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Cellular space-time and quantum field theory

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Il Nuovo Cimento (1955-1965)

Summary

A simple cellular model of space-time is introduced which incorporates a fundamental length in a natural way. The usual concept of field quantities as functions of geometrical points is rejected in favour of the concept of field quantities as functionals of elementary cells. Consequently the usual field equations in the form of partial differential equations are replaced by corresponding partial difference equations. Subsequent quantization yields anS-matrix which is not vitiated by the presence of serious divergences. The basic space-time structure and theS-Matrix (which does not involve derivative coupling) are shown to be integral Lorentz covariant. The semi-convergence or asymptotic behaviour of theS-matrix series is established, and a meaningful Borel-sum of the « gross »S-matrix series is presented.

Riassunto

Si introduce un semplice modello cellulare dello spazio-tempo, comprendente in modo naturale una lunghezza fondamentale. Il concetto usuale delle quantità di campo come funzioni dei punti geometrici viene respinto a favore del concetto delle quantità di campo come funzionali delle celle elementari. Conseguentemente le usuali equazioni di campo sotto forma di equazioni differenziali parziali sono sostituite dalle corrispondenti equazioni alle differenze parziali. La successiva quantizzazione fornisce una matriceS che non è inficiata dalla presenza di divergenze notevoli. Si dimostra che la struttura fondamentale dello spazio-tempo e la matriceS (che non comporta accoppiamenti derivati) sono covarianti per trasformazioni « discrete » di Lorentz, si determina la semi-convergenza o il comportamento asintotico della serie della matriceS, e si presenta una significativa somma di Borel della « maggiorante » della serie della matriceS.

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References

  1. N. Bohr andL. Rosenfeld:Kgl. Dan. Vidensk. Selsk., Mat.-Fys. Medd.,12, no. 8 (1933).

  2. W. Heisenberg:Leipziger Berichte,86, 317 (1934).

    Google Scholar 

  3. E. C. G. Stückelberg:Phys. Rev.,81, 130 (1951).

    Article  ADS  MATH  Google Scholar 

  4. E. C. G. Stückelberg andT. A. Green:Helv. Phys. Acta,24, 153 (1951).

    MATH  Google Scholar 

  5. E. Corinaldesi:Nuovo Cimento,8, 499 (1951);Suppl. Nuovo Cimento,10, 83 (1953).

    Google Scholar 

  6. W. Pauli:Meson Theory of Nuclear Forces (New York, 1946).

  7. G. Chew:Phys. Rev.,94, 1748, 1755 (1954);95, 1669 (1954).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  8. R. Hofstadter:Ann. Rev. Nucl. Sci.,7, 231 (1957).

    Article  ADS  Google Scholar 

  9. D. R. Yennie, M. M. Lévy andD. G. Ravenhall:Rev. Mod. Phys.,29, 144 (1957).

    Article  ADS  Google Scholar 

  10. W. Heisenberg:Zeits. Phys.,65, 4 (1930);Ann. d. Phys.,32, 20 (1938);Zeits. Phys.,120, 513, 673 (1942).

    Article  ADS  MATH  Google Scholar 

  11. V. Ambarzumian andD. Ivanenko:Zeits. Phys.,64, 563 (1930).

    Article  ADS  Google Scholar 

  12. A. March:Zeits. Phys.,104, 93, 161 (1936);105, 620 (1937);106, 49, 291 (1937).

    Article  ADS  Google Scholar 

  13. L. L. Whyte:Brit. Journ. Phil. Sci.,1, 303 (1950);Ann. Sci.,1, 20 (1954).

    MathSciNet  Google Scholar 

  14. L. Silberstein: Univ. of Toronto Studies, Physics Ser. (1936).

  15. H. Snyder:Phys. Rev.,71, 38 (1947);72, 68 (1948).

    Article  ADS  MATH  Google Scholar 

  16. A. Schild:Phys. Rev.,73, 414 (1948);Can. Journ. Math.,1, 29 (1949).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  17. H. S. M. Coxeter andG. J. Whitrow:Proc. Roy. Soc., A201, 417 (1950).

    Article  MathSciNet  ADS  Google Scholar 

  18. E. J. Hellund andK. Tanaka:Phys. Rev.,94, 191 (1954).

    Article  MathSciNet  ADS  Google Scholar 

  19. V. Rojansky:Phys. Rev.,97, 507 (1955).

    Article  MathSciNet  ADS  Google Scholar 

  20. E. L. Hill:Phys. Rev.,100, 1780 (1955).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  21. G. Wataghin:Zeits. Phys.,88, 92 (1934);92, 547 (1935).

    Article  ADS  Google Scholar 

  22. F. Bopp:Ann. d. Phys.,42, 573 (1942).

    MathSciNet  Google Scholar 

  23. R. P. Feynman:Phys. Rev.,74, 939 (1948).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  24. R. E. Peierls andH. McManus:Phys. Rev.,70, 795 (1948).

    Google Scholar 

  25. H. McManus:Proc. Roy. Soc., A195, 323 (1949).

    MathSciNet  ADS  Google Scholar 

  26. C. Bloch:Kgl. Dan. Vidensk. Selsk., Mat.-Fys. Medd.,26, no. 1 (1950);27, no. 8 (1952).

  27. J. Rayski:Phil. Mag.,42, 1289 (1951).

    Article  MathSciNet  MATH  Google Scholar 

  28. P. Kristensen andC. Møller:Kgl. Dan. Vidensk. Selsk., Mat.-Fys. Medd.,27, no. 7 (1952).

  29. M. Chrétien andR. E. Peierls:Proc. Roy. Soc., A223, 468 (1954).

    Article  ADS  Google Scholar 

  30. E. Arnous andW. Heitler:Nuovo Cimento,11, 443 (1959).

    Article  Google Scholar 

  31. E. Arnous, W. Heitler andY. Takahashi:Nuovo Cimento 16, 671 (1960).

    Article  MathSciNet  MATH  Google Scholar 

  32. M. A. Markov:Žurn. Ėksp. Teor. Fiz.,10, 1311 (1940).

    Google Scholar 

  33. H. Yukawa:Phys. Rev.,76, 300 (1949);77, 219, 849 (1950);80, 1047 (1950).

    Article  ADS  MATH  Google Scholar 

  34. A. Pais andG. E. Uhlenbeck:Phys. Rev.,79, 145 (1950).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  35. O. Hara, T. Marumori, Y. Ohnuki andH. Shimodaira:Progr. Theor. Phys.,12, 177 (1954).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  36. Cf. the simple review article byW. A. McKinley:Am. Journ. Phys.,20, 129 (1960).

    Article  ADS  Google Scholar 

  37. W. R. Hamilton:Collected Mathematical Papers, vol.2 (Cambridge, 1940), p. 451.

    Google Scholar 

  38. R. Courant, K. Friedrichs andH. Lewy:Math. Ann.,100, 32 (1928).

    Article  MathSciNet  MATH  Google Scholar 

  39. B. T. Darling:Phys. Rev.,80, 460 (1951);92, 1547 (1953);L. I. Schiff:Phys. Rev.,92, 766 (1953).

    Article  MathSciNet  ADS  Google Scholar 

  40. R. J. Duffin:Duke Math. Journ.,20, 234 (1953);23, 335 (1955).

    Article  MathSciNet  Google Scholar 

  41. E. W. Montroll:Amer. Math. Month.,61, 7 (1954).

    Article  MathSciNet  Google Scholar 

  42. G. Boole:Calculus of Finite Differences, 4th. ed. (New York, 1958).

  43. C. Jordan:Calculus of Finite Differences, 2nd ed. (New York, 1950).

  44. N. E. Nörlung:Vorlesungen über Differenzenrechnung (Ann Arbor, 1945).

  45. W. Heisenberg andW. Pauli:Zeits. Phys.,56, 1 (1929).

    Article  ADS  MATH  Google Scholar 

  46. D. Judge, H. Shimodaira andY. Takahashi:Nuovo Cimento 16, 1139 (1900).

    Article  MathSciNet  Google Scholar 

  47. D. Judge, H. Shimodaira andY. Takahashi:Proc. Roy. Irish Acad., Sect. A (to be published).

  48. C. A. Hurst:Proc. Roy. Soc., A214, 44 (1952);Proc. Camb. Phil. Soc.,48, 625 (1952).

    Article  MathSciNet  ADS  Google Scholar 

  49. F. J. Dyson:Phys. Rev.,85, 631 (1952).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  50. S. Hori:Progr. Theor. Phys.,8, 569 (1952).

    Article  MathSciNet  ADS  Google Scholar 

  51. R. J. Riddell:Phys. Rev.,91, 1243 (1953).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  52. G. N. Watson:Quart. Journ. Math.,10, 266 (1939).

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. Dublin Institute for Advanced Studies, Dublin

    A. Das

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  1. A. Das
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Das, A. Cellular space-time and quantum field theory. Nuovo Cim 18, 482–504 (1960). https://doi.org/10.1007/BF02732721

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  • DOI: https://doi.org/10.1007/BF02732721

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