Foundations of Physics

, Volume 32, Issue 6, pp 815–869

A Unified Mathematical Formalism for the Dirac Formulation of Quantum Mechanics

  • M. Gadella
  • F. Gómez
Article

Abstract

We revise the mathematical implementation of the Dirac formulation of quantum mechanics, presenting a rigorous framework that unifies most of versions of this implementation.

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REFERENCES

  1. 1.
    A. R. Marlow, J. Math. Phys. 6, 919 (1965).Google Scholar
  2. 2.
    J. E. Roberts, J. Math. Phys. 7, 1097 (1966); Commun. Math. Phys. 3, 98 (1966).Google Scholar
  3. 3.
    J. P. Antoine, J. Math. Phys. 10, 53 (1969); 10, 2276 (1969).Google Scholar
  4. 4.
    A. Bohm, Boulder Lectures on Theoretical Physics 1966, Vol. 9A (Gordon &;;; Breach, New York, 1967).Google Scholar
  5. 5.
    O. Melsheimer, J. Math. Phys. 15, 902 (1974); 15, 917 (1974).Google Scholar
  6. 6.
    S. J. L. Eijndhoven and J. de Graaf, A Mathematical Introduction to Dirac's Formalism (North-Holland, Amsterdam, 1986).Google Scholar
  7. 7.
    I. M. Gelfand and G. E. Shilov, Generalized Functions: Spaces of Fundamental and Generalized Functions (Academic, New York, 1968).Google Scholar
  8. 8.
    D. Tjostheim, J. Math. Phys. 16, 766 (1975).Google Scholar
  9. 9.
    K. Maurin, General Eigenfunction Expansions and Unitary Representation of Topological Groups (Polish Scientific Publishers, Warszawa, 1968).Google Scholar
  10. 10.
    J. P. Antoine, “Quantum Mechanics Beyond Hilbert Space,” in Irreversibility and Causality, A. Bohm, H. D. Doebner, and P. Kielanowski, eds. (Springer Lecture Notes in Physics 504) (Springer, Berlin, 1998).Google Scholar
  11. 11.
    C. Foias, Acta Sci. Math. 20, 117 (1959); Compt. Rend. 248, 904 (1959); Compt. Rend. 248, 1105 (1959); Rev. Math. Pures Appl. Acad. Roumaine 7, 241 (1962).Google Scholar
  12. 12.
    H. H. Schaeffer, Topological Vector Spaces (MacMillan, New York, 1977).Google Scholar
  13. 13.
    H. Baumgärtel and M. Wollenberg, Mathematical Scattering Theory (Operator Theory: Advances and Applications 9) (Birkhauser, Basel, 1983).Google Scholar
  14. 14.
    D. R. Yafaev, Mathematical Scattering Theory. General Theory (Translations of Mathematical Monographs 105) (American Mathematical Society, Providence, 1992).Google Scholar
  15. 15.
    J. S. Howland, “Banach space techniques in the perturbation theory of self-adjoint operators with continous spectra,” J. Math. Anal. and Appl. 20, 22–47 (1967).Google Scholar
  16. 16.
    P. A. Retjo, “On gentle perturbations I and II,” Math. Anal. Appl. 17, 453–462 (1967); 20, 145–187 (1967).Google Scholar
  17. 17.
    J. S. Howland, J. Func. Anal. 2, 1 (1968).Google Scholar
  18. 18.
    T. Kato and S. T. Kuroda, in Functional Analysis and Related Fields, F. E. Browder, ed. (Springer, Berlin and New York, 1970).Google Scholar
  19. 19.
    I. M. Gelfand and N. Y. Vilenkin, Generalized Functions: Applications of Harmonic Analysis (Academic, New York, 1964).Google Scholar
  20. 20.
    I. Antoniou and Z. Suchanecki, “Densities of singular measures and generalized spectral decompositions,” in Generalized Functions, Operator Theory and Dynamical Systems, I. Antoniou and G. Lummer, eds. (Chapmann and Hall/CRC, London, 1999).Google Scholar
  21. 21.
    A. Bohm and M. Gadella, Dirac Kets, Gamow Vectors and Gel'fand Triplets, (Springer Lecture Notes in Physics 248) (Springer, New York, 1989).Google Scholar
  22. 22.
    D. Fredricks, Rep. Math. Phys. 8, 277 (1975).Google Scholar
  23. 23.
    M. S. Birman and M. Z. Solomjak, Spectral Theory of Self Adjoint Operators in Hilbert Space (Reidel, Boston, 1987).Google Scholar
  24. 24.
    A. Bohm, Quantum Mechanics: Fundations and Applications (Springer, New York, 1994).Google Scholar
  25. 25.
    A. Bohm, Lett. Math. Phys. 3, 455 (1978); J. Math. Phys. 22, 2813 (1981).Google Scholar
  26. 26.
    I. Prigogine, From Being to Becoming (Freeman, New York, 1980).Google Scholar
  27. 27.
    I. Antoniou and I. Prigogine, Physica 192A, 443 (1993).Google Scholar
  28. 28.
    I. Antoniou and S. Tasaki, Physica A 190, 303–329 (1992); Int. J. Quant. Chem. 46, 424–474 (1993).Google Scholar
  29. 29.
    W. O. Amrein, J. Jauch, and B. Sinha, Scattering Theory in Quantum Mechanics (Benjamin, Reading, 1977).Google Scholar
  30. 30.
    M. Reed and B. Simon, Functional Analysis (Academic, New York, 1972).Google Scholar
  31. 31.
    H. Jarchow, Locally Convex Spaces (Teubner, Stuttgart, 1981).Google Scholar
  32. 32.
    J. Horváth, Topological Vector Space and Distributions (Addison–Wesley, Reading, 1966).Google Scholar
  33. 33.
    J. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton University Press, Princeton, 1955).Google Scholar
  34. 34.
    P. A. M. Dirac, The Principles of Quantum Mechanics (Clarendon, Oxford, 1958).Google Scholar
  35. 35.
    G. Bachman and L. Narici, Functional Analysis (Academic, New York, 1966).Google Scholar
  36. 36.
    G. E. Shilov and B. L. Gurevich, Integral, Measure, and Derivative: A Unified Approach (Prentice Hall, Englewood Cliffs, 1966).Google Scholar
  37. 37.
    B. Sz. Nagy, “Extensions of Linear Transformations in Hilbert Space Which Extends Beyond This Space,” Appendix to Functional Analysis by F. Riesz and B. Sz. Nagy (Ungar, New York, 1960).Google Scholar
  38. 38.
    L. Gårdin, Proc. Natl. Acad. Sci. USA 33, 331 (1947).Google Scholar

Copyright information

© Plenum Publishing Corporation 2002

Authors and Affiliations

  • M. Gadella
    • 1
  • F. Gómez
    • 2
  1. 1.Departamento de Física TeóricaFacultad de CienciasValladolidSpain
  2. 2.Departamento de Análisis MatemáticoFacultad de CienciasValladolidSpain

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