Journal of Statistical Physics

, Volume 134, Issue 5, pp 839–857

Power Series Representations for Bosonic Effective Actions


  • Tadeusz Balaban
    • Department of MathematicsRutgers, The State University of New Jersey
  • Joel Feldman
    • Department of MathematicsUniversity of British Columbia
    • MathematikETH-Zentrum
  • Eugene Trubowitz
    • MathematikETH-Zentrum

DOI: 10.1007/s10955-008-9634-8

Cite this article as:
Balaban, T., Feldman, J., Knörrer, H. et al. J Stat Phys (2009) 134: 839. doi:10.1007/s10955-008-9634-8


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.


Polymer expansionRenormalization group
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© Springer Science+Business Media, LLC 2008