Communications in Mathematical Physics

, Volume 83, Issue 2, pp 213–242 | Cite as

On the regular holonomic character of theS-matrix and microlocal analysis of unitarity-type integrals

  • Takahiro Kawai
  • Henry P. Stapp


The previously proved results that every analytically renormalized Feynman integral is a regular holonomic function suggests that theS-matrix should be locally expressible as an infinite sum of regular holonomic functions. A regularity propertyR is formulated that expresses the condition that theS-matrix be locally expressible near each physical pointp as a convergent sum of regular holonomic functions, with each term enjoying some of the regularity properties of a corresponding Feynman integral. This propertyR holds at every physical pointp that has yet been analyzed by the methods of axiomatic field theory orS-matrix theory. Some analyticity properties of unitarity-type integrals are then examined under the assumption that theS-matrix satisfies propertyR and a weak integrability condition. These results rest heavily on some recently proved properties of regular holonomic functions.


Neural Network Statistical Physic Field Theory Complex System Nonlinear Dynamics 
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  1. 1.
    Sato, M.: Recent developments in hyperfunction theory and its application to physics. In: Lecture Notes in Physics, Vol. 36, p. 13. Berlin, Heidelberg, New York: Springer 1975Google Scholar
  2. 2.
    Kashiwara, M., Kawai, T.: Publ. RIMS, Kyoto Univ.12, Suppl. 131 (1977)Google Scholar
  3. 3.
    Kashiwara, M., Kawai, T.: On holonomic systems of micro-differential equations. III (in press in Publ. RIMS, Kyoto Univ.17)Google Scholar
  4. 4.
    Kawai, T., Stapp, H.P.: Publ. RIMS, Kyoto Univ.12, Suppl. 155 (1977)Google Scholar
  5. 5.
    Stapp, H.P.: J. Math. Phys.9, 1548 (1968)Google Scholar
  6. 5a.
    Iagolnitzer, D.: Commun. Math. Phys.41, 39 (1975)Google Scholar
  7. 5b.
    Kawai, T., Stapp, H.P.: Microlocal study ofS-matrix singularity structure. In: Lecture Notes in Physics, Vol. 39, p. 36. Berlin, Heidelberg, New York: Springer 1975 and [4]Google Scholar
  8. 6.
    Bros, J.: Analytic structure of Green's functions in quantum field theory, in mathematical problems in theoretical physics, K. Osterwalder (ed.). In: Lecture Notes in Physics Vol. 116, p. 166. Berlin, Heidelberg, New York: Springer 1980 and in complex analysis, microlocal calculus and relativistic quantum theory, D. Iagonitzer (ed.). In: Lecture Notes in Physics, Vol. 126, p. 254. Berlin, Heidelberg, New York: Springer 1980Google Scholar
  9. 6a.
    Epstein, H., Glaser, V., Iagolnitzer, D.: Commun. Math. Phys.80, 99 (1981)Google Scholar
  10. 7.
    Landau, L.D.: Nucl. Phys.13, 181 (1959)Google Scholar
  11. 8.
    Stapp, H.P.: Phys. Rev.125, 2139 (1962)Google Scholar
  12. 8a.
    Chew, G.F.:S-matrix theory of strong interactions. New York: Benjamin 1962Google Scholar
  13. 9.
    Stapp, H.P.: Discontinuity formula for multiparticle amplitudes. In: Structural analysis of collision amplitudes, Balian, R., Iagonitzer, D. (eds.). Amsterdam: North-Holland 1976Google Scholar
  14. 10.
    Iagolnitzer, D., Stapp, H.P.: Commun. Math. Phys.57, 1 (1977)Google Scholar
  15. 10a.
    Iagolnitzer, D.: Commun. Math. Phys.77, 254 (1980)Google Scholar
  16. 11.
    Iagolnitzer, D.: Commun. Math. Phys.63, 49 (1978), and Saclay Preprint DPH-T/79/87Google Scholar
  17. 12.
    Stapp, H.P.: J. Math. Phys.8, 1606 (1967)Google Scholar
  18. 13.
    Kashiwara, M., Kawai, T., Stapp, H.P.: Commun. Math. Phys.66, 95 (1979)Google Scholar
  19. 14.
    Kashiwara, M., Kawai, T.: Adv. Math.34, 163 (1979)Google Scholar
  20. 15.
    Sato, M., Kawai, T., Kashiwara, M.: Microfunctions and pseudo-differented equations. In: Lecture Notes in Mathematics, Vol. 287, p. 265. Berlin, Heidelberg, New York: Springer 1973Google Scholar
  21. 16.
    Iagolnitzer, D.: Analytic structure of distributions and essential support theory. In: Structural analysis of collision amplitudes, Part III. Amsterdam: North-Holland 1976Google Scholar
  22. 17.
    Iagolnitzer, D.: Commun. Math. Phys.41, 39 (1975)Google Scholar
  23. 18.
    Zimmermann, W.: Nuovo Cimento21, 249 (1967)Google Scholar
  24. 19.
    Eden, R., Landshoff, P., Olive, D., Polkinghorne, J.: The analyticS-matrix. Cambridge: Cambridge University Press 1966Google Scholar
  25. 20.
    Chandler, C., Stapp, H.P.: J. Math. Phys.10, 826 (1969)Google Scholar

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • Takahiro Kawai
    • 1
    • 2
  • Henry P. Stapp
    • 3
  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.Research Institute for Mathematical SciencesKyoto UniversityKyotoJapan
  3. 3.Lawrence Berkeley LaboratoryUniversity of CaliforniaBerkeleyUSA

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