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Il Nuovo Cimento A (1965-1970)

, Volume 48, Issue 4, pp 997–1007 | Cite as

A reinterpretation of the notion of localization

  • A. J. Kálnay
  • B. P. Toledo
Article

Summary

Several reasons suggest that the difficulties to obtain the position operator in relativistic quantum mechanics are caused by the hypothesis that the measurable values of a component of the position are real numbers instead of regions of space of the order of a Compton wave length. We show how to develop the idea by using nonnormal operators and prove that this is consistent with reasonable requirements of position. The use of non-Hermitian and nonnormal operators in quantum mechanics is discussed.

Keywords

Position Operator Hermitian Operator Relativistic Quantum Mechanic Space Rotation Reasonable Requirement 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Новая интерпретация понятия локализации

Резюме

Высказываются некоторые аргументы, что трудности, связанные с получением оператора положения в релятивистской квантовой механике, обусловлены гипотезой, что измеряемые значания компоненты радиуса-вектора представляют вещественные числа, а не области пространства, порядка Комптоновскои длины волны. Мы показываем, чак развить эту идею, используя ненормальные операторы, и докаываем, что это согласуется с разумными требованиями положения. Обсуждается использование неэрмитовских и ненормальных операторов в квантовой механике.

Riassunto

Molti motivi suggeriscono che le difficoltà nell’ottenere gli operatori di posizione in meccanica quantistica relativistica sono causate dall’ipotesi che i valori misurabili di una componente della posizione siano numeri reali invece che regioni dello spazio dell’ordine della lunghezza d’onda Compton. Si mostra come sviluppare quest’idea usando operatori non normali e si dimostra che ciò è compatibile con ragionevoli esigenze di posizione. Si discute l’uso di operatori non hermitiani e non normali in meccanica quantistica.

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References

  1. (1).
    See for example,L. L. Foldy andS. A. Wouthuysen:Phys. Rev.,78, 29 (1950);H. Bacry:Journ. Math. Phys.,5, 109 (1964);M. Bunge:Nuovo Cimento,1, 977 (1955);P. Beckman:Nuovo Cimento,27, 868 (1963);R. A. Berg:Journ. Math. Phys.,6, 34 (1965);K. Bardakci andR. Acharya:Nuovo Cimento,21, 802 (1961);T. F. Jordan andN. Mukunda:Phys. Rev.,132, 1842 (1963); see also ref. (2,3),T. D. Newton andE. P. Wigner:Rev. Mod. Phys.,21 400 (1949),M. H. L. Pryce:Proc Roy. Soc., A195, 62 (1948).ADSCrossRefGoogle Scholar
  2. (2).
    T. D. Newton andE. P. Wigner:Rev. Mod. Phys.,21, 400 (1949).ADSCrossRefGoogle Scholar
  3. (3).
    M. H. L. Pryce:Proc Roy. Soc., A195, 62 (1948).ADSMathSciNetCrossRefGoogle Scholar
  4. (4).
    It is not Lorentz-invariant in its usual form, but there exists a reformulation of the problem which is Lorentz-invariant, cf.G. N. Fleming:Phys. Rev.,137, B 188 (1965).ADSMathSciNetCrossRefGoogle Scholar
  5. (5).
    T. O. Philips:Phys. Rev.,136, B 893 (1964).ADSMathSciNetCrossRefGoogle Scholar
  6. (6).
    For simplicity we say that position is point-type even if it has noncommuting components, though in this case there is no common eigenstate corresponding to a point of 3-dimensional space.Google Scholar
  7. (7).
    J. A. Gallardo, A. J. Kálnay, B. A. Stec andB. P. Toledo:Nuovo Cimento, to be published.Google Scholar
  8. (8).
    C. Møller: Communication from Dublin Institute for Advance Studies A No. 5, 1949 (unpublished).Google Scholar
  9. (9).
    J. A. Gallardo, A. J. Kálnay, B. A. Stec andB. P. Toledo:Nuovo Cimento,48 A, 1008 (1967).ADSCrossRefGoogle Scholar
  10. (10).
    Notice the difference with Wightman’s localization in a region. Cf. ref. (11).ADSMathSciNetCrossRefGoogle Scholar
  11. (11).
    A. S. Wightman:Rev. Mod. Phys.,34, 845 (1962).ADSMathSciNetCrossRefGoogle Scholar
  12. (12).
    This Section is self-contained. More details will be published elsewhere in the future.Google Scholar
  13. (13).
    P. A. M. Dirac:The Principles of Quantum Mechanics (Oxford, 1947), p. 34.Google Scholar
  14. (14).
    E. C. Kemble:The Fundamental Principles of Quantum Mechanics (New York, 1958), p. 249.Google Scholar
  15. (15).
    H. Margenau:The Nature of Physical Reality (New York, 1950), p. 108.Google Scholar
  16. (16).
    W. E. Thirring:Principles of Quantum Electrodynamics (New York, 1958), p. 112.Google Scholar

Copyright information

© Società Italiana di Fisica 1967

Authors and Affiliations

  • A. J. Kálnay
    • 1
  • B. P. Toledo
    • 1
  1. 1.IMAFUniversidad Nacional de CórdobaCórdobaArgentina

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