Advertisement

Conformal transformations of S-matrix in scalar field theory

theoretical physics

Abstract.

In this paper, three methods for describing the conformal transformations of the S-matrix in quantum field theory are proposed. They are illustrated by applying the algebraic renormalization procedure to the quantum scalar field theory, defined by the LSZ reduction mechanism in the BPHZ renormalization scheme. Central results are shown to be independent of scheme choices and derived to all orders in loop expansions. Firstly, the local Callan-Symanzik equation is constructed, in which the insertion of the trace of the energy-momentum tensor is related to the beta function and the anomalous dimension. With this result, the Ward identities for the conformal transformations of the Green functions are derived. Then the conformal transformations of the S-matrix defined by the LSZ reduction procedure are calculated. Secondly, the conformal transformations of the S-matrix in the functional formalism are related to charge constructions. The commutators between the charges and the S-matrix operator are written in a compact way to represent the conformal transformations of the S-matrix. Lastly, the massive scalar field theory with local coupling is introduced in order to control breaking of the conformal invariance further. The conformal transformations of the S-matrix with local coupling are calculated

Keywords

Green Function Anomalous Dimension Conformal Transformation Beta Function Conformal Invariance 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    H. Epstein, V. Glaser, Ann. Inst. Henri Poincaré 19, 211 (1973)Google Scholar
  2. 2.
    R. Coleman, J. Mandula, Phys. Rev. 159, 1251 (1967)CrossRefMATHGoogle Scholar
  3. 3.
    R. Haag, T. Lopuszanski, M. Sohnius, Nucl. Phys. B 88, 257 (1975)CrossRefGoogle Scholar
  4. 4.
    W. Zimmermann, Commun. Math. Phys. 76, 39 (1980)MathSciNetGoogle Scholar
  5. 5.
    C. Becchi, A. Rouet, R. Stora, Commun. Math. Phys. 42, 127 (1975)Google Scholar
  6. 6.
    C. Becchi, A. Rouet, R. Stora, Ann. Phys. 98, 287 (1976)Google Scholar
  7. 7.
    K. Sibold, Störungtheoretische Renormierung Quantisierung von Eichtheorieen (unpublished, 1993)Google Scholar
  8. 8.
    O. Piguet, S.P. Sorella, Algebraic renormalization: Perturbative renormalization, symmetries and anomalies (Springer 1995)Google Scholar
  9. 9.
    A. Boresch, O. Moritsch, M. Schweda, T. Sommer, H. Zerrouki, S. Emery, Applications of noncovariant gauges in the algebraic renormalization procedure (World Scientific 1998)Google Scholar
  10. 10.
    J.H. Lowenstein, Phys. Rev. D 4, 2281 (1971)CrossRefGoogle Scholar
  11. 11.
    J.H. Lowenstein, Commun. Math. Phys. 24, 1 (1971)MATHGoogle Scholar
  12. 12.
    Y.M.P. Lam, Phys. Rev. D 6, 2145 (1972)CrossRefGoogle Scholar
  13. 13.
    Y.M.P. Lam, Phys. Rev. D 7, 2943 (1973)CrossRefGoogle Scholar
  14. 14.
    W. Zimmermann, Commun. Math. Phys. 11, 208 (1969)Google Scholar
  15. 15.
    W. Zimmermann, Ann. Phys. (N.Y.) 77, 536 (1973)Google Scholar
  16. 16.
    W. Zimmermann, Ann. Phys. (N.Y.) 77, 570 (1973)Google Scholar
  17. 17.
    C.G. Callan, Phys. Rev. D 2, 1541 (1970)CrossRefGoogle Scholar
  18. 18.
    K. Symanzik, Commun. Math. Phys. 18, 227 (1970)MATHGoogle Scholar
  19. 19.
    T. Kugo, I. Ojima, Prog. Theor. Phys. 60, 1869 (1978)MathSciNetMATHGoogle Scholar
  20. 20.
    T. Kugo, I. Ojima, Prog. Theor. Phys. 61, 294 (1979)MathSciNetMATHGoogle Scholar
  21. 21.
    E. Kraus, K. Sibold, Nucl. Phys. B 372, 113 (1992)CrossRefMathSciNetGoogle Scholar
  22. 22.
    E. Kraus, K. Sibold, Nucl. Phys. B 398, 125 (1993)CrossRefMathSciNetGoogle Scholar
  23. 23.
    S. Weinberg, Phys. Rev. 18, 838 (1960)CrossRefGoogle Scholar
  24. 24.
    R. Coleman, R. Jackiw, Ann. Phys. 67, 552 (1971)Google Scholar
  25. 25.
    E. Kraus, C. Rupp, K. Sibold, Eur. Phys. J. C 24, 631 (2002), hep-th/0205013Google Scholar
  26. 26.
    E. Kraus, Nucl. Phys. B 620, 55 (2002), hep-th/0107239MATHGoogle Scholar
  27. 27.
    P.L. White, Class. Quant. Grav. 9, 413 (1992)CrossRefMathSciNetMATHGoogle Scholar
  28. 28.
    P.L. White, Class. Quant. Grav. 9, 1663 (1992)CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin/Heidelberg 2003

Authors and Affiliations

  1. 1.Institut für Theoretische PhysikUniversität LeipzigLeipzigGermany

Personalised recommendations