Journal of Statistical Physics

, Volume 134, Issue 5–6, pp 839–857

Power Series Representations for Bosonic Effective Actions

  • Tadeusz Balaban
  • Joel Feldman
  • Horst Knörrer
  • Eugene Trubowitz
Article

Abstract

We develop a power series representation and estimates for an effective action of the form
$$\ln\frac{\int e^{f(\phi,\psi)}d\mu(\phi)}{\int e^{f(\phi,0)}d\mu(\phi)}$$
Here, f(φ,ψ) is an analytic function of the real fields φ(x),ψ(x) indexed by x in a finite set X, and dμ(φ) is a compactly supported product measure. Such effective actions occur in the small field region for a renormalization group analysis. The customary way to analyze them is a cluster expansion, possibly preceded by a decoupling expansion. Using methods similar to a polymer expansion, we estimate the power series of the effective action without introducing an artificial decomposition of the underlying space into boxes.

Keywords

Polymer expansion Renormalization group 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Balaban, T., Feldman, J., Knörrer, H., Trubowitz, E.: A functional integral representation for many boson systems. I: The partition function. Ann. Henri Poincaré 9, 1229–1273 (2008) MATHCrossRefGoogle Scholar
  2. 2.
    Balaban, T., Feldman, J., Knörrer, H., Trubowitz, E.: A functional integral representation for many boson systems. II: Correlation functions. Ann. Henri Poincaré 9, 1275–1307 (2008) MATHCrossRefGoogle Scholar
  3. 3.
    Balaban, T., Feldman, J., Knörrer, H., Trubowitz, E.: Power series representations for complex bosonic effective actions (2008, preprint) Google Scholar
  4. 4.
    Balaban, T., Feldman, J., Knörrer, H., Trubowitz, E.: Power series representations for complex bosonic effective actions. II. A small field renormalization group flow (2008, preprint) Google Scholar
  5. 5.
    Cammarota, C.: Decay of correlations for infinite range interactions in unbounded spin systems. Commun. Math. Phys. 85, 517–528 (1982) CrossRefADSMathSciNetGoogle Scholar
  6. 6.
    Rivasseau, V.: From Perturbative to Constructive Renormalization. Princeton University Press, Princeton (1991) Google Scholar
  7. 7.
    Rota, G.-C.: On the foundations of combinatorial theory. I. Theory of Möbius functions. Z. Wahrscheinlichkeitstheor. Verw. Geb. 2, 340 (1964) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Salmhofer, M.: Renormalization, An Introduction. Springer, Berlin (1999) MATHGoogle Scholar
  9. 9.
    Seiler, E.: Gauge Theories as a Problem of Constructive Quantum Field Theory and Statistical Mechanics. Springer, Berlin (1982) Google Scholar
  10. 10.
    Simon, B.: The Statistical Mechanics of Lattice Gases, vol. 1. Princeton University Press, Princeton (1993) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Tadeusz Balaban
    • 1
  • Joel Feldman
    • 2
  • Horst Knörrer
    • 3
  • Eugene Trubowitz
    • 3
  1. 1.Department of MathematicsRutgers, The State University of New JerseyPiscatawayUSA
  2. 2.Department of MathematicsUniversity of British ColumbiaVancouverCanada
  3. 3.MathematikETH-ZentrumZürichSwitzerland

Personalised recommendations