Skip to main content
Log in

Renormalization theory in four-dimensional scalar fields (I)

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We present a renormalization group approach to the renormalization theory ofΦ 44 , using techniques that have been introduced and used in previous papers and that lead to very simple methods to bound the coefficients of the effective potential and of the Schwinger functions. The main aim of this paper is to show how one can in this way obtain then!-bounds.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Gallavotti, G.: Memorie dell'Accademia dei LinceiXV, 23 (1978). Ann. Matematica Pura, Appl.CXX, 1–23 (1979)

    Google Scholar 

  2. Benfatto, G., Cassandro, M., Gallavotti, G., Nicolò, F., Olivieri, E., Presutti, E., Scacciatelli, E.: Some probabilistic techniques in field theory. Commun. Math. Phys.59, 143 (1978); Ultraviolet stability in Euclidean scalar field theories. Commun. Math. Phys.71, 95 (1980)

    Google Scholar 

  3. Gawedzki, K., Kupiainen, A.: A rigorous block spin approach to massless lattice theories. Commun. Math. Phys.77, 31 (1980); Renormalization group study of a critical lattice model. I. Convergence to the line of fixed points. Commun. Math. Phys.82, 407 (1981), and II. The correlation functions. Commun. Math. Phys.83, 469 (1982)

    Google Scholar 

  4. Bałaban, T.: The ultraviolet stability-bounds for some latticeσ models and lattice Higgs-Kibble model. In: Proc. of the Intern. Conf. on Math. Phys., Lausanne 1979. Lecture Notes in Physics. Berlin, Heidelberg, New York: Springer 1980

    Google Scholar 

  5. Benfatto, G., Gallavotti, G., Nicolò, F.: On the massive sine-Gordon equation in the first few regions of collapse. Commun. Math. Phys.83, 387 (1982)

    Google Scholar 

  6. Nicolò, F.: On the massive sine-Gordon equation in the higher regions of collapse. Commun. Math. Phys.88, 581 (1983)

    Google Scholar 

  7. Hepp, K.: Proof of the Bogoliubov-Parasiuk theorem on renormalization. Commun. Math. Phys.2, 301 (1966)

    Google Scholar 

  8. de Calan, C., Rivasseau, V.: Local existence of the Borel transform in EuclideanΦ 44 . Commun. Math. Phys.82, 69 (1981)

    Google Scholar 

  9. Polchinski, J.: Renormalization and effective Lagrangians. Nucl. Phys. B231, 269 (1984)

    Google Scholar 

  10. Nelson, E.: In: Constructive quantum field theory. Lecture Notes in Physics. Berlin, Heidelberg, New York: Springer 1973

    Google Scholar 

  11. Gallavotti, G., Rivasseau, V.:Φ 4 field theory in dimension 4: a modern introduction to its unsolved problems. Ann. Inst. Henri Poincaré40, (2), 185 (1984)

    Google Scholar 

  12. Colella, P., Lanford, O.: Sample fields and behaviour for the free Markov random fields. In: Constructive quantum field theory. Lecture Notes in Physics. Berlin, Heidelberg, New York: Springer 1973

    Google Scholar 

  13. Benfatto, G., Gallavotti, G., Nicolò, F.: J. Funct. An. Vol.36, (3), 343 (1980)

    Google Scholar 

  14. Glimm, J.: Boson fields with the:Φ 4: interaction in three dimensions. Commun. Math. Phys.10, 1 (1968)

    Google Scholar 

  15. Glimm, J., Jaffe, A.: Positivity of theϕ 43 Hamiltonian. Fortschr. Phys.21, 327 (1973)

    Google Scholar 

  16. Fröhlich, J.: On the triviality ofλϕ 4 d theories and the approach to the critical point in\(d\mathop > \limits_{( = )} 4\) dimensions. Nucl. Phys. B200, (FS4), 281 (1982)

    Google Scholar 

  17. Aizenman, M.: Proof of the triviality ofϕ 4 d field theory and some mean-field features of Ising models ford>4. Phys. Rev. Lett.47, 1 (1981); Geometric analysis ofΦ 4 fields and Ising models. Parts I and II. Commun. Math. Phys.86, 1 (1982)

    Google Scholar 

  18. Rivasseau, V.: Construction and Borel summability of planar 4-dimensional Euclidean field theory. Commun. Math. Phys.95, 445 (1984)

    Google Scholar 

  19. 't Hooft, G.: On the convergence of planar diagram expansion. Commun. Math. Phys.86, 449 (1982); Rigorous construction of planar diagram field theories in four dimensional Euclidean space. Commun. Math. Phys.88, 1 (1983)

    Google Scholar 

  20. Zimmermann, W.: Convergence of Bogoliubov's method for renormalization in momentum space. Commun. Math. Phys.15, 208 (1969)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Communicated by K. Osterwalder

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gallavotti, G., Nicolò, F. Renormalization theory in four-dimensional scalar fields (I). Commun.Math. Phys. 100, 545–590 (1985). https://doi.org/10.1007/BF01217729

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01217729

Keywords

Navigation