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Dirac Kets, Gamow Vectors, and Gel'fand triplets

Arno Böhm, John Dollard, Manuel Gadella

Published Jan 1, 1989• 140 citations
OpenAlex

Abstract

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.).

Related papers

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

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.

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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
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.
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.