Vector states for single and multiple-pole resonances

The formulation of quantum mechanics in rigged Hilbert spaces is used to study the vector states for resonance states or Gamow vectors. An important part of the work is devoted to the construction of Gamow vectors for resonances that appear as multiple poles on the analytic continuation of theS-matrix,S(E). The kinematical behavior of these vectors is also studied. This construction allow for generalized spectral decompositions of the Hamiltonian and the evolutionary semigroups, valid on certain locally convex spaces. Also a first attempt is made to define the resonance states as densities in an extension of the Liouville space, here called rigged Liouville space.     

Higher order Gamow states with exponential decay

We derive Gamow vectors fromS-matrix poles of higher multiplicity in analogy to the Gamow vectors describing resonances from first-order poles. With these vectors we construct a density operator that describes resonances associated with higher order poles that obey an exponential decay law. It turns out that this operator formed by these higher order Gamow vectors has a unique structure.     

Gamow state vectors as functionals over subspaces of the nuclear space

Exponentially decaying ‘Gamow state’ vectors are obtained from S-matrix poles in the lower half of the second sheet and are defined as functionals over a subspace of the nuclear space Φ. Exponentially growing ‘Gamow state’ vectors are obtained from S-matrix poles in the upper half of the second sheet and are defined as functionals over another subspace of Φ. On functionals over these two subspaces the dynamical group of time development splits into two semigroups.     

Time asymmetric quantum theory – II. Relativistic resonances from S‐matrix poles

Relativistic resonances and decaying states are described by representations of Poincaré transformations, similar to Wigner's definition of stable particles. To associate decaying state vectors to resonance poles of the S‐matrix, the conventional Hilbert space assumption (or asymptotic completeness) is replaced by a new hypothesis that associates different dense Hardy subspaces to the in‐ and out‐scattering states. Then one can separate the scattering amplitude into a background amplitude and one or several “relativistic Breit‐Wigner” amplitudes, which represent the resonances per se. These Breit‐Wigner amplitudes have a precisely defined lineshape and are associated to exponentially decaying Gamow vectors which furnish the irreducible representation spaces of causal Poincaré transformations into the forward light cone.     

Jordan blocks and exponentially decaying higher-order Gamow states

In the framework of the rigged Hilbert space, unstable quantum systems associated with first-order poles of the analytically continued S-matrix can be described by Gamow vectors which are generalized vectors with exponential decay and a Breit-Wigner energy distribution. This mathematical formalism can be generalized to quasistationary systems associated with higher-order poles of the S-matrix, which leads to a set of Gamow vectors of higher order with a non-exponential time evolution. One can define a state operator from the set of higher-order Gamow vectors which obeys the exponential decay law. We shall discuss to what extent the requirement of an exponential time evolution determines the form of the state operator for a quasistationary microphysical system associated with a higher-order pole of the S-matrix. Dedicated to Professor L. P. Horwitz on the occasion of his 65th birthday, October 14, 1995.     

Time Asymmetry and Quantum Theory of Resonances and Decay

These notes review a consistent and exact theory of quantum resonances and decay. Such a theory does not exist in the framework of traditional quantum mechanics and Dirac's formulation. But most of its ingredients have been familiar entities, like the Gamow vectors, the Lippmann-Schwinger (in- and out-plane wave) kets, the Breit-Wigner (Lorentzian) resonance amplitude, the analytically continued S-matrix, and its resonance poles. However, there are inconsistencies and problems with these ingredients: exponential catastrophe, deviations from the exponential law, causality, and recently the ambiguity of the mass and width definition for relativistic resonances. To overcome these problems the above entities will be appropriately defined (as mathematical idealizations). For this purpose we change just one axiom (Hilbert space and/or asymptotic completeness) to a new axiom which distinguishes between (in-)states and (out)observables using Hardy spaces. Then we obtain a consistent quantum theory of scattering and decay which has the Weisskopf-Wigner methods of standard textbooks as an approximation. But it also leads to time-asymmetric semigroup evolution in place of the usual, reversible, unitary group evolution. This, however, can be interpreted as causality for the Born probabilities. Thus we obtain a theoretical framework for the resonance and decay phenomena which is a natural extension of traditional quantum mechanics and possesses the same arrow-of-time as classical electrodynamics. When extended to the relativistic domain, it provides an unambiguous definition for the mass and width of the Z-boson and other relativistic resonances.     

Relativistic Gamow Vectors

Whereas in Dirac quantum mechanics and relativistic quantum field theory one uses Schwartz space distributions, the extensions of the Hilbert space that we propose uses Hardy spaces. The in- and out-Lippmann-Schwinger kets of scattering theory are functionals in two rigged Hilbert space extensions of the same Hilbert space. This hypothesis also allows to introduce generalized vectors corresponding to unstable states, the Gamow kets. Here the relativistic formulation of the theory of unstable states is presented. It is shown that the relativistic Gamow vectors of the unstable states, defined by a resonance pole of the S-matrix, are classified according to the irreducible representations of the semigroup of the Poincaré transformations (into the forward light cone). As an application the problem of the mass definition of the intermediate vector boson Z is discussed and it is argued that only one mass definition leads to the exponential decay law, and that is not the standard definition of the on-the-mass-shell renormalization scheme.     

Complex energies and beginnings of time suggest a theory of scattering and decay

Many useful concepts for a quantum theory of scattering and decay (like Lippmann-Schwinger kets, purely outgoing boundary conditions, exponentially decaying Gamow vectors, causality) are not well defined in the mathematical frame set by the conventional (Hilbert space) axioms of quantum mechanics. Using the Lippmann-Schwinger equations as the takeoff point and aiming for a theory that unites resonances and decay, we conjecture a new axiom for quantum mechanics that distinguishes mathematically between prepared states and detected observables. Suggested by the two signs ±i? of the Lippmann-Schwinger equations, this axiom replaces the one Hilbert space of conventional quantum mechanics by two Hardy spaces. The new Hardy space theory automatically provides Gamow kets with exponential time evolution derived from the complex poles of the S-matrix. It solves the causality problem since it results in a semigroup evolution. But this semigroup brings into quantum physics a new concept of the semigroup time t = 0, a beginning of time. Its interpretation and observations are discussed in the last section.     

Gamow Vectors and Time Asymmetry

We prove that Gamow vectors are important toolsin the quantum theory of irreversibility. We use themathematical formalism of rigged Hilbert spaces. Wediscuss some spectral formulas that include Gamow vectors as well as some results concerningGamow vectors. The role of the time-reversal operator isstudied. The formalism can be applied to formulate asense of irreversibility in cosmology.     

Time asymmetric quantum theory and the ambiguity of the Z-boson mass and width

Relativistic Gamow vectors emerge naturally in a time asymmetric quantum theory as the covariant kets associated to the resonance pole in the second sheet of the analytically continued S-matrix. They are eigenkets of the self-adjoint mass operator with complex eigenvalue and have exponential time evolution with lifetime . If one requires that the resonance width (defined by the Breit-Wigner lineshape) and the resonance lifetime always and exactly fulfill the relation , then one is lead to the following parameterization of in terms of resonance mass and width : . Applying this result to the -boson implies that and $\Gamma_R \approx \Gamma_Z-1.2\mbox{MeV}$ are the mass and width of the {\it Z}-boson and not the particle data values or any other parameterization of the Z-boson lineshape. Furthermore, the transformation properties of these Gamow kets show that they furnish an irreducible representation of the causal Poincaré semigroup, defined as a semi-direct product of the homogeneous Lorentz group with the semigroup of space-time translations into the forward light cone. Much like Wigner's unitary irreducible representations of the Poincaré group which describe stable particles, these irreducible semigroup representations can be characterized by the spin-mass values . Received 8 June 2000 / Published online: 27 November 2000     

An algorithm for separable representations of realistic interactions

A general method for expanding in a separable form a local interaction is developed. The algorithm makes use of the Gamow vectors and momenta, calculated from an eigenvalue problem, and preserves the analytic properties of the original S-matrix. Applications are presented for realistic nucleon-nucleon interactions and phenomenological optical potentials. The agreement between the separable and exact scattering matrices is quite good on and off the energy shell, for a wide range of energies.     

Analytic continuation of scattering data to the region of negative energies for systems that have one and two bound states

An exactly solvable potential model is used to study the possibility of deducing information about the features of bound states for the system under consideration (binding energies and asymptotic normalization coefficients) on the basis of data on continuum states. The present analysis is based on an analytic approximation and on the subsequent continuation of a partial-wave scattering function from the region of positive energies to the region of negative energies. Cases where the system has one or two bound states are studied. The α+d and α+¹²C systems are taken as physical examples. In the case of one bound state, the scattering function is a smooth function of energy, and the procedure of its analytic continuation for different polynomial approximations leads to close results, which are nearly coincident with exact values. In the case of two bound states, the scattering function has two poles—one in the region of positive energies and the other in the region of negative energies between the energies corresponding to the two bound states in question. Padéapproximants are used to reproduce these poles. The inclusion of these poles proves to be necessary for correctly describing the properties of the bound states.     

Resonances and Gamow States in Non-Local Potentials

For a large class of non-local, non separable potentials with non-compact support, the solution of the radial integrodifferential equation may be reduced to the solution of a homogeneous linear integral equation of Fredholm type with a quadratically integrable kernel. In this way we derive expansions of the wave functions and the Green's function of the Schrödinger equation with a non-local potential in terms of bound states, resonant states and a continuum of scattering functions with complex wave number. The rules of normalization, orthogonality and completeness satisfied by the eigenstates of the Schrödinger equation belonging to complex eigenvalues with Im E_n < 0, (Gamow or resonant states) are also derived. Finally, by means of a realistic example, it is shown how to use these expansions to exhibit the resonant behaviour of the differential cross section. Explicit expressions for the transition amplitudes and the partial widths in terms of expectation values of operators computed with Gamow functions are given.     

Oscillating Decay of an Unstable System

We study the medium-time behavior of the survival probability in the frame of the N-level Friedrichs model. The time evolution of an arbitrary unstable initial state is determined. We show that the survival probability may oscillate significantly during the so-called exponential era. This result explains qualitatively the experimental observations of the NaI decay. The Gamow states for N-level Friedrichs model are constructed. The time evolution in terms of the complex spectral representation including the Gamow states is discussed.     

Spectral properties of backward stimulated scattering in liquid carbon disulfide

The spectral structure of backward stimulated scattering from a 10 cm-long CS₂-liquid cell is investigated by using Q-switched 10-ns and 532-nm laser pulses with different spectral linewidths. Under a narrow spectral line (∼0.1 cm⁻¹) pump condition, very strong sharp lines near the pump wavelength (λ ₀) position and the first-order stimulated Raman scattering (λ _s1) position can be observed. However, under a wide line (≈1 cm⁻¹) pump condition, only a strong and superbroadening spectral band can be observed mainly in the red-shift side of the pump wavelength. The different spectral features under these two conditions can be explained by a competition between stimulated Brillouin, Raman, and Rayleigh-Kerr scattering. Under both pump conditions, the broadening spectral distributions are not consistent with the predictions given by stimulated Rayleigh-wing scattering theories, but can be interpreted well utilizing the theoretical model of stimulated Rayleigh-Kerr scattering. Zh. éksp. Teor. Fiz. 112, 1563–1573 (November 1997) Published in English in the original Russian journal. Reproduced here with stylistic changes by the Translation Editor.     

Derivation of Gamow Vectors for Resonances in Cut-Off Potentials

Gamow vectors are rigourously derived for a realistic case including an infinite number of resonant poles. This is the case of resonances produced by cut-off potentials. These are three-dimensional spherically symmetric potentials which vanish outside a bounded region. We solely consider the case of particles without internal structure and work with l = 0.     

Decoherence,correlation, and unstable quantum states in semiclassical cosmology

As almost any S-matrix of quantum theory possesses a set of complex poles (or branch cuts), it is shown using one example that this is the case in quantum field theory in curved space-time. These poles can be transformed into complex eigenvalues, the corresponding eigenvectors being Gamow vectors. This formalism, which is heuristic in ordinary Hilbert space, becomes a rigorous one within the framework of a properly chosen rigged Hilbert space. Then complex eigenvalues produce damping or growing factors and a typical two semigroups structure. It is known that the growth of entropy, decoherence, and the appearance of correlations, occur in the universe evolution, but this fact is demonstrated only under a restricted set of initial conditions. It is proved that the damping factors are mathematical tools that allow one to enlarge the set.     

Dielectric resonances in three-dimensional binary disordered media

Powdered solids often present very specific properties due to their granular nature. Such powders are often obtained by mixing two ingredients in variable proportions: conductor and insulator, or conductor and super-conductor. In a very natural way, these systems are modeled by regular lattices, whose sites or bonds are randomly chosen with given probabilities. It is known that the electrical and optical properties of random bi-dimensional (2D) networks are well described by their conductance's poles (resonances) and residues (amplitudes). The numerical implementation of a spectral method gave the spectral density, the AC conductivity, the multi-fractal properties of the moments for the local electric field (or currents), and spectrum of resonances characteristic of some small clusters (animals). This work extends the spectral method to the three-dimensional (3D) case where the problem is more complicated because the duality property and the corresponding symmetries are broken. As in the 2D-case, the two significant parameters are the ratio of the complex conductances and of both phases, and the probability p (resp. 1-p) of (resp. ). All the resonances lie on the negative real h-axis, i.e. for pure non resistive networks in the AC case. For a static (DC) system, only the value h=0 (corresponding to a binary system with finite and , or and finite) can give a resonance. Some applications are proposed, in particular the ability for small clusters (animals with one, two or three bonds) to present a singular response for well identified frequencies of the incident electromagnetic field. Received 24 March 1999     

Gamow Vectors in a Periodically Perturbed Quantum System

Gamow vectors and resonances play an important role in scattering theory, especially in the physics of metastable states. We study Gamow vectors and resonances in a time-dependent setting using the Borel summation method. In particular, we analyze the behavior of the wave function ψ(x,t) for one dimensional time-dependent Hamiltonian \(H=-\partial_{x}^{2}\pm2\delta(x)(1+2r\cos\omega t)\) where ψ(x,0) is compactly supported.We show that ψ(x,t) has a Borel summable expansion containing finitely many terms of the form \(\sum_{n=-\infty}^{\infty}e^{i^{3/2}\sqrt{-\lambda_{k}+n\omega i}|x|}A_{k,n}e^{-\lambda_{k}t+n\omega it}\), where λ _k represents the associated resonance. This expression defines Gamow vectors and resonances in a rigorous and physically relevant way for all frequencies and amplitudes in a time-dependent model.For small amplitude (|r|?1) there is one resonance for generic initial conditions. We calculate the position of the resonance and discuss its physical meaning as related to multiphoton ionization. We give qualitative theoretical results as well as numerical calculations in the general case.     

A Characteristic Decay Semigroup for the Resonances of Trace Class Perturbations with Analyticity Conditions of Semibounded Hamiltonians

To asymptotic complete scattering systems {M ₊+V,M ₊} on H₊:=L²(R₊,K{\mathcal{H}}_{+}:=L^{2}(\mathbf{R}_{+},{\mathcal{K}}, d λ), where M ₊ is the multiplication operator on H₊{\mathcal{H}}_{+} and V is a trace class operator with analyticity conditions, a decay semigroup is associated such that the spectrum of the generator of this semigroup coincides with the set of all resonances (poles of the analytic continuation of the scattering matrix into the lower half plane across the positive half line), i.e. the decay semigroup yields a “time-dependent” characterization of the resonances. As a counterpart a “spectral characterization” is mentioned which is due to the “eigenvalue-like” properties of resonances.