Putting an Edge to the Poisson Bracket

TL;DR

This paper introduces a new boundary-aware Poisson bracket for Hamiltonian field theories with spatial edges, ensuring differentiability and satisfying the Jacobi identity, with applications to Chern-Simons theory and General Relativity.

Contribution

A novel Poisson bracket formalism that incorporates boundary effects directly, eliminating the need for improvement terms and satisfying fundamental algebraic properties.

Findings

The new Poisson bracket satisfies the Jacobi identity.

The formalism is geometrized on an abstract manifold.

Potential implications for quantization boundary terms.

Abstract

We consider a general formalism for treating a Hamiltonian (canonical) field theory with a spatial boundary. In this formalism essentially all functionals are differentiable from the very beginning and hence no improvement terms are needed. We introduce a new Poisson bracket which differs from the usual ``bulk'' Poisson bracket with a boundary term and show that the Jacobi identity is satisfied. The result is geometrized on an abstract world volume manifold. The method is suitable for studying systems with a spatial edge like the ones often considered in Chern-Simons theory and General Relativity. Finally, we discuss how the boundary terms may be related to the time ordering when quantizing.