Two mathematical problems of canonical quantization. I

The problem of the choice of the representation of canonical commutation relations (CCR) and definition of the Hamiltonian are investigated, in the first part of the study, it is shown that the Hamiltonians investigated in the well-known paper of Araki [1] admit representation in the form of Dirichlet operators. It is shown that an “almost” inverse theorem is also valid. The Dirichlet operators are uniquely determined by measures that at the same time fix the representation of the CCR and characterize the vacuum. For these measures, the concept of a density is introduced in the second part of the study. Methods of calculating generalized densities and measures corresponding to them, i.e., ultimately representations of the CCR and Hamiltonians consistent with them in the sense of Van Hove, are proposed.