Poincare-Cartan form for scalar fields in curved background

Abstract

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 $\varphi$ is a 1-form and the Poincare-Cartan 4-form $\Theta$ is $(*p)\ww dϕ-H$ where the Hamiltonian $H$ is a suitable 4-form made from $\varphi$ 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 $\Omega =-d\Theta$. 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.

arXiv commentLaTeX2e, 18 pages

References in corpus (1)

• Lagrangian in quantum mechanics is a connection one-form