HOME    |    BROWSE    |    SEARCH    |    SUBSCRIPTION   |    SUBMISSION    |    REGISTRATION

Quick Search:
Author: Title/Abstract: Vol./No: Page:
PTP Online Top > Supplement (1961) > No.18

Prog. Theor. Phys. Supplement No.18 (1961) pp. 83-125

[ Full Text PDF : FREE ACCESS (2257K) ]

Wightman Functions, Retarded Functions and Their Analytic Continuations

Huzihiro Araki

Department of Nuclear Engineering, Kyoto University, Kyoto

(Received April 13, 1961)

Abstract:

Recent studies on the analyticity of the general n-point Wightman functions and the retarded functions in quantum field theory are reviewed. According to Wightman, the conventional axiom of quantum field theory is transcribed into an equivalent set of properties of the Wightman functions. It is then shown that the Wightman functions are boundary values of analytic functions in complex coordinate spaces and the linear properties of the Wightman functions are transcribed into an equivalent requirement on domains of analyticity of these analytic functions.
For the study of retarded functions, the contributions of vacuum intermediate states are systematically subtracted from the Wightman functions. The truncated Wightman functions thus obtained have the same linear properties as those of the Wightman functions except for a smaller support in the momentum space. A generalization of θ-function (the step function of Heaviside) to the characteristic function of a convex polyhedral cone in an n-dimensional vector space is defined and is used to define the generalized retarded functions. The properties of the latter which are equivalent to the linear properties of the Wightman functions are stated. It is then shown that the generalized retarded functions are boundary values of analytic functions in complex momentum spaces and a part of the properties of the former are transcribed into an equivalent requirement on domains of analyticity of the latter.
The whole discussion is carried out regardless of the choice between local commutativity and anticommutativity. The problem of obtaining the envelope of holomorphie remains unsolved.


URL : http://ptp.ipap.jp/link?PTPS/18/83/
DOI : 10.1143/PTPS.18.83

[ Full Text PDF : FREE ACCESS (2257K) ] Citation:

References:

  1. H. Lehmann, K. Symanzik and W. Zimmermann, Nuovo Cim. 1 (1955), 205.
  2. H. Lehmann, K. Symanzik and W. Zimmermann, Nuovo Cim. 6 (1957), 319.
  3. V. Glaser, H. Lehmann and W. Zimmermann, Nuovo Cim. 6 (1957), 1122.
  4. K. Nishijima, Prog. Theor. Phys. 17 (1957), 765[PTP].
  5. A. S. Wightman, Phys. Rev. 101 (1956), 860[APS].
  6. D. Hall and A. S. Whitman, Kgl. Dan. Vid. Selsk. Mat. Fys. Medd. 31 (1957), No. 5.
  7. R. Jost, Helv. Phys. Acta 31 (1958), 263.
  8. O. Steinmann, Helv. Phys. Acta 33 (1960), 257.
  9. O. Steinmann, Helv. Phys. Acta 33 (1960), 347.
  10. D. Ruelle, Thése, Université libre de Bruxelles (1959); Nuovo Cim. 19 (1961), 356.
  11. N. Burgoyne, private communication. Also Nuovo Cim. 18 (1960), 342.
  12. H. Araki, J. Math. Phys. 2 (1961), 163[AIP Scitation].
  13. H. Araki, Ann. Phys. 11 (1960), 260[CrossRef].
  14. N. Burgoyne, Nuovo Cim. 8 (1958), 607.
  15. H. Araki, J. Math. Phys. 2 (1961), 267[CrossRef].
  16. G. Källen and A. S. Wightman, Kgl. Dan. Vid. Selsk, Mat. Fys. Skrift. 1 (1958).
  17. S. Brown, private communication.
  18. R. F. Streater, Proc. R. Soc. (London) A 256 (1960), 39.
  19. R. F. Streater, Nuovo Cim. 15 (1960), 937.
  20. R. F. Streater, J. Math. Phys. 1 (1960), 231[AIP Scitation].
  21. K. Symanzik, J. Math. Phys. 1 (1960), 249[CrossRef].
  22. A. S. Wightman, Nuovo Cim. Suppl. 14 (1959), 81, and references therein.
  23. L. Schwartz, Theorie des Distribution I and II (Hermann and Cie, Paris, 1951).
  24. L. Schwartz, Theorie des Distribution I.
  25. L. Schwartz, Theorie des Distribution I, p. 55 §5; pp. 112-3 Example 1.
  26. L. Schwartz, Medd. Lunds Univ. Mat. Seminarium, Suppl. (1952), p. 196.
  27. L. Gärding, private communication.
  28. H. Epstein, J. Math. Phys. 1 (1960), 524[AIP Scitation].
  29. R. Jost, Helv. Phys. Acta 30 (1957), 409.
  30. D. Ruelle, Helv. Phys. Acta 32 (1959), 135.
  31. R. Haag, Phys. Rev. 112 (1958), 669[APS].
  32. This corresponds to Ursell's expansion in statistical mechanics. See H. P. Ursell, Proc. Cambridge Phil. Soc. 23 (1927), 685.
  33. G. Källen, “The analyticity domain of the four point function”, preprint.
  34. R. Jost, Theoretical Physics in the Twentieth Century (Interscience Publishers Inc., New York, 1960), 107; Helv. Phys. Acta 33 (1960), 773.

Citing Article(s) :

  1. Progress of Theoretical Physics Vol. 35 No. 4 (1966) pp. 705-737 :
    Homological Study of the Generalized Singular Functions. I
    Masatsugu Minami
  2. Progress of Theoretical Physics Vol. 51 No. 3 (1974) pp. 912-919 :
    Is the Two-Particle Scattering Amplitude Always a Boundary Value of a Real Analytic Function?
    Noboru Nakanishi

[PTP HOME] [BROWSE] [SEARCH] [SUBSCRIPTION] [SUBMISSION] [REGISTRATION]

Copyright © Progress of Theoretical Physics 2013 All rights reserved.