Poincare Cartan Form for Gauge Fields in Curved Background

The `directly Hamiltonian’ field theory in the extended phase space is applied to gauge fields in curved spacetime background. These fields being differential 1-forms, have canonical momenta which are 2-forms. The Poincare-Cartan 4-forms for matter and gauge fields have to be modified with the exterior derivatives replaced by the covariant derivative for maintaining gauge invariance.