Communications in Mathematical Physics

, Volume 49, Issue 3, pp 233–246

The cluster expansion in statistical mechanics

  • David Brydges
  • Paul Federbush

DOI: 10.1007/BF01608729

Cite this article as:
Brydges, D. & Federbush, P. Commun.Math. Phys. (1976) 49: 233. doi:10.1007/BF01608729


The Glimm-Jaffe-Spencer cluster expansion from constructive quantum field theory is adapted to treat quantum statistical mechanical systems of particles interacting by finite range potentials. The HamiltonianH0+V need be stable in the extended sense thatH0+4V+BN≧0 for someB. In this situation, with a mild technical condition on the potentials, the cluster expansion converges and the infinite volume limit of the correlation functions exists, at low enough density. These infinite volume correlation functions cluster exponentially. We define a class of interacting boson and fermion particle theories with a matter-like potential, 1/r suitably truncated at large distance. This system would collapse in the absence of the exclusion principle—the potential is unstable—but the Hamiltonian is stable. This provides an example of a system for which our method proves existence of the infinite volume limit, that is not covered by the classic work of Ginibre, which requires stable potentials.

One key ingredient is a type of Holder inequality for the expectation values of spatially smeared Euclidean densities, a special interpolation theorem. We also obtain a result on the absolute value of the fermion measure, it equals the boson measure.

Copyright information

© Springer-Verlag 1976

Authors and Affiliations

  • David Brydges
    • 1
  • Paul Federbush
    • 1
  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA