Relativistic resonances and decaying states are described by representations of Poincaré transformations, similar to Wigner's definition of stable particles. To associate decaying state vectors to resonance poles of the S-matrix, the conventional Hilbert space assumption (or asymptotic completeness) is replaced by a new hypothesis that associates different dense Hardy subspaces to the in-and out-scattering states. Then one can separate the scattering amplitude into a background amplitude and one or several "relativistic Breit-Wigner" amplitudes, which represent the resonances per se. These Breit-Wigner amplitudes have a precisely defined lineshape and are associated to exponentially decaying Gamow vectors which furnish the irreducible representation spaces of causal Poincaré transformations into the forward light cone.

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.

As is evident from the multiple values listed in the particle data table for the mass and width of resonances, such as Z°, Δ and ρ, defining these resonance parameters uniquely and unambiguously remains an open problem. This problem is ultimately rooted in the absence of a state vector description of a resonance that has definite properties under spacetime transformations. We show that there exist irreducible representations of the causal Poincaré semigroup that provide such a state vector description to resonances, leading to well-defined mass and width parameters. Generated by an interaction-incorporating Poincaré algebra and characterized by the complex S-matrix pole position and spin of the resonance, these representations synthesize the Bakamjian–Thomas construction of relativistic interactions and the S-matrix description of resonances.

A relativistic resonance which was defined by a pole of the S-matrix, or by a relativistic Breit-Wigner line shape, is represented by a generalized state vector (ket) which can be obtained by analytic extension of the relativistic Lippmann-Schwinger kets. These Gamow kets span an irreducible representation space for Poincaré transformations which, similar to the Wigner representations for stable particles, are characterized by spin (angular momentum of the partial wave amplitude) and complex mass (position of the resonance pole). The Poincaré transformations of the Gamow kets, as well as of the Lippmann-Schwinger plane wave scattering states, form only a semigroup of Poincaré transformations into the forward light cone. Their transformation properties are derived. From these one obtains an unambiguous definition of resonance mass and width for relativistic resonances. The physical interpretation of these transformations for the Born probabilities and the problem of causality in relativistic quantum physics is discussed. µP 2 µ + µν 1 2 J 2 µν is the Nelson operator [8]. But it could also be chosen differently, which we will do later when we will choose two different subspaces called Φ − and Φ + below. 2 Cf. footnote 15 of [1].