Relativistic Gamow vectors

Abstract

The Friedrichs model has often been used in order to obtain explicit formulas for eigenvectors associated to complex eigenvalues corresponding to lifetimes. Such eigenvectors are called Gamow vectors and they acquire meaning in extensions of the conventional Hilbert space of quantum theory to the so-called rigged Hilbert space. In this paper, Gamow vectors are constructed for a solvable model of an unstable relativistic field. As a result, we obtain a time asymmetric relativistic extension of the Fock space. This extension leads to two distinct Poincaré semigroups. The time reversal transformation maps one semigroup to the other. As a result, the usual PCT invariance should be extended. We show that irreversibility as expressed by dynamical semigroups is compatible with the requirements of relativity.

The Friedrichs model and its use in resonance phenomena

“…Now, there are two possible situations. If M < 2m, the functions G + (E, k) and G − (E, k) are analytic with no singularities on the upper and lower half plane respectively [9]. On the other hand, if M ≥ 2m, G + (E, k) has a pole at…”

“…Due to the complexity of this example, we shall limit ourselves to give here a summary of it. Readers interested in a deeper understanding of this example can go to the original paper [9]. Finally, we propose an example of "second quantized Friedrichs model", the Friedrichs model with virtual transitions, which is probably the simplest case of these second quantized models.…”

“…We want to discuss a few models that have been introduced elsewhere [9,8,61,5,6] that are very useful to give us an idea on the construction and behavior of relativistic resonances, not making use of the S matrix formalism (for a study on relativistic resonances in the context of the S matrix see [28,29,31]). Along this section, we are going to present four different situations, which are somehow related with unstable quantum field theory.…”

“…This problem was solved in [9]. The coefficients T (E, E ′ , k), R(E, E ′ , k), t(E, k) and r(E, k) depend on the form factor f (q) in (5.59) and can be written in terms of the Green function [9]:…”

“…In this example, developed in [9], we shall consider an exactly solvable model for relativistic unstable quantum fields. This model considers a quadratic interaction between a local scalar field ϕ with a mass M and a bilocal scalar field ψ with a continuous mass spectrum (we shall explain this terminology soon).…”