Il Nuovo Cimento (1955-1965)

, Volume 16, Issue 4, pp 671–682 | Cite as

On a convergent non-local field theory. — I.

  • E. Arnous
  • W. Heitler
  • Y. Takahashi


A non-local field theory is developed (and formulated in interaction representation) with the view of obtaining a converging theory. At least one special type of form factor is known for which this theoryconverges throughout, but in this and the following paper the form factor is not specified (it may even be aq-number). The view is taken that convergence should take precedence over exact Lorentz-invariance, because present knowledge does not necessarily exclude a violation of Lorentz-invarianceinside the source. It is shown that the total charge is the same as in local theory and is conserved, provided only that the form factor commutes with the field operators. The invariance against groups of transformations is discussed in a general way and for the special case of Lorentz-transformations. It is shown that the Schrödinger equation can always be formulated in an invariant manner, but that this does not imply the invariance of theS-matrix. The conditions for the invariance of theS-matrix are derived. The theory contains the local theory as a special case.


Si sviluppa (e si formula nella rappresentazione delle interazioni) una teoria non-locale del campo allo scopo di ottenere una teoria convergente. Si conosce almeno un tipo speciale di fattore di forma per cui questa teoriaconverge uniformemente, ma in questo scritto e nel seguente il fattore di forma non è specificato (può anche essere un numeroq). Si giudica che la convergenza debba aver la precedenza sull’esatta invarianza di Lorentz, perchè lo stato attuale delle conoscenze non esclude necessariamente una violazione della invarianza di Lorentznell’interno della sorgente. Si dimostra che la carica totale è la stessa che nella teoria locale ed è conservata, purchè almeno il fattore di forma commuti con gli operatori di campo. L’invarianza verso gruppi di trasformazioni viene discussa sia in generale sia per il caso speciale delle trasformazioni di Lorentz. Si dimostra che l’equazione di Schrödinger può sempre scriversi in forma invariante, ma che questo non implica l’invarianza della matriceS. Si deducono le condizioni per l’invarianza della matriceS. La teoria contiene la teoria locale come caso speciale.


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  1. (1).
    Compare for instance:Ch. Møller andP. Kristensen:Kgl. Dan. Vid. Selsk.,27, no. 7 (1952);C. Bloch:Kgl. Dan. Vid. Selsk.,27, no. 8 (1952);G. Wataghin:Nuovo Cimento,5, 689 (1957), and numerous other papers.Google Scholar
  2. (2).
    The following facts can be quoted in this connexion: (i) the π±0 mass difference which is probably mainly an electromagnetic self-energy; (ii) the proton-neutron mass difference which is probably a mixed electromagnetic-mesonic self-energy, ordere 2 f 2,e 2 f 4 … (cfr.R. P. Feynmann andG. Speismann:Phys, Rev.,94, 500 (1954), andL. O’Raifeartaigh, B. Sredniawa andCh. Terreaux:Nuovo Cimento,14, 376 (1959)). Perhaps similar considerations obtain for the K0-K+ mass difference. (iii) The π-nucleon scattering. While perturbation theory based on the first order gave on the whole the correct gross behaviour of the cross section (W. Heitler andH. W. Peng:Proc. Cam. Phil. Soc.,38, 296 (1942)), the quantitative improvement (G. F. Chew:Phys. Rev.,94, 1748, 1755 (1954)), due to the inclusion of higher orders on grounds of the non-relativistic extended source model is very considerable. The same applies to the photo-meson production. It may be remarked that in these calculations thenon-renormalizable p.v. coupling is used which, forsecond order processes is not the non-relativistic limit of the p.s. coupling (contrary to numerous statements made in the literature). Thus the use of a finite size model is imperative. (iv) Although the finite size of the charge and of the magnetic moment distribution of the nucleons is primarily an effect of meson theory (compareM. Slotnick andW. Heitler:Phys. Rev.,75, 1645 (1949)) a quantitative investigation indicates a finite size of the nucleon itself. (L. K. Pandit:Helv. Phys. Acta,31, 379 (1958);Nuovo Cimento,10, 534 (1958)). All these effects lead to a size of the elementary particles of the orderħ/Mc,M = nucleon mass, for all interactions.ADSCrossRefGoogle Scholar
  3. (3).
    E. Arnous andW. Heitler:Nuovo Cimento,11, 443 (1959), cfr. also Part II, Sect.6. In the first paper theC, P. T invariance and causality are also discussed.CrossRefzbMATHGoogle Scholar
  4. (4).
    For details see part II, Sect.6 and a special paper byL. O’Raifeartaigh: to be published.Google Scholar
  5. (5).
    Compare the critical survey byP. S. Farago andL. Jánossy:Nuovo Cimento,5, 1411 (1957).CrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica 1960

Authors and Affiliations

  • E. Arnous
    • 1
  • W. Heitler
    • 2
  • Y. Takahashi
    • 3
  1. 1.Centre National de Recherche ScientifiqueParis
  2. 2.Institut für Theoretische PhysikUniversität
  3. 3.Dublin Institute for Advanced StudiesIreland

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