Tel Aviv University
We have shown in a pevious paper that the relativistic classical and quantum theory of Stueckelberg, Horwitz and Piron (SHP) can be embedded into general relativity. The formulation of the quantum theory in this framework made use of the Fourier transform on the manifold, but a proof was not given for its existence. In this paper I review briefly the embedding of the SHP theory into general relativity and construct a proof of the existence of the Fourier transform on a non-compact, smooth and geodesically complete manifold as well as a proof of the concommitant Parseval-Plancheral relation. We show how the existence of the Fourier transform provides a basis for the Dirac form (bras and kets) of the quantum theory. We furthermore show that the theory of induced representations for the intrinsic spin of a particle can be extended, using a theorem on isomorphisms associated with local diffeomorphisms, to apply to the manifold of general relativity.