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On infrared singularities

  • Takahiro Kawai
  • Henry P. Stapp

Abstract

The traditional separation of infrared divergent part of the S-matrix from a finite remainder ([YFS], [GY]) is effective only at points where the S-matrix is non-singular, as was pointed out in [S2]. This limitation is due primarily to the approximation
$${e^{ikx}} \sim 1\;\;\;(|k| \ll 1),$$
(1)
which is used to replace
$${\delta ^4}(p + k) = \int {{{{d^4}x} \over {{{(2\pi )}^4}}}} {e^{i(p + k)x}}\;\;by\;\;{\delta ^4}(p) = \int {{{{d^4}x} \over {{{(2\pi )}^4}}}} {e^{ipx}}$$
in the demonstration that the infrared divergent terms originating from real photons are cancelled by those originating from virtual photons (e.g. [GY] (3.20) ff). An approximation of this sort seems to be necessary, if we treat the separation of the infrared divergent parts in momentum space. (Cf. Problem A below.) However, the separation can be neatly done in coordinate space, even at singular points of the S-matrix. ([S2]) In view of the fact that a point x in the coordinate space represents the cotangential component of the singularity spectrum of a function on the momentum space (e.g. [KS1],[Sa]) the recipe of Stapp [S2] may be regarded as the microlocalization of the traditional separation of infrared divergences. The core-spirit of microlocal analysis (e.g. [K3]) is to make use of both p-variables and x-variables in the analysis. In fact, to study the infrared finiteness of the remainder terms (the Q-coupling part in the sense of [S2] and [KS4]) p-variables play a central role, while the cancellation of infrared divergent terms (the C-coupling part in the sense of [S2]) proceeds in coordinate space.

Keywords

Singular Point Kernel Function Momentum Space Coordinate Space Virtual Photon 
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.

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References

  1. [GY]
    G. Grammer and D.R. Yennie, Phys. Rev. D8, 4332 (1973).Google Scholar
  2. [I]
    D. Iagolnitzer, The S-matrix, North Holland, Amsterdam, 1978.Google Scholar
  3. [K3]_M. Kashiwara, T. Kawai and T. Kimura, Foundations of Algebraic Analysis, Princeton University Press, 1986.Google Scholar
  4. [KS1]
    T. Kawai and H.P. Stapp, in Lecture Notes in Physics, No.39, ed. H. Araki, Springer, 1975.Google Scholar
  5. [KS2]
    T. Kawai and H.P. Stapp, in Algebraic Analysis (M. Kashiwara and T. Kawai, eds.), vol.1, Academic Press, New York, 1988, pp. 309–330.Google Scholar
  6. [KS3]
    T. Kawai and H.P. Stapp, Ann. Inst. Fourier (Grenoble), 43, 1301 (1993).MathSciNetMATHCrossRefGoogle Scholar
  7. [KS4]
    T. Kawai and H.P. Stapp, Phys. Rev. D52, 2484, 2508 and 2517 (1995).Google Scholar
  8. [Sa]
    M. Sato, in Lect. Notes in Physics, No.39, ed. H. Araki, Springer, 1975.Google Scholar
  9. [S1]
    H.P. Stapp, in Structural Analysis of Collision Amplitudes, ed. R. Balian and D. Iagolnitzer, North-Holland, New York, 1976.Google Scholar
  10. [S2]
    T. Kawai and H.P. Stapp, Phys. Rev. 28D, 1386 (1983).Google Scholar
  11. [YFS]
    D. Yennie, S. Frautschi, and H. Suura, Ann. Phys. (N.Y.) 13, 379 (1961).CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Tokyo 1997

Authors and Affiliations

  • Takahiro Kawai
    • 1
  • Henry P. Stapp
    • 2
  1. 1.Research Institute for Mathematical SciencesKyoto UniversityKyotoJapan
  2. 2.Lawrence Berkeley LaboratoryUniversity of CaliforniaBerkeleyUSA

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