Putting an edge to the Poisson bracket. (English) Zbl 0984.70020

Summary: 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 in general relativity. Finally, we discuss how the boundary terms may be related to the time ordering when quantizing.


70S05 Lagrangian formalism and Hamiltonian formalism in mechanics of particles and systems
70G45 Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics
Full Text: DOI arXiv


[1] DOI: 10.4310/ATMP.1998.v2.n2.a1 · Zbl 0914.53047
[2] Carlip S., Phys. Rev. D 51 pp 632– (1995)
[3] Strominger A., J. High Energy Phys. 2 pp 009– (1998)
[4] G. ’t Hooft,Dimensional Reduction in Quantum Gravity, essay dedicated to A. Salam. Published in Salamfest (1993), gr-qc/9310026; · Zbl 0850.00013
[5] Susskind L., J. Math. Phys. 36 pp 6377– (1995) · Zbl 0850.00013
[6] DOI: 10.1007/BF01217730 · Zbl 0667.57005
[7] DOI: 10.1007/BF01217730 · Zbl 0667.57005
[8] DOI: 10.1016/0003-4916(74)90404-7 · Zbl 0328.70016
[9] Brown J. D., Commun. Math. Phys. 104 pp 207– (1986) · Zbl 0584.53039
[10] Brown J. D., J. Math. Phys. 27 pp 489– (1986) · Zbl 0584.53039
[11] DOI: 10.1016/0370-2693(92)90604-3
[12] DOI: 10.1016/0167-2789(86)90207-1 · Zbl 0638.58044
[13] Soloviev V. O., J. Math. Phys. 34 pp 5747– (1993) · Zbl 0785.70014
[14] V. O. Soloviev, ”Bering’s proposal for boundary contribution to the Poisson bracket,” preprint IC/98/201, hep-th/9901112. · Zbl 0968.37020
[15] D. R. Grigore, ”The variational sequence on finite jet bundle extensions and the Lagrangian formalism,” preprint, dg-ga/9702016.
[16] J. Stasheff, ”Closed string field theory, strong homotopy Lie algebras and the operad action of moduli space,” inPerspectives in Mathematical Physics, talk presented at the Conference on Topics in Geometry and Physics, Los Angeles, 1992, hep-th/9304061, pp. 265–288.
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