The marvelous consequences of hardy spaces in quantum physics

Arno Bohm, Hai Viet Bui

Research output: Chapter in Book/Report/Conference proceedingConference contribution

1 Scopus citations

Abstract

Dynamical differential equations, like the Schrödinger equation for the states, or the Heisenbergequa tion for the observables, need to be solved under boundary conditions. The original boundary condition of von Neumann, the Hilbert space axiom, required that the allowed wave functions are Lebesgue square integrable. This leads by a mathematical theorem of Stone-von Neumann to the unitary group evolution meaning the time t extends over -∞ < t < +<. Physicists do not use Lebesgue integrals but followed a different path usinga lmost exclusively the Dirac formalism and well-behaved (Schwartz) functions. This led the mathematicians to Schwartz-Rigged Hilbert spaces (Gelfand triplets), which are the mathematical core of Dirac's bra-ket formalism. This is insufficient for a theory that includes resonance and decay phenomena, which requires analytic continuation in energy E in order to accommodate exponentially decayingG amow kets, Breit-Wigner (Lorentzian) resonances, and Lippmann-Schwinger kets. This leads to a pair of Rigged Hilbert Spaces of smooth Hardy functions, one representing the prepared states of scatteringe xperiments (preparation apparatus) and the other representingd etected observables (registration apparatus). A mathematical consequence of the Hardy space axiom is that the time evolution is asymmetric given by the semi-group, i.e., t0 ≤ t < +<, with a finite t0. What would the meaningo f that t0 be?

Original languageEnglish (US)
Title of host publicationGeometric Methods in Physics
EditorsPiotr Kielanowski, S. Twareque Ali, Anatol Odzijewicz, Martin Schlichenmaier, Theodore Voronov
PublisherSpringer International Publishing
Pages211-228
Number of pages18
ISBN (Print)9783034804479
DOIs
StatePublished - 2013
Event30th Workshop on Geometric Methods in Physics, 2011 - Bialowieza, Poland
Duration: Jun 26 2011Jul 2 2011

Publication series

NameTrends in Mathematics
Volume59
ISSN (Print)2297-0215
ISSN (Electronic)2297-024X

Conference

Conference30th Workshop on Geometric Methods in Physics, 2011
CountryPoland
CityBialowieza
Period6/26/117/2/11

Keywords

  • Hardy space
  • Rigged Hilbert space
  • Semigroup
  • Time asymmetry
  • Unitary group

ASJC Scopus subject areas

  • Mathematics(all)

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  • Cite this

    Bohm, A., & Bui, H. V. (2013). The marvelous consequences of hardy spaces in quantum physics. In P. Kielanowski, S. T. Ali, A. Odzijewicz, M. Schlichenmaier, & T. Voronov (Eds.), Geometric Methods in Physics (pp. 211-228). (Trends in Mathematics; Vol. 59). Springer International Publishing. https://doi.org/10.1007/978-3-0348-0448-6_17