Non-unitary scattering and capture. II. Quantum dynamical semigroup theory

Robert Alicki

Robert Alicki

S. Wickramasekara

We construct two rigged Hilbert spaces that furnish differentiable representations of the causal Poincaré semigroup. These rigged Hilbert spaces provide the mathematical foundation for a theory of relativistic quasistable states that synthesizes the S-matrix description of resonance scattering with the Bakamjian–Thomas construction for interacting relativistic quantum systems.

Arno Böhma, Hani Kaldass, S. Wickramasekara

In this talk we discuss how to combine aspects of the theory of stable relativistic particles and of the theory of non-relativistic decaying particles into a relativistic theory of unstable particles and resonances. For the theory of stable relativistic particles we use Wigner’s description by unitary irreducible representations (UIR) of the Poincaré group [ 1 ] which has been the foundation for relativistic quantum mechanics and field theory [ 2 ]. For the non-relativistic decaying particles we use a time asymmetric quantum mechanics in which the Hilbert space axiom, that allows only symmetric (unitary group) time evolution, is replaced by time asymmetric boundary conditions which lead to an irreversible , semigroup time evolution [ 3 ]. In the limit in which certain contributions to the energy continuum are neglected, this new quantum mechanics leads to the Weisskopf-Wigner approximate methods [ 4 ] for decay, and to the Breit-Wigner energy distribution [ 5 ] for the resonance lineshape. We thus generalize and combine two fundamental contributions of Wigner into a description of relativistic resonances and decaying states. The central concept of this description is the relativistic Gamow vector, which is a generalized eigenvector (ket) of the self-adjoint invariant square mass operator https://www.w3.org/1998/Math/MathML" display="inline"> M 2 = P μ P μ P μ = P 1 μ + P 2 μ in resonance scattering 1±2→ R →3±4) with a complex eigenvalue, a relativistic Breit-Wigner energy distribution, and an exponential time evolution governed by a semigroup of continuous operators in the space of kets.

Claudio Altafini

We study the quantum Markovian master equation for a two level system in presence of coherent control. We show that its behavior is essentially determined by the complete positivity assumption of the quantum dynamical semigroup and that unitary control cannot modify its irreversibility character. Control theory provides several controllability-type condition's able to capture the peculiarity of such driven system.

G. C. Ghirardi, A. Rimini, T. Weber