On continuous observables in quantum mechanics

The theory of continuous observables in quantum mechanics has been arising mistrust among mathematicians. Von Neumann banished from it the delta-function and such exotic entities as "Hermitian product matrices" with continuous indices and quasi-elements in the form of generalized functions. Thanks to von Neumann the paradigm has been spread according to which the state space of a quantum system must only be Hilbert. De'facto he replaced Copenhagen quantum mechanics by a new theory with new physical consequences. In addition, along with the delta-function, quantum mechanics in many ways lost its heuristic power. Dirac brilliantly anticipated the theory of generalized functions and, in fact, there was no need for such a reform. The report presents the theory of continuous observables in the formal form given to it by Heisenberg and Dirac. At the same time, Dirac's approach is strictly formalized within the framework of the theory of generalized functions. But Hilbert spaces are met here too.