Combining the ideas of causality and the phenomenology of resonances and decaying states, we modify standard quantum theory by changing one of its axioms. The first step was taken decades ago when Dirac kets were given a mathematical meaning as functionals on a Schwartz space, which led to a Gelʼfand triplet in which observables were represented by an algebra of continuous operators. The second step, motivated by the analytic continuation to complex energies for the S-matrix and the two Lippmann-Schwinger kets, distinguishes between prepared in-states-vectors and detected out-vectors. This leads to the pair of Gelʼfand triplets realized by Hardy functions of the upper (for prepared states) and lower (for detected observables) complex energy plane.
Replacing the Hilbert space axiom of quantum mechanics by the Hardy space axiom, one obtains a mathematically consistent theory that unifies resonance and decay phenomena and that has a causal, asymmetric time evolution.
Arno Bohm. Peter Bryant. "Conjecturing The Mathematical Axiom That Provides a Unified Theory of Resonance and Decay and Connects It to Causal Time Evolution." J. Geom. Symmetry Phys. 11 1 - 22, 2008. https://doi.org/10.7546/jgsp-11-2008-1-22