Foundations of the hadronic generalization of the atomic mechanics. II. Modular-isotopic Hilbert space formulation of the exterior strong problem

This paper is devoted to a first formulation of the axiomatic Hilbert space foundations of the branch of the Hadronic Mechanics dealing with the exterior treatment of strong non-Hamiltonian systems, and which admits a Lie-isotopic algebraic character. In particular, we are interested in generalizing the conventional eigenvalue equations of the Atomic Mechanics into the broadest possible equations which are permitted by an associative algebra of opeators on a one-sided, modular, Hilbert space. The objective is made possible by the isotopic generalization of the associative algebra of operators A, B, ... on a Hilbert space, characterized by the product A*B - ATB, where T is a fixed, bounded, and nonsingular operator. By using the T-isotopic product, we first introduce a generalization of the Hermitean conjugate, transpose, Hermitean, skew-Hermitean, unitary, skew-unitary, projection, and exponential operators. We then pass to the isotopic generalization of the notion of determinant, trace, and eigenvalue of a linear operator. Some essential properties of this modular-isotopic formulation of the Hilbert space theory are identified. The results are then applied to Hadronic Mechanics. In particular, we present the hadronic generalization of a number of postulates of Atomic Mechanics, ranging from observables, states, and their time evolution, to total probability, its conservation in time, and the expectation values. We also prove that the isotopic generalization of Schroedinger's equations presented in Paper I are equivalent to the isotopic generalization of Heisenberg's equations