The BEF Symplectic Form: A Lagrangian Perspective

TL;DR

This paper derives the BEF symplectic form from an $L_$-Lagrangian, relates it to the Barnich--Brandt form, and explores its implications for boundary terms and Hamiltonian constructions.

Contribution

It establishes a direct derivation of the BEF symplectic structure from an $L_$-Lagrangian and relates it to existing symplectic forms, clarifying boundary and corner term roles.

Findings

BEF symplectic form derived from $L_$-Lagrangian via covariant phase space.

BEF form coincides with Barnich--Brandt form for second-order theories.

BEF structure encodes boundary conditions and corner term information.

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.