  A Dirac Hamiltonian formulation of d-dimensional Einstein-Hilbert action in
first order form, has shown that as well as secondary first class constraints,
tertiary first class constraints also arise, with an unusual nonlocal Poisson
bracket (PB) algebra among first class constraints [8 ]. This approach is
different from that of ADM in that of the equations of motion which are
independent of the time derivative of fields only those which correspond to
second class constraints (in the sense of the Dirac constraint formalism) are
used to eliminate fields from the action. In this paper, we consider coupling
of a cosmological term, massive scalar fields, Maxwell gauge fields and
Yang-Mills fields to the first order EH action in this formalism, and show that
in spite of the apparent differences with the ADM results in the Hamiltonian
formulation of the first order EH action and its constraint structure, the
generic properties of the ADM Hamiltonian formulation of the first order EH
action in the presence of Bosonic matter are derivable from this novel
Hamiltonian formulation. Addition of a massive scalar field to the EH action
leaves the PB algebra of constraints unaltered, and when the Yang-Mills fields
or Maxwell gauge fields are coupled to the EH action, the PB algebra of the
constraints pertaining to the EH action receives linear contributions from the
generator of the gauge transformations of the action for matter fields, and
those generators form a closed algebra amongst themselves. Moreover, it is
found that for closed spaces, the Hamiltonian of the EH action coupled to
Bosonic matter is weakly zero on the constraint surface defined by the first
class constraints, including the constraints arising from the matter fields.

General Relativity and Quantum Cosmology