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Mathematical Models for Unstable Quantum Systems and Gamow States

Manuel Gadella, Sebastián Fortín, Juan Pablo Jorge, Marcelo Losada

Published Jun 8, 2022• 4 citations

Abstract

We review some results in the theory of non-relativistic quantum unstable systems. We account for the most important definitions of quantum resonances that we identify with unstable quantum systems. Then, we recall the properties and construction of Gamow states as vectors in some extensions of Hilbert spaces, called Rigged Hilbert Spaces. Gamow states account for the purely exponential decaying part of a resonance; the experimental exponential decay for long periods of time physically characterizes a resonance. We briefly discuss one of the most usual models for resonances: the Friedrichs model. Using an algebraic formalism for states and observables, we show that Gamow states cannot be pure states or mixtures from a standard view point. We discuss some additional properties of Gamow states, such as the possibility of obtaining mean values of certain observables on Gamow states. A modification of the time evolution law for the linear space spanned by Gamow shows that some non-commuting observables on this space become commuting for large values of time. We apply Gamow states for a possible explanation of the Loschmidt echo.

DOI: 10.3390/e24060804

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In this review, we present a rigorous construction of an algebraic method for quantum unstable states, also called Gamow states. A traditional picture associates these states to vectors states called Gamow vectors. However, this has some difficulties. In particular, there is no consistent definition of mean values of observables on Gamow vectors. In this work, we present Gamow states as functionals on algebras in a consistent way. We show that Gamow states are not pure states, in spite of their representation as Gamow vectors. We propose a possible way out to the construction of averages of observables on Gamow states. The formalism is intended to be presented with sufficient mathematical rigor.

It is well known that a state with complex energy cannot be the eigenstate of a self-adjoint operator, such as the Hamiltonian. Resonances, i.e., states with exponentially decaying observables, are not vectors belonging to the conventional Hilbert space. One can describe these resonances in an unusual mathematical formalism based on the so-called rigged Hilbert space (RHS). In the RHS, the states with complex energy are denoted as Gamow vectors (GVs), and they model decay processes. We study the GVs of the reversed harmonic oscillator, and we analytically and numerically investigate the unstable evolution of wave packets. We introduce the background function to study initial data that are not composed only by a summation of GVs, and we analyze different wave packets belonging to specific function spaces. Our work furnishes support for the idea that irreversible wave propagation can be investigated using rigged-Hilbert-space quantum mechanics and provides insight for the experimental investigation of irreversible dynamics.

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.

After a summary of the Rigged Hilbert space formulation of quantum mechanics and a brief statement of its advantages over von Neumann’s formulation, a mathematically correct definition of Gamow’s exponentially decaying vectors as generalized energy eigenvectors is suggested. It is shown that exponentially decaying vectors are obtained from the S-matrix poles in the lower half of the second sheet and exponentially growing vectors from the S-matrix poles in the upper half of the second sheet. Decaying ’’state’’ vectors are defined as functionals over half of the space of physical states and growing ’’state’’ vectors are defined as functionals over the other half. On functionals over these subspaces, the dynamical group of time development splits into two semigroups, one for t ≳ 0 and the other for t < 0. The generalized basis system connected with the spectrum of the Hamiltonian is transformed into a new basis system in which the exponentially decaying component of the density matrix is separated.

We introduce a dynamical evolution operator for dealing with unstable physical process, such as scattering resonances, photon emission, decoherence and particle decay. With that aim, we use the formalism of rigged Hilbert space and represent the time evolution of quantum observables in the Heisenberg picture, in such a way that time evolution is nonunitary. This allows to describe observables that are initially noncommutative, but become commutative after time evolution. In other words, a non-abelian algebra of relevant observables becomes abelian when times goes to infinity. We finally present some relevant examples.