Unified Dirac-Von Neumann Formulation of Quantum Mechanics. I. Mathematical Theory

In this paper the results from various areas of mathematical research which are necessary for a consistent unification of the Dirac and von Neumann formulations of quantum mechanics are collected and presented as a single synthesis. For this purpose, direct integral decompositions of Hilbert space must be introduced into Dirac's formulation of spectral theory and representation theory; true unit vectors in the direct integral decomposition spaces replace unnormalizable vectors of infinite length. It then becomes clear that families of modified Dirac projection operators are simply related to the Radon-Nikodym derivative of von Neumann spectral measures. In terms of these mathematical preliminaries a second paper will present the more physical aspects of the resulting unified formulation of quantum mechanics.