  It was generally believed throughout the 20-th century that irreversibility
is a purely classical event without operator counterpart. However, a classical
irreversible system cannot be consistently decomposed into a finite number of
reversible quantum particles (and, vice versa), thus establishing that the
origin of irreversibility is basically unknown at the dawn of the 21-th
century. To resolve this problem, we adopt the historical an- alytic
representation of irreversibility by Lagrange and Hamilton with external terms
in their analytic equations; we show that, when properly written, the brackets
of the time evolution characterize covering Lie-admissible algebras; we show
that the for- malism has a fully consistent operator counterpart given by the
Lie-admissible branch of hadronic mechanics; we identify catastrophic
mathematical and physical inconsis- tencies when irreversible formulations are
treated with the conventional mathematics used for reversible systems; and show
that, when the dynamical equations are treated with a novel irreversible
mathematics, Lie-admissible formulations are fully consistent because invariant
at both the classical and operator level. The case of closed-isolated systems
verifying conventional total conservation laws, yet possessing an irreversible
structure, is treated via the Lie-isotopic subclass of Lie-admissible
formulations. The analysis is conducted for both matter and antimatter at the
classical and operator levels to prevent insidious inconskistenmcies occurring
for the sole study of matter or, separately, antimatter, as known known for the
problem of grand unificationsr.
