An Elementary Canonical Classical and Quantum Dynamics for General Relativity

TL;DR

This paper extends a canonical classical and quantum dynamics framework to general relativity, enabling a covariant formulation of quantum mechanics and gravity, with applications to particle evolution and scattering in curved spacetime.

Contribution

It embeds the SHP theory into GR, preserving canonical structures and developing a covariant quantum framework with new insights into particle dynamics and interactions in curved spacetime.

Findings

Canonical Poisson brackets remain invariant under GR transformations.

A scalar product and Hermitian operators are constructed for the Hilbert space.

Diffeomorphism covariant Newton's law is derived.

Abstract

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, 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; the result (here defined as SHPGR) is generalized to many-body 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 GR 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 state is studied and a scattering theory on the manifold is worked out.