  A novel Dirac Hamiltonian formulation of the first order Einstein-Hilbert
(EH) action, in which algebraic constraints are not solved to eliminate fields
from the action at the Lagrangian level, has been shown to lead to an action
and a constraint structure apparently distinct from the ADM action and the ADM
constraint structure in that secondary first class constraints as well as
tertiary first class constraints arise with an unusual Poisson Bracket (PB)
algebra [24]. By canonical transformations of the fundamental fields we show
how from the tertiary constraints one may derive the Hamiltonian and momentum
constraints. Special attention is paid to the Hamiltonian formulation of the
first order EH action in terms of the variables h, h^i and q^{ij} employed in
[19,20]. It is shown that the variables h and h^i are left undetermined in the
formalism. This fact is used for a proper gauge fixation of the secondary
constraints and reduction to the Faddeev action [19,20]. Considering
invariances of the total action, the generator of the gauge transformations of
the EH Lagrangian action is derived. Using this generator, the explicit form of
the gauge invariance of the field h is obtained, by which the relation between
the gauge functions and the descriptors of the diffeomorphism invariance is
determined in order for the gauge transformations to correspond to
diffeomorphism invariance. By linearizing the novel Hamiltonian formulation of
[24], the Hamiltonian formulation of the first order action for the free spin
two field [4,24] is derived.
