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.

We obtain a set of generalized eigenvectors that provides a generalized spectral decomposition for a given unitary representation of a commutative, locally compact topological group. These generalized eigenvectors are functionals belonging to the dual space of a rigging on the space of square integrable functions on the character group. These riggings are obtained through suitable spectral measure spaces.

Misra-Prigogine-Courbage theory of irreversibility is revisited on the basis of Nagy-Foiaş dilation theory and Halmos-Helson theory of invariant subspaces. Universal models for intrinsically random dynamics are given as well as equivalent conditions to the existence of internal time operators, where innovation processes and Lax-Phillips scattering appear in a natural way.

The purpose of this paper is to show that a generalized Clifford immersion with non-negative Gaussian curvature has constant contact angle, thus extending previous results.

The two centre problem is studied both in the phase plane and in spacetime, assuming first a trajectory collinear, and, as a second case, a circular one through the newtonian attractors, finding a saddle equilibrium. For the latter problem a probably new differential equation is met and solved. Time is then obtained in both cases through elliptic integrals of all kinds and Jacobian functions.

We show that the Bakamjian-Thomas construction for relativistic direct interactions can be extended to describe resonances by irreducible representations of the causal Poincaré semigroup. These representations are generated by a Poincaré algebra that incorporates interactions and are characterized by the complex pole position and spin of the resonance.

We adjoin complete first kind Abelian integrals of genus two to resolve the general sextic equation $c_0 z^6 + c_1 z^5 + \cdots + c_6 = 0$ with simple zeros by genus two theta constants (Thetanullwerten). Using the same formulas, we also resolve each algebraic equation of degree five, four or three. It is shown that the monodromy group of a sextic is isomorphic to the second congruence sub-group $\Gamma (2)$ of the symplectic group ${\rm Sp}_4 ({\mathbb Z})$.