Time asymmetric quantum theory – III. Decaying states and the causal Poincaré semigroup

Abstract

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].

Time asymmetric quantum theory – I. Modifying an axiom of quantum physics

“…There the hypothesis (A V ) will lead to new predictions. This relativistic theory of resonance scattering and decay, in which the time evolution semigroup is generalized to the causal Poincaré semigroup, is the subject of subsequent papers [16,27].…”

“…These generalized eigenvectors we call Gamow vectors or Gamow kets. They describe exponentially decaying resonance states given by first order poles of the S-matrix; their generalization to relativistic physics is the subject of the two subsequent papers [16,27].…”

Absorption in the final state in reactions and decays in the medium

“…As far back as 1959, M. Levy remarked that there does not exist a rigorous theory to which various phenomenological methods of treating resonances and decays can be considered as approximations [19]. Attempts have been made at an axiomatic theory [20,21].…”

Point-form dynamics of quasistable states

“…In a typical scattering experiment, the vector φ + is taken to have evolved from an asymptotic in-state φ in , prepared by a device such as a particle accelerator. On the other hand, 3 The discrepancy in the signs is due the conventions in physics and mathematics. In scattering theory, the in vectors φ + and out vectors ψ − have been defined by their behavior at t → −∞ and t → ∞, respectively.…”