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It was generally believed throughout the 20-th century that irreversibilityis a purely classical event without operator counterpart. However, a classicalirreversible system cannot be consistently decomposed into a finite number ofreversible quantum particles (and, vice versa), thus establishing that theorigin of irreversibility is basically unknown at the dawn of the 21-thcentury. To resolve this problem, we adopt the historical an- alyticrepresentation of irreversibility by Lagrange and Hamilton with external termsin their analytic equations; we show that, when properly written, the bracketsof the time evolution characterize covering Lie-admissible algebras; we showthat the for- malism has a fully consistent operator counterpart given by theLie-admissible branch of hadronic mechanics; we identify catastrophicmathematical and physical inconsis- tencies when irreversible formulations aretreated with the conventional mathematics used for reversible systems; and showthat, when the dynamical equations are treated with a novel irreversiblemathematics, Lie-admissible formulations are fully consistent because invariantat both the classical and operator level. The case of closed-isolated systemsverifying conventional total conservation laws, yet possessing an irreversiblestructure, is treated via the Lie-isotopic subclass of Lie-admissibleformulations. The analysis is conducted for both matter and antimatter at theclassical and operator levels to prevent insidious inconskistenmcies occurringfor the sole study of matter or, separately, antimatter, as known known for theproblem of grand unificationsr.Comment: 37 page

Lie-Admissible Invariant Origin of Irreversibility for Matter and Antimatter at the Classical and Operator Levels