  This paper discusses the implementation of diffeomorphism invariance in
purely Hamiltonian formulations of General Relativity. We observe that, if a
constrained Hamiltonian formulation derives from a manifestly covariant
Lagrangian, the diffeomorphism invariance of the Lagrangian results in the
following properties of the constrained Hamiltonian theory: the diffeomorphisms
are generated by constraints on the phase space so that a) The algebra of the
generators reflects the algebra of the diffeomorphism group. b) The Poisson
brackets of the basic fields with the generators reflects the space-time
transformation properties of these basic fields. This suggests that in a purely
Hamiltonian approach the requirement of diffeomorphism invariance should be
interpreted to include b) and not just a) as one might naively suppose. Giving
up b) amounts to giving up objective histories, even at the classical level.
This observation has implications for Loop Quantum Gravity which are spelled
out in a companion paper. We also describe an analogy between canonical gravity
and Relativistic particle dynamics to illustrate our main point.
