Advertisement

Communications in Mathematical Physics

, Volume 59, Issue 2, pp 143–166 | Cite as

Some probabilistic techniques in field theory

  • G. Benfatto
  • M. Cassandro
  • G. Gallavotti
  • F. Nicolò
  • E. Olivieri
  • E. Presutti
  • E. Scacciatelli
Article

Abstract

We study, in the context of the Markov hyerarchical fields (d=2, 3) the role of the Markov property, of formal renormalization and of formal positivity. We determine upper and lower bounds for the ground state energy and discuss their relation with the perturbation theory series.

Keywords

Neural Network Statistical Physic Field Theory Complex System Lower Bound 
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.
    Nelson, E.: A quartic interaction in two dimensions. In: Mathematical theory of elementary particles (eds. R. Goodman, I. Segal). Cambridge: MIT Press 1966Google Scholar
  2. 1a.
    Glimm, J.: Commun. math. Phys.8, 12 (1968)Google Scholar
  3. 2.
    Gallavotti, G.: Some aspects of the renormalization problems in statistical mechanics and field theory. In: Memoria Accademia dei Lincei. Preprint, IHES 1977 Gallavotti, G.: On the ultraviolet stability in statistical mechanics and field theory. In: Annali di Matematica pura e applicata. Preprint, IHES 1977Google Scholar
  4. 3.
    Guerra, F.: Phys. Rev. Letters28, 1213 (1972)Google Scholar
  5. 3a.
    Glimm, J., Jaffe, A.: Fortschr. Physik21, 327 (1973)Google Scholar
  6. 3b.
    Glimm, J., Jaffe, A., Spencer, T.: The particle structure of the weakly coupledP(φ)2 model and other applications. In: Lecture notes in physics, Vol. 25 (eds. G. Velo, A. Wightman), Berlin-Heidelberg-New York: Springer 1973Google Scholar
  7. 3c.
    Feldman, J.: Commun. math. Phys.37, 93 (1974)Google Scholar
  8. 3d.
    Magnen, J., Seneor, R.: Ann. Inst. Henri Poincaré24, 95 (1976)Google Scholar
  9. 3e.
    Magnen, J.: ThesisGoogle Scholar
  10. 3f.
    Feldman, J., Osterwalder, K.: Ann. Phys.97, 80 (1976)Google Scholar
  11. 4.
    Dobrushin, R.L., Tirozzi, B.: Commun. math. Phys.54, 173–192 (1977)Google Scholar
  12. 5.
    Ruelle, D.: Commun. math. Phys.18, 127 (1970)Google Scholar
  13. 5a.
    Ruelle, D.: Commun. math. Phys.50, 189 (1976)Google Scholar
  14. 6.
    Dinaburg, R., Sinai, Ya.G.: to appearGoogle Scholar
  15. 7.
    Simon, B.: TheP(ϕ)2 euclidean quantum field theory. Princeton: Princeton University Press 1974Google Scholar

Copyright information

© Springer-Verlag 1978

Authors and Affiliations

  • G. Benfatto
    • 1
    • 2
  • M. Cassandro
    • 1
    • 2
  • G. Gallavotti
    • 1
    • 2
  • F. Nicolò
    • 1
    • 2
  • E. Olivieri
    • 1
    • 2
  • E. Presutti
    • 1
    • 2
  • E. Scacciatelli
    • 1
    • 2
  1. 1.Istituto di Matematica dell'Università dell'AquilaRomaItaly
  2. 2.Istituto di Matematica e di Fisica dell'Università di RomaRomaItaly

Personalised recommendations