Putting an Edge to the Poisson Bracket

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.

36 pages, LaTeX. v2: A manifest formulation of the Poisson bracket and some examples are added, corrected a claim in Appendix C, added an Appendix F and a reference. v3: Some comments and references added

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• Hamiltonian surface charges using external sources
• Family of Boundary Poisson Brackets
• Physics-to-gauge conversion at black hole horizons
• Symplectic approach to lightcone QCD
• Product of Boundary Distributions
• A note on the Hamiltonian structure of transgression forms