Advertisement

The Intersection of Brownian Paths as a Case Study of a Renormalization Group Method for Quantum Field Theory

  • Michael Aizenman

Abstract

A new approach is presented for the study of the probability that the random paths generated by two independent Brownian motions in ℝd intersect or, more generally, are within a short distance a of each other. The well known behavior of that function of a-above, below, and at the critical dimension d = 4, as well as further corrections, are derived here by means of a single renormalization group equation. The equation’s derivation is expected to shed some light on the β-function of the λø d 4 quantum field theory.

Keywords

Brownian Motion Renormalization Group Intersection Property Critical Dimension Renormalization Group Equation 
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.
    Kakutani, S.: On Brownian motions in w-space. Proc. Acad. Tokyo 20, 648 (1944)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Dvoretzky, A., Erdös, P., Kakutani, S.: Double points of Brownian paths in w-space. Acta Sci. Math. Szeged 12, 75 (1950)MathSciNetGoogle Scholar
  3. 3.
    Erdös, P., Taylor, S.J.: Some intersection properties of random paths. Acta Math. Acad. Sci. Hung. 11, 231 (1960)MATHCrossRefGoogle Scholar
  4. 4.
    Lawler, G.F.: The probability of intersection of Random walks in four dimensions. Commun. Math. Phys. 86, 539 (1982)MathSciNetADSMATHCrossRefGoogle Scholar
  5. 5.
    Spencer, T.: Private communicationGoogle Scholar
  6. 6.
    Symanzik, K.: Small distance behavior in field theory and power counting. Commun. Math. Phys. 18, 227 (1970) Callan, C.G. Jr.: Broken scale invariance in scalar field theory. Phys. Rev. D2, 1541 (1970) Wilson, K.G.: Anomalous dimension and the breakdown of scale invariance in perturbation theory. Phys. Rev. D2, 1478 (1970)MathSciNetADSMATHCrossRefGoogle Scholar
  7. 7.
    Brezin, E., Le Guillou, J.C., Zinn-Justin, J.: In: Phase transitions and critical phenomena. Domb, C., Green, M.S., (eds.). London, New York, San Francisco: Academic Press 1976 Itzykson, C., Zuber, J.B.: Quantum field theory. New York: McGraw-Hill 1980Google Scholar
  8. 8.
    Symanzik, K.: Euclidean quantum field theory. I. Equations for a scalar model. J. Math. Phys. 7, 510 (1966)MathSciNetADSCrossRefGoogle Scholar
  9. 9.
    Aizenman, M.: Geometric analysis of ϕ4 fields and Ising models. Commun. Math. Phys. 86, 1 (1982)MathSciNetADSMATHCrossRefGoogle Scholar
  10. 10.
    Fröhlich, J.: On the triviality of λϕd4 theories and the approach to the critical point in d > 4 dimensions. Nucl. Phys. B200, [FS4] 281 (1982)ADSCrossRefGoogle Scholar
  11. 11.
    Aragão de Carvalho, C., Caraciollo, S., Fröhlich, J.: Polymers and g\ϕ\4 theory in four dimensions. Nucl. Phys. B215 [FS7], 209 (1983)ADSCrossRefGoogle Scholar
  12. 12.
    Aizenman, M., Graham, R.: On the renormalized coupling constant and the susceptibility in (ϕ44 field theory and the Ising model in four dimensions. Nucl. Phys. B225 [FS9], 261 (1983)MathSciNetADSCrossRefGoogle Scholar
  13. 13.
    Fröhlich, J.: Quantum field theory in terms of Random walks and Random surfaces. Cargèse (1983) lecture notesGoogle Scholar
  14. 14.
    Felder, G., Fröhlich, J.: Intersection properties of simple Random walks: a renormalization group approach. Commun. Math. Phys. 97, 111–124 (1984)ADSCrossRefGoogle Scholar
  15. 15.
    Aizenman, M.: Rigorous studies of critical behavior, to appear in the proceedings of the VIII sitges conference. L. Garrido (ed.), Berlin, Heidelberg, New York: Springer 1984Google Scholar

Copyright information

© Springer-Verlag Berlin, Heidelberg 1985

Authors and Affiliations

  • Michael Aizenman
    • 1
  1. 1.Departments of Mathematics and PhysicsRutgers UniversityNew BrunswickUSA

Personalised recommendations