian by following the covariant phase space approach. We establish a precise relation between the BEF symplectic structure and the Barnich-Brandt symplectic form for general finite-derivative theories. In particular, for theories with second-order equations of motion, the BEF symplectic structure coincides with the Barnich-Brandt construction, thereby explaining the emergence of the canonical corner term in general relativity within the BEF approach. We further argue that the BEF symplectic structure naturally encodes information about generic corner terms and some information about boundary conditions, and we develop a 6A

What carries the argument

The covariant phase space formalism applied to L∞-Lagrangians, which produces the BEF symplectic form and relates it to the Barnich-Brandt form for second-order theories.

Load-bearing premise

Every relevant theory including those with non-local interactions admits a consistent L∞-Lagrangian formulation to which the covariant phase space formalism applies without further restrictions on the boundary or corner data.

What would settle it

Explicitly calculate the symplectic form in general relativity using the BEF prescription derived from its L∞-Lagrangian and verify that it reproduces the known Barnich-Brandt expression including the corner term; a mismatch would disprove the coincidence.

Figures

Figures reproduced from arXiv: 2604.07334 by Georg Stettinger, Mohd Ali.

Figure 1
Figure 1. Figure 1: In this figure, Σ+ and Σ− denote the future and past temporial boundaries, respectively. Γ represents the spatial boundary. The Cauchy surface is shown in beige color, while its boundary is depicted in red, where the corner term C contributes. Here, Ei is the equation-of-motion (EoM) (d, 0)-form 5 and Θ is a (d − 1, 1)-form known as the pre-symplectic potential6 . The multi-index i is running over all dyna… view at source ↗
Figure 2
Figure 2. Figure 2: This figure depicts the generic behavior of the sigmoid function in an appro [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The above figures depict the behaviour of the [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
read the original abstract

In 2025, Bernardes, Erler and Firat proposed a novel, elegant expression for the symplectic form on phase space applicable to non-local theories. We show that this BEF symplectic structure can be derived directly from an $L_\infty$-Lagrangian by following the covariant phase space approach. Moreover, we establish a precise relation between the BEF symplectic structure and the Barnich--Brandt symplectic form for general finite-derivative theories. In particular, we prove that for theories with second-order equations of motion, the BEF symplectic structure coincides with the Barnich--Brandt construction, thereby explaining the emergence of the canonical corner term in general relativity within the BEF approach. We further argue that the BEF symplectic structure naturally encodes information about generic corner terms and some information about boundary conditions. In addition, we develop a general expression for the Hamiltonian in theories in $L_\infty$-form and present several explicit examples illustrating the construction.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper derives the BEF symplectic form directly from an L∞-Lagrangian via the covariant phase space construction. It proves that, for any theory whose equations of motion are second-order, this BEF structure coincides exactly with the Barnich-Brandt symplectic form, thereby accounting for the canonical corner term in general relativity. The manuscript also supplies a general expression for the Hamiltonian in L∞-form, argues that the BEF structure encodes generic corner terms and some boundary-condition information, and illustrates the construction with explicit examples.

Significance. If the central identification holds, the work supplies a Lagrangian origin for the BEF symplectic structure and unifies it with the standard Barnich-Brandt construction for local second-order theories. This clarifies the appearance of the GR corner term inside the BEF framework and extends the covariant phase-space method to non-local theories formulated as L∞-algebras. The provision of a general Hamiltonian and concrete examples strengthens the practical utility of the result.

major comments (2)
  1. [Main derivation of the coincidence (following the covariant phase-space steps)] The proof that the BEF and Barnich-Brandt forms coincide for second-order theories (the central claim) rests on the assumption that an L∞-Lagrangian supplies precisely the same on-shell boundary variations and corner contributions as the ordinary second-order Lagrangian, without additional restrictions. This assumption is load-bearing for the equality and for the explanation of the GR corner term; an explicit check that no implicit conditions on boundary or corner data are introduced by the L∞ structure is required.
  2. [Discussion of corner terms and boundary conditions] The claim that the BEF structure 'naturally encodes information about generic corner terms' needs a precise statement of which corner contributions are captured and which are not; without this, it is unclear whether the encoding is complete or merely partial for the cases where the BEF-Barnich-Brandt match is asserted.
minor comments (2)
  1. [Preliminaries] Notation for the L∞-operations and the precise definition of the covariant phase-space presymplectic current should be introduced with a short self-contained summary, as readers familiar with Barnich-Brandt but not L∞-algebras may otherwise lose the thread.
  2. [Examples] In the examples section, the explicit computation of the symplectic form and Hamiltonian for at least one non-local case would benefit from a side-by-side comparison with the corresponding Barnich-Brandt expressions to illustrate the claimed coincidence.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive overall assessment and for the constructive major comments. We address each point below and indicate where revisions will be made to improve clarity and rigor.

read point-by-point responses

  1. Referee: [Main derivation of the coincidence (following the covariant phase-space steps)] The proof that the BEF and Barnich-Brandt forms coincide for second-order theories (the central claim) rests on the assumption that an L∞-Lagrangian supplies precisely the same on-shell boundary variations and corner contributions as the ordinary second-order Lagrangian, without additional restrictions. This assumption is load-bearing for the equality and for the explanation of the GR corner term; an explicit check that no implicit conditions on boundary or corner data are introduced by the L∞ structure is required.

    Authors: We agree that an explicit verification strengthens the argument. The L∞-Lagrangian is defined so that its Euler-Lagrange equations and their first variations coincide exactly with those of the ordinary second-order Lagrangian; the covariant phase-space construction is then applied verbatim to the L∞-action. Consequently the integration-by-parts identities that generate the boundary and corner terms are identical. In the revised manuscript we will insert a short subsection (or paragraph) that compares the variation formulas term by term, confirming that the L∞-structure introduces no extra non-local boundary contributions or restrictions on corner data when the equations of motion are second-order. This explicit check will be placed immediately before the statement of the coincidence theorem. revision: yes

  2. Referee: [Discussion of corner terms and boundary conditions] The claim that the BEF structure 'naturally encodes information about generic corner terms' needs a precise statement of which corner contributions are captured and which are not; without this, it is unclear whether the encoding is complete or merely partial for the cases where the BEF-Barnich-Brandt match is asserted.

    Authors: We accept the need for greater precision. The BEF symplectic form encodes precisely those corner terms that arise from the standard covariant phase-space variation of the L∞-Lagrangian. For second-order theories these coincide with the Barnich-Brandt corner terms; the encoding is therefore complete with respect to the local dynamics encoded in the Lagrangian. It does not automatically incorporate arbitrary additional boundary conditions or non-standard corner contributions imposed independently of the Lagrangian. In the revised version we will replace the existing sentence with a clearer statement that distinguishes these two classes of contributions and notes the limitation with respect to externally prescribed boundary data. revision: yes

Circularity Check

0 steps flagged

Derivation from L∞-Lagrangian via covariant phase space is self-contained

full rationale

The paper derives the BEF symplectic structure directly from an L∞-Lagrangian input by applying the standard covariant phase space formalism, then proves its coincidence with the Barnich-Brandt form specifically for second-order equations of motion. This identification follows from the on-shell boundary variations supplied by the given Lagrangian without reducing to a fitted parameter, self-definition, or load-bearing self-citation chain. The central result is an explicit relation and proof rather than a renaming or ansatz smuggled from prior author work; the L∞ formulation is treated as an independent starting point to which the established method is applied. No step equates the output to the input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review performed from abstract only; full technical assumptions cannot be audited. The central claim rests on the existence of an L∞-Lagrangian for the theories under consideration and on the applicability of the covariant phase space formalism.

axioms (2)
  • domain assumption Theories of interest admit an L∞-Lagrangian formulation
    Required to start the covariant phase space derivation of the BEF form.
  • domain assumption The covariant phase space approach yields the symplectic form without additional boundary data choices
    Used to obtain the BEF structure and its relation to Barnich-Brandt.

pith-pipeline@v0.9.0 · 5458 in / 1382 out tokens · 49955 ms · 2026-05-10T18:03:46.542731+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

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