On the Mathematical Basis of the Dirac Formulation of Quantum Mechanics

We discuss various attempts to implement mathematically the Dirac formulation of Quantum Mechanics. A first attempt used Hilbert space. This formalization realizes the Dirac formalism if and only if the spectra of the observables under consideration is purely discrete. Therefore, generalized spectral decompositions are needed. These spectral decompositions can be constructed in the framework of rigged Hilbert spaces. We construct generalized spectral decompositions for self-adjoint operators using their spectral measures. We review the previous work by Marlow (in Hilbert spaces), Antoine, Roberts, and Melsheimer and complete it. We show that these generalized spectral decompositions fit well in the framework of a theory constructed by Kato and Kuroda and that all the results can be reproduced in this framework.