Semigroup evolution in Wigner Weisskopf pole approximation with Markovian spectral coupling

TL;DR

This paper explores the relation between Wigner-Weisskopf theory and environment coupling, showing how Markovian spectral coupling can ensure semigroup evolution in multi-channel decay processes.

Contribution

It demonstrates that by generalizing to Markovian spectral coupling, the orthogonality of projectors can be achieved, improving the approximation of decay dynamics.

Findings

Orthogonality of projectors can be ensured regardless of the number of poles.

Background contributions can be suppressed through the generalized theory.

Markovian coupling leads to $z$-independent eigenvectors of $W(z)$.

Abstract

We establish the relation between the Wigner-Weisskopf theory for the description of an unstable system and the theory of coupling to an environment. According to the Wigner-Weisskopf general approach, even within the pole approximation (neglecting the background contribution) the evolution of a total system subspace is not an exact semigroup for the multi-channel decay, unless the projectors into eigesntates of the reduced evolution generator $W(z)$ are orthogonal. In this case these projectors must be evaluated at different pole locations $z_\alpha\neq z_\beta$. Since the orthogonality relation does not generally hold at different values of $z$, for example, when there is symmetry breaking, the semigroup evolution is a poor approximation for the multi-channel decay, even for a very weak coupling. Nevertheless, there exists a possibility not only to ensure the orthogonality of the $W(z)$ projectors regardless the number of the poles, but also to simultaneously suppress the effect of the background contribution. This possibility arises when the theory is generalized to take into account interactions with an environment. In this case $W(z)$, and hence its eigenvectors as well, are {\it independent} of $z$, which corresponds to a structure of the coupling to the continuum spectrum associated with the Markovian limit.