Foundations of the hadronic generalization of the atomic mechanics. III. Bimodular-genotopic Hilbert space formulation of the interior 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 interior strong non-Hamiltonian problem, that is, with the characterization of a particle under external unrestricted strong interactions, and which admits a Lie-admissible algebraic character. In particular, we are interested in extending the modular-isotopic generalization of the eigenvalue equations presented in the adjoining Paper II, into the broadest possible form permitted by a two-sided, bimodular (right and left) characterization of the Hilbert space. For this purpose we point out certain (yet unsolved) technical difficulties related to the preservation of the Hilbert space under a nonassociative algebra of operators. We then point out that these difficulties are absent when the theory is restricted to the original proposal by Santilli, the characterization of a hadron under extenal strong interactions via a birepresentation of a Lie-admissible algebra of operators on a bimodular Hilbert space in which each left and right action is associative-isotopic. In this way we achieve the Lie-admissible generalization of the contents of Paper II via the simple differentiation of the left and right isotopes, with products AT(/sup +/)B and AT(/sup -/)B, where T(/sup + -/) are fixed,more » generally different, bounded, and nonsingular operators. the Lie-isotopic modular formulations of Paper II result to be a particular case of the Lie-admissible bimodular formulations when T(/sup +/) = T(/sup -/) = T. The conventional atomic formulations result to be the simplest possible realization when T(/sup +/) = T(/sup -/) = 1.« less