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). This theory has been 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 relation between representations based on coordinates and momenta is given by Fourier transform; a proof is given here for this functional relation on a manifold. The potential which may occur in the SHP theory emerges as a spacetime scalar mass distribution in GR. Gauge fields, both Abelian and non-Abelian on the quantum mechanical SHPGR Hilbert space in both the single particle and many-body theory, may be generated by phase transformations. Application to the construction of Bekenstein and Sanders in their solution to the lensing problem in TeVeS is discussed.
Part of the book: Progress in Relativity