An elementary canonical classical and quantum dynamics for general relativity

A consistent (off-shell) 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, this theory is embedded into the framework of general relativity (GR), here denoted by SHPGR. The canonical Poisson brackets of the SHP theory remain valid (invariant under local coordinate transformations) on the manifold of GR, and provide the basis for formulating a canonical quantum theory. A scalar product is defined for constructing the Hilbert space and a Hermitian momentum operator defined. The Fourier transform is defined, connecting momentum and coordinate representations. The potential which may occur in the SHP theory emerges as a spacetime scalar mass distribution in GR, and electromagnetism corresponds to a gauge field on the quantum mechanical SHPGR Hilbert space in both the single particle and many-body theory. A diffeomorphism covariant form of Newton’s law is found as an immediate consequence of the canonical formulation of SHPGR. We compute the classical evolution of the off shell mass on the orbit of a particle and the force on a particle and its energy at the Schwarzschild horizon. The propagator for evolution of the one-body quantum state is studied and a scattering theory on the manifold is worked out.