G.P. Pronko

Relativistic Gamow Vectors

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 Poincare 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.

Gamow vectors for degenerate scattering resonances

In this paper we construct Gamow vectors for resonances given by poles of the analytic continuation of the S matrix of any finite order. We study their modes of decay (or growth). We obtain Jordan block structures for the extended Hamiltonians and evolution operators on the subspaces spanned by these Gamow vectors. We perform this study within the context of the rigged Hilbert space extension of quantum theory. We construct an explicit Friedrichs model with a double pole resonance to illustrate the general formulation.

Friedrichs model with virtual transitions. Exact solution and indirect spectroscopy

The Friedrichs-type model of interaction between matter (multilevel system) and radiation including virtual transitions is considered. The canonical Bogolubov transformation diagonalizing the total Hamiltonian is constructed. It is pointed out that the transformation is improper when the discrete part of the spectrum of system is “dissolved” in the continuous one. The new vacuum state for the total Hamiltonian is obtained. The time evolution of the bare vacuum and the bare operators is calculated. Using the exact solution, the result of Passante, Petrosky, and Prigogine [Physica A 218, 437 (1995)] that the transition from the bare vacuum state to the true vacuum leads to the emission of real photons is confirmed. The dressing of the bare vacuum at the presence of resonances is an irreversible process. The relation of the result with the idea of “indirect spectroscopy” is discussed.

The Friedrichs model and its use in resonance phenomena

We present here a relation of different types of Friedrichs models and their use in the description and comprehension of resonance phenomena. We first discuss the basic Friedrichs model and obtain its resonance in the case that this is simple or doubly degenerated. Next, we discuss the model with N levels and show how the probability amplitude has an oscillatory behavior. Two generalizations of the Friedrichs model are suitable to introduce resonance behavior in quantum field theory. We also discuss a discrete version of the Friedrichs model and also a resonant interaction between two systems both with continuous spectrum. In an Appendix, we review the mathematics of rigged Hilbert spaces.

Quantum Zeno and anti-Zeno effects in the Friedrichs model

We analyze the short-time behavior of the survival probability in the frame of the Friedrichs model for different formfactors. We have shown that this probability is not necessary analytic at the time origin. The time when the quantum Zeno effect could be observed is found to be much smaller than usually estimated. We have also studied the anti-Zeno era and have estimated its duration.

Gamow vectors for barrier wells

Abstract We study four examples of Gamow vectors in one-dimensional potential barriers, namely square barriers and delta barriers. We show that resonances appear when the potential has at least two relative maxima and investigate the emergence of double resonances given rise to Gamow–Jordan vectors as well.

Spectrum generating algebras for the free motion in S3

We construct the spectrum generating algebra (SGA) for a free particle in the three-dimensional sphere S3 for both classical and quantum descriptions. In the classical approach, the SGA supplies time-dependent constants of motion that allow to solve algebraically the motion. In the quantum case, the SGA includes the ladder operators that give the eigenstates of the free Hamiltonian. We study this quantum case from two equivalent points of view.

Unstable relativistic quantum fields: two models

We present two models of relativistic interactions in quantum mechanics that produce resonances. In both cases, these resonances are described by poles of the analytic continuations of Green functions in terms of the variable energy. We develop the first model in detail and motivate and describe the second one. We compare the common features between these two models. The analysis is made in the context of quantum field theory.

Classical and quantum integrability in 3D systems

In this paper, we discuss three situations in which complete integrability of a three-dimensional classical system and its quantum version can be achieved under some conditions. The former is a system with axial symmetry. In the second, we discuss a three-dimensional system without spatial symmetry which admits separation of variables if we use ellipsoidal coordinates. In both cases, and as a condition for integrability, certain conditions arise in the integrals of motion. Finally, we study integrability in the three-dimensional sphere and a particular case associated with the Kepler problem in S3.

Non-Locality in Electrodynamics

We investigate the applicability of Hegerfeldts arguments on Quantum nonlocality in Quantum Electrodynamics following the work of Prigogine, Pronko, Petrosky, Ordonez and Karpov. We demonstrate the appearance of nonlocal effects at the level of quantum states. We show, however that the expectation values of some observables spread causally. Therefore the measurement of the nonlocality is questionable. We investigate an approach to classical measurement and conclude that the classical measurement cannot detect the “acausal” effects of the non-locality.