A time asymmetric quantum theory can be achieved if one extends the Hilbert space of ordinary quantum mechanics (of measured systems) to Gel'fand triplets.
Time evolution (and also the relativistic space-time symmetry group) extend to semigroups only, representing irreversibility that originates from causality (boundary) conditions.
Scattering resonances and decaying states are described by Gamow states as elementary objects for which the decay probabilities have time ordering and obey the exponential decay law.
A generalized spectral decomposition (generalization of Dirac's eigenket expansion) allows for a rigged Hilbert space formulation of chaotic maps.
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