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Quantum mechanics beyond hilbert space

  • Quantum Theory Beyond Hilbert Space
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Irreversibility and Causality Semigroups and Rigged Hilbert Spaces

Part of the book series: Lecture Notes in Physics ((LNP,volume 504-504))

Abstract

When describing a quantum mechanical system, it is convenient to consider state vectors that do not belong to the Hilbert space. In the first part of this paper, we survey the various formalisms have been introduced for giving a rigorous mathematical justification to this procedure: rigged Hilbert spaces (RHS), scales or lattices of Hilbert spaces (LHS), nested Hilbert spaces, partial inner product spaces. Then we present three types of applications in quantum mechanics, all of them involving spaces of analytic functions. First we present a LHS built around the Bargmann space, thus giving a natural frame for the Fock-Bargmann (or phase space) representation. Then we review the RHS approach to scattering theory (resonances, Gamow vectors, etc.). Finally, we reformulate the Weinberg-van Winter integral equation approach to scattering in the LHS language, and this allows us to prove that it is in fact a particular case of the familiar complex scaling method.

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  1. Institut de Physique Théorique, Université Catholique de Louvain, B-1348, Louvainla-Neuve, Belgium

    J.-P. Antoine

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  1. J.-P. Antoine
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Arno Bohm Heinz-Dietrich Doebner Piotr Kielanowski

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© 1998 Springer-Verlag

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Antoine, JP. (1998). Quantum mechanics beyond hilbert space. In: Bohm, A., Doebner, HD., Kielanowski, P. (eds) Irreversibility and Causality Semigroups and Rigged Hilbert Spaces. Lecture Notes in Physics, vol 504-504. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0106773

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  • DOI: https://doi.org/10.1007/BFb0106773

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  • Print ISBN: 978-3-540-64305-0

  • Online ISBN: 978-3-540-69725-1

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