Poincare-Cartan form for scalar fields in curved background

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.

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