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Dirac Kets, Gamow Vectors, and Gel'fand triplets : the rigged Hilbert space formulation of quantum mechanics : lectures in mathematical physics at the University of Texas at Austin

Arno Böhm, Manuel Gadella, John Dollard

Published Nov 28, 1989• 59 citations
OpenAlex

Abstract

Dirac's formalism of quantum mechanics was always praised for its elegance. This book introduces the student to its mathematical foundations and demonstrates its ease of applicability to problems in quantum physics. The book starts by describing in detail the concept of Gel'fand triplets and how one can make use of them to make the Dirac heuristic approach rigorous. The results are then deepened by giving the analytic tools, such as the Hardy class function and Hilbert and Mellin transforms, needed in applications to physical problems. Next, the RHS model for decaying states based on the concept of Gamow vectors is presented. Applications are given to physical theories of such phenomena as decaying states and resonances.

Related papers

Dirac Kets, Gamow Vectors, and Gel'fand triplets

Arno Böhm, John Dollard, Manuel Gadella

This book is an expansion of a little booklet on rigged Hilbert spaces and quantum mechanics. The purpose of that book was to present a mathematical foundation of Dirac's bracket formalism. (orig.).

•Jan 1, 1989•140 citations

RIGGED HILBERT SPACE RESONANCES AND TIME ASYMMETRIC QUANTUM MECHANICS

Arno Böhm, Hani Kaldass

The Rigged Hilbert Space (RHS) theory of resonance scattering and decay is reviewed and contrasted with the standard Hilbert space (HS) theory of quantum mechanics. The main difference is in the choice of boundary conditions. Whereas the conventional theory allows for the in-states $ϕ^+$ and the out-states (observables) $ψ^-$ of the S-matrix elements $(ψ^-,ϕ^+)=(ψ^{out},S ϕ^{in})$ any elements of the HS $\H$, $\{ψ^-\}=\{ϕ^+\}(=\H)$, the RHS theory chooses the boundary conditions~: $ϕ^+\inΦ_-\subset\H\subsetΦ_-^\times$, $ψ^-\inΦ_+\subset \H\subset Φ_+^\times$, where $Φ_-$ ($Φ_+$) are Hardy class spaces associated to the lower (upper) half-plane of the second sheet of the analytically continued S-matrix. This can be phenomenologically justified by causality. The two RHS's for states $ϕ^+$ and observables $ψ^-$ provide new vectors which are not in $\H$, e.g. the Dirac-Lippmann-Schwinger kets $|E^{\pm}\inΦ_{\mp}^{\times}$ (solutions of the Lippmann-Schwinger equation with $\pm iε$ respectively) and the Gamow vectors $|E_R-iΓ/2^\pm\inΦ_{\mp}^\times$. The Gamow vectors $|E_R-iΓ/2^-$ have all the properties that one heuristically needs for quasistable states. In addition, they give rise to asymmetric time evolution expressing irreversibility on the microphysical level.

arXiv:quant-ph/9909081•WORLD SCIENTIFIC eBooks•Nov 1, 2000•5 citations

Dirac Formalism and Symmetry Problems in Quantum Mechanics. I. General Dirac Formalism

Jean-Pierre Antoine

Dirac's bra and ket formalism is investigated and incorporated into a complete mathematical theory. First the axiomatic foundations of quantum mechanics and von Neumann's spectral theory of observables are reviewed and several inadequacies are pointed out. These defects then are remedied by extending the usual Hilbert space to a rigged Hilbert space as introduced by Gel'fand, i.e., a triplet Φ⊂H⊂Φ′ where H is a Hilbert space, Φ a dense subspace of H provided with a new (finer) topology, Φ′ the dual of Φ. It is shown that this mathematical structure, together with the Schwartz nuclear theorem, allows us to reproduce Dirac's formalism in a completely rigorous way, without losing its transparency; this makes the theory easier to handle. The temporal evolution of the system and the wave equation are considered. Finally the probabilistic interpretation and the physical aspects of the theory are discussed; Φ is identified with the set of all physically accessible states of the system, Φ′ with the set of all possible experiments (apparatus) to which it can be subjected; this provides a direct connection with Feynman's formulation of quantum mechanics.

Journal of Mathematical Physics•Jan 1, 1969•148 citations

Resonance poles and Gamow vectors in the rigged Hilbert space formulation of quantum mechanics

Arno Böhm

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.

Journal of Mathematical Physics•Dec 1, 1981•175 citations

The Rigged Hilbert space formulation of quantum mechanics and decay processes of higher-order resonances

P. Patuleanu

Resonances and decaying states have been associated with first-order poles of the S-matrix and represented by exponentially decaying Gamow kets using the Rigged Hilbert Space formulation of quantum mechanics. In analogy, higher-order S- matrix poles, represented by a mixture of Gamow-Jordan vectors, are associated with higher-order decaying states. The exact Golden Rule for decaying states associated to a higher-order pole of the S-matrix is calculated. These higher-order decaying states operators also obey the exponential decay law, but they have a more complicated linewidth than the Breit-Wigner energy distribution. This linewidth depends on the interaction V. To test whether these higher-order decaying state operators represent possible physical states, we discuss their linewidth for various assumptions about the matrix elements of V.

•Dec 1, 1999•0 citations

Mathematical Foundations of Time Asymmetric Quantum Mechanics

Manuel Gadella

We review the mathematical tools that are suitable for a formulation of time asymmetry in quantum mechanics. In particular, Hardy functions on a half plane and rigged Hilbert spaces constructed with a subclass of Hardy functions. This time asymmetry often appears in quantum scattering and, in particular, in resonance scattering. We review the construction of Gamow vectors, often considered Gamow states for resonances. A brief summary of the fundamental ideas of time asymmetric quantum mechanics is presented in a last section.

Journal of Physics Conference Series•May 26, 2017•2 citations
Dirac's formalism of quantum mechanics was always praised for its elegance. This book introduces the student to its mathematical foundations and demonstrates its ease of applicability to problems in quantum physics. The book starts by describing in detail the concept of Gel'fand triplets and how one can make use of them to make the Dirac heuristic approach rigorous. The results are then deepened by giving the analytic tools, such as the Hardy class function and Hilbert and Mellin transforms, needed in applications to physical problems. Next, the RHS model for decaying states based on the concept of Gamow vectors is presented. Applications are given to physical theories of such phenomena as decaying states and resonances.
The Rigged Hilbert Space (RHS) theory of resonance scattering and decay is reviewed and contrasted with the standard Hilbert space (HS) theory of quantum mechanics. The main difference is in the choice of boundary conditions. Whereas the conventional theory allows for the in-states ϕ+ϕ^+ϕ+ and the out-states (observables) ψ−ψ^-ψ− of the S-matrix elements (ψ−,ϕ+)=(ψout,Sϕin)(ψ^-,ϕ^+)=(ψ^{out},S ϕ^{in})(ψ−,ϕ+)=(ψout,Sϕin) any elements of the HS $\H$, {ψ−}={ϕ+}(=)˝\{ψ^-\}=\{ϕ^+\}(=\H){ψ−}={ϕ+}(=)˝​, the RHS theory chooses the boundary conditions~: ϕ+∈Φ−⊂⊂˝Φ−×ϕ^+\inΦ_-\subset\H\subsetΦ_-^\timesϕ+∈Φ−​⊂⊂˝​Φ−×​, ψ−∈Φ+⊂⊂˝Φ+×ψ^-\inΦ_+\subset \H\subset Φ_+^\timesψ−∈Φ+​⊂⊂˝​Φ+×​, where Φ−Φ_-Φ−​ (Φ+Φ_+Φ+​) are Hardy class spaces associated to the lower (upper) half-plane of the second sheet of the analytically continued S-matrix. This can be phenomenologically justified by causality. The two RHS's for states ϕ+ϕ^+ϕ+ and observables ψ−ψ^-ψ− provide new vectors which are not in $\H$, e.g. the Dirac-Lippmann-Schwinger kets ∣E±∈Φ∓×|E^{\pm}\inΦ_{\mp}^{\times}∣E±∈Φ∓×​ (solutions of the Lippmann-Schwinger equation with ±iε\pm iε±iε respectively) and the Gamow vectors ∣ER−iΓ/2±∈Φ∓×|E_R-iΓ/2^\pm\inΦ_{\mp}^\times∣ER​−iΓ/2±∈Φ∓×​. The Gamow vectors ∣ER−iΓ/2−|E_R-iΓ/2^-∣ER​−iΓ/2− have all the properties that one heuristically needs for quasistable states. In addition, they give rise to asymmetric time evolution expressing irreversibility on the microphysical level.
Dirac's bra and ket formalism is investigated and incorporated into a complete mathematical theory. First the axiomatic foundations of quantum mechanics and von Neumann's spectral theory of observables are reviewed and several inadequacies are pointed out. These defects then are remedied by extending the usual Hilbert space to a rigged Hilbert space as introduced by Gel'fand, i.e., a triplet Φ⊂H⊂Φ′ where H is a Hilbert space, Φ a dense subspace of H provided with a new (finer) topology, Φ′ the dual of Φ. It is shown that this mathematical structure, together with the Schwartz nuclear theorem, allows us to reproduce Dirac's formalism in a completely rigorous way, without losing its transparency; this makes the theory easier to handle. The temporal evolution of the system and the wave equation are considered. Finally the probabilistic interpretation and the physical aspects of the theory are discussed; Φ is identified with the set of all physically accessible states of the system, Φ′ with the set of all possible experiments (apparatus) to which it can be subjected; this provides a direct connection with Feynman's formulation of quantum mechanics.
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.
Resonances and decaying states have been associated with first-order poles of the S-matrix and represented by exponentially decaying Gamow kets using the Rigged Hilbert Space formulation of quantum mechanics. In analogy, higher-order S- matrix poles, represented by a mixture of Gamow-Jordan vectors, are associated with higher-order decaying states. The exact Golden Rule for decaying states associated to a higher-order pole of the S-matrix is calculated. These higher-order decaying states operators also obey the exponential decay law, but they have a more complicated linewidth than the Breit-Wigner energy distribution. This linewidth depends on the interaction V. To test whether these higher-order decaying state operators represent possible physical states, we discuss their linewidth for various assumptions about the matrix elements of V.
We review the mathematical tools that are suitable for a formulation of time asymmetry in quantum mechanics. In particular, Hardy functions on a half plane and rigged Hilbert spaces constructed with a subclass of Hardy functions. This time asymmetry often appears in quantum scattering and, in particular, in resonance scattering. We review the construction of Gamow vectors, often considered Gamow states for resonances. A brief summary of the fundamental ideas of time asymmetric quantum mechanics is presented in a last section.