2011/04/27 by Pankaj Sharan, Sharan, Pankaj · 1 citation
Physics and Astronomy · #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #High Energy Physics - Theory (hep-th) #gr-qc #hep-th
paper · pdf · doi:10.48550/arxiv.1104.5095
LaTeX2e, 18 pages
arxiv created 2011/04/27 · arxiv updated 2011/04/28
Poincare-Cartan form for scalar field is constructed as a differential 4-form in a `directly Hamiltonian' formalism which does not use a Lagrangian. The canonical momentum p of a scalar field ϕ is a 1-form and the Poincare-Cartan 4-form Θ is (*p)\ww dϕ-H where the Hamiltonian H is a suitable 4-form made from ϕ and p using the Hodge star operator defined by the Riemannian metric of the background spacetime. An allowed field configuration is a 4-dimensional surface in the 9-dimensional extended phase space such that its tangent vectors annihilate Ω=-dΘ. Relation of this to variational principle, symmetry fields and conserved quantities is worked out. Observables are defined as differential 4-forms constructed from field and momenta smeared with appropriate test functions. A bracket defined by Peierls long ago is found to be the suitable candidate for quantization.