Il Nuovo Cimento (1955-1965)

, Volume 32, Issue 2, pp 317–322 | Cite as

Note on the formulation of gauge-invariance conditions inS-matrix theories

  • M. E. Mayer


Gauge-invariance of theS-matrix is formulated in terms of a functional derivative condition with respect to a vanishing classical 4-vector field depending on macroscopic space-time co-ordinates (or a family of such fields for the case of non-Abelian gauge groups). This condition can be used to derive identities of the Ward-Takahashi type for the coefficient functions of expansions of theS-matrix in terms of asymptotic field operators. The connection of gauge theories (especially electrodynamics) with macroscopic space-time and with the asymptotic momentum-space behaviour of scattering amplitudes is briefly discussed.


Si dà una formulazione dell’invarianza di gauge della matriceS in funzione di una condizione di derivata funzionale rispetto a un campo quadrivettoriale classico tendente a zero dipendente da coordinate spazio-temporali macroscopiche (o ad una famiglia di tali campi nel caso di gruppi di gauge non abeliani). Questa condizione può essere usata per derivare identità del tipo di Ward-Takahashi per le funzioni coefficienti degli sviluppi della matriceS in funzione di operatori di campo asintotici. Si discute brevemente la connessione delle teorie di gauge (specialmente elettrodinamiche) con lo spazio-tempo macroscopico e con il comportamento delle ampiezze di scattering nello spazio dei momenti asintotico.


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  1. (1).
    An incomplete list of papers devoted to this aspect of gauge-invariance is,e.g.:L. D. Landau andI. M. Khalatnikov:Žurn. Ėksp. Teor. Fiz.,29, 89 (1955);N. N. Bogoliubov andD. V. Shirkov:Introduction to the Theory of Quantized Fields, (Moscow 1957; New York, 1959)Ch. VII;L. Evans, G. Feldman andP. T. Matthews:Ann. Phys. (N.Y.),13, 268 (1961);H. Rollnik:Zeits. Phys.,161, 370 (1961).Google Scholar
  2. (3).
    Seee.g. H. Ekstein:Phys. Rev.,120, 1917 (1960) andNuovo Cimento,23, 606 (1962). To the author’s knowledge, doubled ray-representations of the Lorentz group were apparently first used byI. M. Gel’fand andM. Tseitlin:Žurn. Ėksp. Teor. Fiz.,36, 1109 (1956) in an attempt to solve the tau-theta paradox. However, such representations seem to have been known toBargmann, Wightman andWigner (unpublished notes on the Lorentz group). A brief discussion can be found inWightman’s lectures inRelations de dispersion et particules élémentaires (ed. byde Witt andOmnès) (New York, 1960), p. 163, and also in the book byGel’fand, Minlos andShapiro:Representations of the rotation group and of the Lorentz group (in Russian) (Moscow, 1958).MathSciNetADSCrossRefGoogle Scholar
  3. (4).
    Seee.g. M. E. Mayer:Nuovo Cimento,11, 760 (1959), where references to previous work can be found. An exhaustive bibliography on the gauge approach to vector mesons can be found in the critical paper ofOgievetskii andPolubarinov:Proceedings of the 1962 Conference on High-Energy Physics at CERN (Geneva, 1962).CrossRefMATHGoogle Scholar
  4. (5).
    See their book quoted in ref. (1), chapter XIII.Google Scholar
  5. (7).
    K. Nishijima:Phys. Rev.,119, 485 (1960);122, 248 (1961).MathSciNetADSCrossRefMATHGoogle Scholar
  6. (8).
    Private communication (1957). See also in this connectionB. V. Medvedev andM. I. Polivanov:Žurn. Ėksp. Teor. Fiz.,41, 1130 (1960) and ref. (7).Google Scholar

Copyright information

© Società Italiana di Fisica 1964

Authors and Affiliations

  • M. E. Mayer
    • 1
  1. 1.Department of PhysicsBrandeis UniversityWaltham

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