An Elementary Canonical Classical and Quantum Dynamics for General Relativity

A consistent canonical classical and quantum dynamics in the framework of special relativity was formulated by Stueckelberg in 1941, and generalized to many body theory by Horwitz and Piron in 1973 (SHP). In this paper, using local coordinate transformations, following the original procedure of Einstein, this theory is embedded into the framework of general relativity (GR) both for potential models (where the potential appears as a spacetime mass distribution with dimension of mass) and for electromagnetism (emerging as a gauge field on the quantum mechanical Hilbert space). The canonical Poisson brackets of the SHP theory remain valid (invariant under local coordinate transformations) on the manifold of GR, and provide the basis, following Dirac's quantization procedure, for formulating a quantum theory. The theory is developed both for one and many particles.