The Intersection of Brownian Paths as a Case Study of a Renormalization Group Method for Quantum Field Theory
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.
Unable to display preview. Download preview PDF.
- 5.Spencer, T.: Private communicationGoogle Scholar
- 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.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
- 13.Fröhlich, J.: Quantum field theory in terms of Random walks and Random surfaces. Cargèse (1983) lecture notesGoogle Scholar
- 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