Relativistic dynamics of quasistable states. I. Perturbation theory for the Poincaré group

Abstract

We propose a theory of resonances by combining the S-matrix approach with the Bakamjian–Thomas (BT) construction. Characterization of resonances by the poles of the S-matrix has many advantages. Foremost among them is perhaps the gauge invariance of the definitions of resonance mass and width, a problem with which some definitions based on field theoretical approaches suffer. The BT construction provides a general framework for constructing Poincaré generators for an interacting quantum system. While much of what we develop here can be cast in the language of quantum field theory, in the spirit of BT construction, which does not assume the existence of local field mediating interactions, we will work at the fundamental level of an interacting Poincaré algebra. Our construction shows that a subset of this Poincaré algebra integrates to a representation of the semigroup of causal transformations of relativistic space-time. These representations are characterized by the spin and S-matrix complex pole position of the resonance. The state vectors that transform under these representations also show an exact exponential decay, the signature of a decaying state. In this sense, the semigroup representations developed here tie together resonances and decaying states into a single theoretical description.