A Hamiltonian approach to quantum gravity

In general relativity the post-Newtonian Hamiltonian approach is widely used for calculations of relativistic gravitational effects. This approach is usually assumed to be just an approximation to the rigorous general-relativistic theory of the dynamical curved space-time. The difficulties of reconciling this point of view with quantum mechanics are well-known. In this paper we explore a simple approach to quantum gravity, which is based on Wigner-Dirac theory of unitary representations of the Poincaré group in the Hilbert space of states. In the instant form of Dirac’s dynamics, interaction terms are present in the Hamiltonian and in the ”center of energy” observable, so that commutators of 10 symmetry generators satisfy Poincaré Lie algebra relationships. Despite lack of symmetry between time and space coordinates and the use of instantaneous action-at-a-distance forces, this approach does not contradict any experimental data or the principles of relativity and causality. PACS numbers: 03.70.+k, 04.50.Kd, 04.60.-m