Poincare-Cartan form for scalar fields in curved background

TL;DR

This paper develops a geometric Hamiltonian formalism for scalar fields in curved spacetime using the Poincare-Cartan form, connecting it to variational principles, symmetries, and quantization.

Contribution

It introduces a Lagrangian-free Hamiltonian approach with a differential 4-form for scalar fields in curved backgrounds, linking to observables and Peierls bracket.

Findings

Constructed Poincare-Cartan 4-form for scalar fields

Defined observables as differential 4-forms with test functions

Identified Peierls bracket as suitable for quantization

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