The exponential decay (or growth) of resonances provides an arrow of time which is described as the semigroup time evolution of Gamow vector in a new formulation of quantum mechanics. Another direction of time follows from the fact that a state must first be prepared before observables can be measured in it. Applied to scattering experiments, this produces another quantum mechanical arrow of time. The mathematical statements of these two arrows of time are shown to be equivalent. If the semigroup arrow is interpreted as microphysical irreversibility and if the arrow of time from the prepared in-state to its effect on the detector of a scattering experiment is interpreted as causality, then the equivalence of their mathematical statements implies that causality and irreversibility are interrelated.

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Gamow vectors are generalized eigenvectors (kets) of self-adjoint Hamiltonians with complex eigenvalues (ER∓iΓ/2) describing quasistable states. In the relativistic domain this leads to Poincaré semigroup representations which are characterized by spin j and by complex invariant mass square s = sR = MR − i 2 ΓR 2 . Relativistic Gamow kets have all the properties required to describe relativistic resonances and quasistable particles with resonance mass MR and lifetime /ΓR.

Many useful concepts for a quantum theory of scattering and decay (like Lippmann-Schwinger kets, purely outgoing boundary conditions, exponentially decaying Gamow vectors, causality) are not well defined in the mathematical frame set by the conventional (Hilbert space) axioms of quantum mechanics. Using the Lippmann-Schwinger equations as the takeoff point and aiming for a theory that unites resonances and decay, we conjecture a new axiom for quantum mechanics that distinguishes mathematically between prepared states and detected observables. Suggested by the two signs ±iǫ of the Lippmann-Schwinger equations, this axiom replaces the one Hilbert space of conventional quantum mechanics by two Hardy spaces. The new Hardy space theory automatically provides Gamow kets with exponential time evolution derived from the complex poles of the S-matrix. It solves the causality problem since it results in a semigroup evolution. But this semigroup brings into quantum physics a new concept of the semigroup time t = 0, a beginning of time. Its interpretation and observations are discussed in the last section.

A new parametrization of the Kobayashi-Maskawa (KM) matrix is proposed. It is based on the eigenvalues and the eigenvectors of the KM matrix. In this parametrization the experimental data for the KM matrix (including CP violation) can be reproduced with only two angles. This suggests that CP violation is intimately connected with the Cabibbo-type rotations. We also predict that | V u t,/V C b I -0.128 ±0.001.